Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

Actuarial Modelling of 

Claim Counts 

Risk Classification, Credibility and 

Bonus-Malus Systems 

Michel Denuit 

Institut de Statistique, Université Catholique de Louvain, Belgium 

Xavier Maréchal 

Reacfin, Spin-off of the Université Catholique de Louvain, Belgium 

Sandra Pitrebois 

Secura, Belgium 

Jean-François Walhin 

Fortis, Belgium


Claim Counts


Claim Counts 

Risk Classification, Credibility and 


Michel Denuit 

Institut de Statistique, Université Catholique de Louvain, Belgium 


Reacfin, Spin-off of the Université Catholique de Louvain, Belgium 


Secura, Belgium 


Fortis, Belgium

Copyright © 2007 

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Anniversary Logo Design: Richard J. Pacifico 

Library of Congress Cataloging in Publication Data 

Actuarial Modelling of Claim Counts : Risk Classification, Credibility and Bonus-Malus Systems / 

Michel Denuit [et al.]. 

p. cm. 

Includes bibliographical references and index. 

ISBN 978-0-470-02677-9 (cloth) 

1. Insurance, Automobile—Rates—Europe. 2. Automobile insurance claims—Europe. 

I. Denuit, M. (Michel) 

HG9970.2.A25 2007 

368 ′ .092094—dc22 

2007019885 

British Library Cataloguing in Publication Data 

A catalogue record for this book is available from the British Library 

ISBN-13 978-0-470-02677-9 

Typeset in 10/12pt Times by Integra Software Services Pvt. Ltd, Pondicherry, India 

Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire 

This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which 

at least two trees are planted for each one used for paper production.

Contents 

Foreword 

Preface 

Notation 

xiii 

xv 

xxv 

Part I Modelling Claim Counts 1 

1 Mixed Poisson Models for Claim Numbers 3 

1.1 Introduction 3 

1.1.1 Poisson Modelling for the Number of Claims 3 

1.1.2 Heterogeneity and Mixed Poisson Model 4 

1.1.3 Maximum Likelihood Estimation 4 

1.1.4 Agenda 5 

1.2 Probabilistic Tools 5 

1.2.1 Experiment and Universe 5 

1.2.2 Random Events 5 

1.2.3 Sigma-Algebra 6 

1.2.4 Probability Measure 6 

1.2.5 Independent Events 7 

1.2.6 Conditional Probability 7 

1.2.7 Random Variables and Random Vectors 8 

1.2.8 Distribution Functions 8 

1.2.9 Independence for Random Variables 9 

1.3 Poisson Distribution 10 

1.3.1 Counting Random Variables 10 

1.3.2 Probability Mass Function 10 

1.3.3 Moments 10 

1.3.4 Probability Generating Function 11 

1.3.5 Convolution Product 12 

1.3.6 From the Binomial to the Poisson Distribution 13 

1.3.7 Poisson Process 17

vi 

Actuarial Modelling of Claim Counts 

1.4 Mixed Poisson Distributions 21 

1.4.1 Expectations of General Random Variables 21 

1.4.2 Heterogeneity and Mixture Models 22 

1.4.3 Mixed Poisson Process 25 

1.4.4 Properties of Mixed Poisson Distributions 26 

1.4.5 Negative Binomial Distribution 28 

1.4.6 Poisson-Inverse Gaussian Distribution 31 

1.4.7 Poisson-LogNormal Distribution 33 

1.5 Statistical Inference for Discrete Distributions 35 

1.5.1 Maximum Likelihood Estimators 35 

1.5.2 Properties of the Maximum Likelihood Estimators 37 

1.5.3 Computing the Maximum Likelihood Estimators with the 

Newton–Raphson Algorithm 40 

1.5.4 Hypothesis Tests 41 

1.6 Numerical Illustration 44 

1.7 Further Reading and Bibliographic Notes 46 

1.7.1 Mixed Poisson Distributions 46 

1.7.2 Survey of Empirical Studies Devoted to Claim Frequencies 46 

1.7.3 Semiparametric Approach 47 

2 Risk Classification 49 


2.1.1 Risk Classification, Regression Models and Random Effects 49 

2.1.2 Risk Sharing in Segmented Tariffs 50 

2.1.3 Bonus Hunger and Censoring 51 

2.1.4 Agenda 52 

2.2 Descriptive Statistics for Portfolio A 52 

2.2.1 Global Figures 52 

2.2.2 Available Information 52 

2.2.3 Exposure-to-Risk 53 

2.2.4 One-Way Analyses 54 

2.2.5 Interactions 58 

2.2.6 True Versus Apparent Dependence 59 

2.3 Poisson Regression Model 62 

2.3.1 Coding Explanatory Variables 62 

2.3.2 Loglinear Poisson Regression Model 64 

2.3.3 Score 64 

2.3.4 Multiplicative Tariff 65 

2.3.5 Likelihood Equations 66 

2.3.6 Interpretation of the Likelihood Equations 67 

2.3.7 Solving the Likelihood Equations with the Newton–Raphson 

Algorithm 67 

2.3.8 Wald Confidence Intervals 69 

2.3.9 Testing for Hypothesis on a Single Parameter 69 

2.3.10 Confidence Interval for the Expected Annual Claim Frequency 70 

2.3.11 Deviance 71 

2.3.12 Deviance Residuals 72 

2.3.13 Testing a Hypothesis on a Set of Parameters 72 

2.3.14 Specification Error and Robust Inference 72 

2.3.15 Numerical Illustration 73

Contents 

vii 

2.4 Overdispersion 79 

2.4.1 Explanation of the Phenomenon 79 

2.4.2 Interpreting Overdispersion 79 

2.4.3 Consequences of Overdispersion 80 

2.4.4 Modelling Overdispersion 80 

2.4.5 Detecting Overdispersion 81 

2.4.6 Testing for Overdispersion 82 

2.5 Negative Binomial Regression Model 83 


2.5.2 Numerical Illustration 85 

2.6 Poisson-Inverse Gaussian Regression Model 86 



2.7 Poisson-LogNormal Regression Model 87 



2.8 Risk Classification for Portfolio A 89 

2.8.1 Comparison of Competing models with the Vuong Test 89 

2.8.2 Resulting Risk Classification for Portfolio A 90 

2.9 Ratemaking using Panel Data 90 

2.9.1 Longitudinal Data 90 

2.9.2 Descriptive Statistics for Portfolio B 92 

2.9.3 Poisson Regression with Serial Independence 94 

2.9.4 Detection of Serial Dependence 97 

2.9.5 Estimation of the Parameters using GEE 101 

2.9.6 Maximum Likelihood in the Negative Binomial Model for Panel Data 105 

2.9.7 Maximum Likelihood in the Poisson-Inverse Gaussian Model for Panel Data 106 

2.9.8 Maximum Likelihood in the Poisson-LogNormal Model for Panel Data 107 

2.9.9 Vuong Test 109 

2.9.10 Information Criteria 110 

2.9.11 Resulting Classification for Portfolio B 110 


2.10.1 Generalized Linear Models 111 

2.10.2 Nonlinear Effects 112 

2.10.3 Zero-Inflated Models 112 

2.10.4 Fixed Versus Random Effects 113 

2.10.5 Hurdle Models 113 

2.10.6 Geographic Ratemaking 114 

2.10.7 Software 116 

Part II Basics of Experience Rating 119 

3 Credibility Models for Claim Counts 121 


3.1.1 From Risk Classification to Experience Rating 121 

3.1.2 Credibility Theory 121 

3.1.3 Limited Fluctuation Theory 122 

3.1.4 Greatest Accuracy Credibility 122 

3.1.5 Linear Credibility 123 

3.1.6 Financial Equilibrium 123

viii 


3.1.7 Combining a Priori and a Posteriori Ratemaking 123 

3.1.8 Loss Function 124 

3.1.9 Agenda 124 

3.2 Credibility Models 124 

3.2.1 A Simple Introductory Example: the Good Driver / Bad Driver 

Model 124 

3.2.2 Credibility Models Incorporating a Priori Risk Classification 126 

3.3 Credibility Formulas with a Quadratic Loss Function 128 

3.3.1 Optimal Least-Squares Predictor 128 

3.3.2 Predictive Distribution 129 

3.3.3 Bayesian Credibility Premium 130 

3.3.4 Poisson-Gamma Credibility Model 131 

3.3.5 Predictive Distribution and Bayesian Credibility Premium 132 


3.3.7 Discrete Poisson Mixture Credibility Model 135 

3.3.8 Discrete Approximations for the Heterogeneous Component 136 


3.4 Credibility Formulas with an Exponential Loss Function 149 

3.4.1 Optimal Predictor 149 

3.4.2 Poisson-Gamma Credibility Model 151 



3.5 Dependence in the Mixed Poisson Credibility Model 155 

3.5.1 Intuitive Ideas 155 

3.5.2 Stochastic Order Relations 156 

3.5.3 Comparisons of Predictive Distributions 156 

3.5.4 Positive Dependence Notions 157 

3.5.5 Dependence Between Annual Claim Numbers 157 

3.5.6 Increasingness in the Linear Credibility Model 158 


3.6.1 Credibility Models 158 

3.6.2 Claim Count Distributions 159 

3.6.3 Loss Functions 159 

3.6.4 Credibility and Regression Models 159 

3.6.5 Credibility and Copulas 160 

3.6.6 Time Dependent Random Effects 161 

3.6.7 Credibility and Panel Data Models 162 

3.6.8 Credibility and Empirical Bayes Methods 163 

4 Bonus-Malus Scales 165 


4.1.1 From Credibility to Bonus-Malus Scales 165 

4.1.2 The Nature of Bonus-Malus Scales 166 

4.1.3 Relativities 166 

4.1.4 Bonus-Malus Scales and Markov Chains 166 

4.1.5 Financial Equilibrium 167 

4.1.6 Agenda 167 

4.2 Modelling Bonus-Malus Systems 168 

4.2.1 Typical Bonus-Malus Scales 168 

4.2.2 Characteristics of Bonus-Malus Scales 169

Contents 

ix 

4.2.3 Trajectory 170 

4.2.4 Transition Rules 171 

4.3 Transition Probabilities 172 

4.3.1 Definition 172 

4.3.2 Transition Matrix 173 

4.3.3 Multi-Step Transition Probabilities 174 

4.3.4 Ergodicity and Regular Transition Matrix 176 

4.4 Long-Term Behaviour of Bonus-Malus Systems 176 

4.4.1 Stationary Distribution 176 

4.4.2 Rolski–Schmidli–Schmidt–Teugels Formula 179 

4.4.3 Dufresne Algorithm 182 

4.4.4 Convergence to the Stationary Distribution 183 

4.5 Relativities with a Quadratic Loss Function 184 

4.5.1 Relativities 184 

4.5.2 Bayesian Relativities 185 

4.5.3 Interaction between Bonus-Malus Systems and a Priori Ratemaking 189 

4.5.4 Linear Relativities 191 

4.5.5 Approximations 193 

4.6 Relativities with an Exponential Loss Function 194 

4.6.1 Bayesian Relativities 194 

4.6.2 Fixing the Value of the Severity Parameter 196 

4.6.3 Linear Relativities 196 


4.7 Special Bonus Rule 200 

4.7.1 The Former Belgian Compulsory System 200 

4.7.2 Fictitious Levels 200 

4.7.3 Determination of the Relativities 200 


4.7.5 Linear Relativities for the Belgian Scale 207 

4.8 Change of Scale 208 

4.8.1 Migration from One Scale to Another 208 

4.8.2 Kolmogorov Distance 208 

4.8.3 Distances between the Random Effects 209 


4.9 Dependence in Bonus-Malus Scales 213 


Part III Advances in Experience Rating 217 

5 Efficiency and Bonus Hunger 219 


5.1.1 Pure Premium 219 

5.1.2 Statistical Analysis of Claim Costs 219 

5.1.3 Large Claims and Extreme Value Theory 220 

5.1.4 Measuring the Efficiency of the Bonus-Malus Scales 220 

5.1.5 Bonus Hunger and Optimal Retention 220 

5.1.6 Descriptive Statistics for Portfolio C 221 

5.2 Modelling Claim Severities 222 

5.2.1 Claim Severities in Motor Third Party Liability Insurance 222

x 


5.2.2 Determining the Large Claims with Extreme Value Theory 223 

5.2.3 Generalized Pareto Fit to the Costs of Large Claims 227 

5.2.4 Modelling the Number of Large Claims 229 

5.2.5 Modelling the Costs of Moderate Claims 230 

5.2.6 Resulting Price List for Portfolio C 236 

5.3 Measures of Efficiency for Bonus-Malus Scales 240 

5.3.1 Loimaranta Efficiency 240 

5.3.2 De Pril Efficiency 242 

5.4 Bonus Hunger and Optimal Retention 246 

5.4.1 Correcting the Estimations for Censoring 246 

5.4.2 Number of Claims and Number of Accidents 249 

5.4.3 Lemaire Algorithm for the Determination of Optimal Retention Limits 251 


5.5.1 Modelling Claim Amounts in Related Coverages 255 

5.5.2 Tweedie Generalized Linear Model 255 

5.5.3 Large Claims 256 

5.5.4 Alternative Approaches to Risk Classification 257 

5.5.5 Efficiency 257 

5.5.6 Optimal Retention Limits and Bonus Hunger 257 

6 Multi-Event Systems 259 


6.2 Multi-Event Credibility Models 260 

6.2.1 Dichotomy 260 

6.2.2 Multivariate Claim Count Model 260 

6.2.3 Bayesian Credibility Approach 261 

6.2.4 Summary of Past Claims Histories 262 

6.2.5 Variance-Covariance Structure of the Random Effects 263 

6.2.6 Variance-Covariance Structure of the Annual Claim Numbers 263 

6.2.7 Estimation of the Variances and Covariances 264 

6.2.8 Linear Credibility Premiums 264 

6.2.9 Numerical Illustration for Portfolio A 268 

6.3 Multi-Event Bonus-Malus Scales 270 

6.3.1 Types of Claims 270 

6.3.2 Markov Modelling for the Multi-Event Bonus-Malus Scale 273 

6.3.3 Determination of the relativities 274 

6.3.4 Numerical Illustrations 274 


7 Bonus-Malus Systems with Varying Deductibles 277 


7.2 Distribution of the Annual Aggregate Claims 278 

7.2.1 Modelling Claim Costs 278 

7.2.2 Discretization 279 

7.2.3 Panjer Algorithm 281 

7.3 Introducing a Deductible Within a Posteriori Ratemaking 284 

7.3.1 Annual Deductible 284 

7.3.2 Per Claim Deductible 285 

7.3.3 Mixed Case 285

Contents 

xi 

7.4 Numerical Illustrations 286 

7.4.1 Claim Frequencies 286 

7.4.2 Claim Severities 286 

7.4.3 Annual Deductible 287 

7.4.4 Per Claim Deductible 288 

7.4.5 Annual Deductible in the Mixed Case 289 

7.4.6 Per Claim Deductible in the Mixed Case 289 


8 Transient Maximum Accuracy Criterion 293 


8.1.1 From Stationary to Transient Distributions 293 

8.1.2 A Practical Example: Creating a Special Scale for New Entrants 293 

8.1.3 Agenda 295 

8.2 Transient Behaviour and Convergence of Bonus-Malus Scales 295 

8.3 Quadratic Loss Function 297 

8.3.1 Transient Maximum Accuracy Criterion 297 

8.3.2 Linear Scales 302 

8.3.3 Financial Balance 305 

8.3.4 Choice of an Initial Level 307 

8.4 Exponential Loss Function 308 

8.5 Numerical Illustrations 308 

8.5.1 Scale −1/Top 308 

8.5.2 −1/+2 Scale 315 

8.6 Super Bonus Level 319 

8.6.1 Mechanism 319 

8.6.2 Initial Distributions 319 

8.6.3 Transient Relativities 319 


9 Actuarial Analysis of the French Bonus-Malus System 325 


9.2 French Bonus-Malus System 326 

9.2.1 Modelling Claim Frequencies 326 

9.2.2 Probability Generating Functions of Random Vectors 327 

9.2.3 CRM Coefficients 327 

9.2.4 Computation of the CRMs at Time t 328 

9.2.5 Global CRM 329 

9.2.6 Multivariate Panjer and De Pril Recursive Formulas 330 

9.2.7 Analysis of the Financial Equilibrium of the French Bonus-Malus System 333 


9.3 Partial Liability 338 

9.3.1 Reduced Penalty and Modelling Claim Frequencies 338 

9.3.2 Computations of the CRMs at Time t 338 

9.3.3 Financial Equilibrium 340 

9.3.4 Numerical Illustrations 341 


Bibliography 345 

Index 355

Foreword 

Belgium has a long and distinguished history in actuarial science. One of its leading centres 

in the area is the Institut des Sciences Actuarielles at l’Université Catholique de Louvain 

(UCL). Since its tender beginnings in the 1970s, the Institute has grown to critical mass and 

now boasts an internationally renowned faculty conducting research and education in a broad 

range of actuarial subjects – newish ones in the interface of insurance and finance as well as 

more traditional ones that used to form the core of insurance mathematics. Among the latter 

is risk classification and experience rating in general insurance, which is the subject matter 

of the present book. This is an area of applied statistics that has been fetching tools from 

various kits of theoretical statistics, notably empirical Bayes, regression, and (generalized) 

linear models. However, the complexity of the typical application, featuring unobservable 

risk heterogeneity, imbalanced design, and nonparametric distributions, inspired independent 

fundamental research under the label ‘credibility theory’, now a cornerstone in contemporary 

insurance mathematics. Quite naturally, the present book is a tribute to Florian (Etienne) De 

Vylder, who was a Professor at UCL and one of the greatest minds in insurance mathematics 

and, in particular, credibility theory. The book grew out of years of studies by a collective 

of researchers based at UCL and its industrial environment. The lead author, Michel Denuit, 

is one of the most prolific researchers in contemporary actuarial science, who has publicized 

widely in actuarial and statistical journals on topics in risk theory and actuarial science and 

related basic disciplines. Also Jean-François Walhin is a well established researcher with a 

long list of publications in the scientific actuarial press. Together with their (even) younger 

co-authors Xavier Maréchal and Sandra Pitrebois, they have formed a team that is well 

placed to write a comprehensive reference text on risk classification and premium rating. 

Their combined expertise covers all theory areas that are at the base of the topic – risk theory 

and insurance mathematics, but also modern statistics and scientific computation. The team’s 

total contribution to theoretical research in the subject matter of the book is substantial, and 

it is merged with sound practical insights gained through commitment to applicability and 

also through career experience outside the purely academic walks of life. 

The book will be welcomed by practitioners and researchers who need a broad introduction 

to the titular subject area or an update aided by modern statistical methodology for complex

xiv 


models and high-dimensional data. The book may also serve as a textbook at graduate level 

further to an introduction to basic principles explained in simple models. 

The opening Chapters 1 and 2 present basic notions of risk and risk characteristics and 

their theoretical representation in stochastic models with fixed and random effects and more 

or less specified classes of distributions. Anticipating the orientation of the book, emphasis is 

placed on parametric models for the number of claims. This gives clarity to the exposition and 

also sets a suitable framework for discussion of model choice and model calibration that goes 

way beyond what is usually found in conventional tutorials. Poisson conditional distributions 

with varying exposures are merged with different mixing distributions on the individual 

proportional hazards, and there are extensions to generalized linear (regression) models, time 

trends, and spatial patterns. Statistical calibration is carried out with maximum likelihood 

methods but also with alternative schemes like generalized estimating functions. Ample 

numerical examples with authentic data gives real life to the theoretical ideas throughout. 

Claim counts remain a main theme, but the remainder of the book nevertheless presents 

a wealth of material, partly based on recent research by the authors: credibility theory, 

Bayes estimation with exponential loss, bonus-malus systems in a number of variations, 

elements of heavy-tailed distributions, bonus hunger and other ‘behavioural’ problems related 

to individual experience rating, optimal design of bonus-malus systems for aggregates of 

sub-portfolios, and much more. The final chapter is devoted to a carefully conducted case 

study of the French bonus-malus system. 

I would like to thank the authors for soliciting my views on a draft version of the book and 

for inviting my preface to their work. Most of all I would like to thank them for undertaking 

the formidable task of collecting and making accessible to a wide readership an area of 

actuarial science that has undergone great changes over the past few decades while remaining 

essential to decision making in insurance. 

Ragnar Norberg 

London, March 2007

Preface 

Motor Insurance 

This book is devoted to the analysis of the number of claims filed by an insured driver over 

time. Property and liability motor vehicle coverage is broadly divided into first and third 

party coverage. First party coverage provides protection in the event the vehicle owner is 

responsible for the accident and protects him and his property. Third party coverage provides 

protection in the event the vehicle owner causes harm to another party, who recovers their 

cost from the policyholder. First party coverages may include first party injury benefits such 

as medical expenses, death payments and comprehensive coverages. 

A third party liability coverage is required in most countries for a vehicle to be allowed 

on the public road network. The compulsory motor third party liability insurance represents 

a considerable share of the yearly nonlife premium collection in developed countries. This 

share becomes even more prominent when first party coverages are considered (such as 

medical benefits, uninsured or underinsured motorist coverage, and collision and other than 

collision insurance). Moreover, large data bases recording policyholders’ characteristics as 

well as claim histories are maintained by insurance companies. The economic importance 

and the availability of detailed information explain why a large body of the nonlife actuarial 

literature is devoted to this line of business. 

Tort System Versus No Fault System 

The liability insurance provides coverage to the policyholder if, as the driver of a covered 

vehicle, the policyholder injures a third party’s property. If the policyholder is sued with 

respect to negligence for such bodily injury or property damage, the insurer will provide 

legal defense for the policyholder. If the policyholder is found to be liable, the insurer will 

pay, on behalf of the policyholder, damages assessed against the policyholder. 

In the tort system, the insurer indemnifies the claim only if it believes the insured was at 

fault in the accident or the third party sues the insured and proves that he/she was at fault 

in the accident. Needless to say, a large part of the premium income is consumed by legal 

fees, court costs and insurers’ administration expenses in such a system. Because of this, 

several North-American jurisdictions have implemented a no-fault motor insurance system.

xvi 


Even in a pure no-fault motor environment, the police still ask which driver was at fault (or 

the degrees to which the drivers shared the fault) because at-fault events cause the insurance 

premium to rise at the next policy renewal. 

Insurance Ratemaking 

Cost-based pricing of individual risks is a key actuarial ratemaking principle. The price 

charged to policyholders is an estimate of the future costs related to the insurance coverage. 

The pure premium approach defines the price of an insurance policy as the ratio of the 

estimated costs of all future claims against the coverage provided by the insurance policy 

while it is in effect to the risk exposure, plus expenses. 

The property/casualty ratemaking is based on a claim frequency distribution and a loss 

distribution. The claim frequency is defined as the number of incurred claims per unit 

of earned exposure. The exposure is measured in car-year for motor third party liability 

insurance (the rate manual lists rates per car-year). The average loss severity is the average 

payment per incurred claim. Under mild conditions, the pure premium is then the product 

of the average claim frequency times the average loss severity. The loss models for motor 

insurance are reviewed in Chapters 1–2 (frequency part) and 5 (claim amounts). 

In liability insurance, the settlement of larger claims often requires several years. Much of 

the data available for the recent accident years will therefore be incomplete, in the sense that 

the final claim cost will not be known. In this case, loss development factors can be used to 

obtain a final cost estimate. The average loss severity is then based on incurred loss data. In 

contrast to paid loss data (that are purely objective, representing the actual payments made 

by the company), incurred loss data include subjective reserve estimates. The actuary has to 

carefully analyse the large claims since they represent a considerable share of the insurer’s 

yearly expenses. This issue will be discussed in Chapter 5, where incurred loss data will be 

analysed and appropriately modelled. 

Risk Classification 

Nowadays, it has become extremely difficult for insurance companies to maintain cross 

subsidies between different risk categories in a competitive market. If, for instance, females 

are proved to cause significantly fewer accidents than males and if a company disregarded 

this variable and charged an average premium to all policyholders regardless of gender, most 

of its female policyholders would be tempted to move to another company offering better 

rates to female drivers. The former company is then left with a disproportionate number of 

male policyholders and insufficient premium income to pay for the claims. 

To avoid lapses in a competitive market, actuaries have to design a tariff structure that 

will fairly distribute the burden of claims among policyholders. The policies are partitioned 

into classes with all policyholders belonging to the same class paying the same premium. 

Each time a competitor uses an additional rating factor, the actuary has to refine the partition 

to avoid losing the best drivers with respect to this factor. This explains why so many factors 

are used by insurance companies: this is not required by actuarial theory, but instead by 

competition among insurers. 

In a free market, insurance companies need to use a rating structure that matches the 

premiums for the risks as closely as possible, or at least as closely as the rating structures used

Preface 

xvii 

by competitors. This entails using virtually every available classification variable correlated 

to the risks, since failing to do so would mean sacrificing the chance to select against 

competitors, and incurring the risk of suffering adverse selection by them. It is thus the 

competition between insurers that leads to more and more partitioned portfolios, and not 

actuarial science. This trend towards more risk classification often causes social disasters: 

bad drivers (or more precisely, drivers sharing the characteristics of bad drivers) do not find 

a coverage for a reasonable price, and are tempted to drive without insurance. Note also that 

even if a correlation exists between the rating factor and the risk covered by the insurer, 

there may be no causal relationship between that factor and risk. Requiring that insurance 

companies establish such a causal relationship to be allowed to use a rating factor is subject 

to debate. 

Property and liability motor vehicle insurers use classification plans to create risk classes. 

The classification variables introduced to partition risks into cells are called a priori variables 

(as their values can be determined before the policyholder starts to drive). Premiums for 

motor liability coverage often vary by the territory in which the vehicle is garaged, the use 

of the vehicle (driving to and from work or business use) and individual characteristics (such 

as age, gender, occupation and marital status of the main driver of the vehicle, for instance, 

if not precluded by legislation or regulatory rules). If the policyholders misrepresent any of 

these classification variables in their declaration, they are subject to loss of coverage when 

they are involved in a claim. There is thus a strong incentive for accurate reporting of risk 

characteristics. 

As explained in Chapter 2, it is convenient to achieve a priori classification with the 

help of generalized regression models. The method can be roughly summarized as follows: 

One risk classification cell is chosen as the base cell. It normally has the largest amount 

of exposure. The rate for the base cell is referred to as the base rate. Other rate cells are 

defined by a variety of risk classification variables, such as territory and so on. For each 

risk classification variable, there is a vector of differentials, with the base cell characteristics 

always assigned 100 %. 

In this book, we make extensive use of the generalized linear models (better known 

under the acronym GLM) developed after Nelder & Wedderburn (1972). These authors 

discovered that regression models with a response distribution belonging to the exponential 

family of probability distributions shared the same characteristics. Members of this family 

include Normal, Binomial, Poisson, Gamma and Inverse Gaussian distributions that have 

been widely used by actuaries to model the number of claims, or their severities. Working 

in the exponential family allows the actuary to relax the very restrictive hypotheses behind 

the Normal linear regression model, namely: 

• the response variable takes on the theoretical shape of a Normal distribution; 

• the variance is constant over individuals; 

• the fitted values are obtained from linear combinations of the explanatory variables (called 

linear predictors, or scores). 

Specifically, the Normal distribution can be replaced with another member of the exponential 

family, heteroscedasticity can be allowed for, and fitted values can be obtained from a 

nonlinear transformation (called the link function) of linear predictors. Efficient algorithms

xviii 


are available under most statistical packages to estimate the regression parameters by 

maximum likelihood. 

Pay-As-You-Drive System 

Every kilometer travelled by a vehicle transfers risk to its insurer: the total cost of the 

coverage thus increases kilometer by kilometer. This is why several authors, including 

Butler (1993) suggested charging a cents-per-kilometer based premium; the car-kilometer 

should be adopted as the exposure unit instead of the car-year that is currently used. Motor 

insurance companies are adopting a new scheme called ‘pay as you drive’ (henceforth 

referred to as PAYD for the sake of brevity). Under a PAYD system, a driver pays for every 

kilometer driven at a rate varying from a premium to use the busiest roads at peak hours to 

a lower rate for the rural roads. 

Several insurance companies (including the pioneering company Norwich Union, 

http://www.norwichunion.com/pay-as-you-drive/) have now started to offer a motor 

insurance policy under a PAYD system after successful pilot schemes involving thousands of 

motorists. With PAYD systems, drivers are provided with in-car Global Positioning System 

(GPS) devices coupled with maps, enabling the insurance company to calculate insurance 

premiums for each journey, depending on time of day, type of road and distance travelled. 

A ‘black box’ is installed in the car and receives signals from GPS technology to determine 

the vehicle’s current position, speed, and time and direction driven. The black box then 

acts as a wireless modem to transmit these inputs through standard mobile phone networks 

to the insurer. The insurer sends a monthly bill to the customer based on vehicle usage, 

including time of day, type of road and distance travelled. Historical data then provide 

detailed information of how, when and where cars are actually used, and whether accidents 

and claims can be identified with particular factors. Moreover, the tracker detects speed 

infringements and more generally, the aggressiveness behind the wheel. Dangerous driving 

habits could lead to higher premiums for car insurance, increasing road safety. In addition 

to the static measures of risk, such as the driver’s age, dynamic measures, such as speed, 

time of day, and location, are used to give the best possible overall risk assessment. 

The generalization of the PAYD system is also expected to change motorists’ attitudes: 

like petrol, insurance is bought on a pay-as-you-drive basis, and people think of their 

insurance costs as related to their actual use of their vehicle. Several North-American studies 

demonstrate that PAYD systems could reduce motoring by more than 10 %. The PAYD 

rating system is expected to decrease congestion and pollution (since the busier roads usually 

attract the higher rates). 

Experience Rating 

The trend towards more classification factors has lead the supervising authorities to exclude 

from the tariff structure certain risk factors, even though they may be significantly correlated 

to losses. Many states consider banning classification based on items that are beyond the 

control of the insured, such as gender or age. The resulting inadequacies of the a priori 

rating system can be corrected for by using the past number of claims to reevaluate future 

premiums. This is much in line with the concept of fairness: as it will be seen from Chapter 2, 

a priori ratemaking penalizes individuals who ‘look like’ bad drivers (even if they are in

Preface 

xix 

reality excellent drivers who will never cause any accident) whereas experience rating uses 

the individual claim record to adjust the amount of premium. Actuarial credibility models 

make a balance between the likelihood of being an unlucky good driver (who suffered a 

claim) and the likelihood of being a truly bad driver (who should suffer an increase in 

the premium paid to the insurance company for coverage). It seems fair to correct the 

inadequacies of the a priori system by using an adequate experience rating plan; such a 

‘crime and punishment’ system may be more acceptable to policyholders than seemingly 

arbitrary a priori classifications. 

Moreover, many important factors cannot be taken into account in the a priori risk 

classification. Think for instance of swiftness of reflexes, drinking habits or respect for the 

highway code. Consequently, tariff cells are still quite heterogeneous despite the use of many 

classification variables. This heterogeneity can be modelled by a random effect in a statistical 

model. It is reasonable to believe that the hidden characteristics are partly revealed by the 

number of claims reported by the policyholders. Several empirical studies have shown that, 

if insurers were allowed to use only one rating variable, it should be some form of merit 

rating: the best predictor of the number of claims incurred by a driver in the future is not 

age or vehicle type but past claims history. Hence the adjustment of the premium from the 

individual claims experience in order to restore fairness among policyholders as explained 

in Chapter 3. In that respect, the allowance of past claims in a rating model derives from an 

exogeneous explanation of serial correlation for longitudinal data. In this case, correlation is 

only apparent and results from the revelation of hidden features in the risk characteristics. 

It is worth mentioning that serial correlation for claim numbers can also receive an 

endogeneous explanation. In this framework, the history of individuals modifies the risk they 

represent; this mechanism is termed ‘true contagion’, referring to epidemiology. For instance, 

a car accident may modify the perception of danger behind the wheel and lower the risk of 

reporting another claim in the future. Experience rating schemes also provide incentives to 

careful driving and should induce negative contagion. Nevertheless, the main interpretation 

for automobile insurance is exogeneous, since positive contagion (that is, policyholders who 

reported claims in the past being more likely to produce claims in the future than those 

who did not) is always observed for numbers of claims, whereas true contagion should be 

negative. 


In many European and Asian countries, as well as in North-American states or provinces, 

insurers use experience rating in order to relate premium amounts to individual past claims 

experience in motor insurance. Such systems penalize insured drivers responsible for one 

or more accidents by premium surcharges (or maluses) and reward claim-free policyholders 

by awarding them discounts (or bonuses). Such systems are called no-claim discounts, 

experience rating, merit rating, or bonus-malus systems. 

Discounts for claim-free driving have been awarded in the United Kingdom as early 

as 1910. At that time, they were intended as an inducement to renew a policy with the 

same company rather than as a reward for prudent driving. The first theoretical treatments 

of bonus-malus systems were provided in the pioneering works of Grenander (1957a,b). 

The first ASTIN colloquium held in France in 1959 was exclusively devoted to no-claim 

discounts in insurance, with particular reference to motor business.

xx 


There are various bonus-malus systems used around the world. A typical form of no-claim 

bonus in the United Kingdom is as follows: 

one claim-free year 25 % discount 

two claim-free years 40 % discount 

three claim-free years 50 % discount 

four claim-free years 60 % discount. 

Drivers earn an extra year of bonus for each year they remain without claims at fault up to a 

maximum of four years, but lose two years bonus each time they report a claim at fault. In 

such a system, maximum bonus is achieved in only a few years and the majority of mature 

drivers have maximum bonus. 

Bonus-malus systems used in Continental Europe are often more elaborate. Bonus-malus 

scales consist of a finite number of levels, each with its own relativity (or relative premium). 

The amount of premium paid by a policyholder is then the product of a base premium with 

the relativity corresponding to the level occupied in the scale. New policyholders have access 

to a specified level. After each year, the policy moves up or down according to transition 

rules of the bonus-malus system. If a bonus-malus system is in force, all policies in the same 

tariff class are partitioned according to the level they occupy in the bonus-malus scale. In 

this respect, the bonus-malus mechanism can be considered as a refinement of a priori risk 

evaluation splitting each risk class into a number of subcategories according to individual 

past claims histories. 

As explained in Chapter 4, bonus-malus systems can be modelled using (conditional) 

Markov chains provided they possess a certain memoryless property that can be summarized 

as follows: the knowledge of the present level and of the number of claims of the present 

year suffices to determine the level to which the policy is transferred. In other words, the 

bonus-malus system satisfies the famous Markov property: the future (the level for year t +1) 

depends on the present (the level for year t and the number of accidents reported during year 

t) and not on the past (the claim history and the levels occupied during years 1 2t−1). 

This allows us to determine the optimal relativities in Chapter 4 using an asymptotic criterion 

based on the stationary distribution, and in Chapter 8 using transient distributions. Several 

performance measures for bonus-malus systems are reviewed in Chapters 5 and 8. 

During the 20th century, most European countries imposed a uniform bonus-malus system 

on all the companies operating in their territory. In 1994, the European Union decreed that 

all its member countries must drop their mandatory bonus-malus systems, claiming that such 

systems reduced competition between insurers and were in contradiction to the total rating 

freedom implemented by the Third Directive. Since that date, Belgium, for instance, dropped 

its mandatory system, but all companies operating in Belgium still apply the former uniform 

system (with minor modifications for the policyholders occupying the lowest levels in the 

scale). In other European countries, however, insurers compete on the basis of bonus-malus 

systems. This is the case for instance in Spain and Portugal. 

However, the mandatory French system is still in force. Quite surprisingly, the European 

Court of Justice decided in 2004 that both the French and Grand Duchy of Luxembourg 

mandatory bonus-malus systems were not contrary to the rating freedom imposed by the 

European legislation. The French law thus still imposes on the insurers operating in France 

a unique bonus-malus system. That bonus-malus system is not based on a scale. Instead the

Preface 

xxi 

French bonus-malus system uses the concept of an increase-decrease coefficient (coefficient 

de réduction-majoration in French). More precisely, the French bonus-malus system implies 

a malus of 25 % per claim and a bonus of 5 % per claim-free year. So each policyholder 

is assigned a base premium and this base premium is adapted according to the number of 

claims reported to the insurer, multiplying the premium by 1.25 each time an accident at fault 

is reported to the company, and by 0.95 per claim-free year. The French-type bonus-malus 

systems will be studied in Chapter 9. 

Actuarial and Economic Justifications for Bonus-Malus Systems 

Bonus-malus systems allow premiums to be adapted for hidden individual risk factors 

and to increase incentives for road safety, by taking into consideration the past claim 

record. This can be justified by asymmetrical information between the insurance company 

and the policyholders. Asymmetric information arises in insurance markets when firms 

have difficulties in judging the riskiness of those who purchase insurance coverage. There 

are mainly two aspects of this phenomenon: adverse selection and moral hazard. Adverse 

selection occurs when the policyholders have a better knowledge of their claim behaviour than 

the insurer does. Policyholders take advantage of information about their driving patterns, 

known to them but unknown to the insurer. In the context of compulsory motor third party 

liability insurance, adverse selection is not a significant problem compared to moral hazard 

when the insurance companies charge similar amounts of premium to all policyholders. 

Things are more complicated in a deregulated environment with companies using many 

risk classification factors. Since very heterogeneous driving behaviours are observed among 

policyholders sharing the same a priori variables, adverse selection cannot be avoided. For 

all the related coverages, such as comprehensive damages for instance, adverse selection 

always plays an important role. 

Considering adverse selection in the vein of Rotschild and Stiglitz, individuals partly reveal 

their underlying risk through the contract they choose, a fact that has to be taken into account 

when setting an adequate tariff structure. In the presence of unobservable heterogeneity, 

riskier agents will choose a more comprehensive coverage and low risk insurance applicants 

have an interest in signalling their quality, by selecting high deductibles (excesses) for 

instance. 

It is interesting to compare economists’ and actuaries’ approaches to experience rating. 

In the economic literature, discounts and penalties are introduced mainly to counteract the 

inefficiency which arises from moral hazard. In the actuarial literature, the main purpose is 

to better assess the individual risk so that everyone will pay, in the long run, a premium 

corresponding to his own claim frequency. Actuaries are thus more interested in adverse 

selection than moral hazard. 

Cost of Claims 

The vast majority of bonus-malus systems in force around the world penalize the number 

of at-fault accidents reported to the company, and not their amounts. A severe accident 

involving bodily injuries is penalized in the same way as a fender-bender. The reason to 

base motor risk classification on just claim frequencies is the long delay to access the cost of

xxii 


bodily injury and other severe claims. Not incorporating claim sizes in bonus-malus systems 

and a priori risk classification requires an (implicit) assumption of independence between 

the random variables ‘number of claims’ and ‘cost of a claim’, as well as the belief that the 

latter does not depend on the driver’s characteristics. This means that the actuarial practice 

considers that the cost of an accident is, for the most part, beyond the control of a driver: a 

cautious driver reduces the number of accidents, but for the most part cannot control the cost 

of these accidents (which is largely independent of the mistake that caused it). This belief 

will be challenged in Chapter 5. 

The penalty induced by the majority of bonus-malus systems being independent of the 

claim amount, policyholders have to decide whether it is profitable or not to report small 

claims (in order to avoid an increase in premium). Cheap claims are likely to be defrayed 

by the policyholders themselves, and not to be reported to the company. This phenomenon 

is known as the hunger for bonus and censors claim amounts and claim frequencies. In 

Chapter 5, a statistical model is specified, that takes into account the fact that only ‘expensive’ 

claims are reported to the insurance company. Retention limits for the policyholders are 

determined using the Lemaire algorithm. 

In a few bonus-malus systems, however, reporting a ‘severe’ claim (typically, a claim 

with bodily injuries) entails a more severe penalty than reporting a ‘minor’ claim (typically, 

a claim with material damage only). In the system in force in Japan before 1993, claims 

involving bodily injuries were penalized by four levels, while claims with property damage 

only were penalized by only two levels. Bonus-malus systems using different types of events 

to update premium amount will be examined in Chapter 6. 

In Chapter 7, we examine an innovative system using variable deductibles rather than 

premium relativities. It differs from the systems studied in preceding chapters in that 

it mixes elements of both a conventional bonus-malus system and a set of deductibles 

depending on the level occupied in the bonus-malus scale. The first system is a conventional 

discount system with loss of discount in the case where a claim at fault is reported. 

The second system also has a variable discount scale, which can increase with claim-free 

experience. However, there is no stepback of the discount on claim, only a stepback of the 

deductible. 

Aims of This Book 

About ten years after the seminal book ‘Bonus-Malus Systems in Automobile Insurance’ 

by Professor Jean Lemaire, we aim to offer a comprehensive treatment of the various 

experience rating systems applicable to automobile insurance and their relationships with 

risk classification. 

We hope that the present book will be useful for students in actuarial science, actuaries 

(both practitioners and in academia) and more generally for all the persons involved in 

technical problems inside insurance companies or consulting firms. For the first time, systems 

taking into account the exogeneous information are presented in an actuarial textbook. 

Many numerical illustrations carried out with advanced statistical softwares allow for a deep 

understanding of the concepts. 

The present book is the result of a close and fruitful collaboration between the Institute of 

Actuarial Science of the Université Catholique de Louvain, Louvain-la-Neuve, Belgium, its 

spin-off consulting firm Reacfin SA and the reinsurance company Secura, based in Brussels.

Preface 

xxiii 

This collaboration brings together academic expertise and practical experience to provide 

efficient solutions to motor ratemaking. 

Software 

The numerical illustrations presented in this book use SAS R (standing for Statistical Analysis 

System), a powerful software package for the manipulation and statistical analysis of data. 

SAS R is widely used in the insurance industry and practicing actuaries should be familiar 

with it. Among the large range of modules that can be added to the basic system (known 

as SAS R /BASE), we concentrate on the SAS R /STAT module. When no built-in procedures 

were available, we have coded programs in the SAS R /IML environment. 

The computations of bonus-malus scales are performed with the software BM-Builder 

developed by Reacfin. This is a computer solution running on SAS R that enables creation 

of a new bonus-malus scale by choosing the number of levels, the transition rules, etc. This 

scale, tailored to the insurer’s portfolio, is financially balanced. 

Here and there, comments about software available to perform the analyses detailed in this 

book will be provided to help the readers interested in practical implementation. Appropriate 

references to the websites of the providers are given for further information. 

Acknowledgements 

The present text originated from a series of lectures by Michel Denuit and Jean-François 

Walhin to Masters students in actuarial science in different universities (including UCL, 

Louvain-la-Neuve, Belgium; UCBL, Lyon, France; ULP, Strasbourg, France; and INSEA, 

Rabat, Morocco). Both Michel Denuit and Jean-François Walhin would like to thank the 

students, who have worked through the nonlife ratemaking courses over the past years and 

supplied invaluable reactions and comments. 

Training sessions with insurance professionals provided practical insights in the contents 

of the lectures. The feedback we received from short course audiences in Bucharest, Niort, 

Paris, and Warsaw, helped to improve the presentation of the topic. 

The authors’ own research in this area has benefited at various stages from discussions 

or collaborations with esteemed colleagues, including Jean-Philippe Boucher, Arthur 

Charpentier, Christophe Crochet, Jan Dhaene, Montserrat Guillén, Philippe Lambert, Stefan 

Lang, José Paris, Christian Partrat, Jean Pinquet, and Richard Verrall. 

We gratefully acknowledge the financial support of the Communauté française de 

Belgique under contract ‘Projet d’Actions de Recherche Concertées’ ARC 04/09-320, of the 

Région Wallonne under project First Spin-off ‘ActuR&D # 315481’, of Secura, the Belgian 

reinsurance company, and of the Banque Nationale de Belgique under grant ‘Risk measures 

and Economic capital’. 

We would like to express our deepest gratitude to Professor Ragnar Norberg for kindly 

accepting to preface this book, as well as for his careful reading of a previous version of 

this manuscript and for the numerous resulting comments. Any errors or omission, however, 

remain the responsibility of the authors. Professor Norberg’s pioneering works are among 

the most influential contributions to credibility theory and bonus-malus systems. It is a real 

honour that, thirty years after his seminal work appeared in the Scandinavian Actuarial

xxiv 


Journal which integrates bonus-malus scales in the framework of Markov chains, Professor 

Norberg introduces the present work. 

As always, it has been a pleasure to work with Wendy Hunter and Susan Barclay, Project 

Editors; Simon Lightfoot, Publishing Assistant; Kathryn Sharples, Commissioning Editor in 

Statistics and Mathematics; and Kelly Board and Sarah Kiddle, Content Editors, Engineering 

and Statistics at John Wiley & Sons, Ltd and Sunita Jayachandran at Integra Software 

Services Pvt. Ltd. 

Last but not least, we apologize to our families for the time not spent with them during 

the preparation of this book, and we are very grateful for their understanding. 

Michel Denuit 




Louvain-la-Neuve and Brussels, January 2007.

Notation 

Here are a few words on the notation and terminology used throughout the book. For the 

most part, the notation used in this book conforms to what is usual in mathematical statistics 

as well as nonlife insurance mathematics. 

The real line − + is denoted as . The half positive real line is + = 0 +. 

The set of the nonnegative integers is = 0 1 2. The real n-dimensional space is 

denoted as n , and n is the set of all the n-tuples of nonnegative integers. A point of n 

is an n-dimensional vector with real coordinates. It is represented by a bold letter x; the 

ith component of x is x i , i = 1 2n. All the vectors are tacitly assumed to be column 

vectors, that is, 

⎛ ⎞ 

x 1 

⎜ 

x = ⎝ 

⎟ 

⎠ 

x n 

A superscript ‘T’ is used to indicate transposition. Hence, x T is a row vector with components 

x 1 x n . The dimension of x is denoted as dimx. 

The matrices are denoted by a capital letter in boldface, for instance M; M T denotes 

the transposition of M. The vector of ones, that is 1 11 T , will be denoted by e. 

The identity matrix (with entries 1 on the main diagonal and 0 elsewhere) is denoted by I. 

The determinant of the matrix M is denoted as detM, its inverse by M −1 . 

We denote as oh a function of h that tends to 0 faster than the identity, that is, such that 

oh 

lim 

h↘0 h 

= 0 

Intuitively, oh is negligible when h becomes sufficiently small. 

The factorial of the positive integer n is denoted as n! and defined by n!=nn − 11. 

By convention, 0!=1. The binomial coefficient ( n 

k) 

denotes the number of different possible 

combinations of k items from n different items:

xxvi 


( n 

k) 

= 

( ) 

n! n 

k!n − k! = 

n − k 

Note that the binomial coefficient is sometimes denoted as Cn k , especially in the Frenchwritten 

mathematical literature, but here we adhere to the more standard notation ( n 

k) 

. The 

Gamma function · is defined as 

x = 

∫ 

0 

t x−1 exp−tdt 

x > 0 

As for any positive integer n, we have n = n−1!, the Gamma function can be considered 

as an interpolation of the factorials defined for positive integers. Integration by parts shows 

that x + 1 = xx for any positive real x. When a and b are positive real numbers, the 

definition of the binomial coefficient is extended to positive integers as 

( a a + 1 

= 

b) 

a − b + 1b + 1 

The incomplete Gamma function · · is defined as 

t = 1 

t 

∫ 

0 

x t−1 exp−xdx t ≥ 0 

A real-valued random variable is denoted by a capital letter, for instance X. The 

mathematical expectation operator is denoted as E·. For instance, EX is the expectation 

of the random variable X. The variance is VX, given by VX = EX 2 − EX 2 .A 

random vector is denoted by a bold capital letter, for instance X = X 1 X n T . Matrices 

should not be confused with random vectors (the context will make this clear). The variancecovariance 

matrix of X has covariances CX i X j = EX i X j − EX i EX j outside the 

main diagonal (that is, for i ≠ j) and the variances VX i along the main diagonal. 

The probability distributions used in this book are summarized next: 

• the Bernoulli distribution with parameter 0

Notation 

xxvii 

• the Negative Binomial distribution with parameters a>0 and >0, denoted as 

ina , has probability mass function 

( a + k − 1 

pk = 

k 

)( a 

a + 

) a ( ) k 

k∈ 

a + 

• the Normal distribution with parameters ∈ and 2 > 0, denoted as or 2 , has 

probability density function 

fx = 1 ( 

√ 2 exp − 1 ) 

2 x − 2 2 x∈ 

• the LogNormal distribution with parameters ∈ and 2 > 0, denoted as N or 2 , 

has probability density function 

( 

1 

fx = 

x √ 2 exp − 1 

) 

2 lnx − 2 2 x∈ + 

• the Negative Exponential distribution with parameter >0, denoted as xp, has 


fx = exp−x x ∈ + 

• the Gamma distribution with parameters >0 and >0, denoted as am , has 


fx = x−1 exp−x 

x∈ + 

 

• the Inverse Gaussian distribution, with parameters >0 and >0, denoted as 

au , has probability density function 

( 

 

fx = √ exp − 1 ) 

2x 

3 2x x − 2 x∈ + 

• the Pareto distribution, with parameters >0 and >0, denoted as ar , has 


fx = 

 

x + +1 x∈ + 

• the Uniform distribution, with parameters a

Part I 

Modelling Claim 

Counts 

Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus-Malus Systems 

S. Pitrebois and J.-F. Walhin © 2007 John Wiley & Sons, Ltd 

M. Denuit, X. Maréchal,

1 

Mixed Poisson Models for Claim 

Numbers 

1.1 Introduction 

1.1.1 Poisson Modelling for the Number of Claims 

In view of the economic importance of motor third party liability insurance in industrialized 

countries, many attempts have been made in the actuarial literature to find a probabilistic 

model for the distribution of the number of claims reported by insured drivers. This chapter 

aims to introduce the basic probability models for count data that will be applied in motor 

insurance. References to alternative models are gathered in the closing section to this chapter. 

The Binomial distribution is the discrete probability distribution of the number of successes 

in a sequence of n independent yes/no experiments, each of which yields success with 

probability q. Such a success/failure experiment is also called a Bernoulli experiment or 

Bernoulli trial. Two important distributions arise as approximations of Binomial distributions. 

If n is large enough and the skewness of the distribution is not too great (that is, q is not 

too close to 0 or 1), then the Binomial distribution is well approximated by the Normal 

distribution. When the number of observations n is large, and the success probability q 

is small, the corresponding Binomial distribution is well approximated by the Poisson 

distribution with mean = nq. The Poisson distribution is thus sometimes called the law 

of small numbers because it is the probability distribution of the number of occurrences of 

an event that happens rarely but has very many opportunities to happen. The parallel with 

traffic accidents is obvious. 

The Poisson distribution was discovered by Siméon-Denis Poisson (1781–1840) and 

published in 1838 in his work entitled Recherches sur la Probabilité des Jugements en 

Matières Criminelles et Matière Civile (which could be translated as ‘Research on the 

Probability of Judgments in Criminal and Civil Matters’). Typically, a Poisson random 




4 Actuarial Modelling of Claim Counts 

variable is a count of the number of events that occur in a certain time interval or spatial 

area. For example, the number of cars passing a fixed point in a five-minute interval, or the 

number of claims reported to an insurance company by an insured driver in a given period. 

A typical characteristic associated with the Poisson distribution is certainly equidispersion: 

the variance of the Poisson distribution is equal to its mean. 

1.1.2 Heterogeneity and Mixed Poisson Model 

The Poisson distribution plays a prominent role in modelling discrete count data, mainly 

because of its descriptive adequacy as a model when only randomness is present and the 

underlying population is homogeneous. Unfortunately, this is not a realistic assumption 

to make in modelling many real insurance data sets. Poisson mixtures are well-known 

counterparts to the simple Poisson distribution for the description of inhomogeneous 

populations. Of special interest are populations consisting of a finite number of homogeneous 

sub-populations. In these cases the probability distribution of the population can be regarded 

as a finite mixture of Poisson distributions. 

The problem of unobserved heterogeneity arises because differences in driving behaviour 

among individuals cannot be observed by the actuary. One of the well-known consequences 

of unobserved heterogeneity in count data analysis is overdispersion: the variance of the 

count variable is larger than the mean. Apart from its implications for the low-order 

moment structure of the counts, unobserved heterogeneity has important implications for the 

probability structure of the ensuing mixture model. The phenomena of excesses of zeros as 

well as heavy upper tails in most insurance data can be seen as an implication of unobserved 

heterogeneity (Shaked’s Two Crossings Theorem will make this clear). It is customary to 

allow for unobserved heterogeneity by superposing a random variable (called a random 

effect) on the mean parameter of the Poisson distribution, yielding a mixed Poisson model. 

In a mixed Poisson process, the annual expected claim frequency itself becomes random. 

1.1.3 Maximum Likelihood Estimation 

All the models implemented in this book are parametric, in the sense that the probabilities are 

known functions depending on a finite number of (real-valued) parameters. The Binomial, 

Poisson and Normal models are examples of parametric distributions. The first step in the 

analysis is to select a reasonable parametric model for the observations, and then to estimate 

the underlying parameters. The maximum likelihood estimator is the value of the parameter 

(or parameter vector) that makes the observed data most likely to have occurred given the 

data generating process assumed to have produced the observations. All we need to derive the 

maximum likelihood estimator is to formulate statistical models in the form of a likelihood 

function as a probability of getting the data at hand. The larger the likelihood, the better the 

model. 

Maximum likelihood estimates have several desirable asymptotic properties: consistency, 

efficiency, asymptotic Normality, and invariance. The advantages of maximum likelihood 

estimation are that it fully uses all the information about the parameters contained in the data 

and that it is highly flexible. Most applied maximum likelihood problems lack closed-form 

solutions and so rely on numerical maximization of the likelihood function. The advent of 

fast computers has made this a minor issue in most cases. Hypothesis testing for maximum

Mixed Poisson Models for Claim Numbers 5 

likelihood parameter estimates is straightforward due to the asymptotic Normal distribution 

of maximum likelihood estimates and the Wald and likelihood ratio tests. 

1.1.4 Agenda 

Section 1.2 briefly reviews the basic probability concepts used throughout this chapter (and 

the entire book), for further reference. Notions including probability spaces, random variables 

and probability distributions are made precise in this introductory section. 

In Section 1.3, we recall the main probabilistic tools to work with discrete distributions: 

probability mass function, distribution function, probability generating function, etc. Then, 

we review some basic counting distributions, including the Binomial and Poisson laws. 

Section 1.4 is devoted to mixture models to account for unobserved heterogeneity. Mixed 

Poisson distributions are discussed, including Negative Binomial (or Poisson-Gamma), 

Poisson-Inverse Gaussian and Poisson-LogNormal models. 

Section 1.5 presents the maximum likelihood estimation method. Large sample properties 

of the maximum likelihood estimators are discussed, and testing procedures are described. 

The large sample properties are particularly appealing to actuaries who usually deal with 

tens of thousands of observations in insurance portfolios. 

Section 1.6 gives numerical illustrations on the basis of a Belgian motor third party liability 

insurance portfolio. The observed claim frequency distribution is fitted using the Poisson 

distribution and various mixed Poisson probability distributions, and the optimal model is 

selected on the basis of appropriate goodness-of-fit tests. 

The final Section, 1.7, concludes Chapter 1 by providing suggestions for further reading 

and bibliographic notes about the models proposed in the actuarial literature for the annual 

number of claims. 

1.2 Probabilistic Tools 

1.2.1 Experiment and Universe 

Many everyday statements for actuaries take the form ‘the probability of A is p’, where A 

is some event (such as ‘the total losses exceed the threshold E 1 000 000’ or ‘the number of 

claims reported by a given policyholder is less than two’) and p is a real number between zero 

and one. The occurrence or nonoccurrence of A depends upon the chain of circumstances 

under consideration. Such a particular chain is called an experiment in probability; the result 

of an experiment is called its outcome and the set of all outcomes (called the universe) is 

denoted by . 

The word ‘experiment’ is used here in a very general sense to describe virtually any 

process for which all possible outcomes can be specified in advance and for which the actual 

outcome will be one of those specified. The basic feature of an experiment is that its outcome 

is not definitely known by the actuary beforehand. 

1.2.2 Random Events 

Random events are subsets of the universe associated with a given experiment. A random 

event is the mathematical formalization of an event described in words. It is random since


we cannot predict with certainty whether it will be realized or not during the experiment. 

For instance, if we are interested in the number of claims incurred by a policyholder of 

an automobile portfolio during one year, the experiment consists in observing the driving 

behaviour of this individual during an annual period, and the universe is simply the set 

0 1 2of the nonnegative integers. The random event A = ‘the policyholder reports 

at most one claim’ is identified with the subset 0 1 ⊂ . 

As usual, we use A∪B and A∩B to represent the union and the intersection, respectively, 

of any two subsets A and B of . The union of sets is defined to be the set that contains 

the points that belong to at least one of the sets. The intersection of sets is defined to be the 

set that contains the points that are common to all the sets. These set operations correspond 

to the ‘or’ and ‘and’ between sentences: A ∪ B is the event which is realized if A or B is 

realized and A ∩ B is the event realized if A and B are simultaneously realized during the 

experiment. We also define the difference between sets A and B, denoted as A B, asthe 

set of elements in A but not in B. Finally, A is the complementary event of A, defined as 

A; it is the set of points of that do not belong to A. This corresponds to the negation: 

A is realized if A is not realized during the experiment. In particular, =∅, where ∅ is the 

empty set. 

1.2.3 Sigma-Algebra 

One needs to specify a family of events to which probabilities can be ascribed in a 

consistent manner. The family has to be closed under standard operations on sets; indeed, 

given two events A and B in , we want A∪B, A∩B and A to still be events (i.e. still belong 

to ). Technically speaking, this will be the case if is a sigma-algebra. Recall that a family 

of subsets of the universe is called a sigma-algebra if it fulfills the three following 

properties: (i) ∈ , (ii) A ∈ ⇒ A ∈ , and (iii) A 1 A 2 A 3 ∈ ⇒ ⋃ i≥1 A i ∈ . 

The three properties (i)-(iii) are very natural. Indeed, (i) means that itself is an event 

(it is the event which is always realized). Property (ii) means that if A is an event, the 

complement of A is also an event. Finally, property (iii) means that the event consisting in 

the realization of at least one of the A i s is also an event. 

1.2.4 Probability Measure 

Once the universe has been equipped with a sigma-algebra of random events, a 

probability measure Pr can be defined on . The knowledge of Pr allows us to discuss the 

likelihood of the occurrence of events in . To be specific, Pr assigns to each random event 

A its probability PrA; PrA is the likelihood of realization of A. Formally, a probability 

measure Pr maps to 0 1, with Pr = 1, and is such that given A 1 A 2 A 3 ∈ 

which are pairwise disjoint, i.e., such that A i ∩ A j =∅if i ≠ j, 

[ ] 

⋃ 

Pr A i = ∑ PrA i 

i≥1 i≥1 

this technical property is usually referred to as the sigma-additivity of Pr. 

The properties assigned to Pr naturally follow from empirical evidence: if we were allowed 

to repeat an experiment a large number of times, keeping the initial conditions as equal


as possible, the proportion of times that an event A occurs would behave according to the 

definition of Pr. Note that PrA is then the mathematical idealization of the proportion of 

times A occurs. 

1.2.5 Independent Events 

Independence is a crucial concept in probability theory. It aims to formalize the intuitive 

notion of ‘not influencing each other’ for random events: we would like to give a precise 

meaning to the fact that the realization of an event does not decrease nor increase the 

probability that the other event occurs. Formally, two events A and B are said to be 

independent if the probability of their intersection equals the product of their respective 

probabilities, that is, if PrA ∩ B = PrAPrB. 

This definition is extended to more than two events as follows. The events in a family 

of events are independent if for every finite sequence A 1 A 2 A k of events in , 

[ ] 

⋂ k k∏ 

Pr A i = PrA i (1.1) 

i=1 

The concept of independence is very important in assigning probabilities to events. For 

instance, if two or more events are regarded as being physically independent, in the sense 

that the occurrence or nonoccurrence of some of them has no influence on the occurrence 

or nonoccurrence of the others, then this condition is translated into mathematical terms 

through the assignment of probabilities satisfying Equation (1.1). 

i=1 

1.2.6 Conditional Probability 

Independence is the exception rather than the rule. In any given experiment, it is often 

necessary to consider the probability of an event A when additional information about the 

outcome of the experiment has been obtained from the occurrence of some other event B. 

This corresponds to intuitive statements of the form ‘if B occurs then the probability of A is 

p’, where B can be ‘March is rainy’ and A ‘the claim frequency in motor insurance increases 

by 5 %’. This is called the conditional probability of A given B, and is formally defined as 

follows. If PrB > 0 then the conditional probability PrAB of A given B is defined to be 

PrAB = 

PrA ∩ B 

(1.2) 

PrB 

The definition of conditional probabilities through (1.2) is in line with empirical evidence. 

Repeating a given experiment a large number of times, PrAB is the mathematical 

idealization of the proportion of times A occurs in those experiments where B did occur, 

hence the ratio (1.2). 

It is easily seen that A and B are independent if, and only if, 

PrAB = PrAB = PrA (1.3)


Note that this interpretation of independence is much more intuitive than the definition given 

above: indeed the identity expresses the natural idea that the realization or not of B does not 

increase nor decrease the probability that A occurs. 

1.2.7 Random Variables and Random Vectors 

Often, actuaries are not interested in an experiment itself but rather in some consequences of 

its random outcome. For instance, they are more concerned with the amounts the insurance 

company will have to pay than with the particular circumstances which give rise to the 

claims. Such consequences, when real-valued, may be thought of as functions mapping 

into the real line . 

Such functions are called random variables provided they satisfy certain desirable 

properties, precisely stated in the following definition: A random variable X is a measurable 

function mapping to the real numbers, i.e., X→ is such that X −1 − x ∈ 

for any x ∈ , where X −1 − x = ∈ X ≤ x. In other words, the measurability 

condition X −1 − x ∈ ensures that the actuary can make statements like ‘X is less 

than or equal to x’ and quantify their likelihood. Random variables are mathematical 

formalizations of random outcomes given by numerical values. An example of a random 

variable is the amount of a claim associated with the occurrence of an automobile accident. 

A random vector X = X 1 X 2 X n T is a collection of n univariate random variables, 

X 1 , X 2 , , X n , say, defined on the same probability space Pr. Random vectors are 

denoted by bold capital letters. 

1.2.8 Distribution Functions 

In many cases, neither the universe nor the function X need to be given explicitly. 

The practitioner has only to know the probability law governing X or, in other words, its 

distribution. This means that he is interested in the probabilities that X takes values in 

appropriate subsets of the real line (mainly intervals). 

To each random variable X is associated a function F X called the distribution function 

of X, describing the stochastic behaviour of X. Of course, F X does not indicate what is the 

actual outcome of X, but shows how the possible values for X are distributed (hence its 

name). More precisely, the distribution function of the random variable X, denoted as F X ,is 

defined as 

F X x = PrX −1 − x ≡ PrX ≤ x x ∈ 

In other words, F X x represents the probability that the random variable X assumes a 

value that is less than or equal to x. IfX is the total amount of claims generated by some 

policyholder, F X x is the probability that this policyholder produces a total claim amount 

of at most E x. The distribution function F X corresponds to an estimated physical probability 

distribution or a well-chosen subjective probability distribution. 

Any distribution function F has the following properties: (i) F is nondecreasing, i.e. 

Fx ≤ Fy if x


i.e. lim h↘0 Fx+h = Fx, and (v) Pra


1.3 Poisson Distribution 

1.3.1 Counting Random Variables 

A discrete random variable X assumes only a finite (or countable) number of values. The 

most important subclass of nonnegative discrete random variables is the integer case, where 

each observation (outcome) is an integer (typically, the number of claims reported to the 

company). More precisely, a counting random variable N is valued in 0 1 2. Its 

stochastic behaviour is characterized by the set of probabilities p k k= 0 1assigned 

to the nonnegative integers, where p k = PrN = k. The (discrete) distribution of N associates 

with each possible integer value k = 0 1 2 the probability p k that it will be the observed 

value. The distribution must satisfy the two conditions: 

p k ≥ 0 for all k and 

+∑ 

k=0 

p k = 1 

i.e. the probabilities are all nonnegative real numbers lying between zero (impossibility) and 

unity (certainty), and their sum must be unity because it is certain that one or other of the 

values will be observed. 

1.3.2 Probability Mass Function 

In discrete distribution theory the p k s are regarded as values of a mathematical function, i.e. 

p k = pk k = 0 1 2 (1.4) 

where p· is a known function depending on a set of parameters . The function p· 

defined in (1.4) is usually called the probability mass function. Different functional forms 

lead to different discrete distributions. This is a parametric model. 

The distribution function F N → 0 1 of N gives for any real threshold x, the probability 

for N to be smaller than or equal to x. The distribution function F N of N is related to the 

probability mass function via 

x 

∑ 

F N x = p k x∈ + 

k=0 

where p k is given by Expression (1.4) and where x denotes the largest integer n such that 

n ≤ x (it is thus the integer part of x). Considering (1.4), F N also depends on . 

1.3.3 Moments 

There are various useful and important quantities associated with a probability distribution. 

They may be used to summarize features of the distribution. The most familiar and widely 

used are the moments, particularly the mean 

EN = 

+∑ 

k=0 

kp k


which is given by the sum of the products of all the possible outcomes multiplied by their 

probability, and the variance 

VN = EN − EN 2 = 

+∑ 

k=0 

( 

k − EN 

) 2pk 

 

which is given by the sum of the products of the squared differences between all the 

outcomes and the mean, multiplied by their probability. Expanding the squared difference 

in the definition of the variance, it is easily seen that the variance can be reformulated as 

VN = E [ N 2 − 2N EN + EN 2] = EN 2 − EN 2 

which provides a convenient way to compute the variance as the difference between the 

second moment EN 2 and the square EN 2 of the first moment EN. The mean and the 

variance are commonly denoted as and 2 , respectively. Considering (1.4), both EN and 

VN are functions of , that is, 

EN = and VN = 2 

The mean is used as a measure of the location of the distribution: it is an average of 

the possible outcomes 0 1weighted by the corresponding probabilities p 0 p 1 The 

variance is widely used as a measure of the spread of the distribution: it is a weighted average 

of the squared distances between the outcomes 0 1and the expected value EN. Recall 

that E· is a linear operator. From the properties of E·, it is easily seen that the variance 

V· is shift-invariant and additive for independent random variables. 

The degree of asymmetry of the distribution of a random variable N is measured by its 

skewness, denoted as N. The skewness is the third central moment of N , normalized by 

its variance raised to the power 3/2 (in order to get a number without unit). Precisely, the 

skewness of N is given by 

N = EN − EN3 

 

VN 3/2 

For any random variable N with a symmetric distribution the skewness N is zero. 

Positively skewed distributions tend to concentrate most of the probability mass on small 

values, but the remaining probability is stretched over a long range of larger values. 

There are other related sets of constants, such as the cumulants, the factorial moments, 

the factorial cumulants, etc., which may be more convenient to use in some circumstances. 

For details about these constants, we refer the reader, e.g., to Johnson ET AL. (1992). 

1.3.4 Probability Generating Function 

In principle all the theoretical properties of the distribution can be derived from the probability 

mass function. There are, however, several other functions from which exactly the same 

information can be derived. This is because the functions are all one-to-one transformations


of each other, so each characterizes the distribution. One particularly useful function is the 

probability generating function, which is defined as 

N z = Ez N = 

+∑ 

k=0 

p k z k 0


The probability generating function of N 1 + N 2 is easily obtained from 

N1 +N 2 

z = Ez N 1+N 2 

= Ez N 1 

Ez N 2 

= N1 

z N2 

z 

since the mutual independence of N 1 and N 2 ensures that 

Ez N 1+N 2 

= 

= 

+∑ 

+∑ 

k 1 =0 k 2 =0 

+∑ 

k 1 =0 

z k 1+k 2 

PrN 1 = k 1 PrN 2 = k 2 

z k 1 

PrN 1 = k 1 

= Ez N 1 

Ez N 2 

 

+∑ 

k 2 =0 

z k 2 

PrN 2 = k 2 

Summing random variables thus corresponds to a convolution product for probability mass 

functions and to regular products for probability generating functions. An expansion of 

N1 

N2 

· as a series in powers of z then gives the probability mass function of N 1 + N 2 , 

usually in a much easier way than computing the convolution product of the probability 

mass functions of N 1 and N 2 . 

1.3.6 From the Binomial to the Poisson Distribution 

Bernoulli Distribution 

The Bernoulli distribution is an extremely simple and basic distribution. It arises from what is 

known as a Bernoulli trial: a single observation is taken where the outcome is dichotomous, 

e.g., success or failure, alive or dead, male or female, 0 or 1. The probability of success is 

q. The probability of failure is 1 − q. 

If N is Bernoulli distributed with success probability q, which is denoted as N ∼ erq, 

we have 

⎧ 

⎪⎨ 1 − q if k = 0 

pkq = q if k = 1 

⎪⎩ 

0 otherwise. 

There is thus just one parameter: the success probability q. The mean is 

and the variance is 

The probability generating function is 

EN = 0 × 1 − q + 1 × q = q (1.6) 

VN = EN 2 − q 2 = q − q 2 = q1 − q (1.7) 

N z = 1 − q × z 0 + q × z 1 = 1 − q + qz (1.8) 

It is easily seen that N 0 = p0q and 

N ′ 0q = p1q, as it should be.


Binomial Distribution 

The Binomial distribution describes the outcome of a sequence of n independent Bernoulli 

trials, each with the same probability q of success. The probability that success is the outcome 

in exactly k of the trials is 

( n 

pkn q = q 

k) 

k 1 − q n−k k= 0 1n (1.9) 

and 0 otherwise. Formula (1.9) defines the Binomial distribution. There are now two 

parameters: the number of trials n (also called the exponent, or size) and the success 

probability q. Henceforth, we write N ∼ inn q to indicate that N is Binomially 

distributed, with size n and success probability q. 

Moments of the Binomial Distribution 

The mean of N ∼ inn q is 

EN = 

n∑ 

k=1 

= nq 

n! 

k − 1!n − k! qk 1 − q n−k 

n∑ 

PrM = k − 1 = nq (1.10) 

where M ∼ inn − 1q. Furthermore, with M as defined before, 

so that the variance is 

EN 2 = 

n∑ 

k=1 

= nq 

k=1 

n! 

k − 1!n − k! kqk 1 − q n−k 

n∑ 

k PrM = k − 1 

k=1 

= nn − 1q 2 + nq 

VN = EN 2 − nq 2 = nq1 − q (1.11) 

We immediately observe that the Binomial distribution is underdispersed, i.e. its variance is 

smaller than its mean : VN = nq1 − q ≤ EN = nq. 

Probability Generating Function and Closure under Convolution for the 

Binomial Distribution 

The probability generating function of N ∼ inn q is 

( ) n∑ n 

N z = qz k 1 − q n−k = 1 − q + qz n (1.12) 

k=0 

k 

Note that Expression (1.12) is the Bernoulli probability generating function (1.8), raised to 

the nth power. This was expected since the Binomial random variable N can be seen as the


sum of n independent Bernoulli random variables with equal success probability q. This also 

explains why (1.10) is equal to n times (1.6) and why (1.11) is equal to n times (1.7) (in the 

latter case, since the variance is additive for independent random variables). 

From (1.12), we also see that having independent random variables N 1 ∼ inn 1 q and 

N 2 ∼ inn 2 q, the sum N 1 + N 2 is still Binomially distributed. This comes from the fact 

that the probability generating function of N 1 + N 2 is 

N1 +N 2 

z = N1 

z N2 

z = 1 − q + qz n 1+n 2 

so that N 1 + N 2 ∼ inn 1 + n 2 q. Note that this is not the case if the success probabilities 

differ. 

Limiting Form of the Binomial Distribution 

When n becomes large, (1.9) may be approximated by a Normal distribution according to 

the De Moivre–Laplace theorem. The approximation is much improved when a continuity 

correction is applied. The Poisson distribution can be obtained as a limiting case of the 

Binomial distribution when n tends to infinity together with q becoming very small. 

Specifically, let us assume that N n ∼ inn /n and let n tend to +. The probability 

mass at 0 then becomes 

( 

PrN n = 0 = 1 − ) n 

→ exp− as n →+ 

n 

To get the probability masses on the positive integers, let us compute the ratio 

from which we conclude 

PrN n = k + 1 

PrN n = k 

= 

n−k 

k+1 n 

1 − n 

→ 

as n →+ 

k + 1 

lim PrN n = k = exp− k 

k= 0 1 2 

n→+ k! 

Poisson Distribution 

The Poisson random variable takes its values in 0 1and has probability mass function 

pk = exp− k 

k= 0 1 (1.13) 

k! 

Having a counting random variable N , we denote as N ∼ oi the fact that N is Poisson 

distributed with parameter . The Poisson distribution occupies a central position in discrete 

distribution theory analogous to that occupied by the Normal distribution in continuous 

distribution theory. It also has many practical applications. 

The Poisson distribution describes events that occur randomly and independently in space 

or time. A classic example in physics is the number of radioactive particles recorded by 

a Geiger counter in a fixed time interval. This property of the Poisson distribution means 

that it can act as a reference standard when deviations from pure randomness are suspected. 

Although the Poisson distribution is often called the law of small numbers, there is no need


for = nq to be small. It is the largeness of n and the smallness of q = /n that are 

important. However most of the data sets analysed in the literature show a small frequency. 

This will be the case with motor data sets in insurance applications. 

Moments of the Poisson Distribution 

If N ∼ oi, then its expected value is given by 

Moreover, 

EN = 

+∑ 

k=1 

= exp− 

k exp− k 

k! 

+∑ k+1 

k=0 

k! 

= (1.14) 

EN 2 = 

+∑ 

k=1 

so that the variance of N is equal to 

= exp− 

k 2 exp− k 

k! 

+∑ 

k=0 

k + 1 k+1 

= + 2 

k! 

VN = EN 2 − 2 = (1.15) 

Considering Expressions (1.14) and (1.15), we see that both the mean and the variance of 

the Poisson distribution are equal to , a phenomenon termed as equidispersion. 

The skewness of N ∼ oi is 

N = √ 1 (1.16) 

 

Clearly, N decreases with . For small values of the distribution is very skewed 

(asymmetric) but as increases it becomes less skewed and is nearly symmetric by = 15. 

Probability Generating Function and Closure Under Convolution for the 

Poisson Distribution 

The probability generating function of the Poisson distribution has a very simple form. 

Coming back to the Equation (1.5) defining N and replacing the p k s with their expression 

(1.13) gives 

+∑ 

N z = exp− zk = exp ( z − 1 ) (1.17) 

k=0 

k! 

This shows that the Poisson distribution is closed under convolution. Having independent 

random variables N 1 ∼ oi 1 and N 2 ∼ oi 2 , the probability generating function of 

the sum N 1 + N 2 is 

N1 +N 2 

z = N1 

z N2 

z = exp ( 1 z − 1 ) exp ( 2 z − 1 ) = exp ( 1 + 2 z − 1 ) 

so that N 1 + N 2 ∼ oi 1 + 2 .


The sum of two independent Poisson distributed random variables is also Poisson 

distributed, with parameter equal to the sum of the Poisson parameters. This property 

obviously extends to any number of terms, and the Poisson distribution is said to be closed 

under convolution (i.e. the convolution of Poisson distributions is still Poisson). 

1.3.7 Poisson Process 

Definition 

Recall that a stochastic process is a collection of random variables Nt t ∈ indexed by 

a real-valued parameter t taking values in the index set . Usually, represents a set of 

observation times. In this book, we will be interested in continuous-time stochastic processes 

where = + . 

A stochastic process Nt t ≥ 0 is said to be a counting process if t ↦→ Nt is rightcontinuous 

and Nt−Nt− is 0 or 1. Intuitively speaking, Nt represents the total number 

of events that have occurred up to time t. Such a process enjoys the following properties: 

(i) Nt ≥ 0, (ii) Nt is integer valued, (iii) if s0if 

(i) the process has stationary increments, that is, 

PrNt + − Nt = k = PrNs + − Ns = k 

for any integer k, instants s ≤ t and increment >0. 

(ii) the process has independent increments, that is, for any integer k>0 and instants 

0 ≤ t 0


claims in a sufficiently small time interval is negligible when compared to the probability 

that he reports zero or only one claim. 

Link with the Poisson Distribution 

The Poisson process is intimately linked to the Poisson distribution, as precisely stated in 

the next result. 

Property 1.1 For any Poisson process, the number of events in any interval of length t is 

Poisson distributed with mean t, that is, for all s t ≥ 0, 

PrNt + s − Ns = n = exp−t tn n= 0 1 2 

n! 

Proof Without loss of generality, we only have to prove that Nt ∼ oit. For any 

integer k, let us denote p k t = PrNt = k, t ≥ 0. The announced result for k = 0 comes 

from 

p 0 t + t = PrNt = 0 and Nt + t − Nt = 0 

= PrNt = 0 PrNt + t − Nt = 0 

= p 0 tp 0 t 

= p 0 t ( 1 − t + ot ) 

where the joint probability factors into two terms since the increments of a Poisson process 

are independent random variables. This gives 

p 0 t + t − p 0 t 

t 

Taking the limit for t ↘ 0 yields 

=−p 0 t + ot p 

t 0 t 

d 

dt p 0t =−p 0 t 

This differential equation with the initial condition p 0 0 = 1 admits the solution 

p 0 t = exp−t (1.18) 

which is in fact the oit probability mass function evaluated at the origin. 

For k ≥ 1, let us write 

p k t + t = PrNt + t = k 

= PrNt + t = kNt = k PrNt = k 

+ PrNt + t = kNt = k − 1 PrNt = k − 1 

k∑ 

+ PrNt + t = kNt = k − j PrNt = k − j 

j=2


= PrNt + t − Nt = 0 PrNt = k 

+ PrNt + t − Nt = 1 PrNt = k − 1 

k∑ 

+ PrNt + t − Nt = j PrNt = k − j 

j=2 

Since the increments of a Poisson process are independent random variables, we can write 

k∑ 

p k t + t = p 0 tp k t + p 1 tp k−1 t + p j tp k−j t 

j=2 

= 1 − tp k t + tp k−1 t + ot 

This gives 

p k t + t − p k t 

t 

Taking the limit for t ↘ 0 yields as above 

= p k−1 t − p k t + ot 

t 

d 

dt p kt = p k−1 t − p k t k ≥ 1 (1.19) 

Multiplying by z k each of the equation (1.19), and summing over k gives 

+∑ 

k=0 

( d 

dt p kt) 

z k = z 

+∑ 

k=0 

p k tz k − 

+∑ 

k=0 

p k tz k (1.20) 

Denoting as t the probability generating function of Nt, equation (1.20) becomes 

With the condition 0 z = 1, Equation (1.21) has solution 

 

t tz = z − 1 t z (1.21) 

t z = exptz − 1 

where we recognize the oit probability generating function (1.17). This ends the proof. 

□ 

When the hypotheses behind a Poisson process are verified, the number N1 of claims 

hitting a policy during a period of length 1 is Poisson distributed with parameter . So, 

a counting process Nt t ≥ 0, starting from N0 = 0, is a Poisson process with rate 

>0if


(i) The process has independent increments 

(ii) The number of events in any interval of length t follows a Poisson distribution with 

mean t (therefore it has stationary increments), i.e. 

PrNt + s − Ns = k = exp−t tk k= 0 1 2 

k! 

Exposure-to-Risk 

The Poisson process setting is useful when one wants to analyse policyholders that have 

been observed during periods of unequal lengths. Assume that the claims occur according to 

a Poisson process with rate . If the policyholder is covered by the company for a period of 

length d then the number N of claims reported to the company has probability mass function 

PrN = k = exp−d dk k= 0 1 

k! 

that is, N ∼ oid. In actuarial studies, d is referred to as the exposure-to-risk. We see 

that d simply multiplies the annual expected claim frequency in the Poisson model. 

Time Between Accidents 

The Poisson distribution arises for events occurring randomly and independently in time. 

Indeed, denote as T 1 T 2 the times between two consecutive accidents. Assume further 

that these accidents occur according to a Poisson process with rate . Then, the T k s are 

independent and identically distributed and 

PrT k >t= PrT 1 >t= PrN t = 0 = exp−t 

so that T 1 T 2 have a common Negative Exponential distribution. 

Note that in this case, the equality 

PrT k >s+ tT k >s= PrT k >s+ t 

PrT k >s 

= PrT k >t 

holds for any s and t ≥ 0. It is not difficult to see that this memoryless property is related 

to the fact that the increments of the process Nt t ≥ 0 are independent and stationary. 

Assuming that the claims occur according to a Poisson process is thus equivalent to assuming 

that the time between two consecutive claims has a Negative Exponential distribution. 

Nonhomogeneous Poisson Process 

A generalization of the Poisson process is obtained by letting the rate of the process 

vary with time. We then replace the constant rate by a function t ↦→ t of time t 

and we define the nonhomogeneous Poisson process with rate ·. The Poisson process 

defined above (with a constant rate) is then termed as the homogeneous Poisson process. 

A counting process Nt t ≥ 0 starting from N 0 = 0 is said to be a nonhomogeneous 

Poisson process with rate ·, where t ≥ 0 for all t ≥ 0, if it satisfies the following 

conditions:


(i) The process Nt t ≥ 0 has independent increments, and 

(ii) 

⎧ 

⎪⎨ 1 − th + oh if k = 0 

PrNt + h − Nt = k = th + oh if k = 1 

⎪⎩ 

oh if k ≥ 2 

The only difference between the nonhomogeneous Poisson process and the homogeneous 

Poisson process is that the rate may vary with time, resulting in the loss of the stationary 

increment property. 

For any nonhomogeneous Poisson process Nt t ≥ 0, the number of events in the 

interval s t, s ≤ t, is Poisson distributed with mean 

that is 

ms t = 

∫ t 

s 

udu 

PrNt − Ns = k = exp ( − ms t )( ms t ) k 

k= 0 1 

k! 

In the homogeneous case, we obviously have ms t = t − s. 

1.4 Mixed Poisson Distributions 

1.4.1 Expectations of General Random Variables 

Mixed Poisson distributions involve expectations of Poisson probabilities with a random 

parameter. Therefore, we need to be able to compute expectations with respect to general 

distribution functions. 

Continuous probability distributions are widely used in probability and statistics when 

the underlying random phenomenon is measured on a continuous scale. If the distribution 

function is a continuous function, the associated probability distribution is called a continuous 

distribution. Note that in this case, 

PrX = x = lim 

h↘0 

Prx


The interpretation of the probability density function is that 

Prx ≤ X ≤ x + h ≈ fxh for small h>0 

That is, the probability that a random variable, with an absolutely continuous probability 

distribution, takes a value in a small interval of length h is given by the probability density 

function times the length of the interval. 

A general type of distribution function is a combination of the discrete and (absolutely) 

continuous cases, being continuous apart from a countable set of exception points 

x 1 x 2 x 3 with positive probabilities of occurrence, causing jumps in the distribution 

function at these points. Such a distribution function F X can be represented as 

F X x = 1 − pF c 

X x + pF d 

X x x ∈ (1.22) 

for some p ∈ 0 1, where F c 

X is a continuous distribution function and F d 

X 

distribution function with support d 1 d 2 . 

Let us assume that F X is of the form (1.22) with 

pF d 

X t = ∑ ( 

) 

F X d n − F X d n − = ∑ PrX = d n 

d n ≤t 

d n ≤t 

is a discrete 

where d 1 d 2 denotes the set of discontinuity points and 

Then, 

∫ t 

1 − pF c 

X t = F X t − pF d 

X t = f c 

X xdx 

− 

EX = ∑ ) 

d n 

(F X d n − F X d n − + 

n≥1 

∫ + 

− 

xf c 

X xdx (1.23) 

= 

∫ + 

− 

xdF X x 

where the differential of F X , denoted as dF X , is defined as 

{ 

FX d n − F X d n − if x = d n 

dF X x = 

f c 

X xdx otherwise 

This unified notation allows us to avoid tedious repetitions of statements like ‘the proof 

is given for continuous random variables; the discrete case is similar’. A very readable 

introduction to differentials and Riemann–Stieltjes integrals can be found in Carter & Van 

Brunt (2000). 

1.4.2 Heterogeneity and Mixture Models 

Definition 

Mixture models are a discrete or continuous weighted combination of distributions aimed 

at representing a heterogeneous population comprised of several (two or more) distinct subpopulations. 

Such models are typically used when a heterogeneous population of sampling


units consists of several sub-populations within each of which a relatively simpler model 

applies. The source of heterogeneity could be gender, age, geographical area, etc. 

Discrete Mixtures 

In order to define a mixture model mathematically, suppose the distribution of N can be 

represented by a probability mass function of the form 

PrN = k = pk = q 1 p 1 k 1 +···+q p k (1.24) 

where = q T T T , q T = q 1 q , T = 1 . The model is usually referred 

to as a discrete (or finite) mixture model. Here j is a (vector) parameter characterizing the 

probability mass function p j · j and the q j s are mixing weights. 

Example 1.1 A particular example of finite mixture is the zero-inflated distribution. It has 

been observed empirically that counting distributions often show excess of zeros against the 

Poisson distribution. In order to accommodate this feature, a combination of the original 

distribution p k k= 0 1(be it Poisson or not) together with the degenerate distribution 

with all probability concentrated at the origin, gives a finite mixture with 

PrN = 0 = + 1 − p 0 

PrN = k = 1 − p k k= 1 2 

A mixture of this kind is usually referred to as zero-inflated, zero-modified or as a distribution 

with added zeros. 

Model (1.24) allows each component probability mass function to belong to a different 

parametric family. In most applications, a common parametric family is assumed and thus 

the mixture model takes the following form 

pk = q 1 pk 1 +···+q pk (1.25) 

which we assume to hold in the sequel. The mixing weight q can be regarded as a discrete 

probability function over , describing the variation in the choice of across the population 

of interest. 

This class of mixture models includes mixtures of Poisson distributions. Such a mixture 

is adequate to model count data (number of claims reported to an insurance company, 

number of accidents caused by an insured driver, etc.) where the components of the 

mixture are Poisson distributions with mean j . In that respect, (1.25) means that there 

are categories of policyholders, with annual expected claim frequencies 1 2 , 

respectively. The proportion of the portfolio in the different categories is q 1 q 2 q , 

respectively. Considering a given policyholder, the actuary does not know to which category 

he belongs, but the probability that he comes from category j is q j . The probability mass 

function of the number of claims reported by this insured driver is thus a weighted average 

of the probability mass functions associated with the k categories.


Continuous Mixtures 

Multiplying the number of categories in (1.25) often leads to a dramatic increase in the 

number of parameters (the q j s and the j s). For large , it is therefore preferable to switch 

to a continuous mixture, where the sum in (1.25) is replaced with an integral with respect to 

some simple parametric continuous probability density function. 

Specifically, if we allow to be continuous with probability density function g·, the 

finite mixture model suggested above is replaced by the probability mass function 

∫ 

pk = pkgd 

which is often referred to as a mixture distribution. When g· is modelled without parametric 

assumptions, the probability mass function p· is a semiparametric mixture model. Often in 

actuarial science, g· is taken from some parametric family, so that the resulting probability 

mass function is also parametric. 

Mixed Poisson Model for the Number of Claims 

The Poisson distribution often poorly fits observations made in a portfolio of policyholders. 

This is in fact due to the heterogeneity that is present in the portfolio: driving abilities vary 

from individual to individual. Therefore it is natural to multiply the mean frequency of 

the Poisson distribution by a positive random effect . The frequency will vary within the 

portfolio according to the nonobservable random variable . Obviously we will choose 

such that E = 1 because we want to obtain, on average, the frequency of the portfolio. 

Conditional on , we then have 

PrN = k = = pk = exp− k k= 0 1 (1.26) 

k! 

where p· is the Poisson probability mass function, with mean . The interpretation we 

give to this model is that not all policyholders in the portfolio have an identical frequency 

. Some of them have a higher frequency ( with ≥ 1), others have a lower frequency 

( with ≤ 1). Thus we use a random effect to model this empirical observation. 

The annual number of accidents caused by a randomly selected policyholder of the portfolio 

is then distributed according to a mixed Poisson law. In this case, the probability that a 

randomly selected policyholder reports k claims to the company is obtained by averaging the 

conditional probabilities (1.26) with respect to . In general, is not discrete nor continuous 

but of mixed type. The probability mass function associated with mixed Poisson models is 

defined as 

PrN = k = E [ pk ] = 

∫ 

0 

exp− k dF 

k! (1.27) 

where F denotes the distribution function of , assumed to fulfill F 0 = 0. The mixing 

distribution described by F represents the heterogeneity of the portfolio of interest; dF 

is often called the structure function. It is worth mentioning that the mixed Poisson model 

(1.27) is an accident-proneness model: it assumes that a policyholder’s mean claim frequency 

does not change over time but allows some insured persons to have higher mean claim 

frequencies than others. We will say that N is mixed Poisson distributed with parameter 

and risk level , denoted as N ∼ oi when it has probability mass function (1.27).


Remark 1.1 Note that a better notation would have been oi F instead of 

oi since only the distribution function of matters to define the associated 

Poisson mixture. We have nevertheless opted for oi for simplicity. 

Note that the condition E = 1 ensures that when N ∼ oi 

EN = 

∫ +∑ 

0 

k=0 

= E = 

k exp− k dF 

k! 

or, more briefly, 

[ ] 

EN = E EN = E = (1.28) 

In (1.28), E· means that we take an expected value considering as a constant. We 

then average with respect to all the random components, except . Consequently, E· is 

a function of . Given , N is Poisson distributed with mean so that EN = . The 

mean of N is finally obtained by averaging EN with respect to . The expectation of N 

given in (1.28) is thus the same as the expectation of a oi distributed random variable. 

Taking the heterogeneity into account by switching from the oi to the oi 

distribution has no effect on the expected claim number. 

1.4.3 Mixed Poisson Process 

The Poisson processes are suitable models for many real counting phenomena but they are 

insufficient in some cases because of the deterministic character of their intensity function. 

The doubly stochastic Poisson process (or Cox process) is a generalization of the Poisson 

process when the rate of occurrence is influenced by an external process such that the 

rate becomes a random process. So, the rate, instead of being constant (homogeneous 

Poisson process) or a deterministic function of time (nonhomogeneous Poisson process) 

becomes itself a stochastic process. The only restriction on the rate process is that it 

has to be nonnegative. Mixed Poisson distributions are linked to mixed Poisson processes 

in the same way that the Poisson distribution is associated with the standard Poisson 

process. 

Specifically, let us assume that given = , Nt t ≥ 0 is a homogeneous Poisson 

process with rate . Then Nt t ≥ 0 is a mixed Poisson process, and for any s t ≥ 0, 

the probability that k events occur during the time interval s t is 

PrNt + s − Ns = k = 

= 

∫ 

0 

∫ 

0 

PrNt + s − Ns = k = dF 

exp−t tk dF 

k! 

that is, Nt + s − Ns ∼ oit . Note that, in contrast to the Poisson process, mixed 

Poisson processes have dependent increments. Hence, past number of claims reveal future 

number of claims in this setting (in contrast to the Poisson case).


1.4.4 Properties of Mixed Poisson Distributions 

Moments and Overdispersion 

If N ∼ oi then its second moment is 

so that 

EN 2 = 

∫ + 

0 

+ 2 2 dF = E + 2 E 2 

VN = E + 2 E 2 − 2( E ) 2 

= + 2 V (1.29) 

It is then easily seen that the variance of N exceeds its mean, that is, 

VN = + 2 V ≥ = EN (1.30) 

Therefore, unless is degenerated in 1, we observe that mixed Poisson distributions are 

overdispersed: the variance exceeds the mean. The skewness can be expressed as 

( 

) 

1 

N = 

3VN − 2EN + 

VN − EN 2 

√ (1.31) 

VN 3/2 V EN 

Shaked’s Two Crossings Theorem 

Recall that EX ≥ EX for any random variable X and convex function . This 

inequality, known as the Jensen inequality, ensures that if N ∼ oi then 

PrN = 0 = 

∫ 

0 

( 

exp−dF ≥ exp − 

∫ 

0 

) 

dF = exp− 

showing that mixed Poisson distributions have an excess of zeros compared to Poisson 

distributions with the same mean. This is in line with empirical studies, where actuaries often 

observe more policyholders producing 0 claims than the number predicted by the Poisson 

model. 

The following result that has been proved by Shaked (1980) reinforces this straightforward 

conclusion. 

Property 1.2 Let N be mixed Poisson distributed with mean EN = . Then there exist 

two integers 0 ≤ k 0


Shaked’s Two Crossings Theorem tells us (i) that the mixed Poisson distribution has an 

excess of zeros compared to the Poisson distribution with the same mean and (ii) that the 

mixed Poisson distribution has a thicker right tail than the Poisson distribution with the 

same mean. 

Probability Generating Function 

The probability generating function of Poisson mixtures is closely related to the moment 

generating function of the underlying random effect. Moment generating functions are a 

widely used tool in many statistics texts, and also in actuarial mathematics. They serve 

to prove statements about convolutions of distributions, and also about limits. Recall that 

the moment generating function of the nonnegative random variable X, denoted as M X ,is 

given by 

M X t = EexptX t > 0 

It is interesting to mention that M X characterizes the probability distribution of X, i.e. the 

information contained in F X and M X is equivalent. 

If there exists h>0 such that M X t exists and is finite for 0


From (1.33), we see that the knowledge of the mixed Poisson distribution oi 

is equivalent to the knowledge of F . The mixed Poisson distributions are thus identifiable, 

that is, having N 1 ∼ oi 1 and N 2 ∼ oi 2 then N 1 and N 2 are identically 

distributed if, and only if, 1 and 2 are identically distributed. 

1.4.5 Negative Binomial Distribution 

Gamma Distribution 

Recall that a random variable X is distributed according to the two-parameter Gamma 

distribution, which will henceforth be denoted as X ∼ am , if its probability density 

function is given by 

fx = x−1 exp−x 

x>0 (1.34) 

 

Note that when = 1, the Gamma distribution reduces to the Negative Exponential one 

(which is denoted as X ∼ xp) with probability density function 

fx = exp−x 

x > 0 

The distribution function F of X can be expressed in terms of the incomplete Gamma 

function. Specifically, if X ∼ am , then Fx = x. 

Probability Mass Function 

The Negative Binomial distribution is a widely used alternative to the Poisson distribution for 

handling count data when the variance is appreciably greater than the mean (this condition, 

known as overdispersion, is frequently met in practice, as discussed above). 

There are several models that lead to the Negative Binomial distribution. A classic example 

arises from the theory of accident proneness which was developed after Greenwood & 

Yule (1920). This theory assumes that the number of accidents suffered by an individual 

is Poisson distributed, but that the Poisson mean (interpreted as the individual’s accident 

proneness) varies between individuals in the population under study. If the Poisson mean 

is assumed to be Gamma distributed, then the Negative Binomial is the resultant overall 

distribution of accidents per individual. 

Specifically, completing (1.26)–(1.27) with ∼ ama a, that is, with probability 

density function 

f = 1 

a aa a−1 exp−a > 0 (1.35) 

yields the Negative Binomial probability mass function 

( ) 

a + k − 1 ···a a a ( ) d k 

PrN = k = 

k! a + d a + d 

( ) 

a + k a a ( ) d k 

= k= 0 1 2 

ak! a + d a + d


where is the annual expected claim number and d is the length of the observation period 

(the exposure-to-risk). The probability mass function can be expressed using the generalized 

binomial coefficient: 

( ) 

a + k a a ( ) d k 

PrN = k = 

ak + 1 a + d a + d 

( )( ) a + k − 1 a a ( ) d k 

= 

k= 0 1 2 

k a + d a + d 

Henceforth, we write N ∼ ina d to indicate that N obeys the Negative Binomial 

distribution with parameters a and d. This model has been applied to retail purchasing, 

absenteeism, doctor’s consultations, amongst many others. 

Moments 

If X ∼ am , its mean is EX = / and its variance is VX = / 2 .IfN ∼ 

ina d then the mean is EN = d and the variance is VN = d+d 2 /a according 

to (1.29). It can be shown that / √ V = 2, in (1.31) for the Negative Binomial 

distribution. 

Probability Generating Function 

If X ∼ am , its moment generating function is 

( 

Mt = 1 − 

) t − 

if t< (1.36) 

The probability generating function of N ∼ ina d is 

( 

) 

a 

a 

N z = 

(1.37) 

a − dz − 1 

This result comes from (1.33) together with (1.36). 

True and Apparent Contagion 

Apparent contagion arises from the recognition that sampled individuals come from a 

heterogeneous population in which individuals have a constant but different propensity to 

experience accidents. A given individual may have a high (or low) propensity for accidents 

but occurrence of an accident does not make it more (or less) likely that another accident will 

occur. However, aggregation across heterogeneous individuals may generate a misleading 

statistical finding which suggests that occurrence of an accident increases the probability of 

another accident; the observed but persistent heterogeneity can be misinterpreted as due to 

a strong serial dependence. 

True contagion refers to dependence between the occurrences of successive events. The 

occurrence of an event, such as an accident or illness, may change the probability of 

subsequent occurrences of similar events. True positive contagion implies that the occurrence 

of an event shortens the expected waiting time to the next occurrence of the event. 

The alleged phenomenon of accident proneness can be interpreted in terms of true 

contagion as suggesting that an individual who has experienced an accident is more likely


to experience another accident. In a longitudinal setting, actual and future outcomes are 

directly influenced by past values, and this causes a substantial change over time in the 

corresponding distribution. 

Since with event count data we only observe the total number of events at the end 

of the period, contagion, like heterogeneity, is an unobserved, within-observation process. 

For research problems where both heterogeneity and contagion are plausible, the different 

underlying processes are not distinguishable with aggregate count data because they both 

lead to the same probability distribution for the counts. One can still use this distribution to 

derive fully efficient and consistent estimates, but this analysis will only be suggestive of 

the underlying process. 

Poisson Limiting Form 

The Negative Binomial distribution has a Poisson limiting form if V = 1 → 0. This result 

a 

can be recovered from the sequence of the probability generating functions, noting that 

( 

lim 

a↗ 

a 

a − d1 − z 

) a ( 

= lim 1 − d ) −a 

a↗ a 1 − z = exp−d1 − z 

that is seen to converge to the probability generating function of the Poisson distribution 

with parameter d. 

Derivation as a Compound Poisson Distribution 

A different type of heterogeneity occurs when there is clustering. If it is assumed that 

the number of clusters is Poisson distributed, but the number of individuals in a cluster is 

distributed according to the Logarithmic distribution, then the overall distribution is Negative 

Binomial. In an actuarial context, this amounts to recognizing that several vehicles can 

be involved in the same accident, each of the insured drivers filing a claim. Therefore, a 

single accident may generate several claims. If the number of claims per accident follows 

a Logarithmic distribution, and the number of accidents over the time interval of interest 

follows a Poisson distribution, then the total number of claims for the time interval can be 

modelled with the Negative Binomial distribution. 

Let us formally establish this result. Recall that the random variable M has a Logarithmic 

distribution if 

PrM = k = 

k 

−k ln1 − 

k= 1 2 

where 0


The random variable N just defined has a compound Poisson distribution. The probability 

generating function of a compound distribution is given by 

N z = Ez M 1+···+M K 

 

= 

= 

+∑ 

k=0 

+∑ 

k=0 

PrK = kEz M 1+···+M k 

 

PrK = k ( M z ) k 

= K 

( 

M z ) (1.38) 

Note that formula (1.38) is true in general for compound distributions. Replacing K and 

M with their expressions gives the probability generating function of N 

( 

) 

N z = exp − 1 − M z 

( ) 1 − −/ ln1− 

= 

 

1 − z 

It can be checked that the probability generating function N corresponds to the probability 

generating function (1.37) of a Negative Binomial distribution with d = 1, a =−/ ln1− 

and =−/1 − ln1 − . 

1.4.6 Poisson-Inverse Gaussian Distribution 

There is no reason to restrict ourselves to the Gamma distribution for , except perhaps 

mathematical convenience. In fact, any distribution with support in the half positive real line 

is a candidate to model the stochastic behaviour of . Here, we discuss the Inverse Gaussian 

distribution. 

Inverse Gaussian Distribution 

The Inverse Gaussian distribution is an ideal candidate for modelling positive, right-skewed 

data. Recall that a random variable X is distributed according to the Inverse Gaussian 

distribution, which will be henceforth denoted as X ∼ au , if its probability density 

function is given by 

fx = 

( 

 

√ exp − 1 ) 

2x 

3 2x x − 2 x>0 (1.39) 

If X ∼ au then the mean is EX = and the variance is VX = . The moment 

generating function is given by 

∫ + 

( 

 

Mt = √ exp − 1 

) 

0 2x 

3 2x x − 2 + tx dx


( ) ∫ + 

( 

 

= exp √ 

 

exp − 1 ( 

x 2 1 − 2t + 2)) dx 

0 2x 

3 2x 

Making the change of variable = x √ 1 − 2t yields 

⎛ 

⎞ 

( ) ∫ + 

 

1 

Mt = exp √ 

⎝− 

0 

2 √ 

 

2 √ 2 + 2 ⎠ d 

1−2t 

1−2t 3 exp 

( ( √ ) ) 

= exp 1 − 1 − 2t (1.40) 

 

For the last three decades, the Inverse Gaussian distribution has gained attention in 

describing and analyzing right-skewed data. The main appeal of Inverse Gaussian models 

lies in the fact that they can accommodate a variety of shapes, from highly skewed to 

almost Normal. Moreover, they share many elegant and convenient properties with Gaussian 

models. In applied probability, the Inverse Gaussian distribution arises as the distribution of 

the first passage time to an absorbing barrier located at a unit distance from the origin in a 

Wiener process. 

Poisson-Inverse Gaussian Distribution 

Let us now complete (1.26)–(1.27) with ∼ au1, that is, 

( 

1 

f = √ exp − 1 ) 

2 

3 2 − 12 >0 (1.41) 

The probability mass function is given by 

PrN = k = 

∫ 

0 

exp−d dk 

k! 

( 

1 

√ exp − 1 ) 

2 

3 2 − 12 d (1.42) 

The probability mass function can be expressed using modified Bessel functions of the 

second kind. Bessel functions have some useful properties that can be used to compute the 

Poisson-Inverse Gaussian probabilities and to find the maximum likelihood estimators, for 

instance. 

Moments and Probability Generating Function 

Considering (1.28) and (1.29), we have 

EN = and VN = + 2 

It can be shown that / √ V = 3 in (1.31) for the Poisson-Inverse Gaussian distribution. 

Therefore the skewness of a Poisson-Inverse Gaussian distribution exceeds the skewness of 

the Negative Binomial distribution having the same mean and the same variance. 

Setting = 1 and = , the probability generating function of N can be obtained from 

(1.33) together with (1.40), which gives 

( 1 

( 

N z = exp 1 − √ 1 − 2z − 1) )


Computation of the Probability Mass Function 

The probability mass at the origin is 

( 1 

( 

N 0 = PrN = 0 = exp 1 − √ 1 + 2) ) 

 

Now, taking the derivatives of N with respect to t, and evaluating it at 0 gives the probability 

mass function for positive integers. Specifically, 

′ N 

0 = PrN = 1 

 

∣ 

= √ N z 

1 − 2z − 1 

= 

 

√ 

1 + 2 

PrN = 0 

∣ 

z=0 

and 

′ N 

0 = 2PrN = 2 

2 

 

∣ 

= ( ) 3/2 

N z∣ + √ ′ N z ∣∣z=0 

1 − 2z − 1 z=0 1 − 2z − 1 

= 

= 

2 

 

( ) 3/2 

PrN = 0 + √ PrN = 1 

1 + 2 1 + 2 

√ ( ) 

2 1 + 2 

2 

( ) 3/2 

PrN = 1 + √ PrN = 0 

1 + 2 

1 + 2 

= 

2 

PrN = 1 + PrN = 0 

1 + 2 1 + 2 

In general, we have the following recursive formula 

PrN = n = 

2 

1 + 2 

+ 

( 

1 − 3 ) 

PrN = n − 1 

2n 

2 

PrN = n − 2 (1.43) 

1 + 2nn − 1 

valid for n = 2 3 4, which allows us to compute the probability mass function of the 

Poisson-Inverse Gaussian distribution. The formal proof of (1.43) is based on properties of 

the modified Bessel function. 

1.4.7 Poisson-LogNormal Distribution 

In addition to the Gamma and Inverse Gaussian distributions to model , the LogNormal 

distribution is often used in biostatistical studies.


LogNormal Distribution 

Recall that a random variable X is Normally distributed with mean and variance 2 , 

denoted as X ∼ or 2 , if its distribution function is 

where 

( x − 

) 

Fx = 

 

x = 1 √ 

2 

∫ x 

− 

exp−y 2 /2 dy (1.44) 

Now, a random variable X is LogNormally distributed with parameters and (notation 

X ∼ or 2 )iflnX is Normally distributed with mean and variance 2 , that is, if 

its probability density function is given by 

fx = 

( 

1 

√ exp − 1 

) 

2x 2 ln x − 2 2 

If X ∼ or 2 , then its mean is 

) 

EX = exp 

( + 2 

 

2 

and its variance 

VX = exp ( 2 + 2)( exp 2 − 1 ) 

x>0 

Poisson-LogNormal Distribution 

Taking =− 2 /2 (to ensure that E = 1), the probability density function of ∼ 

or− 2 /2 2 is 

( 

1 

f = 

√ 2 exp − ln + ) 

2 /2 2 

>0 (1.45) 

2 2 

The probability mass function of the Poisson-LogNormal distribution is given by 

PrN = k = 1 

√ d k 

2 k! 

∫ 

Coming back to (1.28) and (1.29), we easily see that 

0 

exp−d k−1 exp 

(− ln + ) 

2 /2 2 

d 

2 2 

EN = and VN = + 2( exp 2 − 1 ) 

It can be shown that / √ V = 2 + exp 2 in (1.31) for the Poisson-LogNormal 

distribution. Therefore the skewness of a Poisson-LogNormal distribution exceeds the 

skewness of the Poisson-Inverse Gaussian distribution having the same mean and the same 

variance.


1.5 Statistical Inference for Discrete Distributions 

1.5.1 Maximum Likelihood Estimators 

Maximum likelihood is a method of estimation and inference for parametric models. The 

maximum likelihood estimator is the value of the parameter (or parameter vector) that makes 

the observed data most likely to have occurred given the data generating process assumed 

to have produced the variable of interest. 

The likelihood of a sample of observations is defined as the joint density of the data, with 

the parameters taken as variable and the data as fixed (multiplied by any arbitrary constant 

or function of the data but not of the parameters). Specifically, let N 1 N 2 N n be a set 

of independent and identically distributed outcomes with probability mass function p· 

where is a vector of parameters. The likelihood function is the probability of observing 

the data N 1 = k 1 N n = k n , that is, 

= 

n∏ 

pk i 

i=1 

The key idea for estimation in likelihood problems is that the most reasonable estimate is 

the value of the parameter vector that would make the observed data most likely to occur. 

The implicit assumption is of course that the data at hand are reliable. More formally we 

seek a value of that maximizes . The maximum likelihood estimator of is the 

random variable ̂ for which the likelihood is maximum, that is 

̂ ≥ for all 

It is usually simpler mathematically to find the maximum of the logarithm of the likelihood 

L = ln = 

n∑ 

ln pk i 

rather than the likelihood itself. The function L is usually referred to as the log-likelihood. 

Because the logarithm is a monotonic transformation, the log-likelihood will be maximized 

at the same parameter value that maximizes the likelihood (although the shape of the loglikelihood 

is different from that of the likelihood). 

When working with counting variables, it is often easier to use the observed frequencies 

i=1 

f k = #observations equal to k k = 0 1 2 (1.46) 

In other words, f k is the number of times that the value k has been observed in the sample. 

Denoting the largest observation as 

the log-likelihood becomes 

k max = max k i 

i=1n 

k∑ 

max 

L = f k ln pk 

k=0


We may solve analytically for the maximum likelihood estimator. To maximize any regular 

function, we find the value of the parameters that makes the first derivatives of the function 

with respect to the parameters equal to zero. The first derivative of the log-likelihood is 

called Fisher’s score, and is denoted by 

U j = 

j 

L j = 1dim (1.47) 

Then one can find the maximum likelihood estimator by setting the score to zero, i.e. by 

solving the system of equations 

U j = 0 j= 1dim 

We also check the second derivatives to ensure that this is a maximum. 

Example 1.2 Assume that policyholder i has been observed during a period d i and produced 

k i claims. Assuming that the annual number of claims is Poisson distributed with mean 

(here, is the annual expected claim frequency, so that policyholder i is expected to produce 

d i claims during the observation period, under the condition of the Poisson process); the 

log-likelihood is 

( n∑ 

L = ln exp−d i d ) 

i k i 

i=1 

k i ! 

n∑ n∑ ( ) n∑ 

=− d i + k i ln + ln di − ln k i ! 

i=1 

i=1 

i=1 

n∑ 

n∑ 

=− d i + ln k i + constant 

Setting the first derivative of L with respect to equal to 0 gives 

The second derivative is 

i=1 

i=1 

n∑ 

− d i + 1 ∑ n∑ 

n 

k 

i=1 

i = 0 ⇒ ̂ i=1 

= 

k i 

∑ n 

i=1 

i=1 d 

i 

− 1 2 

n∑ 

k i < 0 for any 

i=1 

so that ̂ indeed corresponds to the maximum of L. The estimated annual expected claim 

frequency ̂ is thus obtained as the ratio of the total number of claims to the total exposureto-risk. 

It is important to note here that the total number of claims is not divided by the 

number of policies, because of unequal risk exposure.


Example 1.3 Assume, as above, that policyholder i has been observed during a period d i 

and produced k i claims. Assuming that the annual number of claims filed by policyholder i 

is Negative Binomially distributed with mean d i , the log-likelihood is 

La = 

n∑ 

k i −1 

∑ 

i=1 j=0 

n∑ 

n∑ 

lna + j + na ln a − a + k i lna + d i + ln k i + constant 

The maximum likelihood estimators for a and solve 

k 

 

n∑ i −1 

a La = ∑ 1 

n∑ 

n∑ 

i=1 j=0 

a + j + n ln a + n − a + k 

lna + d i − 

i 

= 0 

i=1 

i=1 

a + d i 

 

n∑ 

La =− a + k 

d i 

i + 1 n∑ 

k 

a + d i i = 0 

i=1 

i=1 

i=1 

These equations do not possess explicit solutions, and must be solved numerically. A 

convenient choice is to use the Newton–Raphson algorithm (see Section 1.5.3). Initial 

values for the parameters are obtained by the method of moments. Specifically, the moment 

estimator of is simply 

i=1 

ˆ = 

∑ n 

i=1 k i 

∑ n 

i=1 d i 

 

which is the maximum likelihood estimate of in the homogeneous Poisson case. For the 

variance, we start from VN i = EN i + d i 2 where = V i . The empirical analogue 

is given by 

ˆ = 

∑ n 

i=1 

(k i − d i 2 − d i 

) 

∑ n 

i=1 d i 2 

from which we easily deduce an estimator for a in the Negative Binomial case. 

1.5.2 Properties of the Maximum Likelihood Estimators 

Maximum likelihood estimators enjoy a number of convenient properties that are discussed 

below. It is important to note that these are asymptotic properties, i.e. properties that hold 

only as the sample size becomes infinitely large. It is impossible to say in general at what 

point a sample is large enough for these properties to apply, but the majority of actuarial 

applications involve large data sets so that actuaries generally trust in the large sample 

properties of the maximum likelihood estimators. 

Consistency 

First, maximum likelihood estimators are consistent. There are several definitions of 

consistency, but an intuitive version is that as the sample size gets large the estimator is 

increasingly likely to fall within a small region around the true value of the parameter. This


is called convergence in probability and is defined more formally as follows: A consistent 

estimator T j for some parameter j computed from a sample of size n is one for which 

lim 

n↗ PrT j − j ≥c = 0 

for all positive c. This will henceforth be denoted as T j → proba j as n ↗+. A consistent 

estimator is thus an estimator that converges to the population parameter as the sample size 

goes to infinity. Consistency is an asymptotic property. 

Asymptotic Normality 

Any estimator will vary across repeated samples. We must be able to calculate this 

variability in order to express our uncertainty about a parameter value and to make statistical 

inferences about the parameters. This variability is measured by the variance-covariance 

matrix of the estimators. This matrix provides the variances for each parameter on the 

main diagonal while the off-diagonal elements estimate the covariances between all pairs of 

parameters. 

The asymptotic variance-covariance matrix ̂ 

for maximum likelihood estimators ̂ is 

the inverse of what is called the Fisher information matrix . Element ij of is given 

by 

[ ] [ ] 

 

2 

 

2 

−E ln L =−nE ln pN 

i j i 1 

j 

[ 

= nE ln pN 

1 ] 

ln pN 

i 1 

j 

∑ 

= n pk ln pk ln pk 

i j 

k=0 

Thus, ̂ 

= ( ) −1 

. 

An insight into why this makes sense is that the second derivatives measure the rate of 

change in the first derivatives, which in turn determines the value of the maximum likelihood 

estimate. If the first derivatives are changing rapidly near the maximum, then the peak of 

the likelihood is sharply defined and the maximum is easy to see. In this case, the second 

derivatives will be large and their inverse small, indicating a small variance of the estimated 

parameters. If on the other hand the second derivatives are small, then the likelihood function 

is relatively flat near the maximum and so the parameters are less precisely estimated. The 

inverse of the second derivatives will produce a large value for the variance of the estimates, 

indicating low precision of the estimates. 

The distribution of ̂ is usually difficult to obtain. Therefore we resort to the following 

asymptotic theory: Under mild regularity conditions (including that the true value of the 

parameter must be interior to the parameter space, that the log-likelihood function must be 

thrice differentiable, and that the third derivatives must be bounded) that are usually fulfilled, 

the maximum likelihood estimator ̂ has approximately in large samples a multivariate 

Normal distribution with mean equal to the true parameter and variance-covariance matrix 

given by the inverse of the information matrix.


Recall that having a n × n positive definite matrix M and a real vector , the random 

vector X = X 1 X 2 X n T is said to have the multivariate Normal distribution with mean 

and variance-covariance matrix M if its probability density function is of the form 

f X x = 

1 

√ 

(− 

2n detM exp 1 ) 

2 x − T M −1 x − x ∈ n (1.48) 

Henceforth, we indicate that the random vector X has the multivariate Normal distribution 

with probability density function (1.48) as X ∼ or M. A convenient characterization 

of the multivariate Normal distribution is as follows: X ∼ or M if, and only if, any 

random variable of the form ∑ n 

i=1 iX i with ∈ R n , has the univariate Normal distribution. 

Coming back to the properties of the maximum likelihood estimator ̂, we have that 

̂ is approximately or ̂ 

distributed (1.49) 

that is, the distribution function of ̂ can be approximated by integrating the Normal 


f S = 

1 

√ 

2 dim det̂ 

( 

exp 

− 1 2 S − T −1 S − 

̂ 

) 

 

S ∈ dim 

Attribute (1.49) says that maximum likelihood estimators converge in distribution to a 

Normal with mean equal to the population value of the parameter and variance-covariance 

matrix equal to the inverse of the information matrix. This means that regardless of the 

distribution of the variable of interest the maximum likelihood estimator of the parameters 

will have a multivariate Normal distribution. Thus, a variable may be Poisson distributed, 

but the maximum likelihood estimate of the Poisson mean will be asymptotically Normally 

distributed, and likewise for any distribution. Note however that in the Poisson case, the exact 

distribution of the maximum likelihood estimator of the parameter derived in Example 1.2 

can easily be derived from the stability of the Poisson family under convolution. 

Invariance 

A natural question is how the parameterization of a likelihood affects the resulting 

inference. Maximum likelihood has the property that any transformation of a parameter 

can be estimated by the same transformation of the maximum likelihood estimate of 

that parameter. This provides substantial flexibility in how we parameterize our models 

while guaranteeing that we will get the same result if we start with a different 

parameterization. 

The invariance property can be stated formally as follows: If = t, where t· is a 

one-to-one transformation, then the maximum likelihood estimator of is t̂. In particular, 

the maximum likelihood estimator of pk is simply pk̂, that is, 

̂pk = pk̂


1.5.3 Computing the Maximum Likelihood Estimators with the 

Newton–Raphson Algorithm 

Calculation of the maximum likelihood estimators often requires iterative procedures. Let 

H denote the Hessian (or matrix of second derivatives) of the log-likelihood function, with 

elements 

H ij = 

2 

i j 

L 

= 

i 

U j (1.50) 

k∑ 

max 

2 

=− f k ln pk 

i j 

k=0 

for i j = 1dim. For ⋆ close enough to ̂, a first-order Taylor expansion gives 

0 = Û ≈ U ⋆ + H ⋆ 

(̂ − 

⋆) 

yielding 

̂ ≈ ⋆ − H −1 ⋆ U ⋆ 

Starting from an appropriate initial value 0 , the Newton–Raphson algorithm is based on 

the recurrence relation 

̂r+1 = ̂ 

) 

r −1 

− H 

(̂r U 

(̂r) (1.51) 

This result provides the basis for an iterative approach for computing the maximum 

likelihood estimator known as the Newton–Raphson technique. Given a trial value, we use 

(1.51) to obtain an improved estimate and repeat the process until the elements of the vector 

of first derivatives are sufficiently close to zero. 

This procedure tends to converge quickly if the log-likelihood is well-behaved in a 

neighbourhood of the maximum and if the starting value is reasonably close to the maximum 

likelihood estimator. 

Remark 1.2 (Fisher Scoring) Noting that =−EH, an alternative procedure is 

to replace minus the Hessian by its expected value, i.e. minus the Fisher information matrix. 

The resulting procedure takes as an improved estimate 

̂r+1 ≈ ̂ 

) 

r −1 

+ 

(̂r U 

(̂r) 

and is known as Fisher Scoring.


1.5.4 Hypothesis Tests 

Sample Distribution of Individual Parameters 

Standard hypothesis tests about parameters in maximum likelihood models are handled quite 

easily, thanks to the asymptotic Normal distribution of the maximum likelihood estimator. 

Specifically, we use the fact that 

̂j − j 

̂j 

is approximately or0 1 

where the standard deviation ̂j 

of ̂ j is the square root of the jth diagonal element of 

̂ 

= −1 

Such tests will be useful in Chapter 2 to select the relevant risk factors. 

In practice, often involves unknown parameters so that it is estimated by the jth 

̂j 

element of ̂̂j 

̂̂ 

= ̂ −1 

In such a case, ̂ j − j is approximately Student’s distributed with n − 1 degrees 

/̂̂j 

of freedom. This is the familiar z-score for a standard Normal variable developed in all 

introductory statistics classes. The Normality of maximum likelihood estimates means that 

our testing of hypotheses about the parameters is as simple as calculating the z-score and 

finding the associated p-value from a table or by calling a software function. 

The hypothesis test is based on Student’s t-distribution. However, because the maximum 

likelihood properties are all asymptotic we are unable to address the finite sample distribution. 

Asymptotically, the Student’s t-distribution converges to the Normal as the degrees for 

freedom grow, so that using or0 1 p-values in the maximum likelihood test is the same 

as the t-test as long as the number of cases is large enough. Specifically, 

̂j − j 

̂̂j 

is approximately or0 1 (1.52) 

if the sample size n is large enough. 

In addition to the test of hypotheses about a single parameter, there are three classical 

tests that encompass hypotheses about sets of parameters as well as one parameter at a time: 

the likelihood ratio, Wald, and Lagrange multiplier tests. All are asymptotically equivalent, 

but they differ in the ease of implementation depending on the particular case. Here, we will 

present Wald, likelihood ratio and Vuong tests, as well as the Score test. 

Likelihood Ratio Test 

This test is based on a comparison of maximized likelihoods for nested models. Specifically, 

the null hypothesis H 0 corresponds to a constrained model with dim − j parameters, 

whereas the alternative H 1 corresponds to the full model with dim parameters. Most of


the time, the test is performed with j = 1, so that we compare the full model to a simpler 

one with one parameter less. 

Let ˜ be the maximum likelihood estimator under H 0 , and let ̂ be the maximum likelihood 

estimator under H 1 . The likelihood ratio test is based on the ratio of the likelihoods between 

a full and a restricted (or reduced) nested model with fewer parameters. The restricted model 

must be nested within (i.e., be a subset of) the full model. The likelihood ratio test statistic is 

T = 2ln ̂ ( 

) 

˜ = 2 L̂ − L˜ 

The evidence against H 0 will be strong when T is large. 

The Chi-square distribution plays a prominent role in likelihood ratio tests. Recall that the 

Gamma distribution with = /2 and = 1/2 for some positive integer is known as the 

Chi-square distribution with degrees of freedom (which is denoted as 2 ), with associated 


fx = x/2−1 exp−x/2 

( ) 

 

2 2 

/2 

x>0 

If X ∼ 2 then its mean is , and its variance 2. It is useful to recall that the 2 distribution 

is closely related to the Normal distribution. Specifically, the 2 arises as the distribution of 

the sum of independent squared or0 1 random variables. 

Under H 0 , the test statistic T is approximatively Chi-square distributed with degrees 

of freedom equal to the number of parameters in the full model minus the number of 

parameters in the restricted model (that is, with j degrees of freedom) when the sample size 

n is sufficiently large (and additional mild regularity conditions are fulfilled). Note that the 

likelihood ratio test requires us to perform two maximum likelihood estimations, one under 

H 0 and another one under H 1 . When the largest model H 1 is misspecified (that is, the data 

have not been generated by this probability model), the likelihood ratio statistic is no longer 

necessarily Chi-square distributed under H 0 . 

Unfortunately, there are cases where regularity conditions do not hold for T to be 

approximately j 

2 distributed under H 0 . In particular this happens when a constrained 

parameter is on the boundary of the parameter space, e.g., testing Poisson versus Negative 

Binomial. Here Poisson is a particular case of Negative Binomial when the latter has a 

parameter on its boundary space. In this case, the limiting distribution of the statistic T 

becomes a mixture of Chi-square distributions. We refer the reader to Titterington ET AL. 

(1985) for more details about these situations. 

Wald Tests 

The Wald test provides an alternative to the likelihood ratio test that requires the estimation 

of only the full model, not the restricted model. The logic of the Wald test is that if the 

restrictions are correct then the unrestricted parameter estimates should be close to the value 

hypothesized under the restricted model. 

The Wald test is based on the distribution of a quadratic form of the weighted sum of 

squared Normal deviates, a form that is known to be Chi-square distributed. Specifically, 

using (1.49), we can test H 0 = 0 versus H 1 ≠ 0 with the statistic


W = ̂ − 0 ̂̂ − 0 

which is approximately 2 dim distributed under H 0, in large samples. The test statistic can 

be interpreted as a measure of the distance between the maximum likelihood estimator ̂ 

and the hypothesized value 0 . The Wald test leads to the rejection of H 0 in favor of H 1 if 

̂ is too far from 0 . Note that the Wald test suffers from the same problems as likelihood 

ratio tests when 0 lies on the boundary of the parametric space. 

Sometimes the calculation of the expected information is difficult, and we may use the 

observed information instead. 

Score Tests 

Using the asymptotic theory, we have that U is approximately or0 distributed. 

Therefore we can test H 0 = 0 versus H 1 ≠ 0 with the statistic 

Q = U 0 I −1 0 U 0 

which is approximately 2 dim distributed under H 0, in large samples. 

The advantage of the score test is that the calculation of the maximum likelihood estimator 

̂ is bypassed. Moreover, it remains applicable even if 0 lies on the boundary of the 

parametric space. 

Vuong Test 

The Chi-square approximation to the distribution of the likelihood ratio test statistic is valid 

only for testing restrictions on the parameters of a statistical model (i.e., H 0 and H 1 are 

nested hypotheses). With non-nested models, we cannot make use of likelihood ratio tests for 

model comparison. In this case, information criteria like AIC or (S)BIC are useful, as well 

as the Vuong test for non-nested models. Recall that the AIC (Akaike Information Criteria), 

is given by 

AIC =−2L̂ + 2 dim 

and the BIC (Bayesian Information Criteria), is given by 

BIC =−2L̂ + lnn dim 

Both criteria are equal to minus two times the maximum log-likelihood, penalized by a 

function of the number of observations and sample size. 

Vuong (1989) proposed a likelihood ratio-based statistic for testing the null hypothesis 

that the competing models are equally close to the true data generating process against the 

alternative that one model is closer. Consider two statistical models given by the probability 

mass functions p· and q· with dim = dim, and define the likelihood ratio 

statistic for the model p· against q· as 

k∑ 

max 

LR̂ n ̂ n = f k ln pk̂ n 

qk̂ n 

k=0


where ̂ n and ̂ n are the maximum likelihood estimators in each model based on the sample 

k 1 k n and f k is defined as in (1.46). 

If both models are strictly non-nested (so that standard likelihood ratio tests do not apply) 

then under H 0 

LR̂ n ̂ n 

̂ n 

√ n 

is approximately or0 1 distributed 

where 

̂ n = 1 n 

( 

n∑ 

i=1 

ln pk î n 

qk i ̂ n 

) 2 

− 

( 

1 

n 

n∑ 

i=1 

) 

ln pk 2 

î n 

 

qk i ̂ n 

This provides a very simple test for model selection. Specifically, the actuary chooses a 

critical value z from the or0 1 distribution for some significance level . If the value 

of the test statistic is higher than z then he rejects the null hypothesis that the models are 

equivalent in favour of p· being better than q·. If the test statistic is smaller than −z 

then he rejects the null hypothesis in favour of q· being better than p·. Finally, if the 

test statistic is between −z and z then we cannot discriminate between the two competing 

models given the data. 

The test statistic can be adjusted if the competing models do not have the same number 

of parameters, i.e. dim ≠ dim (which is not the case in this chapter). 

1.6 Numerical Illustration 

Here, we consider a Belgian motor third party liability insurance portfolio observed during 

the year 1997 (henceforth referred to as Portfolio A). The observed claim distribution is 

given in Table 1.1. A thorough description of this portfolio is deferred to Section 2.2. 

We see from Table 1.1 that the total exposure is not equal to the number of policies due 

to the fact that some policies have not been in force during the full observation period (12 

months). Some of them have been cancelled before the end of the observation period. Others 

have been written after the start of the observation period. 

Let us now fit the observations to the Poisson, the Negative Binomial, the Poisson-Inverse 

Gaussian and the Poisson-LogNormal distributions. The results are summarized below: 

Table 1.1 Observed claim distribution in Portfolio A. 

Number of claims Number of policies Total exposure (in years) 

0 12962 10 54594 

1 1369 1 18713 

2 157 13466 

3 14 1108 

4 3 252 

Total 14505 11 88135


Poisson the maximum likelihood estimate of the Poisson mean is ̂ = 01462. The 95 % 

confidence interval for is (0.1395;0.1532). The log-likelihood of the Poisson model is 

−5579.339. 

Negative Binomial the maximum likelihood estimate of the mean is ̂ = 01474 and 

the dispersion parameter â = 0889. The variance of the random effect is estimated as 

̂V = 1/â = 11253. The respective 95 % confidence intervals are (0.1402;0.1551) for 

and (0.8144;1.4361) for V. The log-likelihood of the Negative Binomial model is 

-5534.36, which is better than the Poisson log-likelihood. 

Poisson-Inverse Gaussian the maximum likelihood estimation of the mean is ̂ = 01475, 

and the variance of the random effect is estimated to ̂V = ̂ = 11770. The respective 

95 % confidence intervals are (0.1402;0.1552) for and (0.8258;1.5282) for V. The 

log-likelihood of the Poisson-Inverse Gaussian model is −553428, which is better than the 

Poisson log-likelihood and almost equivalent to the Negative Binomial log-likelihood. 

Poisson-LogNormal the maximum likelihood estimation of the mean is ̂ = 01476, and 

̂ 2 = 07964. The variance of the random effect is estimated to ̂V = 12175. The respective 

95 % confidence intervals are (0.1403;0.1553) for and (0.6170;0.9758) for 2 . The loglikelihood 

of the Poisson-LogNormal model is −553444, which is better than the Poisson 

log-likelihood and almost equivalent to the Negative Binomial and Poisson-Inverse Gaussian 

log-likelihoods. 

The results have been obtained with the help of the SAS R procedure GENMOD for the 

Poisson and Negative Binomial distributions (details will be given in the next chapter) and 

by a direct maximization of the log-likelihood using the Newton–Raphson procedure (coded 

in the SAS R environment IML) in the Poisson-Inverse Gaussian and Poisson-LogNormal 

cases. 

It is interesting to note that the values of ̂ are different in the Poisson and mixed Poisson 

models. If all the risk exposures were equal then these values would have been the same in 

all cases. 

Let us now compare the Poisson fit to Portfolio A with each of the mixed Poisson fits. 

To this end, we use a likelihood ratio test, with an adjusted Chi-square approximation (since 

the Poisson case is at the border of the mixed Poisson family). Comparing the Poisson fit to 

any of the three mixed Poisson models leads to a clear rejection of the former one: 

Poisson against Negative Binomial 

less than 10 −10 . 

likelihood ratio test statistic of 89.95, with a p-value 

Poisson against Poisson-Inverse Gaussian 

p-value less than 10 −10 . 

likelihood ratio test statistic of 90.12, with a 

Poisson against Poisson-LogNormal likelihood ratio test statistic of 89.80, with a p-value 

less than 10 −10 . 

The rejection of the Poisson assumption in favour of a mixed Poisson model is interpreted 

as a sign that the portfolio is composed of different types of drivers (i.e. the portfolio is 

heterogeneous). 

Now, comparing the three mixed Poisson models with the Vuong test gives:


Negative Binomial against Poisson-Inverse Gaussian Vuong test statistic equal to -0.1086, 

with p-value 91.36 %. 

Poisson-LogNormal against Negative Binomial 

with p-value 96.54 %. 

Vuong test statistic equal to −00435, 

Poisson-LogNormal against Poisson-Inverse Gaussian Vuong test statistic equal to 

−03254, with p-value 74.48 %. 

We cannot discriminate between the three competing models given the data, and they all 

fit the model equally well. 

Remark 1.3 (Chi-Square Goodness-of-Fit Tests) In many papers appearing in the 

actuarial literature devoted to the analysis of claim numbers, as well as in most empirical 

studies, Chi-square goodness-of-fit tests are performed to select the optimal model. However, 

this approach neglects the exposures-to-risk (acting as if all the policies were in the portfolio 

for the whole year). We do not rely on Chi-square goodness-of-fit tests here since they do 

not allow for unequal risk exposures. Note that the vast majority of papers appearing in the 

actuarial literature disregard risk exposures (and proceed as if all the risk exposures were 

equal to 1). 

1.7 Further Reading and Bibliographic Notes 

1.7.1 Mixed Poisson Distributions 

Mixed Poisson distributions are often used to model insurance claim numbers. The statistical 

analysis of counting random variables is described in much detail in Johnson ET AL. (1992). 

An excellent introduction to statistical inference is provided by Franklin (2005). In the 

actuarial literature, Klugman ET AL. (2004) provide a good account of statistical inference 

applied to insurance data sets, and in particular the analysis of counting random variables. 

Generating functions are described in Kendall & Stuart (1977) and Feller (1971). 

The axiomatic approach for which the (mixed) Poisson distribution is the counting 

distribution for a (mixed) Poisson process is presented in Grandell (1997). Mixture models 

are discussed in Lindsay (1995). See also Titterington ET AL. (1985). Let us mention the 

work by Karlis (2005), who applied the EM algorithm for maximum likelihood estimation 

in mixed Poisson models. 

1.7.2 Survey of Empirical Studies Devoted to Claim Frequencies 

Kestemont & Paris (1985), using mixtures of Poisson processes, defined a large class of 

probability distributions and developed an efficient method for estimating their parameters. 

For the six data sets in Gossiaux & Lemaire (1981), they proposed a law depending on 

three parameters and they always obtained extremely good fits. As particular cases of the 

laws introduced in Kestemont & Paris (1985), we find the ordinary Poisson distribution, 

the Poisson-Inverse Gaussian distribution, and the Negative Binomial distribution. 

Tremblay (1992) used the Poisson-Inverse Gaussian distribution. Willmot (1987) 

compared the Poisson-Inverse Gaussian distribution to the Negative Binomial one and 

concluded that the fits are superior with the Poisson-Inverse Gaussian in all the six cases


studied by Gossiaux & Lemaire (1981). See also the paper by Besson & Partrat (1990). 

Ruohonen (1987) considered a model for the claim number process. This model is a mixed 

Poisson process with a three-parameter Gamma distribution as the structure function and is 

compared with the two-parameter Gamma model giving the Negative Binomial distribution. 

He fitted his model to some data that can be found in the actuarial literature and the results 

were satisfying. Panjer (1987) proposed the Generalized Poisson-Pascal distribution (in fact, 

the Hofmann distribution), which includes three parameters, for the modelling of the number 

of automobile claims. The fits obtained were satisfactory, too. Note that the Pólya-Aeppli, the 

Poisson-Inverse Gaussian and the Negative Binomial are special cases of this distribution. 

Consul (1990) tried to fit the same six data sets by the Generalized Poisson distribution. 

Although the Generalized Poisson law is not rejected by a Chi-square test, the fits obtained 

by Kestemont & Paris (1985), for instance, are always better. Furthermore, Elvers (1991) 

reported that the Generalized Poisson distribution did not fit the data observed in a motor 

third party liability insurance portfolio very well. Islam & Consul (1992) suggested the 

Consul distribution as a probabilistic model for the distribution of the number of claims in 

automobile insurance. These authors approximated the chance mechanism which produces 

vehicle accidents by a branching process. They fit the model to the data sets used by Panjer 

(1987) and by Gossiaux & Lemaire (1981). Note that this model deals only with cars in 

accidents. Consequently, the zero-class has to be excluded. The fitted values seem good. 

However, this has to be considered cautiously, due to the comments by Sharif & Panjer 

(1993) who found serious flaws embedded in the fitting of the Consul model. In particular, 

the very restricted parameter space and some theoretical problems in the derivation of the 

maximum likelihood estimators. They refer to other simple probability models, such as the 

Generalized Poisson-Pascal or the Poisson-Inverse Gaussian, whose fits were found quite 

satisfying. 

Denuit (1997) demonstrated that the Poisson-Goncharov distribution introduced by 

Lefèvre & Picard (1996) provides an appropriate probability model to describe the annual 

number of claims incurred by an insured motorist. Estimation methods were proposed, and 

the Poisson-Goncharov distribution successfully fitted the six observed claims distributions 

in Gossiaux & Lemaire (1981), as well as other insurance data sets. 

1.7.3 Semiparametric Approach 

Traditionally, actuaries have assumed that the distribution of values among all drivers 

is well approximated by a parametric distribution, be it Gamma, Inverse Gaussian or 

LogNormal. However, there is no particular reason to believe that F belongs to some 

specified parametric family of distributions. Therefore, it seems interesting to resort to a 

nonparametric estimator for F . 

There have been several attempts to estimate the structure function in a mixed Poisson 

model nonparametrically. Most of them include the annual claim frequency in the random 

effect, and thus work with ˜ = . Assuming that ˜ has a finite number of support points 

and that its probability distribution is uniquely determined by its moments, Tucker (1963) 

suitably precised by Lindsay (1989a,b) suggested estimation of the support points of ˜ and 

the corresponding probability masses by solving a moment system. This estimator was then 

smoothed by Carrière (1993b) using a mixture of LogNormal distribution functions where 

all the parameters are estimated by a method of moments.


In a seminal paper, Simar (1976) gave a detailed description of the nonparametric 

maximum likelihood estimator of F˜, 

as well as an algorithm for its computation. The 

nonparametric maximum likelihood estimator has a discrete distribution, and Simar (1976) 

obtained an upper bound for the size of its support. 

Walhin & Paris (1999) showed that, although the nonparametric maximum likelihood 

estimator is powerful for evaluation of functionals of claim counts, it is not suitable for 

ratemaking, because it is purely discrete. For this reason, Denuit & Lambert (2001) 

proposed a smoothed version of the nonparametric maximum likelihood estimator. This 

approach is somewhat similar to the route followed by Carrière (1993b), who proposed to 

smooth the Tucker-Lindsay moment estimator with a LogNormal kernel. 

Young (1997) applied nonparametric density estimation techniques to estimate F˜. 

Because the actuary only observes claim numbers and not the conditional mean, an estimation 

of the underlying risk parameter relating to the ith policy of the portfolio is the average claim 

number x i (i.e. the total number of claims generated by this policy divided by the length 

of the exposure period). Therefore, given a kernel K, Young (1997) suggested estimating 

dF by 

d̂F˜t = 

n∑ 

i=1 

( ) 

w i t − xi 

K 

h i h i 

in which h i is a positive parameter called the bandwidth and w i is a weight (taken to be the 

number of years the ith policy is in force divided by the total number of policy-years for the 

collective). Young (1997) suggested using the Epanechnikov kernel and determined the h i s 

in order to minimize the mean integrated squared error (by reference to a Normal prior).

2 

Risk Classification 


2.1.1 Risk Classification, Regression Models and Random Effects 

Motor ratemaking is essentially about classifying policies according to their risk 

characteristics. The classification variables are called a priori variables (as their values can 

be determined before the policyholder starts to drive). In motor insurance, they include the 

age, gender and occupation of the policyholders, the type and use of their car, the place 

where they reside and sometimes even the number of cars in the household, marital status, 

or the colour of the vehicle. 

These observable risk characteristics are typically seen as nonrandom covariates. Other risk 

characteristics are unobservable and must be seen as unknown parameters or, in the vein of 

credibility theory, latent variables with a common distribution. The literature about premium 

rating in motor insurance comprises two mainstream approaches: (i) the first one disregards 

observable covariates altogether and lumps all the individual characteristics into random 

latent variables and (ii) the second one disregards random individual risk characteristics and 

tries instead to catch all relevant individual variations by covariates. Chapter 1 adopted the 

first approach. The present chapter combines both views, employing contemporary, advanced 

data analysis. 

If the data are subdivided into risk classes determined by a priori variables, actuaries work 

with figures which are small in exposure and claim numbers. Therefore, simple averages will 

be suspect and regression models are needed. Regression analyses the relationship between 

one variable and another set of variables. This relationship is expressed as an equation that 

predicts a response variable (the expected number of claims filed by a given policyholder) 

from a function of explanatory variables and parameters (involving a linear combination 

of these explanatory variables and parameters, called a linear predictor). The parameters 

are estimated so that a measure of the goodness-of-fit is optimized (the log-likelihood, in 





most cases). Actuaries use regression techniques to predict the expected number of claims 

knowing some information about the policyholders, vehicles and types of contract. It is worth 

mentioning that even with all the covariates included here, there still remain substantial risk 

differentials between individual drivers (due to hidden characteristics, like temper and skill, 

aggressiveness behind the wheel, knowledge of the highway code, etc.). Random effects 

are added to the linear predictor on the score scale to take this residual heterogeneity into 

account, reconciling the two approaches (i)–(ii) mentioned above. 

In nonlife business, the pure premium is the expected cost of all the claims that 

policyholders will file during the coverage period (under the assumption of the Law of Large 

Numbers). The computation of this premium relies on a statistical model incorporating all 

the available information about the risk. The technical tariff aims to evaluate as accurately as 

possible the pure premium for each policyholder via regression techniques. It is well-known 

that market premiums may differ from those computed by actuaries; see, e.g., Coutts (1984) 

for a discussion. In that respect, the overall market position of the company compared to its 

competitors with regard to growth and pricing is crucial. 

Sometimes, motor ratemaking is performed on panel data. Bringing several observation 

periods together has some advantages: it increases the sample size and avoids granting too 

much importance to a single calendar year (during which the particular weather conditions 

could have increased or decreased the number of traffic accidents, for instance). However, 

this induces some dependence in the data, since observations relating to the same policyholder 

across time are expected to be correlated. The analysis of correlated data with Poisson 

marginals arising from repeated measurements can be performed with the help of Generalized 

Estimating Equations (GEEs). GEEs provide a practical method with reasonable statistical 

efficiency to analyse such panel data. GEEs also give initial values for maximum likelihood 

procedures in models for longitudinal data. 

2.1.2 Risk Sharing in Segmented Tariffs 

The following discussion is inspired by the paper by De Wit & Van Eeghen (1984). 

Consider a portfolio of n policies from motor third party liability insurance. The random 

variable Y i models a quantity of actuarial interest for policy i (for instance the amount of a 

claim, the aggregate claim amount in one period or the number of accidents at fault reported 

by policyholder i during one period). In order to explain the outcomes of Y i , the actuary has 

observable covariates Xi 

T = X i1 X i2 at his disposal (e.g., age, gender and occupation 

of policyholder i, the place where he resides, type and use of his car). However, Y i also 

depends on a sequence of unknown characteristics Zi 

T = Z i1 Z i2 (e.g., annual mileage, 

accuracy of judgment, aggressiveness behind the wheel, drinking behaviour, etc.). Some of 

these quantities are unobservable, others cannot be measured in a cost efficient way. 

The ‘true’ premium for policyholder i is EY i X i Z i . It is the function g of X i and Z i 

that is the ‘closest’ to Y i , in the sense that EY i − gX i Z i 2 is minimum for gX i Z i = 

EY i X i Z i . If the insurer charges EY i X i Z i to policyholder i, then the policyholders pay 

premiums that absorb the inter-individual variations (that is, the variations of the premiums 

due to the modifications in personal characteristics X i and Z i , the magnitude of which are 

quantified by V [ EY i X i Z i ] ). The company covers the purely random intra-individual risks 

(that is, the random fluctuations of Y i , which are quantified by the variance E [ VY i X i Z i ] 

of the outcomes of Y i once the personal characteristics X i and Z i have been fixed). Risk

Risk Classification 51 

sharing can be summarized as follows: using the variance decomposition formula and then 

taking expectations gives: 

VY i = E [ VY i X i Z i ] 

+ V [ EY i X i Z i ] 

 

} {{ } } {{ } 

→insurer 

→policyholder 

Of course, since the elements of Z i are unknown to the insurer, the situation described 

above is purely theoretical. Since the company only knows X i , the insurer can only charge 

EY i X i . The risk sharing is now 

VY i = E [ VY i X i ] 

+ V [ EY i X i ] 

 

} {{ } } {{ } 

→insurer →policyholder 

The part of the variance supported by the insurer is now larger, since residual heterogeneity 

remains with the company. To see this, let us write 

[ ] [ ] [ [ 

∣ 

E VY i X i = E VY i X i Z i + E V EY i X i Z i ∣X i 

]] 

The first term in this sum, i.e. E [ VY i X i Z i ] , represents the purely random fluctuations of 

the risk and is supported by the insurance company in application of the very basic principle 

of insurance. On the contrary, the second term represents the variations of the expected 

claims due to the unknown risk characteristics Z i . This quantity should be corrected by an 

experience rating mechanism (as discussed in Chapters 3 and 4). 

We can now clearly see the link existing between a priori and a posteriori ratemaking. The 

idea behind experience rating is that past claims experience reveals the hidden features Z i . 

Let Yi 

← denote the past claims experience available about Y i . The idea is that the information 

contained in X i Yi 

← becomes comparable to X i Z i as time goes on. Therefore, the a 

posteriori premium is EY i X i Yi 

← . Experience rating is based on a ‘crime and punishment’ 

mechanism: claim-free policyholders are rewarded by premium discounts called bonuses, 

whereas policyholders reporting one or more accidents at fault are penalized by premium 

surcharges called maluses. 

In a priori ratemaking, the actuary aims to identify the best predictors X i and to compute 

the risk premium EY i X i .Ina posteriori ratemaking, the actuary purposes to compute 

premium corrections according to past claims history Yi 

← in order to reflect the unavailable 

information contained in Z i . A posteriori ratemaking techniques are discussed in the next 

chapter. 

2.1.3 Bonus Hunger and Censoring 

In this chapter, we fit statistical models for the number of claims subject to the existing 

rules adopted by the insurance company to penalize reported claims. Due to bonus-malus 

mechanisms, some claims are not filed to the company, because policyholders think it is 

cheaper for them to defray the third party (or to pay for their own costs in first party 

coverages) to avoid premium surcharges. The data are thus censored in a complicated 

way, and the conclusions of the actuarial analysis are valid only if the existing rules


are kept unchanged. Analysing insurance data, the actuary is able to draw conclusions 

about the number of claims filed by policyholders subject to a specific a posteriori 

ratemaking mechanism. The actuary is not able to draw any conclusions about the number 

of accidents caused by these insured drivers. We will come back to this important issue in 

Chapter 5. 

2.1.4 Agenda 

This chapter is devoted to a priori ratemaking, and focusses on claim frequencies. To make 

the discussion more concrete, we analyse the statistics from a couple of motor insurance 

portfolios. We first work with cross-sectional data (i.e. data gathered during one year of 

observation) from a Belgian motor third party liability insurance portofolio observed during 

the year 1997 (called Portfolio A, already used in Chapter 1). We also show how it is 

possible to build a ratemaking on the basis of panel data. To this end, we use another Belgian 

portfolio (called Portfolio B) for which the data have been collected during 3 years (from 

1997 to 1999). 

To fix the ideas, in Section 2.2 we present the data observed during the year 1997 and the 

different explanatory variables available for Portfolio A. We give a detailed description of 

all the variables and we have a first look at their influence on the risk borne by the insurer. 

Then, in Section 2.3, we show how it is possible to build an a priori ratemaking thanks to 

a Poisson regression. We illustrate the technique on the data from Portfolio A. Section 2.4 

addresses the problem of overdispersion. A random effect is added to the covariates to 

account for residual heterogeneity (vector Z i in the preceding discussion). We examine 

three classical models: the Poisson-Gamma (or Negative Binomial) model, the Poisson- 

Inverse Gaussian model and the Poisson-LogNormal model. These are extensions of the 

corresponding models presented in Chapter 1, to incorporate exogenous information about 

policyholders. 

In Section 2.9, we develop ratemaking techniques using panel data. Portfolio B that has 

been observed during three consecutive years is used for the numerical illustrations. GEE and 

maximum likelihood are used to estimate the parameters involved in models for longitudinal 

data. The final Section 2.10 offers an extensive discussion of topics not covered in this 

chapter, together with appropriate references. 

2.2 Descriptive Statistics for Portfolio A 

2.2.1 Global Figures 

The data relate to a Belgian motor third party liability insurance portfolio observed during 

the year 1997. The data set (henceforth referred to as Portfolio A, for brevity) comprises 

14 505 policies. The observed claim number distribution in the portfolio has been described 

in Table 1.1. The observed mean claim frequency for Portfolio A is 14.6 %. 

2.2.2 Available Information 

The following information is available on an individual basis: in addition to the number of 

claims filed by each policyholder (variable Nclaim) and the exposure-to-risk from which


these claims originate (i.e. the number of days the policy has been in force during 1997, 

variable Expo), we know 

Age : Policyholder’s age (four categories: 1 = ‘between 18 and 24’, 2 = ‘between 25 and 

30’, 3 = ‘between 31 and 60’, 4 = ‘more than 60’) 

Gender : Policyholder’s gender (two categories: 1 = ‘woman’, 2 = ‘man’) 

District : kind of district where the policyholder lives (two categories: 1 = ‘urban’, 2 = 

‘rural’) 

Use : Use of the car (two categories: 1 = ‘private use, i.e. leisure and commuting’, and 2 = 

‘professional use’) 

Split : premium split (two categories: 1 = ‘premium paid once a year’ and 2 = ‘premium 

split up’). 

In practice, insurers have at their disposal much more information about their 

policyholders. Here, we focus on these few explanatory variables for pedagogical purposes, 

to ease the exposition of ideas. 

We see that all the explanatory variables listed above are categorical, i.e. they can be 

used to partition the portfolio into homogeneous classes with respect to the variables. Such 

explanatory variables are called factors, each factor having a number of levels. In practice, 

there are also continuous covariates. We explain in the last section of this chapter how to 

deal with such explanatory variables. 

2.2.3 Exposure-to-Risk 

The majority of policies are in force during the whole year. However, in some cases, the 

observation period does not last for the entire year. This is the case, for instance, for new 

policyholders entering the portfolio during the observation period, and in case of policy 

cancellations. It is also common in practice to start a new period if some changes occur 

in the observable characteristics of the policies (for instance, the policyholder moves from 

a rural to an urban area and the company uses the rating variable District). The policy 

is then represented as two different lines in the data base, and observations are recorded 

separately for the two periods (the policy number allows the actuary to track these changes). 

Note that independence is lost in this case, and allowance for panel data is preferable (as 

in Section 2.9). This variety of situations is taken into account in the Poisson process, by 

multiplying the annual expected claim frequency by the length of the observation period, as 

explained in Chapter 1. 

In Portfolio A, the average coverage period is 298.98 days. Figure 2.1 gives an idea of the 

distribution of the exposure-to-risk in the portfolio. About 65 % of the policies have been 

observed during the whole year 1997. Considering the distribution of the exposure-to-risk, 

we see that policy issuances and lapses are randomly spread over the year. The distribution 

of the policies in force during less than one year is roughly uniform over [0,365]. 

It is worth mentioning that the policies that are just issued often differ from those in the 

portfolio. This is why it may be preferable to conduct a separate analysis for this type of 

policy.


10000 

9000 

8000 

Number of policyholders 

7000 

6000 

5000 

4000 

3000 

2000 

1000 

0 

1 2 3 4 5 6 7 8 9 10 11 12 

Months 

Figure 2.1 Exposure-to-risk in Portfolio A. 

2.2.4 One-Way Analyses 

Age 

The age structure of the portfolio is described in Figure 2.2. Most policyholders are middleaged 

as 6722 insured drivers (representing 46.4 % of the portfolio) are between 31 and 60. 

Only 802 insured drivers (representing 5.8 % of the portfolio) are over 60. The young drivers 

represent 15.3 % of the portfolio (2295 policyholders) and the remaining 4686 insured drivers 

(32.5 % of the portfolio) are between 25 and 30. 

In the preliminary descriptive analysis, the actuary considers the marginal impact of each 

rating factor. The possible effect of the other explanatory variables is thus disregarded. 

Let us assume for a while that the claim frequencies only depend on Age. If the 

occurrence of the claims filed by the policyholders conforms with a Poisson process, the 

number N i of claims reported by policholder i obeys the Poisson distribution with mean 

d i Agei , where d i is the exposure-to-risk (i.e., the length of the coverage period) for 

policyholder i, Agei is the age category to which policyholder i belongs (1, 2, 3 or 

4) and the j s, j = 1 2 3 4, are the annual expected claim frequencies for the 4 age 

classes.



7000 

6722 

.220 .213 

6000 

.210 

.200 

.190 

.180 

.170 

5000 

.160 

.155 

4686 

.150 

.140 

.130 

4000 

.123 

.120 

.110 

.108 

3000 

.100 

.090 

2295 

.080 

.070 

2000 

.060 

1000 

0 

18–24 25–30 31–60 

802 

>60 

.050 

.040 

.030 

.020 

.010 

.000 

18–24 25–30 31–60 >60 

Age 

Age 

Figure 2.2 Composition of Portfolio A with respect to Age (left panel) and observed annual claim 

frequencies according to Age (right panel). 

Annual claim frequency 

Assuming that the numbers of claims filed by the policyholders in the portfolio are 

independent random variables, the likelihood then becomes 

1 2 3 4 = 

∝ 

n∏ 

i=1 

) ki 

( ) 

(d i Agei 

exp − d i Agei 

k i ! 

( 

4∏ 

exp − j 

j=1 

∑ 

iAgei=j 

) 

∑ 

iAgei=j k 

d i 

i 

j 

where k i denotes the observed number of claims for policyholder i, and ‘∝’ reads ‘is 

proportional to’. Differentiating L 1 2 3 4 = ln 1 2 3 4 with respect to j 

and setting the derivative equal to 0 gives 

− 

∑ 

iAgei=j 

d i + 1 j 

∑ 

iAgei=j 

k i = 0


The maximum likelihood estimator of j is then obtained from 

̂j = 

= 

∑ 

iAgei=j k i 

∑ 

iAgei=j d i 

# of claims filed by policyholders in age category j 

total exposure-to-risk (in year) for age category j 

Of course, the reliability of ̂ j depends on the magnitude of the exposure-to-risk appearing 

in the denominator. 

We see in Figure 2.2 that the observed annual claim frequency decreases with age. The 

young drivers are riskier as their observed annual claim frequency is 21.3 %. Old drivers are 

safer with an observed annual claim frequency of 10.8 %. We notice that the policyholders 

aged between 25 and 30, with an observed annual claim frequency of 15.5 %, tend to 

report more claims than the policyholders aged between 31 and 60 (observed annual claim 

frequency of 12.3 %). 

The analysis conducted in this paragraph is often referred to as a one-way analysis: the 

effect of Age on claim frequencies is studied without taking account of the effect of other 

variables. The major flaw with one-way analyses is that they can be distorted by correlations. 

For instance, one can imagine that the majority of young policyholders split the payment 

of the insurance premiums (for budget reasons). If more claims are filed by young drivers, 

a one-way analysis of Split may show higher claim frequencies for drivers having split 

their premium payment. However, this may result from the fact that such drivers are in 

general the high-risk young policyholders. Premium differentials based on one-way analyses 

of Split and Age would double-count the effect of Age. Multivariate methods (such as the 

Poisson regression approach discussed below) adjust for correlations between explanatory 

variables. The correlations existing between explanatory variables explain why the policies 

are not uniformly distributed over risk classes but cluster in some specified highly populated 

classes. 

Gender 

It is common to include the gender of the main driver in the actuarial ratemaking. Note 

however that some states have banned the use of this rating factor (as well as of age, for 

instance). The reason is that age and gender are out of the policyholders’ control, in contrast 

to many other covariates (like the power of the car, or the driving area). The latter may thus 

be freely used for ratemaking purposes, but some limitations are needed for the former. 

In Portfolio A, there are 9358 male policyholders (representing 64.7 % of the portfolio) 

and 5147 female policyholders (representing 35.3 % of the portfolio). Figure 2.3 suggests a 

higher annual claim frequency for males (observed annual claim frequency of 15.2 %) than 

for females (observed annual claim frequency of 13.6 %). 

District 

Figure 2.4 gives the distribution of the policyholders according to the district where they 

live. We see that 8664 policyholders (representing 59.8 % of the portfolio) live in an urban 

area and 5841 policyholders (representing 40.2 % of the portfolio) live in a rural one. The 

urban policyholders have a larger observed annual claim frequency (15.7 %) than the rural 

ones (13.0 %).



10000 

9000 

8000 

7000 

6000 

5000 

4000 

3000 

2000 

1000 

0 

9358 

5147 


.160 

.152 

.150 

.140 

.136 

.130 

.120 

.110 

.100 

.090 

.080 

.070 

.060 

.050 

.040 

.030 

.020 

.010 

Male Female 

.000 

Male Female 

Gender 

Gender 

Figure 2.3 Composition of Portfolio A with respect to Gender (left panel) and observed annual claim 

frequencies according to Gender (right panel). 

Use 

The majority of the policyholders (12 745 insured drivers, representing 88.0 % of the 

portfolio) use their vehicle only for leisure and commuting. There are 1760 professional 

users (representing 12.0 % of the portfolio). Figure 2.5 indicates that the influence of the use 

of the vehicle on the number of claims is almost negligible, as the annual observed claim 

frequencies are almost equal for the two categories of insured drivers: 14.6 % for private 

users and 14.3 % for professional users. 

Premium Split 

Figure 2.6 indicates that 10 568 policyholders (representing 74.5 % of the portfolio) pay 

their premium once a year. The remaining 3937 policyholders (representing 25.5 % of the 

portfolio) pay their premium twice a year, thrice a year or on a monthly basis. Figure 2.6 

also shows the influence of the premium split on the number of claims: it can be seen that 

splitting the premium payment is associated with a considerable increase in the observed 

annual claim frequency (from 12.4 % to 21.1 %).


9000 

8000 

7000 

8664 

.160 

.150 

.140 

.130 

.120 

.130 

.157 


6000 

5000 

4000 

3000 

5841 


.110 

.100 

.090 

.080 

.070 

.060 

.050 

2000 

.040 

.030 

1000 

.020 

.010 

0 

Rural Urban 

District 

.000 

Rural Urban 

District 

Figure 2.4 Composition of Portfolio A with respect to District (left panel) and observed annual claim 

frequencies according to District (right panel). 

2.2.5 Interactions 

So far, only the marginal effect of each observed covariate on the claim frequency has 

been assessed. Besides these one-way analyses, it is also important to account for possible 

interactions. Often Gender and Age interact, in the sense that the effect of Age on the average 

claim frequency is different for males than for females. Typically, young male drivers are 

more dangerous than young female drivers (but the higher risk associated with young male 

drivers may be due to higher annual mileage, or to other risk factors correlated to the fact 

of being a young male). Formally, two explanatory variables are said to interact when the 

effect of one factor varies depending on the levels of the other factor. Multivariate models 

allow for investigation into interaction effects. 

This phenomenon can be seen from Figure 2.7. The observed annual claim frequency for 

young males (ages 18–24) peaks at 23.8 %, whereas young females (ages 18–24) have an 

observed annual claim frequency similar to males aged 25–30. Both genders become more 

similar for categories 31–60 and over 60. We have thus detected an Age–Gender interaction 

in Portfolio A. 

Note that standard regression models do not automatically account for interaction (in 

contrast to correlations between covariates, for which the estimated regression coefficients


13000 

12000 

12745 

.150 

.140 

.147 

.143 

11000 

.130 


10000 

9000 

8000 

7000 

6000 

5000 

4000 

3000 

2000 

1760 


.120 

.110 

.100 

.090 

.080 

.070 

.060 

.050 

.040 

.030 

.020 

1000 

.010 

0 

Private Professional 

Use of car 

.000 

Private Professional 

Use of car 

Figure 2.5 Composition of Portfolio A with respect to Use (left panel) and observed annual claim 

frequencies according to Use (right panel). 

are adjusted). The reason is as follows: interactions cannot be rendered by linear combinations 

of the covariates. To account for interactions, nonlinear functions of the covariates (products) 

are needed, as explained in Example 2.3 below. In ANOVA terminology, one would speak 

of interaction when the effects are not just additive. The actuary needs to identify the existing 

interactions at the preliminary exploratory stage, and then define new ratemaking factors 

combining the levels of the two interacting variables. Inserting these new factors in the 

regression model then allows us to account for interaction. 

2.2.6 True Versus Apparent Dependence 

The descriptive analysis conducted so far suggests that some observed characteristics may 

influence the number of claims reported to the company. It is nevertheless important to 

realize the kind of relationship just evidenced: the actuary has to keep in mind that he 

has not established any causal relationship so far, but only that some correlations seem 

to exist between the rating factors and the number of claims. Such correlations may have 

been produced by a causal relationship, but could also result from confounding effects. The 

following simple example illustrates this situation.



0 

Not split Split 

11000 

.220 

10568 

.210 

10000 

.200 

.190 

9000 

.180 

.170 

8000 

.160 

.000 

.150 

7000 

.140 

.130 

6000 

.120 

.110 

5000 

.100 

4000 

3000 

2000 

1000 

3957 

.090 

.080 

.070 

.060 

.050 

.040 

.030 

.020 

.010 

Premium split 


.211 

.124 

Not split Split 

Premium split 

Figure 2.6 Composition of Portfolio A with respect to Split (left panel) and observed annual claim 

frequencies according to Split (right panel). 

Example 2.1 Assume that living in rural or urban areas (observed covariate District) does 

not influence claim occurrences but the driving experience (hidden characteristic) does. 

Specifically, let us assume that 

PrN ≥ 1inexperienced, rural = PrN ≥ 1inexperienced, urban 

= PrN ≥ 1inexperienced = 015 

PrN ≥ 1experienced, rural = PrN ≥ 1experienced, urban 

= PrN ≥ 1experienced = 005 

The portfolio comprises 50 % inexperienced drivers, but 

and 

Prinexperiencedurban = 1 − Prexperiencedurban = 09 

Prinexperiencedrural = 1 − Prexperiencedrural = 01



5000 .240 

.230 

.220 

4286 

.210 

.200 

4000 

.190 

.180 

.170 

.160 

2972 

.150 

3000 

.140 

.130 

2436 

.120 

.110 

2000 

.100 

1714 

.090 

1494 

.080 

.070 

.060 

1000 .050 

801 

606 

.040 

.030 

196 

.020 

.010 

0 

.000 

F 

18–24 

F 

25–30 

F 

31–60 

F 

>60 

M 

18–24 

M 

25–30 

Interaction Age–Gender 

M 

31–60 

M 

>60 


.165 

F 

18–24 

.140 

F 

25–30 

.127 

F 

31–60 

.112 

F 

>60 

.238 

M 

18–24 

.164 

M 

25–30 

Interaction Age–Gender 

.120 

M 

31–60 

.107 

M 

>60 

Figure 2.7 Composition of Portfolio A with respect to the Age–Gender interaction (left panel) and 

observed annual claim frequencies according to the Age–Gender interaction (right panel). 

In other words, there is a majority of experienced drivers in rural areas (90 %) whereas in 

urban areas, the majority of drivers are inexperienced. 

Clearly, 

whereas 

PrN ≥ 1urban = PrN ≥ 1inexperienced, urban Prinexperiencedurban 

+ PrN ≥ 1experienced, urban Prexperiencedurban 

= 014 

PrN ≥ 1rural = PrN ≥ 1inexperienced, rural Prinexperiencedrural 

+ PrN ≥ 1experienced, rural Prexperiencedrural 

= 006 

Even if District does not influence the number of claims once the driving experience has been 

accounted for, unconditionally there is some dependence between District and the number 

of claims filed by policyholders. The univariate analysis will detect a rural/urban effect, but 

the latter should disappear in the multivariate analysis (taking experience and the variable 

District into account). Therefore, we cannot say after a marginal analysis that living in an 

urban area causes an increase in the average number of claims: both variables are related to 

driving experience.


The reader has always to keep in mind that it is often not possible to disentangle a true 

effect of a rating factor from an apparent effect resulting from correlation with unobservable 

characteristics. 

2.3 Poisson Regression Model 

2.3.1 Coding Explanatory Variables 

All the explanatory variables presented above are categoric (or nominal). A categoric variable 

with k levels partitions the portfolio into k classes (for instance, 4 classes for Age in Portfolio 

A). It is coded with the help of k − 1 binary variables, being all zero for the reference level. 

The reference level is usually selected as the most populated class in the portfolio. The 

following example illustrates this coding methodology. 

Example 2.2 In Portfolio A, reference levels are ‘31–60’ for Age, ‘Male’ for Gender, 

‘Urban’ for District, ‘Premium paid once a year’ for Split and ‘Private’ for Use. Policyholder 

i is then represented by a vector of dummies giving the values of 

{ 1 if policyholder i is less than 24 

x i1 = 

0 otherwise 

{ 1 if policyholder i is 25–30 

x i2 = 

0 otherwise 

{ 1 if policyholder i is over 60 

x i3 = 

0 otherwise 

{ 1 if policyholder i is a female 

x i4 = 

0 otherwise 

{ 1 if policyholder i lives in a rural area 

x i5 = 

0 otherwise 

{ 1 if policyholder i splits his premium payment 

x i6 = 

0 otherwise 

{ 1 if policyholder i uses his car for professional reasons 

x i7 = 

0 otherwise. 

The results are interpreted with respect to the reference class (for which all the x ij s 

are equal to 0) corresponding to a man aged between 31 and 60, living in an urban area, 

paying the premium once a year and using the car for private purposes only. The sequence 

(1,0,0,0,0,0,1) represents a man aged less than 24, living in an urban area, paying the premium 

once a year and using the car for professional reasons. 

In case two covariates interact, a new variable is created by combining the levels of each 

of the interacting covariates, as shown in the following example.


Example 2.3 Figure 2.7 suggests that the variables Age and Gender interact in Portfolio 

A. It is thus not possible to represent accurately the effect of being a male policyholder 

(compared with being a female policyholder) in terms of a single multiplier, nor can the 

effect of Age be represented by a single multiplier. The relevant explanatory variables are not 

x i1 , x i2 , x i3 , and x i4 but rather x i1 x i4 , x i2 x i4 , x i3 x i4 , 1 − x i1 x i2 x i3 x i4 , x i1 1 − x i4 , x i2 1 − x i4 , 

and x i3 1 − x i4 (denoted henceforth as x 

ij ′ ). 

To reflect the situation accurately, it is thus necessary to consider multipliers dependent 

on the combined levels of Age and Gender. To this end, a variable Gender ∗ Age is created, 

with levels ‘Female 18–24’, ‘Female 25–30’, ‘Female 31–60’, ‘Female over 60’, ‘Male 

18–24’, ‘Male 25–30’, ‘Male 31–60’ and ‘Male over 60’. This new variable possesses 8 

levels, and is coded by means of 7 dummies, being all 0 for the reference level (taken as 

‘Male 31–60’). Specifically, the explanatory variables x i1 to x i4 in Example 2.2 are replaced 

with 

{ 1 if policyholder i is a female less than 24 

x ′ i1 = 0 otherwise 

{ 1 if policyholder i is a female aged 25–30 


{ 1 if policyholder i is a female aged 31–60 


{ 1 if policyholder i is a female over 60 


{ 1 if policyholder i is a male less than 24 


{ 1 if policyholder i is a male aged 25–30 


{ 1 if policyholder i is a male over 60 


Rather than declaring Age and Gender as two explanatory variables coded with the help of 

x i1 to x i4 , a combined Age ∗ Gender variable is declared and is coded with the help of the 

covariates x 

i1 ′ to x′ i7 . 

The part of the linear predictor involving Age and Gender is 1 x i1 +···+ 4 x i4 without 

interaction, and becomes ′ 1 x′ i1 +···+′ 7 x′ i7 

if allowance is made for interaction. If Age and 

Gender indeed interact then the values of these two linear combinations differ from each 

other, whereas they collapse in the absence of interaction. 

Allowing for interaction thus dramatically increases the number of explanatory variables. 

Introducing Age and Gender separately requires 4 binary variables whereas allowing 

for the Age–Gender interaction requires 7 dummies. As a consequence, the number 

of parameters to be estimated increases accordingly. We will see below that grouping 

some levels of the combined variable accounting for interaction is nevertheless often 

possible.


2.3.2 Loglinear Poisson Regression Model 

In Poisson regression, we have a collection of independent Poisson counts whose means are 

modelled as nonnegative functions of covariates. Specifically, let N i , i = 1 2n,bethe 

number of claims reported by policyholder i and d i be the corresponding risk exposure (the 

N i s are assumed to be independent). All the observable characteristics (the a priori variables 

presented in Section 2.2 for Portfolio A, say) related to this policyholder are summarized 

into the vector xi 

T = x i1 x ip . Poisson regression is a technique analogous to linear 

regression except that errors no longer follow a Normal distribution but the randomness 

in the model is described by a Poisson distribution. We first assume that the conditional 

expectation of N i given x i is of the form 

( 

) 

p∑ 

EN i x i = d i exp 0 + j x ij i= 1 2n (2.1) 

j=1 

where T = 0 1 p is the vector of unknown regression coefficients. The 

explanatory variables enter the model in the linear combination 0 + ∑ p 

j=1 jx ij , where 0 

acts as an intercept and j is the coefficient indicating the weight given to the jth covariate. 

The Poisson regression model consists in stating that N i is Poisson distributed with mean 

given by Expression (2.1), that is 

( ( 

)) 

p∑ 

N i ∼ oi d i exp 0 + j x ij i= 1 2n 

2.3.3 Score 

The quantity 

j=1 

score i = 0 + 

p∑ 

j x ij 

is called the score (or linear predictor in statistics) because it allows the actuary to rank the 

policyholders from the least to the most dangerous. The claim frequency for policyholder i is 

d i expscore i ; its annual claim frequency being expscore i . Increasing the score thus means 

that the associated average annual claim frequency increases. The use of the exponential link 

function ensures the claim frequency is positive even if the score is negative. 

Let us denote as ̂ 0 ̂ 1 ̂ p the estimators of the regression coefficients 

0 1 p . In a statistical sense, 

̂i = d i exp ( ( 

) 

) p∑ 

ŝcore i = di exp ̂0 + ̂j x ij 

is the predicted expected number of claims for policyholder i. Prediction in this sense does 

not refer to ‘predicting the future’ (called forecasting by statisticians) but rather to guessing 

the expected number of claims (i.e., the response) from the values of the regressors in an 

observation taken under the same circumstances as the sample from which the regression 

equation was estimated. 

j=1 

j=1


2.3.4 Multiplicative Tariff 

When the a priori variables x ij s are coded by means of binary variables (so that each 

policyholder is represented by a vector of 0s and 1s), the intercept 0 represents the risk 

associated to the reference class. Then, the annual claim frequency i associated with 

characteristics x i is of the following multiplicative form 

i = exp 0 ∏ 

exp j 

jx ij =1 

where exp 0 is the annual claim frequency corresponding to the reference class and 

exp j models the impact of the jth ratemaking variable. The intercept 0 represents the risk 

associated with the reference class. If j > 0, being in the class coded by the jth explanatory 

variable increases the score and thus the annual claim frequency. Conversely, if j < 0, the 

score and the annual claim frequency decrease when x ij = 1. With the exponential link, high 

(respectively low) scores mean high (respectively low) claim frequencies. The following 

example makes this clear. 

Example 2.4 

of the form 

Here, 

Let us continue Example 2.2. In Portfolio A, the score for policyholder i is 

score i = 0 + 1 x i1 + 2 x i2 +···+ 7 x i7 

etc 

exp 0 = annual claim frequency for men aged 31–60, living in an 

urban area, paying the premium once a year, using the 

car for private purposes 

exp 0 + 1 = annual claim frequency for men less than 24, living in an 



exp 0 + 2 = annual claim frequency for men aged 25–30, living in an 



exp 0 + 3 = annual claim frequency for men over 60, living in an 



exp 0 + 4 = annual claim frequency for women aged 31–60, living in an 


car for private purposes


2.3.5 Likelihood Equations 

Let k i be the number of claims filed by policyholder i during the observation period. The 

likelihood associated with these observations equals 

where 

n∏ 

n∏ 

= PrN i = k i x i = exp− i k i 

i 

k i ! 

i=1 

i=1 

i = d i expscore i = expln d i + score i 

The maximum likelihood estimator ̂ of maximizes : ̂ is the value of the regression 

coefficients that makes the observations the most plausible. 

Remark 2.1 (Grouping Data) The maximum likelihood estimators obtained for individual 

or grouped data are identical. Let us prove it formally. To this end, let group be the 

likelihood obtained after a grouping in risk classes. We will show below that 

∝ group 

In words, the likelihood group based on grouped data is proportional to the likelihood 

based on individual data, so that the corresponding maximum likelihood estimates will 

coincide. 

Let s 1 s q be the q possible values for the score, say, and let us define 

d •j = 

∑ 

iscore i =s j 

d i and k •j = 

∑ 

iscore i =s j 

k i for j = 1q 

In words, d •j is the total risk exposure for risk class j (corresponding to the value s j of the 

score) and k •j is the total number of claims recorded for the same risk class. Then 

q∏ ∏ 

= exp− i k i 

i 

j=1 

k 

iscore i =s j i ! 

⎛ ⎞ 

q∏ 

∝ exp ⎝− 

∑ 

i 

⎠ ( ) k•j 

exps j d •j 

j=1 

iscore i =s j 

⎛ 

q∏ 

∑ 

= exp ⎝− exps j ⎠ ( ) k•j 

exps j d •j 

j=1 

iscore i =s j 

d i 

⎞ 

∝ 

q∏ 

exp ( ) ( ) k•j 

exps j d •j 

− exps j d •j 

k •j ! 

j=1 

= group 

Maximizing or group then gives the same maximum likelihood estimator ̂.


The computation is easier if we change the likelihood to the log-likelihood which is then 

given by 

L = ln = 

n∑ 

i=1 

( 

− ln k i !+k i ln i − i 

) 

(2.2) 

The maximum likelihood estimators ̂ 0 and the ̂ j s are the solutions of the following 

likelihood equations that are obtained by making the first derivatives of the log-likelihood 

with respect to the regression coefficients equal to zero: 

 

0 

L = 0 ⇔ 

n∑ 

k i − i = 0 (2.3) 

i=1 

 

j 

L = 0 ⇔ 

n∑ 

x ij k i − i = 0 j = 1p (2.4) 

i=1 

2.3.6 Interpretation of the Likelihood Equations 

Equation (2.3) has an obvious interpretation: the fitted total number of claims ∑ n 

i=1̂ i is 

equal to the observed total number of claims ∑ n 

i=1 k i. Therefore, provided that an intercept 

0 is included in the score, the total claim number predicted by the regression model equals 

its observed counterpart. Note that this equality holds for the observation period and not 

necessarily for the future, when the ratemaking will be implemented in practice. In other 

words, we cannot be sure that ∑ n 

i=1̂ i claims will be filed in the future, just that the actual 

number of claims should be close to ∑ n 

i=1̂ i if the yearly number of claims filed to the 

company remains stable over time. 

The interpretation of the second likelihood Equation (2.4) is as follows: In Example 2.2 

with Portfolio A, Equation (2.4) for j = 4 gives 

∑ 

females 

k i = ∑ 

females 

Therefore, the model fits exactly the total number of claims filed by female policyholders. 

There is no cross-subsidies between men and women. The conclusion is similar for the other 

values of j. For j = 1 2 3 for instance, the Equations (2.4) thus ensure that the sum of all 

the claims reported for each age category is exactly reproduced by the model. 

̂i 

2.3.7 Solving the Likelihood Equations with the Newton–Raphson 

Algorithm 

The likelihood equations do not admit explicit solutions and must therefore be solved 

numerically. Let U be the gradient vector of the log-likelihood L = ln defined


in (1.47). Let us define ˜x i as the vector x i of explanatory variables for policyholder i 

supplemented with a unit first component, that is, ˜x i = 1 x T i T . Then, considering (2.3)– 

(2.4), U is given by 

U = 

n∑ 

˜x i k i − i (2.5) 

i=1 

in the Poisson regression model. Let H be the Hessian matrix of L defined in (1.50). 

Specifically, H is given by 

H =− 

n∑ 

˜x i˜x T i i (2.6) 

i=1 

in the Poisson regression model. The maximum likelihood estimator ̂ j of the parameters 

j then solves U = 0. 

The approach used to solve the likelihood equations is the Newton–Raphson algorithm 

(1.51). Starting from an appropriate ̂ 0 , the Newton–Raphson algorithm is based on the 

following iteration 

̂r+1 = ̂ 

) 

r −1 

− H 

(̂r U 

(̂r) 

(2.7) 

= ̂ r + 

( ) n∑ −1 

˜x i˜x T ̂ 

r n∑ ( 

i i ˜x i k i − ̂ ) r 

i 

i=1 

i=1 

for r = 0 1 2, where ̂ i 

r 

= di exp˜x T i ̂ r . Appropriate starting values are given by 

̂ 0 

0 

= ln 

1 

n 

n∑ 

k i and ̂ 

0 

j = 0 for j = 1p. 

i=1 

Note that these starting values are equal to the values of the regression coefficients when no 

segmentation is in force. Therefore, final values close to the starting ones indicate that the 

portfolio is quite homogeneous. 

Remark 2.2 (Iterative Least-Squares) It is possible to interpret the Newton–Raphson 

approach (2.7) in terms of iterative least-squares. Specifically, it is possible to rewrite 

the iterative algorithm (2.7) in the Poisson model in such a way that ̂r+1 

appears as the maximum likelihood estimator in a linear model with adjusted dependent 

variables. Fitting the Poisson regression model by maximum likelihood thus boils down 

to estimating the regression parameter in a sequence of linear models, with adjusted 

responses and explanatory variables. This is particularly interesting since the numerical 

aspects of estimation in a linear model are well-known and have been optimized for 

decades.


2.3.8 Wald Confidence Intervals 

The asymptotic variance-covariance matrix ̂ 

of the maximum likelihood estimator ̂ of 

the regression coefficients vector is the inverse of the Fisher information matrix. This 

matrix can be estimated by 

̂̂ 

= 

( ) n∑ −1 

˜x i˜x T ̂ i i where ̂ i = d i expŝcore i 

i=1 

We know from (1.49) that provided the sample size is large enough ̂− is approximately 

or0 ̂̂ distributed. It is thus possible to compute confidence intervals at level 1− for 

each of the j s. These intervals are of the form 

[̂j − z /2̂̂j 

̂ 

] 

j + z /2̂̂j 

(2.8) 

where ̂ 2̂j 

is the estimated variance of ̂ j , given by the element j j of ̂̂. 

Remark 2.3 (Confidence Intervals for the j s: the Likelihood Ratio Method) The 

confidence interval (2.8) is based on the large sample properties of the maximum likelihood 

estimator ̂. Other methods for constructing such a confidence interval are available. One 

such method is based on the profile likelihood for j that is defined as 

j j = max 

0 j−1 j+1 p 

If ̂ is the maximum likelihood estimator of , we have that 2 ( L̂ − L j j ) is 

approximately 1 2 provided j is the true parameter value. A confidence interval at level 

1 − for j is then given by 

{ ∣ ∣∣Lj 

j j ≥ L̂ − 1 } 

2 2 11− 

 

where 2 11− is the 1 − th quantile of the 2 1 distribution. 

2.3.9 Testing for Hypothesis on a Single Parameter 

It is often interesting to check the validity of the null hypothesis H 0 : j = 0 against the 

alternative H 1 : j ≠ 0. If the jth explanatory variable is dichotomous (think for instance 

of gender), then failing to reject H 0 suggests that this variable is not relevant to explaining 

the expected number of claims. If the jth explanatory variable is coded by means of a set 

of binary variables, then the nullity of the regression coefficient associated with one of the 

binary variables means that the corresponding level can be grouped with the reference level. 

In such a case, equality between the regression coefficients should also be tested to decide 

about the optimal grouping of the levels. Hypotheses involving a set of regression parameters 

will be examined in Section 2.3.13.


Considering (1.52), the easiest test statistic for H 0 against H 1 is certainly 

T = 

̂j 

̂̂j 

which is approximately or0 1 under H 0 , provided the sample size is large enough. 

Alternatively, T 2 is approximately Chi-square with one degree of freedom. Rejection of H 0 

occurs when T is large in absolute values, or when T 2 is large. In this case, j is significantly 

different from 0, and the associated characteristic has a significant impact in ratemaking. 

Note that, as always with hypothesis testing, statistical significance is the resultant of two 

effects: firstly, the distance between the true parameter value and the hypothesized value, 

and secondly, the number of observations (or more precisely, the amount of information 

contained in the data). Even a hypothesis that is approximately true (and useful as a working 

hypothesis) will be rejected with a sufficiently large sample. Conversely, any hypothesis may 

fail to be rejected (and accepted as a working hypothesis) as long as the actuary has only 

scanty data at his disposal. The reader should keep this in mind in the numerical illustrations 

worked out in this book. 

If the explanatory variables are correlated (as it is usually the case in actuarial studies), 

it becomes difficult to disentangle the effects of one explanatory variable from another, and 

the parameter estimates may be highly dependent on which explanatory variables are used in 

the model. If the explanatory variables are strongly correlated then the maximum likelihood 

estimators will have a large variance. The actuary should then reduce the set of regressors. 

2.3.10 Confidence Interval for the Expected Annual Claim Frequency 

It is possible to build a confidence interval for the annual claim frequency. Recall that 

the multivariate Normal distribution has the following useful invariance property. Let C 

be a given n × n matrix with real entries and let b be a n-dimensional real vector. If 

X ∼ or M then Y = CX + b is orC + b CMC T . The variance of the predicted 

score, ŝcore i = ˜x T i ̂, is thus given by 

which is estimated by 

Vŝcore i = ˜x T i ̂˜x i 

̂ Vŝcore i = ˜x T i ̂̂˜x i 

As the maximum likelihood estimator ̂ is approximately Gaussian when the number of 

policies is large, ŝcore i is also Gaussian and an approximate confidence interval at level 

1 − for the annual claim frequency can be computed as 

[ 

exp 

(̂T˜x i − z /2 

√˜x i T ̂̂˜x 

) 

i exp 

(̂T˜x i + z /2 

√˜x i T ̂̂˜x 

)] 

i


2.3.11 Deviance 

Let ̂ be the model likelihood, i.e. 

̂ = 

n∏ 

i=1 

exp−̂ i ̂ k i 

i 

k i ! 

Note that the maximal value of ↦→ exp− k /k! is obtained for = k. Therefore, the 

Poisson likelihood is maximum with expected claim frequencies equal to the observed 

number of claims. The maximal likelihood possible under the Poisson assumption is then 

k = 

n∏ 

i=1 

exp−k i kk i 

i 

k i ! 

This is the likelihood of the saturated model, predicting the observed number of claims for 

each insured driver (there are thus as many parameters as observations). This model just 

replicates the observed data. 

The deviance Dk̂ is defined as the likelihood ratio test statistic for the current model 

against the saturated model, that is, 

Dk̂ =−2ln ̂ ( 

) 

k = 2 ln k − ln ̂ 

( ) ( 

∏ n 

∏ n 

= 2ln 

= 2 

i=1 

( 

n∑ 

i=1 

exp−k kk i 

i 

i k i ! 

k i ln k i 

̂i 

− k i −̂ i 

− 2ln 

) 

i=1 

) 

exp−̂ ̂ k 

i 

i 

i k i ! 

where y ln y = 0 for y = 0 by convention. It measures the distance of the model likelihood 

to the saturated model replicating the observed data. The smaller the deviance, the better the 

current model. 

When an intercept 0 is included in the linear predictor, (2.3) allows us to simplify the 

deviance as 

Dk̂ = 2 

n∑ 

i=1 

k i ln k i 

̂i 

 

Provided the data have been grouped as much as possible and the model is correct, D is 

approximately 2 n−dim 

distributed (where n is now the number of classes in the portfolio). 

The model is considered as inappropriate if D obs is ‘too large’, that is, if 

D obs > 2 n−dim1−


2.3.12 Deviance Residuals 

The deviance residuals in the Poisson model are given by: 

r D i 

= √ 2signk i − ̂ i 

√ 

k i ln k i 

̂i 

− k i − ̂ i 

Summing the ri D 2 gives the deviance Dk̂. The deviance residual ri 

D is thus the signed 

square root of the contribution of policyholder i to the deviance Dk̂. 

A plot for individual data is often uninformative in motor insurance (because of the 

few observed values for the k i s, the deviance residuals ri 

D are always structured, being 

concentrated along curves corresponding to 0 claim, 1 claim, 2 claims, etc.). A plot of the 

residuals against fitted frequencies ̂ i for grouped data helps to check the adequacy of the 

model. 

2.3.13 Testing a Hypothesis on a Set of Parameters 

We would like to test the null hypothesis 

{ 

H0 = 0 = 0 1 2 q T 

H 1 = 1 = 0 1 2 q q+1 p T 

Let D 0 be the deviance of the Poisson regression model under H 0 , and D 1 be the deviance 

under H 1 . The test statistic is 

( 

= D 0 − D 1 = 2 Lk − L̂ 0 

= 2 

) 

( 

L̂ 1 − L̂ 0 

( 

) 

− 2 Lk − L̂ 1 

) 

≈ d 2 p−q 

Note that is a likelihood ratio test statistic. The null hypothesis H 0 is rejected in favour of 

H 1 if obs is ‘too large’, that is, if 

obs > 2 p−q1− 

2.3.14 Specification Error and Robust Inference 

According to the asymptotic theory related to generalized linear models, the estimator ̂ 

obtained by maximizing the Poisson likelihood remains consistent and efficient provided 

the mean and variance of the model are correctly specified (even if the underlying data 

generating process is not Poisson). Moreover, in order to obtain consistency, only the correct 

specification of the mean function is required, that is, only (2.1) has to be valid. And ̂ 

remains Normally distributed in all cases. 

Let us now assume that EN i x i = d i exp T 0 x i holds true for some 0 (i.e. the conditional 

mean is correctly specified), but N i is not Poisson distributed (N i is in reality Negative 

Binomial, for instance). In this case, there is a specification error, in that inference conducted


with the Poisson likelihood is based on a false distributional assumption. The Poisson 

maximum likelihood estimator ̂ remains nevertheless consistent for the true parameter 0 , 

i.e. ̂ → proba 0 as the sample size n →+. This explains why the Poisson regression 

model is so useful: it continues to give reliable estimations for the annual expected claim 

frequency even if the true model is not Poisson, provided the sample size is large enough. 

However, the variances of the ̂ j s are mis-estimated. Inference must then be based on the 

robust information matrix estimate of the variance-covariance matrix of ̂ that is based on the 

empirical estimate of the observed information. Specifically, because of the misspecification, 

the asymptotic variance-covariance matrix of ̂ is now given by 

where 

i=1 

In practice, it will be estimated by 

̂ 

= −1 

n∑ 

n∑ 

= ˜x i˜x T i d i exp T 0˜x i and = ˜x i˜x T i VN ix i 

i=1 

̂̂ 

= ̂ −1̂̂ (2.9) 

where 

n∑ 

̂ = ˜x i˜x T̂ n∑ 

i i and ̂ = ˜x i˜x T i ̂ i − k i 2 

i=1 

with ̂ i = d i exp̂ T˜x i . Let us point out that ̂ − remains approximately Normally 

distributed with mean 0 and covariance matrix ̂, 

provided the sample size is large enough. 

i=1 

2.3.15 Numerical Illustration 

Within SAS R , the GENMOD procedure can be used to fit Poisson regression models. This 

procedure supports the Normal, Binomial, Poisson, Gamma, Inverse Gaussian, Negative 

Binomial and Multinomial distributions, in the framework of generalized linear models. A 

typical use of the GENMOD procedure is to perform Poisson regression with a log link 

function. This type of model is usually called a loglinear model. 

The logarithm of the exposure-to-risk is used as an offset, that is, a regression variable 

with a constant coefficient of 1 for each observation. A log linear relationship between the 

mean and the explanatory factors is specified by the log link function. The log link function 

ensures that the mean number of insurance claims predicted from the fitted model is positive. 

The results obtained from the Poisson regression for Portfolio A presented in Section 2.2 

are shown in Table 2.1. Table 2.1 is similar to the ‘Analysis of Parameter Estimates’ table 

produced by the GENMOD procedure. Such a table summarizes the results of the iterative 

parameter estimation process. For each parameter in the model, the GENMOD procedure 

displays columns with the parameter name, the degrees of freedom associated with the 

parameter, the estimated parameter value, the standard error of the parameter estimate, the


Table 2.1 Results of the Poisson regression for the model with the 5 explanatory variables, 

Portfolio A. 

Variable Level Coeff Std error Wald 95 % conf limit Chi-sq Pr>Chi-sq 

Intercept −22131 00582 −23271 −20991 144740 60 years −00010 02350 −04616 04596 000 09967 

Gender ∗ Age Male 18–24 years 06429 00797 04867 07990 6510


used). The ratio of the deviance to the number of degrees of freedom should be close to 

1 to indicate goodness-of-fit. Here, we obtain 05357 (the deviance is equal to 7764.83). 

Note however that the data should be grouped to make the Chi-square approximation more 

reliable, so that we cannot use this statistic at the present stage. 

An important aspect of insurance ratemaking with generalized regression models is the 

selection of explanatory variables in the model. Changes in goodness-of-fit statistics are 

often used to evaluate the contribution of subsets of explanatory variables to a particular 

model. One strategy for variable selection is to fit a sequence of models, beginning with 

a simple model with only an intercept term, and then include one additional explanatory 

variable in each successive model. The importance of the additional explanatory variable 

can be measured by the difference in fitted log-likelihoods between successive models. 

Asymptotic tests computed by the GENMOD procedure enable the actuary to assess the 

statistical significance of the additional term (this is called Type I analysis in SAS R ). 

Another strategy (adopted here) consists of starting from a model incorporating all the 

available information, and then excluding the irrelevant explanatory variables. To this end, 

the GENMOD procedure generates a Type 3 analysis (analogous to Type III sums of squares 

in the GLM procedure). A Type 3 analysis does not depend on the order in which the terms 

for the model are specified (in contrast to the Type 1 analysis). 

Type 3 analysis compares the complete model (that is the model which includes all the 

specified variables) with the different submodels obtained by deleting one of the explanatory 

variables. It enables the actuary to test the relevance of one variable taking all the others 

into account. It roughly corresponds to the backward approach: at each step, we exclude the 

variable with the largest p-value until no more can be excluded (i.e. until all the p-values 

are smaller than a fixed threshold, generally 5 %). Note that the Type 3 analysis works with 

the variables and not with the levels of these variables. Indeed, it is possible to obtain a 

relevant variable for which some levels are not relevant. The results of the Type 3 analysis 

are as follows: 

Source DF Chi-square Pr > Chi-sq 

Gender ∗ Age 7 7516


The option ‘Estimate’ of GENMOD (which is similar to the option ‘Contrast’) can be used 

to assess the relevance of grouping the levels of the interaction Age–Gender 2 by 2. This 

option computes likelihood ratio statistics for user-defined contrasts, that is, linear functions 

of the parameters, and p-values based on their asymptotic Chi-square distributions. Here, the 

option ‘Estimate’ is used to test for the equality of the regression coefficients (that is, for H 0 : 

j1 

= j2 

against H 1 : j1 ≠ j2 

for all the combinations j 1 j 2 corresponding to binary variables 

coding Age ∗ Gender). The test statistic is based on the ratio of the likelihoods corresponding 

to the models under H 0 and under H 1 . This grouping process can be summarized as follows: 

Step 1 the reference level ‘Male 31–60’ is first merged with ‘Female >60’ (p-value of 

9967 %); 

Step 2 then ‘Male 25–30’ and ‘Female 18–24’ are grouped together (p-value of 8585 %); 

Step 3 ‘Male 31–60’, ‘Male > 60’ and ‘Female > 60’ are grouped together (p-value of 

6613 %); 

Step 4 ‘Male 31–60’, ‘Male > 60’, ‘Female 31–60’ and ‘Female > 60’ are grouped together 

(p-value of 3517 %); 

Step 5 finally ‘Male 25–30’, ‘Female 18–24’ and ‘Female 25–30’ are grouped together 

(p-value of 1239 %). 

The level ‘Male 18–24’ cannot be grouped with other levels. After grouping, the 

variable Age ∗ Gender resulting from the interaction of Age with Gender has three levels: 

‘Males 18–24’, ‘Females 18–30 and Males 25–30’, and ‘Males and Females over 30’. As 

expected, it accounts for the extra risk of young male drivers (aged between 18 and 24), and 

of the young drivers, whereas the reference level 3 is assigned to drivers over 30. 

The fit of the final model is shown in Table 2.2. The log-likelihood is now equal to 

−54842 and the Type 3 analysis gives the following results: 

Source DF Chi-square Pr>Chi-sq 



Table 2.2 Results of the Poisson regression for the final model, Portfolio A. 


Intercept −21975 00466 −22888 −21062 222551


5 

Individual data 

4 

Deviance residuals 

3 

2 

1 

0 

–1 

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 

Predicted value 

1 

Grouped data 

0 

Deviance residuals 

–1 

–2 

–3 

–4 

0 100 200 300 

Predicted value 

Figure 2.8 Deviance residuals against fitted annual claim frequencies for individual data (top panel) 

and grouped by risk classes (bottom panel), Portfolio A. 

The significance of the explanatory variables is clearly apparent (even if the Chi-square 

test statistics are smaller than in the regular Poisson case, and the corresponding p-values 

larger).


Table 2.3 Results of the robust Poisson regression for the final model, Portfolio A. 

Variable Level Coeff Std error Wald 95 % conf limit Z Pr > Z 

Intercept −21975 00487 −22930 −21020 −4510


In the class C 1 ∪ C 2 , the expected claim number equals 

m = p 1 m 1 + p 2 m 2 

where p 1 and p 2 denote the respective weights of C 1 and C 2 (the ratio of the class exposure 

to the sum of the exposures of both classes, say). Considering the variance of the number of 

claims in C 1 ∪ C 2 , it can be decomposed as the average of the conditional variances 2 1 and 

2 2 plus the variance of the conditional means m 1 and m 2 (that is, the weighted sum of their 

squared difference with respect to the grand mean m). The variance thus becomes 

p 1 2 1 + p 2 2 2 

+p 

} {{ } 1 m 1 − m 2 +p 

} {{ } 2 m 2 − m 2 >m 

} {{ } 

=m 

>0 

>0 

which exceeds the mean. Hence, omitting relevant ratemaking variables induces 

overdispersion. 

2.4.3 Consequences of Overdispersion 

As mentioned in Section 2.3.14, misspecification of the variance function does not affect the 

consistency of ̂, but leads to misspecification of the asymptotic variance-covariance matrix 

of ̂. As a result, we have a loss of efficiency. 

Overdispersion leads to underestimates of standard errors and overestimates of Chi-square 

statistics (as demonstrated in Table 2.3), which in turn may imply artificial statistical 

significance for the parameters. Consequently, some explanatory variables may become not 

significant after overdispersion has been accounted for. In practice, failing to account for 

overdispersion might produce too many risk classes in the portfolio. 

2.4.4 Modelling Overdispersion 

Many explanatory variables are unknown to the insurance company or cannot be incorporated 

in the price list (for legal, moral or economic reasons). There are thus unobservable 

characteristics Z i that may influence the number of claims filed by policyholder i as explained 

in Section 2.1.2. Of course, some Z ij s may be correlated with the observable characteristics 

X i . To remove these correlations, we could think of first regressing the Z i sontheX i s, with 

a linear regression model 

p∑ 

Z ij = 0 + k X ik + ij 

Then, the score becomes 

0 + 

p∑ 

j=1 

dimZ i 

∑ 

j X ij + j Z ij = 0 + 

j=1 

for appropriate ˜ 0 , ˜ j s and ˜ i . 

= ˜ 0 + 

k=1 

p∑ 

j=1 

dimZ i 

∑ 

j X ij + j 

( 0 + 

p∑ 

˜j X ij +˜ i 

j=1 

j=1 

p∑ ) 

k X ik + ij 

k=1


The unobserved heterogeneity (when correlated to observable characteristics) thus modifies 

the regression coefficients: the true effect of X i on N i becomes an apparent effect ˜. 

Hence, the estimated jth regression coefficient does not only represent the effect of the 

jth covariate on the number of claims, but also accounts for the effect of all the hidden 

characteristics Z i correlated with the jth observable one. This is why the ̂ j s may strongly 

depend on which covariates are included in the model. Moreover, there remains an error 

term ˜ i representing the influence of the hidden variables on N i , corrected for the effect of 

the observed risk factors X i . 

For these reasons, we now consider a mixed Poisson model 

( ( 

)) 

p∑ 

N i ∼ oi exp 0 + j x ij + i i= 1 2n (2.10) 

j=1 

where the random variable i represents the residual effect of the hidden characteristics. 

Therefore, the heterogeneity is taken into account by assuming that the number of accidents 

is Poisson distributed with mean varying from one policyholder to another. 

Note that some hidden characteristics are correlated to those in X i (e.g. the hidden annual 

mileage and the observable use of the vehicle). The random variable i in (2.10) models the 

effect of hidden characteristics that is not already explained by X i . Since i accounts for a 

residual effect, we will consider in the remainder of this book that i is independent from 

X i . The price to pay is that the estimated regression coefficient j does not only express the 

effect of the jth regressor, but also the effect of all the hidden characteristics correlated with 

the jth regressor. This is important when the actuary tries to interpret the resulting price list. 

The policyholders have different accident proneness because of observable characteristics 

taken into account in the price list and hidden characteristics to be corrected a posteriori. 

The heterogeneity is taken into account by assuming that the number of accidents is Poisson 

distributed with mean varying from one policyholder to another. The annual claim frequency 

becomes a random variable i i where i = exp i models the oscillations around the grand 

mean i (with E i = 1). We can now write N i ∼ Poi i i where i = d i exp˜x T i . 

As in (1.29), we have 

VN i = i + 2 i V i> i = EN i (2.11) 

so that any mixed Poisson regression model induces overdispersion. 

2.4.5 Detecting Overdispersion 

Residual heterogeneity remains considerable in the risk classes despite the use of many a 

priori variables. Indeed, many explanatory variables are unknown to the insurance company 

and cannot be incorporated in the price list. Let us denote as ̂m k the empirical mean claim 

number of risk class k, and ̂ 

k 

2 the associated variance. In order to graphically test the 

Poisson mixture assumption, we have plotted the points ̂m k ̂ 

k 2 and also the bisecting line. 

The plot is displayed in Figure 2.9. We observe that ̂m k < ̂ 

k 

2 in numerous risk classes, 

indicating that the homogeneous Poisson model is inappropriate. We also see that most of 

the observed pairs ̂m k ̂ 

k 2 lie above the bisecting line, thus supporting overdispersion. The 

three points below the 45-degree line correspond to classes with just a few policies (36, 13 

and 12, respectively).


0.8 

Overdispersion of the data 

Empirical variances 

0.6 

0.4 

0.2 

0.0 

0.0 0.1 0.2 0.3 0.4 

Empirical means 

Figure 2.9 Mean–Variance pairs for the risk classes, Portfolio A. 

A quadratic curve (without intercept) has been fitted to the mean–variance couples by 

weighted least-squares (the weights being the exposures of the risk classes). The high value 

of the R 2 coefficient (86.41 %) supports the quality of the fit. This shows that equation (2.11) 

is supported by the data, and provides empirical evidence for a mixed Poisson model. 

2.4.6 Testing for Overdispersion 

The graphical test of the previous section is an easy way of detecting overdispersion. But 

many statistical tests for the overdispersion assumption have been developed in the literature. 

Testing for overdispersion can be done by testing for the Poisson distribution against a mixed 

Poisson model. One problem with standard specification tests (such as likelihood ratio tests) 

occurs when the null hypothesis is on the boundary of the parameter space, as explained in 

Chapter 1. When a parameter is bounded by the H 0 hypothesis, the estimate is also bounded 

and the asymptotic Normality of the maximum likelihood estimator no longer holds under 

H 0 . Consequently, a correction must be made. 

Alternatively, testing for overdispersion can be based on the variance function. According 

to (2.11), the variance function of a heterogeneity model is of the form: 

VN i = i + 2 i (2.12) 

with = V i being the variance of the random effect. Therefore, we have to test the null 

hypothesis H 0 : = 0 against H 1 : >0. The following score statistics can be used to test the 

Poisson distribution against heterogeneity models with a variance function of the form (2.12): 

T 1 = 

∑ n 

i=1 

(k i −̂ i 2 − k i 

) 

√ 

2 ∑ n 

i=1̂ 2 i


T 2 = 

∑ n 

i=1 

√ 

∑n 

i=1 

(k i −̂ i 2 − k i 

) 

( 

k i −̂ i 2 − k i 

) 2 

 

and 

T 3 = √ 

1 

n 

∑ n 

i=1 

(k i −̂ i 2 − k i 

) 

∑ 

( 

) 2 

√ 

n ∑n 

i=1̂ −2 

i k i −̂ i 2 − k i i=1̂ 2 i 

All these test statistics are or0 1 distributed. For Portfolio A, T 1 = 918, T 2 = 613 and 

T 3 = 438. All the p-values are less than 10 −4 leading to the rejection of the null hypothesis 

(and thus the Poisson model) in favour of the mixed Poisson model. 

2.5 Negative Binomial Regression Model 

2.5.1 Likelihood Equations 

Overdispersion is taken into account by the inclusion of a random effect, representing 

an unknown relative risk level. More precisely, assume that 1 n are independent 

ama a distributed random variables, i.e. the common probability density function 

of the i s is given by (1.35). In this case, E i = 1 and V i = 1/a. 

Note that the assumptions made about the i s are rather strong. Their common 

distribution means that the effect of hidden variables does not depend on observable 

ones. 

Conditional on the observable characteristics summarized in the vector x i and on the 

random effect i = , the annual claim number caused by policyholder i conforms to the 

oi i law. In other words, i is the expected claim frequency for policyholder i (based 

on x i ) and is the relative risk level for this policyholder (if


The maximum likelihood estimator for and a solve 

( 

) 

 

n∑ 

La= a + k 

˜x i k i − i 

i = 0 

a + i 

i=1 

Note that these equations are similar to the ones obtained in the Poisson case except that i 

is now replaced with i a + k i /a + i . 

Remark 2.4 Let us give an intuitive explanation for the ratio a + k i /a + i involved 

in the Negative Binomial likelihood equations. The joint probability density function of 

N i i equals 

( ) ki 

i i 1 

exp− i i 

k i ! a aa a−1 

i 

exp−a i 

∝ exp − i i k i+a−1 

i exp−a i 

The probability density function of i given N i = k i is 

exp − i a + i a+k i−1 

i 

∫ + 

0 

exp − a + i a+k i−1 

d 

= exp − i a + i a+k a + 

i−1 i a+k i 

i 

a + k i 

so that i given N i = k i follows the am a + k i a+ i distribution. Therefore, 

E i N i = k i = a + k i 

a + i 

and the maximum likelihood estimators in the Negative Binomial regression model solve 

n∑ 

i=1 

) 

x i 

(k i − i E i N i = k i = 0 

Compared to the Poisson likelihood equations, the predicted expected claim number i is 

replaced with its update i E i N i = k i based on the information contained in the number 

k i of claims filed by policyholder i. 

As already shown in Section 2.3.7, it is possible to solve the Negative Binomial likelihood 

equations with the help of the Newton–Raphson iterative procedure. Starting values for the 

Newton–Raphson iterative procedure are usually obtained as follows: The Poisson maximum 

likelihood estimator ̂ is known to be consistent, so that we keep it as a reasonable starting 

value. We still have to find an initial estimate for = V i = 1/a. So, we first compute 

the variance of N i which is given by 

VN i = EN i + ( d i expscore i ) 2


and then write the empirical analogue of the last relation 

n∑ 

i=1 

( (ki 

− d i expscore i ) 2 

− ki − ( d i expscore i ) 2 

 

) 

= 0 

Therefore, the estimator of is given by 

1 

â =̂ = ∑ n 

i=1 

( (ki 

− d i expŝcore i ) 2 

− ki 

) 

∑ n 

i=1 

( 

di expŝcore i ) 2 

 

where ŝcore i =˜x T i ̂, ̂ being the Poisson maximum likelihood estimator for . The estimators 

̂ and ̂ are consistent in the Poisson mixture model, and are thus good starting values for 

finding the maximum likelihood estimators. 


The Negative Binomial regression with categorical variables can be performed with the 

SAS R /STAT procedure GENMOD which corrects the estimations for overdispersion. The 

final model for Portfolio A is shown in Table 2.4. The interpretation of the different columns 

is the same as in Section 2.3.15. Compared with the Poisson fit, we see that the estimated 

j s are very similar, but the standard errors are larger in the Negative Binomial case. 

The estimation of the parameter a by the method of moments is given by â = 12401 

whereas the maximum likelihood estimator is equal to â = 1065. The log-likelihood is equal 

to −54485. The variance-covariance matrix of the estimated regression coefficients and the 

dispersion parameter is 

̂̂ 

= 

⎛ 

⎞ 

0002424 −0001537 −0001472 −0001042 −0001033 −0001537 0000033 

−0001537 0003249 0001692 −0000080 −0000286 0000824 0000022 

−0001472 0001692 0006073 −0000216 −0000537 0001077 0000319 

−0001042 −0000080 −0000216 0002859 0000217 −0000018 0000008 

 

⎜ 

⎝ −0001033 −0000286 −0000537 0000217 0003038 0000688 0000200 ⎟ 

⎠ 

−0001537 0000824 0001077 −0000018 0000688 0006785 −0000019 

0000033 0000022 0000319 0000008 0000200 −0000019 0020850 

The Type 3 analysis gives the following results: 




Table 2.4 Results of the Negative Binomial regression for the final model, Portfolio A. 


Intercept −21963 00492 −22928 −20998 199008


Table 2.5 Results of the Poisson Inverse-Gaussian regression, Portfolio A. 


Intercept −21962 00494 −22930 −20995 19794



The GENMOD procedure of SAS R does not support the Poisson-LogNormal model. We 

have computed the regression coefficients with the aid of the NLMIXED procedure of SAS R . 

The results are given in Table 2.6 and the estimation of 2 is equal to 07064 which leads to 

̂ = ̂V i = 1027 

for the variance of the random effect. We observe that this value is greater than those obtained 

in the Poisson-Inverse Gaussian model and in the Negative Binomial model. The resulting 

regression coefficients are similar to those obtained previously. The log-likelihood is equal 

to −54481 and is intermediate between the log-likelihoods of the Negative Binomial model 

and of the Poisson-Inverse Gaussian model (which is the maximum). The variance-covariance 

matrix of the estimated regression coefficients and ̂ 2 is 

̂̂ 

= 

⎛ 

⎞ 

0002438 −0001546 −0001474 −0001047 −0001036 −0001543 0000075 

−0001546 0003270 0001707 −0000082 −0000287 0000831 0000064 

−0001474 0001707 0006104 −0000224 −0000548 0001088 0000354 

−0001047 −0000082 −0000224 0002875 0000218 −0000022 −0000050 

 

⎜ 

⎝ −0001036 −0000287 −0000548 0000218 0003056 0000687 0000188 ⎟ 

⎠ 

−0001543 0000831 0001088 −0000022 0000687 0006825 0000058 

0000075 0000064 0000354 −0000050 0000188 0000058 0008056 

The Type 3 analysis for the final model gives 




2.8 Risk Classification for Portfolio A 

So far, we have several competing models for the observed claim frequencies in Portfolio 

A. This section purposes to compare these models in order to select the optimal one. 

2.8.1 Comparison of Competing models with the Vuong Test 

Let us consider two non-nested competing models for the number of claims, with respective 

probability mass functions p·x and q·x , where x is a vector of explanatory 

variables, and and include the regression parameters as well as some dispersion 

coefficients . The corresponding log-likelihoods are 

L p = 

L q = 

n∑ 

ln pk i x i 

i=1 

n∑ 

ln qk i x i 

In this regression context, the test statistic proposed by Vuong (1989) is 

i=1 

T LRNN = L p̂ − L q ̂ 

√ n 

(2.14) 

where 

2 = 1 n 

( 

n∑ 

i=1 

ln pk ix i ̂ 

qk i x i ̂ 

) 2 

− 

( 

1 

n 

n∑ 

i=1 

) 

ln pk 2 

ix i ̂ 

(2.15) 

qk i x i ̂ 

is the estimate of the variance of the log-likelihood difference. None of the model has to 

be true: the test is aimed at selecting the model that is the closer to the true conditional 

distribution. The null hypothesis of the test is that the two models are equivalent. Under the 

null hypothesis, the test statistic is asymptotically Normally distributed. Rejection in favour 

of p happens when T LRNN >c or in favour of q if T LRNN < −c, where c represents the 

or 0 1 critical value for some significance level. If T LRNN ≤c then the null hypothesis 

is not rejected and the Vuong test cannot discriminate between the two models, given the data. 

Now, comparing the three mixed Poisson models with the Vuong test gives: 

Negative Binomial against Poisson-Inverse Gaussian 

to −0754544 leading to a p-value of 4506 %. 

the value of the test statistic is equal 

Poisson-LogNormal against Negative Binomial 

to −0470894 leading to a p-value of 6378 %. 

the value of the test statistic is equal to 

Poisson-LogNormal against Poisson-Inverse Gaussian the value of the test statistic is 

equal to to −0216702 leading to a p-value of 8284 %. 

These results do not enable us to distinguish between the three models which are 

therefore statistically equivalent. The Negative Binomial model will be used for the numerical


illustrations involving Portfolio A in the next chapters. The reason is that, in this case, explicit 

expressions are available, providing a deeper insight in the mechanisms behind experience 

rating systems. 

2.8.2 Resulting Risk Classification for Portfolio A 

Table 2.7 gives the resulting price list obtained with the Negative Binomial model of 

Table 2.4. A ‘Yes’ indicates the presence of the characteristic corresponding to the column. 

The final a priori ratemaking contains 23 classes. Table 2.7 gives the estimated expected 

annual claim frequencies obtained from the Negative Binomial regression model, and the 

relative importance of each risk class. 

Note that there is another way to present the results displayed in Table 2.7. The idea is to 

start from the annual expected claim frequency of the reference class, estimated to 

exp̂ 0 = 1112 % 

according to Table 2.4, and then to apply correction coefficients. Specifically, the annual 

expected claim frequency of a given policyholder is simply obtained from 

1112 % 

⎧ 

exp06399 = 190 if the policyholder is a male aged between 18 and 24 

⎪⎨ 

exp02363 = 127 if the policyholder is a female aged between 18 and 30 

× 

or a male aged between 25 and 30 

⎪⎩ 

1 

otherwise 

{ exp−01805 = 083 if the policyholder lives in a rural district 

× 

1 

otherwise 

{ exp04783 = 161 if the policyholder splits the premium payment 

× 

1 

otherwise 

{ exp02145 = 124 if the policyholder uses the car for professional purposes 

× 

1 

otherwise 

2.9 Ratemaking using Panel Data 

2.9.1 Longitudinal Data 

Actuaries often pool several observation periods to determine the price list (the main goal 

being to increase the size of the data base). The serial dependence arising from the fact that 

the same individuals are followed and produce correlated claim numbers should prevent the 

actuaries from using classical statistical techniques (which assume independence). 

During the observation period, n policies have been in the portfolio, each one observed 

during T i periods. Let N it be the number of claims reported by policyholder i during year t, 

i = 1 2n, t = 1 2T i . Such motor insurance data have a panel structure: typically, 

n is large whereas the T i s are small. 

Let d it be the length of observation period t for policyholder i. Usually, d it = 1, but there 

are a variety of situations where this is not the case. Indeed, a new period of observation

Table 2.7 A priori risk classification for Portfolio A (Negative Binomial regression). 

Female 18–30 

Male 25–30 

Gender–age Use of the car Premium split District Exp. annual 

claim freq. (%) 

Male 18–24 Others Private Professional Annual Split Rural Urban 

Weights 

(%) 

Yes No No Yes No Yes No Yes No 1176 1049 

Yes No No Yes No Yes No No Yes 1408 1396 

Yes No No Yes No No Yes Yes No 1897 398 

Yes No No Yes No No Yes No Yes 2272 705 

Yes No No No Yes Yes No Yes No 1457 076 

Yes No No No Yes Yes No No Yes 1746 122 

Yes No No No Yes No Yes Yes No 2351 013 

Yes No No No Yes No Yes No Yes 2816 014 

No Yes No Yes No Yes No Yes No 1761 293 

No Yes No Yes No Yes No No Yes 2109 299 

No Yes No Yes No No Yes Yes No 2840 152 

No Yes No Yes No No Yes No Yes 3402 242 

No Yes No No Yes Yes No Yes No 2182 007 

No Yes No No Yes Yes No No Yes 2614 009 

No Yes No No Yes No Yes Yes No 3520 002 

No No Yes Yes No Yes No Yes No 928 1338 

No No Yes Yes No Yes No No Yes 1112 1973 

No No Yes Yes No No Yes Yes No 1498 294 

No No Yes Yes No No Yes No Yes 1794 661 

No No Yes No Yes Yes No Yes No 1151 372 

No No Yes No Yes Yes No No Yes 1378 517 

No No Yes No Yes No Yes Yes No 1856 025 

No No Yes No Yes No Yes No Yes 2223 044


starts as soon as some policy characteristics are modified (think for instance of a policyholder 

house moving for a company using postcode as rating factor, a policyholder’s wedding for 

a company using marital status, or simply the policyholder buying a new car). Moreover, 

in the year the policy is issued and in the one it is possibly cancelled the length of the 

observation period is generally less than unity. 

We face a nested structure: each policyholder generates a sequence N i = 

N i1 N i2 N iTi T of claim numbers. It is reasonable to assume independence between the 

series N 1 N 2 N n , but this assumption is very questionable inside the N i s. Regarding a 

priori ratemaking, the dependence between the components of each N i is a nuisance (in the 

statistical sense). This means that, at this stage, we are not interested in accurately modelling 

this dependence, but we must take it into account when estimating the regression coefficients. 

The idea now is to incorporate in the N it s exogenous information (like age, gender, power 

of the car, and so on) summarized in the vectors x it ; to this end, we resort to a regression 

model for longitudinal data. 

The distributional assumption for the random component of the regression model has to 

account for the non-negativity of the data, as well as their integer values. We begin with 

Poisson regression and assume that the N it s conform to the Poisson distribution with a 

mean that can be written as an exponential function of a linear combination 0 + ∑ p 

j=1 jx itj 

of the explanatory variables x it , with unknown regression coefficients to be estimated 

from the data. Despite its prevalence as a starting point in the analysis of count data, the 

Poisson specification is often inappropriate because of unobserved heterogeneity and failure 

of the independence assumption if the data consist in repeated observations on the same 

policyholders. A convenient way to take this phenomenon into account is to introduce a 

random effect into the model. 

Remark 2.5 Before embarking on a panel analysis pooling together the observations 

relating to several years, it is interesting to first work year by year to assess the stability of 

the effect of each rating variable on the annual expected claim frequency. Specifically, the 

vector of the regression coefficients is estimated on the basis of each calendar year and the 

components are checked for their stability over time. Only stable coefficients are interesting 

for the purpose of ratemaking. Rating factors with unstable regression coefficients should be 

excluded from the risk classification scheme. In some cases, a time trend is visible for some 

estimated regression coefficients (this is typically true for the intercept 0 ). A time effect 

can then be incorporated into the model to account for coefficients with trends. 

2.9.2 Descriptive Statistics for Portfolio B 

The analysis in this section is based on an insurance portfolio containing 20 354 policies and 

observed during 3 years (from 1997 to 1999). We have 45 350 observations available as not 

all the policies have been in force for 3 years. For each policy and for each year, we know 

the exposure-to-risk, the number of claims filed and some other explanatory variables: 

Gender: Policyholder’s gender (male–female) 

Age: Policyholder’s age (18–22, 23–30 and over 30) 

Power: The power of the vehicle (less than 66 kW, 66–110 kW and more than 110 kW) 

Size: The size of the city where the policyholder lives (large, middle or small), and 

Colour: The colour of the vehicle (red or other).


Frequency 

0.36 

0.34 

0.32 

0.30 

0.28 

0.26 

0.24 

0.22 

0.20 

0.18 

0.16 

0.14 

0.12 

0.10 

0.08 

0.06 

0.04 

0.02 

0.00 

0–3 3–6 6–9 9–12 12–15 15–18 18–21 21–24 24–27 27–30 30–33 33–36 

Months 

Figure 2.10 Exposure-to-risk in Portfolio B. 

The observed mean annual claim frequency is 184 %. Figures 2.11 to 2.15 display the 

histograms giving, for each explanatory variable, the distribution of the portfolio between 

the different levels of the variable and, for each of these levels, the observed mean annual 

claim frequency. 

Figure 2.10 gives the distribution of the exposure-to-risk in the portfolio. About 34 % of 

the policies have been in force during 3 years. The exposures for policies in force for less 

than three years are roughly uniformly distributed accross the triennium. 

The age structure of the portfolio is described in Figure 2.11. Most policyholders (60.6 % 

of the portfolio) are older than 30. Only 3.0 % of the policyholders are less than 22 whereas 

36.4 % are between 23 and 30. We see that the annual claim frequency decreases with 

the age of the policyholders. In this portfolio, the young drivers are rather risky as their 

observed annual claim frequency is 30.8 %. Drivers over 30 have an observed annual claim 

frequency of 16.3 %. Finally, the drivers aged between 23 and 30 have an observed annual 

claim frequency of 20.8 %. 

Figure 2.12 suggests a higher annual claim frequency for males (18.8 %) than for females 

(17.7 %). The portfolio is comprised of 64.0 % male policyholders and 36.0 % female 

policyholders. 

Figure 2.13 gives the distribution of the policyholders according to the size of the city 

where they live. We see that 31.6 % of the policyholders live in a large city, 35.0 % in a 

middle-sized city and the remaining 33.4 % in a small city. The annual claim frequency 

seems to increase with the size of the city (going from 16.6 % for the small cities, to 17.8 % 

for the middle-sized cities and to 21.0 % for the large cities). 

We see, in Figure 2.14, that most of the cars (65.6 % of the portfolio) have a power smaller 

than 66 kW, 31.2 % have a power included in the 66–110 kW range and only 3.2 % of the 

cars hold an engine with a power greater than 110 kW. We notice that the most powerful 

cars are the least risky (with an annual claim frequency of 16.6 %) whereas the cars with an 

engine power range between 66 kW and 110 kW are the riskiest (with an observed annual 

claim frequency of 18.7 %). Finally, the least powerful cars have an annual claim frequency 

of 18.3 %.



30000 

20000 

10000 

0 

16507 

1361 

18–22 23–30 

Age 

0.32 0.308 

27482 

0.30 

0.28 

0.26 

0.24 


0.22 0.208 

0.20 

0.18 

0.163 

0.16 

0.14 

0.12 

0.10 

0.08 

0.06 

0.04 

0.02 

>30 

0.00 

18–22 23–30 >30 

Age 

Figure 2.11 Composition of Portfolio B with respect to Age (left panel) and observed annual claim 

frequencies according to Age (right panel). 

Figure 2.15 shows that the colour of the car has nearly no influence on the number of 

claims. We can notice that 10.1 % of the cars are red. 

2.9.3 Poisson Regression with Serial Independence 

In this subsection, we assume that the N it s are independent for the different values of 

i and t. With the help of the SAS R /STAT procedure GENMOD, we have obtained the 

results displayed in Table 2.8 for the model with the 5 explanatory variables presented in 

Subsection 2.9.2. The results obtained from a Type 3 analysis are as follows: 


Gender 1 485 00276 

Age 2 17356



30000 

20000 

10000 

0 

29024 

0.19 

0.188 

0.18 

0.177 

0.17 

0.16 

0.15 

0.14 

0.13 

0.12 

16326 

0.11 

0.10 

0.09 

0.08 

0.07 

0.06 

0.05 

0.04 

0.03 

0.02 

0.01 

Female Male 

0.00 

Female Male 

Gender 


Gender 

Figure 2.12 Composition of Portfolio B with respect to Gender (left panel) and observed annual 

claim frequencies according to Gender (right panel). 

The variable Colour is not relevant, as the p-value is 56.98 % according to the Type 3 

analysis. The variable Power could then also be removed (p-value of 11.21 % after having 

removed the variable Colour from the regression model) but since this variable is common 

in the tariff of insurers, we will try an interaction between the variables Age and Power. 

The interaction between Age and Power is illustrated in Figure 2.16 (where ‘LP’ stands 

for large power, that is, over 110 kW, ‘MP’ stands for medium power, that is, between 66 

and 110 kW, and ‘SP’ for small power, that is, less than 66 kW). The results are given in 

Table 2.9. The Type 3 analysis is as follows: 


Gender 1 486 0.0275 

Age ∗ Power 8 18001



16000 

15000 

14000 

13000 

12000 

11000 

10000 

9000 

8000 

7000 

6000 

5000 

4000 

3000 

2000 

1000 

0 

14331 

15872 

0.210 

0.21 

15147 

0.20 

0.19 

0.178 

0.18 

0.17 

0.166 

0.16 

0.15 

0.14 

0.13 

0.12 

0.11 

0.10 

0.09 

0.08 

0.07 

0.06 

0.05 

0.04 

0.03 

0.02 

0.01 

Large Middle Small 

0.00 

Large Middle Small 

Size of city 

Size of city 


Figure 2.13 Composition of Portfolio B with respect to Size of the city (left panel) and observed 

annual claim frequencies according to Size of the city (right panel). 

Nevertheless, several levels of the interaction between Age and Power can be grouped 

together. As already explained in Section 2.3.15, the option ‘Estimate’ of GENMOD can be 

used to assess the relevance of grouping the levels of the interaction Age–Power 2 by 2. The 

following groups have been defined: 

Group 1 Age > 30 and any power 

Group 2 Age 23–30 and any power as well as Age 18–22 and power > 110 kW 

Group 3 Age 18–22 and power < 110 kW 

The final model is shown in Table 2.10. The variance-covariance matrix of the estimated 

regression coefficients is 

̂̂ 

= 

⎛ 

⎞ 

0003428 −0000244 −0003022 −0003028 −0000441 −0000468 

−0000244 0000679 0000012 0000006 −0000004 0000007 

−0003022 0000012 0003350 0003064 −0000084 −0000050 

⎜ −0003028 0000006 0003064 0003436 −0000059 −0000045 

 

⎟ 

⎝ −0000441 −0000004 −0000084 −0000059 0000938 0000511 ⎠ 

−0000468 0000007 −0000050 −0000045 0000511 0000964



30000 

29750 

20000 

14149 

10000 

1451 


0.19 0.187 

0.183 

0.18 

0.17 

0.156 

0.16 

0.15 

0.14 

0.13 

0.12 

0.11 

0.10 

0.09 

0.08 

0.07 

0.06 

0.05 

0.04 

0.03 

0.02 

0.01 

0 

0.00 

66–110 kw 110 kw 66–110 kw 110 kw 

Power of car 

Power of car 

Figure 2.14 Composition of Portfolio B with respect to Power (left panel) and observed annual claim 

frequencies according to Power (right panel). 

Results of the Type 3 analysis are as follows: 


Gender 1 650 00108 




50000 

40000 

30000 

20000 

10000 

0 

0.19 0.184 

0.180 

0.18 

0.17 

40770 

0.16 

0.15 

0.14 

0.13 

0.12 

0.11 

0.10 

0.09 

0.08 

0.07 

0.06 

0.05 

0.04 

0.03 

4580 

0.02 

0.01 

0.00 

Other Red Other Red 

Colour of car 

Colour of car 

Annual claims frequency 

Figure 2.15 Composition of Portfolio B with respect to Colour of the car (left panel) and observed 

annual claim frequencies according to Colour of the car (right panel). 

Table 2.8 Results of the Poisson regression for the model with 5 variables and serial independence, 

Portfolio B. 


Intercept −19242 00302 −19833 −18650 406354



17000 

16000 

15000 

14000 

13000 

12000 

11000 

10000 

9000 

8000 

7000 

6000 

5000 

4000 

3000 

2000 

1000 

0 

LP LP LP MP MP MP SP SP 

18–22 23–30 >30 18–22 23–30 >30 18–22 23–30 

Interaction Age–Power 

SP 

>30 


0.36 

0.34 

0.32 

0.30 

0.28 

0.26 

0.24 

0.22 

0.20 

0.18 

0.16 

0.14 

0.12 

0.10 

0.08 

0.06 

0.04 

0.02 

0.00 

0.000 

LP 

18–22 

0.253 

LP 

23–30 

0.158 

0.341 

0.218 

0.171 

0.302 

0.204 

LP MP MP MP SP SP 

>30 18–22 23–30 >30 18–22 23–30 

Interaction Age–Power 

0.160 

SP 

>30 

Figure 2.16 Composition of Portfolio B with respect to the Age–Power interaction (left panel) and 

annual claim frequencies according to the Age–Power interaction (right panel). 

Table 2.9 Results of the Poisson regression for model with interactions between Age and Power, 

Portfolio B. 


Intercept −19248 00312 −19859 −18637 381603 −166022 6143464 −120576 1202437 000 09978 

110 kW 

Age ∗ Power 17–22 

07596 01333 04983 10208 3247 30 

00547 00357 −00152 01246 235 01250 

66–110 kW 

Age ∗ Power > 30 < 66 kW 0 0 0 0 

Size of the city Large 02556 00306 01956 03156 6962


Table 2.10 Results of the Poisson regression for the final model, Portfolio B. 


Intercept −12473 00585 −13620 −11325 45383


These results are then corrected with a multiplying factor computed thanks to a Poisson 

regression with ‘number of claims of the previous year’ as the single explanatory variable. 

We use the sum of the logarithm of the estimated annual expected claim frequency obtained 

when assuming serial independence and of the logarithm of the exposure-to-risk as an offset. 

Specifically, the expected number of claims for policyholder i during period t, t = 2 3, is 

now of the form 

d it exp 

( 

̂0 + 

) 

p∑ 

̂j x itj exp˜ 0 +˜ 1 N it−1 

j=1 

where the ̂ j s are those of Table 2.10 and where the parameters ˜ 0 and ˜ 1 have to be 

estimated by Poisson regression. The offset is 

ln d it +̂ 0 + 

p∑ 

̂j x itj 

Results of the Poisson regression on the model incorporating the past claims are as follows: 

j=1 

Variable Coeff ˜ Std error Wald 95 % conf limit Chi-sq Pr>Chi-sq 

Intercept −00412 00180 −0.0766 −00059 523 00222 

N t−1 03241 00370 02517 03966 7688 Chi-sq 

N t−1 1 68.66


The variance-covariance matrix of the N it s in the Poisson model with serial independence is 

given by 

⎛ 

⎞ 

i1 0 ··· 0 

0 i2 ··· 0 

A i = ⎜ 

⎝ 

 

 

⎟ 

⎠ 

0 0 ··· iTi 

In fact this matrix does not take the overdispersion and the serial dependence of the data 

into account. Noting that 

 

EN i = A i X i 

it is possible to transform (2.16) in order to let A i explicitly appear in the likelihood equations. 

This gives 

n∑ 

i=1 

( ) 

T 

EN i A ( −1 

i n i − EN i ) = 0 (2.17) 

The principle of Generalized Estimating Equations (GEE) is to find a suitable variancecovariance 

matrix to insert in Equation (2.17) instead of A i based on serial independence. 

This matrix should take the overdispersion and the serial dependence into account. A possible 

form of this matrix could be 

V i = A 1/2 

i R i A i 

1/2 

where the ‘working’ correlation matrix R i takes the serial dependence between the 

components of N i into account and depends on a parameter . The overdispersion is also 

taken into account as VN it = it exceeds EN it = it provided >1. 

The idea behind the GEE method is then to replace A i by V i in (2.17) and to compute the 

estimator of as the solution of 

n∑ 

i=1 

( ) 

T 

EN i V ( −1 

i n i − EN i ) = 0 (2.18) 

The resulting estimator is consistent whatever the choice of the matrix R i but the precision 

will be much better if R i is close to the true correlation matrix of N i . Equation (2.18) 

is solved thanks to a modified version of the Fisher scoring method for and a moment 

estimation for and . The iterative procedure is as follows: 

1. Compute an initial estimate of assuming independence. 

2. Compute the current ‘working’ correlation matrix based on standardized residuals, current 

and the assumed structure of R i . 

3. Estimate the covariance matrix V i . 

4. Update . 

Note that GEE is not a likelihood based method of estimation, so that inferences based on 

likelihoods are not possible in this case.


Modelling Dependence with the ‘Working Correlation Matrix’ 

The ‘working’ correlation matrix R i takes into account the dependence between the 

observations corresponding to the same policyholder. The form of this matrix must be 

specified and depends on the parameter vector . 

If R i = I, (2.18) gives exactely the likelihood equations (2.16) under the assumption 

of independence. The SAS R /STAT procedure GENMOD supports the following structures 

of the ‘working’ correlation matrix: fixed (user-specific correlation matrix, not estimated 

from the data but specified by the actuary), independent (R i = I, giving the estimates 

of obtained under serial dependence), m-dependent (correlation equal to t for lags 

t = 1m, and 0 for higher lags), exchangeable (constant correlation , whatever the 

lag), unstructured (each correlation coefficient jk , between observations made at times j 

and k, is estimated from the data), and AR1 (autoregressive of order 1, with a correlation 

coefficient equal to t at lag t). 

Numerical Example 

The GEE approach can be performed thanks to the procedure GENMOD of SAS R . The 

‘Repeated’ statement of GENMOD invokes the GEE method, specifies the correlation structure, 

and controls the displayed output from the GEE model. Initial parameter estimates for iterative 

fitting of the GEE model are computed as in a Poisson regression model, as described 

previously. Results of the initial model fit are displayed as part of the generated output 

of SAS R . Statistics for the initial model fit such as parameter estimates, standard errors, 

deviances, and Pearson Chi-squares do not apply to the GEE model, and are only valid 

for the initial model fit. The SAS R parameter estimates table contains parameter estimates, 

standard errors, confidence intervals, Z-scores, and p-values for the parameter estimates. 

The ‘Repeated’ statement specifies the covariance structure of multivariate responses for 

GEE model fitting in the GENMOD procedure. In addition, the ‘Repeated’ statement controls 

the iterative fitting algorithm used in GEE and specifies optional output. Other GENMOD 

procedure statements are used in the same way as they are for ordinary Poisson regression 

models to specify the regression model for the mean of the responses. 

The statement ‘SUBJECT = subject-effect’ identifies subjects in the input data set. The 

subject-effect can be a single variable, an interaction effect, a nested effect, or a combination. 

Each distinct value, or level, of the effect identifies a different subject, or cluster. Responses 

from different subjects are assumed to be independent, and responses within subjects are 

assumed to be correlated. A subject-effect must be specified, and variables used in defining 

the subject-effect must be listed in the CLASS statement. In actuarial applications, the policy 

number is typically used as subject-effect. 

The same variables as for the model where the serial independence was assumed are kept 

in the final model. The results are given in Table 2.12 that is similar to the SAS R outputs 

for GEEs. The estimation of the ‘working correlation matrix’ gives 

⎛ 

⎝ 

1 00493 00460 

00493 1 00493 

00460 00493 1 

and ̂ = 13419. 

Since the GEE apprach is not based on a likelihood function, we cannot use the large 

sample approximations for the estimated variance-covariance matrix ̂̂ of the estimated 

⎞ 

⎠


regression coefficients presented in Chapter 1. The model-based estimation of ̂ 

in the 

GEE case is given by ̂ 

= −1 

GEE where 

GEE = 

n∑ 

i=1 

( ) 

T ( ) 

EN −1 

i V i 

EN i 

Here, −1 

GEE 

is the GEE-equivalent of the inverse of the Fisher information matrix. It is a 

consistent estimator of the covariance matrix of ̂ if the mean structure and the ‘working’ 

correlation matrix are correctly specified. Then, 

̂ 

= −1 

GEE GEE −1 

GEE 

is the robust estimate of the covariance matrix of ̂, where 

GEE = 

n∑ 

i=1 

( ) 

T ( ) 

EN i V −1 −1 

i CN i V i 

EN i 

The robust estimate for ̂ 

is consistent even if the ‘working’ correlation matrix is 

misspecified. In computing, the covariance matrix CN i of N i is replaced with 

ĈN i = N i − ÊN iN i − ÊN i T 

The robust estimated variance-covariance matrix of the estimated regression coefficients is 

̂̂ 

= 

⎛ 

⎜ 

⎝ 

0003910 −0000359 −0003418 −0003393 −0000509 −0000559 

−0000359 0000801 0000138 0000081 −0000054 −0000015 

−0003418 0000138 0003755 0003395 −0000089 −0000034 

−0003393 0000081 0003395 0003806 −0000051 −0000029 

−0000509 −0000054 −0000089 −0000051 0001100 0000595 

−0000559 −0000015 −0000034 −0000029 0000595 0001132 

If we compare Tables 2.10 (where the serial independence was assumed) and 2.12 (where 

the serial dependence is taken into account), we see that the standard errors are systematically 

larger in the GEE approach as serial dependence induces overdispersion. 

Generalized score tests for Type III contrasts are computed for GEE models (Wald tests 

are also available). Results of the Type 3 analysis are as follows: 

⎞ 

 

⎟ 

⎠ 


Gender 1 575 00165 



Table 2.12 Results of the Poisson regression with a GEE approach, Portfolio B. 

Variable Level Coeff Std error 95 % conf limit Z Pr > Z 

Intercept −12506 00625 −13731 −11280 −2000


Table 2.13 Results of the Negative Binomial regression model with panel data, Portfolio B. 

Variable Level Coeff Std error 95 % conf limit t Pr > t 

Intercept −12277 00646 −13542 −11011 −1901


PrN i1 = k i1 N iTi = k iTi 

( ) 

∫ + Ti 

( 

∏ 

1 

= PrN it = k it i = √ exp − 1 ) 

0 

2 

3 2 − 12 d 

t=1 

Again, numerical methods are needed to obtain the solutions and the SAS R /STAT procedure 

NLMIXED can be used to maximize the log-likelihood. 

The fit of the Poisson-Inverse Gaussian model is described in Table 2.14. We get 

̂ = 05575. The variance-covariance matrix of the estimated regression coefficients and 

dispersion parameter ̂ is 

⎛ 

⎞ 

0004216 −0000289 −0003723 −0003709 −0000534 −0000567 0000115 

−0000289 0000830 −0000002 −0000004 0000000 0000009 −0000002 

−0003723 −0000002 0004118 0003755 −0000097 −0000058 −0000027 

̂̂ 

= 

−0003709 −0000004 0003755 0004193 −0000069 −0000052 0000011 

 

⎜ −0000534 0000000 −0000097 −0000069 0001152 0000616 0000030 ⎟ 

⎝ −0000567 0000009 −0000058 −0000052 0000616 0001169 −0000010 ⎠ 

0000115 −0000002 −0000027 0000011 0000030 −0000010 0002492 

The log-likelihood is equal to −19 6437 and Type 3 analysis gives 


Gender 1 54 00201 




( ) 

∫ + ∏ Ti 

= exp− it it k it 

1 

( 

0 

k it ! √ 2 exp − ln + 2 /2 2 ) 

d 

2 2 

t=1 

The integral has no closed-form solution so that it is not possible to derive the log-likelihood 

equations. Therefore, numerical procedures are needed to solve the integral and to find 

maximum likelihood estimates (the NLMIXED procedure of SAS R /STAT, for instance). 

Now, 

EN it = it and VN it = it + ( exp 2 − 1 ) it 2 

The fit of the Poisson-LogNormal model is described in Table 2.15. We get ̂ 2 = 04581. 

The variance-covariance matrix of the estimated regression coefficients and dispersion 

parameter ̂ 2 is 

̂̂ 

= 

⎛ 

⎞ 

0004225 −0000291 −0003729 −0003714 −0000536 −0000569 0000091 

−0000291 0000832 −0000001 −0000004 −0000001 0000009 −0000003 

−0003729 −0000001 0004125 0003760 −0000097 −0000058 −0000022 

−0003714 −0000004 0003760 0004199 −0000069 −0000052 0000006 

 

⎜ −0000536 −0000001 −0000097 −0000069 0001155 0000618 0000022 

⎟ 

⎝ −0000569 0000009 −0000058 −0000052 0000618 0001173 −0000007 ⎠ 

0000091 −0000003 −0000022 0000006 0000022 −0000007 0001193 

The log-likelihood is −19 6423 and Type 3 analysis gives 


Gender 1 52 00226 



2.9.9 Vuong Test 

In the framework of panel data, the test proposed in Section 2.8.1 is no longer valid. This 

test can be modified by using the conditional log-likelihood defined, for policyholder i and 

for the Negative Binomial model, as 

l NB 

i 

k it k i1 k it−1 = ln PrN it = k it N i1 = k i1 N it−1 = k it−1 

PrN 

= ln i1 = k i1 N it = k it 

PrN i1 = k i1 N it−1 = k it−1 

( ∫ ∏ 

1 t 

0 s=1 

= ln 

exp− ) 

is is k is f d 

∫ 

k it ! ∏ t−1 

s=1 exp− 

is is k is f d 

0 

where f is given by (1.35). Therefore, for panel data, the log-likelihood can be equivalently 

rewritten as 

L = 

T n∑ ∑ i 

l NB 

i 

k it k i1 k it−1 

i=1 t=1 

where for t = 1, li 

NB is simply the probability PrN i1 = k i1 . The quantities li 

PIG and li 

PLN are 

defined equivalently for i s following the Inverse Gaussian and LogNormal distributions, 

substituting for f the probability density functions (1.41) and (1.45), respectively. 

Golden (2003) extended the test proposed by Vuong (1989) to the serial case. It can 

be applied here to compare the different mixed Poisson alternatives provided the panel is 

balanced (that is, the T i s are equal for all the policyholders). In the application to Portfolio 

B, we keep the policies in the portfolio for three periods (that is, those for which T i = T = 3). 

There are 9894 such policies out of a total of 20 354 policies in the complete portfolio. The 

test statistic for comparing the Negative Binomial model to the Poisson-Inverse Gaussian 

model is then 

T ext = 

∑ n ∑ T 

i=1 t=1 

( 

l NB 

i 

) 

k it k i1 k it−1 − li 

PIG k it k i1 k it−1 

√ 

nT 

with the variance defined as 

n∑ 

2 = 1 n 

( 

× 

T∑ 

T∑ ( 

i=1 t=1 s=1 

l NB 

i 

l NB 

i 

) 

k it k i1 k it−1 − l PIG 

i 

k it k i1 k it−1 

k is k i1 k is−1 − l PIG 

i 

k is k i1 k is−1 

This expression for the variance is close to the one obtained in the independent case, except 

that covariances between correlated observations appear here. Note that several assumptions 

have to be checked before the test can be formally performed. We refer the reader to Golden 

(2003) for more details. Here, we proceed as in French & Jones (2004) and apply the test 

directly. The following results are obtained: 

)


• when we compare the Poisson-Gamma model with the Poisson-Inverse Gaussian model, 

the value of the test statistic is equal to −07988 leading to a p-value of 4244 %. 

• when we compare the Poisson-Gamma model with the Poisson-LogNormal model, the 

value of the test statistic is equal to −07688 leading to a p-value of 4420 %. 

• when we compare the Poisson-Inverse Gaussian model with the Poisson-LogNormal 

model, the value of the test statistic is equal to −05922 leading to a p-value of 

5537 %. 

Therefore, the three models are not statistically different. Applying the same testing procedure 

to the policies that are in the portfolio for the first two years (this means 12 202 policies) 

yields the same conclusion. 

2.9.10 Information Criteria 

Non-nested models are often compared using likelihood-based criteria, including the 

well-known AIC, for instance. Since the competing models have the same number of 

parameters, it is enough to examine the respective log-likelihoods. A commonly used 

rule-of-thumb consists in considering that two models are significantly different if the 

difference in the log-likelihoods exceeds five (corresponding to a difference in AICs of 

more than ten, as discussed in Burnham & Anderson (2002)). This means here that 

the Poisson-Inverse Gaussian and Poisson-LogNormal models are significantly better than 

the Negative Binomial model. Considering the maximum of the log-likelihood, we chose 

the Poisson-LogNormal model for Portfolio B. The same conclusion is obtained using 

different criteria often used in practice. For instance, Raftery (1995) suggested that a 

model significantly outperfoms a competitor if the difference in their respective BIC values 

exceeds 5. 

2.9.11 Resulting Classification for Portfolio B 

Table 2.16 gives the resulting price list obtained with the Poisson-LogNormal model 

described in Table 2.15. The final a priori ratemaking contains 18 classes. This ratemaking 

will be used throughout the text for the examples involving Portfolio B. 

There is another way to present the results displayed in Table 2.16. The annual expected 

claim frequency is obtained from the reference class, with 

exp̂ 0 = 2950 % 

according to Table 2.15. Applying correction coefficients, we then obtain the annual expected 

claim frequency of any policyholder. Specifically, it is obtained from the multiplicative 

formula


Table 2.16 

A priori risk classification for Portfolio B (Poisson-LogNormal regression model). 

Age–Power Size of the city Gender Annual 

claim freq. (%) 

Group 1 Group 2 Group 3 Large Middle Small Female Male 

Weights 

(%) 

No No Yes No No Yes No Yes 2950 082 

No No Yes No No Yes Yes No 2759 043 

No No Yes No Yes No No Yes 3173 063 

No No Yes No Yes No Yes No 2968 037 

No No Yes Yes No No No Yes 3833 044 

No No Yes Yes No No Yes No 3586 031 

No Yes No No No Yes No Yes 1960 788 

No Yes No No No Yes Yes No 1833 464 

No Yes No No Yes No No Yes 2108 832 

No Yes No No Yes No Yes No 1972 466 

No Yes No Yes No No No Yes 2547 689 

No Yes No Yes No No Yes No 2382 399 

Yes No No No No Yes No Yes 1519 1259 

Yes No No No No Yes Yes No 1420 702 

Yes No No No Yes No No Yes 1634 1381 

Yes No No No Yes No Yes No 1528 724 

Yes No No Yes No No No Yes 1974 1268 

Yes No No Yes No No Yes No 1846 730 

2950 % 

{ exp−00669 = 094 if the policyholder is a female driver 

× 

1 

otherwise 

⎧ 

exp−06640 = 051 if the policyholder belongs to Group 1 with respect to 

⎪⎨ 

the combined variable Age ∗ Power 

× exp−04089 = 066 if the policyholder belongs to Group 2 with respect to 

the combined variable Age ⎪⎩ 

∗ Power 

1 

otherwise 

⎧ 

⎨ exp02620 = 130 if the policyholder resides in a large city 

× exp00731 = 108 if the policyholder resides in a middle-sized city 

⎩ 

1 

otherwise 


2.10.1 Generalized Linear Models 

After decades dominated by statistically unsophisticated models, it is now common practice 

since Mc Cullagh & Nelder (1989) and Brockman & Wright (1992) to achieve a priori 

classification with the help of generalized linear models (GLMs). They are so called because 

they generalize the classical linear models based on the Normal distribution.


Risk classification techniques for claim counts have been the topic of many papers 

appearing in the actuarial literature. Early references include ter Berg (1980b) and 

Albrecht (1983a,b,c). Dionne & Vanasse (1989, 1992) used a Negative Binomial 

regression model, while Dean, Lawless & Willmot (1989) used a Poisson-Inverse 

Gaussian distribution to fit the number of claims. Cummins, Dionne, McDonnald & 

Pritchett (1990) applied the GB2 family of distributions in modelling claim counts. 

Ter Berg (1996) considered the Generalized Poisson distribution of Consul (1990) and 

incorporated explanatory variables with the help of a loglinear model. There are now several 

textbooks devoted to the statistical analysis of count data. Let us mention Cameron & 

Trivedi (1998) and Winkelmann (2003). Before the Poisson regression became popular 

among actuaries, claims data were often analysed using logistic regression; see, e.g., 

Beirlant, Derveaux, De Meyer, Goovaerts, Labies & Maenhoudt (1991). 

Separate analyses are usually conducted for claim frequencies and costs, including 

expenses, to arrive at a pure premium. With the noticeable exception of Jorgensen 

& Paes de Souza (1994), all the actuarial analyses of the pure premium so far have 

examined frequencies and severities separately. This approach is particularly relevant in 

motor insurance, where the risk factors influencing the two components of the pure premium 

are usually different. 

2.10.2 Nonlinear Effects 

GLMs however only deal with categorical risk factors in an efficient way. The main drawback 

of GLMs is that covariate effects are modelled in the form of a linear predictor. GLMs are 

too restrictive if nonlinear effects of continuous covariates are present. Continuous covariates 

can efficiently enter GLMs only if they are suitably transformed to reflect their true effect 

on the score scale. However, it is not always clear how the variables should be transformed. 

It has been common practice in insurance companies to model possibly nonlinear effects 

of a covariate by polynomials. However, it is well known to statisticians that polynomials 

are often not flexible enough to capture the variability of the data particularly when the 

polynomial degree is small (see, e.g., Fahrmeir & Tutz (2001), Chapter 5). For larger 

degrees the flexibility of polynomials increases but at the cost of possibly high variability 

of resulting estimates particularly at the left and right extreme values of the covariate. 

A more flexible approach for modelling nonlinear effects can be based on piecewise 

polynomials. More specifically, the unknown functions are approximated by polynomial 

splines, which may be regarded as piecewise polynomials with additional regularity 

conditions (see, e.g., de Boor, 1978). We refer the interested reader to Denuit & Lang 

(2004) for an overview of the existing approaches. 

2.10.3 Zero-Inflated Models 

Insurance data usually include a relatively large number of zeros (no claim). Deductibles 

and no claim discounts increase the proportion of zeros, since small claims are not reported 

by insured drivers. Zero-inflated models, including the Zero-Inflated Poisson (ZIP) model, 

account for this phenomenon. 

ZIP models can be considered as a mixture of a zero point mass and a Poisson distribution 

and were first used to study soldering defects on print wiring boards (Lambert, 1992). To


account for overdispersion in the Poisson part, generalizations of the model are possible and 

include the Zero-Inflated Negative Binomial (ZINB) distribution. See Yip & Yau (2005) for 

an application to insurance claim count data. 

Other than the zero-inflated models, parametric methods such as the mixture of 

distributions can be used to model the claim frequency distribution with extra zeros. 

Hürlimann (1990) discussed the use of several pseudo compound Poisson distributions in 

modelling the claim count data. To test for a Poisson mixture, a test statistic was proposed 

by Carrière (1993a). 

2.10.4 Fixed Versus Random Effects 

The mixed Poisson distribution is often used to account for unknown characteristics of the 

driver, influencing the number of accidents reported to the company. When panel data are 

available, these hidden features can alternatively be captured by an individual heterogeneity 

term that is constant over time (the standard reference for panel data is Hsiao (2003); the 

particular case of count variables is treated in Cameron & Trivedi (1998)). Boucher & 

Denuit (2006) compared the two approaches with emphasis on the actual meaning of the 

estimated parameters in a mixed Poisson regression when random effects and covariates are 

correlated. In such a case, parameter estimates should be seen as the apparent effects of the 

covariates on the frequency. Keeping this in mind allows for a better understanding of the 

resulting price list. 

The results obtained by Boucher & Denuit (2006) legitimate the use of random effects 

models even if there exists a correlation between the regressors and the heterogeneity. The 

parameter estimates do not identify the impact of these regressors on the premium but only 

the apparent effects. Since this is usually the focus for the actuary in ratemaking, there 

is no problem with this interpretation. However, such a correlation clearly indicates that a 

correction should be done to obtain a more accurate model. In particular, the apparent high 

risk of young drivers should deserve some attention. The analysis conducted by Boucher 

& Denuit (2006) shows that the fixed effects are very heterogeneous for these individuals. 

Instead of penalizing these insureds in the a priori ratemaking, an appropriate bonus-malus 

scheme could be designed. Merit rating systems improve the fairness of the tariff in that 

respect. We will come back to this issue in Chapter 8. 

2.10.5 Hurdle Models 

Boucher, Denuit & Guillén (2006) presented and compared different risk classification 

models for the annual number of claims reported to the insurer. Generalized heterogeneous, 

zero-inflated, hurdle and compound frequency models are applied to a sample of an 

automobile portfolio of a major company operating in Spain. 

The hurdle models are widely used in connection with health care demands. An application 

to credit scoring is proposed in Dionne, Artis & Guillén (1996). With health care demand, 

it is generally accepted that the demand for certain types of health care services depends on 

two processes: the decisions of the individual and those of the health care provider. See, e.g. 

Pohlmeier & Ulrich (1995) or Santos Silva & Windmeijer (2001). The hurdle model 

also possesses a natural interpretation for the number of reported claims. A reason for the 

good fit of the zero-inflated models is certainly the reluctance of some insureds to report their


accident (since they would then be penalized by some bonus-malus scheme implemented by 

the insurer). It is reasonable to believe that the behaviour of the insureds is not the same 

when they have already reported a claim. This suggests that two processes govern the total 

number of claims, as in the hurdle model. 

2.10.6 Geographic Ratemaking 

It is common in motor insurance to let the risk premium per unit exposure vary with 

geographic area when all other risk factors are held constant. Most companies have adopted 

a risk classification according to the geographical zone where the policyholder lives (urban / 

nonurban for instance, or a more accurate split of the country according to Zip codes). The 

spatial variation may be related to geographic factors (e.g. traffic density or proximity to 

arterial roads) or to socio-demographic factors. In such cases it will be desirable to estimate 

the spatial variation in risk premium and to price accordingly. Spatial postcode methods 

for insurance rating attempt to extract information which is in addition to that contained 

in standard factors (like age or gender for instance). Often, claim characteristics tend to be 

similar in neighbouring postcode areas (after other factors have been accounted for). The idea 

of postcode rating models is to exploit this spatial smoothness by allowing for information 

transfer to and from neighbouring regions. 

Following Boskov & Verrall (1994) and Brouhns, Denuit, Masuy & Verrall 

(2002), the risk associated with each district can be assessed with the help of statistical 

models for spatial data. The techniques used for geographic ratemaking are closely related 

to those used in disease mapping by epidemiologists. Figure 2.17 is taken from Brouhns, 

Denuit, Masuy & Verrall (2002). It displays the raw exposures e i for each area in 

Belgium (the number of policy-years, in our case) whilst Figure 2.18 shows crude claim rates 

(observed number of claims divided by the corresponding expected number of claims given 

the characteristics of the policyholders living in each area). However, the latter map is at 

best difficult to interpret and can even be seriously misleading because the crude claim rates 

tend to be far more extreme in regions with smaller risk exposures (see Figures 2.17–2.18; 

the high rates in the north-west of Belgium (West Flanders) or in the south of the country 

correspond to districts for which we have less policyholders). Hence regions with the least 

reliable data will typically draw the main visual attention. This is one reason why it is 

difficult in practice to attempt any smoothing or risk assessment ‘by eye’. 

Whereas epidemiologists and environmetricians have been interested in spatial models for a 

long time, the actuarial literature is rather poor in respect of ratemaking methods incorporating 

geographic components. Taylor (1989) used two-dimensional splines on a plane linked to 

the map of the region by a transformation chosen to match the features of the specific region. 

He applied this method to a data set from Sydney, Australia. Boskov & Verrall (1994) 

highlighted some deficiencies in Taylor’s model, and provided an alternative treatment which 

made use of the Gibbs sampler to implement a Bayesian revision of the observation in each 

area. The main advantage of the Bayesian framework is that it recognizes the magnitudes of 

sampling error and incorporates the concept of smoothing over neighbouring areas. Taylor 

(1996) adopted a similar point of view and applied Whittaker graduation (a widely accepted 

actuarial technique which has also been shown to have a Bayesian interpretation). Dixon, 

Kelsey & Verrall (2000) proposed an extension of the Boskov & Verrall (1994) model 

including weighting factors accounting for distances between regions.


Raw exposure 

(policyholders) 

48 84 157 318 621 

Figure 2.17 

Map of Belgium with exposures-to-risk. 

A key step in model specification is the definition of neighbours, i.e. those areas whose 

claim rates are correlated with that of a given area. A traditional definition of neighbours 

includes all areas contiguous to a given area. The methodology applied in Brouhns, Denuit, 

Masuy & Verrall (2002) is as follows. In a first stage, available explanatory variables 

are incorporated to the policyholders’ claim frequencies with the help of a (mixed) Poisson 

regression model. In a second stage, the data are aggregated by districts and overdispersion 

is accounted for by introduction of a random effect (split into a spatially structured part and 

a spatially unstructured one). The Boskov and Verrall model is then used to recover the 

spatial structure of the claims pattern that can be used to design the geographical ratemaking 

strategy of the company. 

In order to figure out the global claim pattern, Figure 2.19 displays the geographical 

risk variation, and serves as basis for the determination of the rating areas. Clearly, several 

regions with high, medium and low values of geographic risk emerge. This map can thus 

serve to design different rating areas, which in turn become a categorical variable that may 

enter the (mixed) Poisson regression model.


Crude claim rates 0.0695 0.0893 0.113 0.1325 0.1554 

Figure 2.18 

Map of Belgium with crude rates. 

The approach however proceeds in two steps: first a regression is performed with all 

the covariates except the spatial ones to get the expected claim number of each area, and 

then the Boskov and Verrall model recovers the spatial claim pattern. In order to avoid the 

preprocessing of the data to remove the effect of all risk factors other than the spatial ones, 

Dimakos & Rattalma (2002) proposed a fully Bayesian approach to nonlife ratemaking. 

This approach still relies on GLMs and thus suffers from the drawbacks mentioned in 

Section 2.10.2: continuous covariates such as policyholders’ age enter linearly into the model 

(on the score scale) whereas it is now well established that the effect of some continuous 

variables is far from linear (typically, convex for policyholders’ age). 

Statistical modelling tools to perform space-time analysis of insurance data have been 

proposed by Denuit & Lang (2004) and Fahrmeir, Lang & Spies (2003). This approach 

enables the actuary to explore spatial and temporal effects simultaneously with the impact 

of other covariates. Bayesian generalized additive models provide a broad and flexible 

framework for regression analyses in realistically complex situations with cross-sectional, 

longitudinal and spatial data. All effects, as well as smoothing parameters, are regarded as 

random and are assigned appropriate priors. 

2.10.7 Software 

SAS R has been used throughout this chapter. The readers interested in the use of this software 

to perform statistical analyses are referred to Der & Everitt (2002) for a very readable


Geographic risk –0.1463 –0.0975 –0.0311 –0.0415 –0.1141 

Figure 2.19 

Estimation of the geographically structured risk. 

introduction. For more information, we refer the interested reader to the SAS R website, 

http://www.sas.com/. 

A number of software packages specific to the insurance industry have been developed, 

some of them based on SAS R . We mention a few of them hereafter, without being exhaustive. 

Note also that the authors did not test these tools, which are not available free of charge, so 

that we could not evaluate their relative performances. 

The Tricast suite has been developed using the support software from SAS R and 

includes actuarial tools to create rating models. For more information, the interested 

reader may email tricast@tricast-group.com. The consulting firm Watson Wyatt offers the 

Pretium R system, which integrates with SAS R . For more details, we refer the reader to 

http://www.watsonwyatt.com/. EMB has developed several computer tools for insurance 

rating. The Emblem R software can be used to model claims experience and the Classifier TM 

software allows one to assess and categorize geographically distributed risk. For more details, 

see http://www.emb.co.uk/. 

Academic researchers and research and development actuaries appreciate the software 

system ‘R’. This is a free language and environment for statistical computing and graphics. 

R is a GNU project which is similar to the S language and environment which was developed 

at Bell Laboratories (formerly AT&T, now Lucent Technologies) by John Chambers and 

colleagues. R can be considered as a different implementation of S. There are some important 

differences, but much code written for S runs unaltered under R. R is available as Free 

Software under the terms of the Free Software Foundation’s GNU General Public License


in source code form. It compiles and runs on a wide variety of UNIX platforms and similar 

systems (including FreeBSD and Linux), Windows and MacOS. For more details, we refer 

the interested reader to http://www.r-project.org/. A good reference about the use of R for 

topics related to this chapter is certainly Faraway (2006). Readers completely new to R are 

referred to Dalgaard (2002) for an introduction.

Part II 

Basics of Experience 

Rating 




3 

Credibility Models for Claim 

Counts 


3.1.1 From Risk Classification to Experience Rating 

We have seen in Chapter 2 how to partition a heterogeneous portfolio into more homogeneous 

classes with all policyholders belonging to the same class paying the same premium. 

However, tariff cells are still quite heterogeneous despite the use of many a priori variables. 

The expected claim frequency for the tariff cell is designed to reflect the average experience 

of the entire group. If the experience of a policy is consistently better (or worse) than the 

average experience of the group, the insurance company may consider adapting the amount 

of premium to be charged for this policy. Of course, this requires a model which can separate 

random variation from signal in the historical data to indicate whether this policy is of better 

(or worse) quality compared to the group average. 

It is reasonable to believe that the hidden features (unobserved risk characteristics that 

have been modelled by a random effect in the mixed Poisson regression model) are revealed 

by the number of claims reported by the policyholders over the successive insurance periods. 

Hence the adjustment of the premium based on the individual claims experience in order 

to restore fairness among policyholders. The allowance for the history of the policyholder 

in a rating model thus derives from interpretation of serial correlation for longitudinal data 

resulting from hidden features in the risk distribution. 

3.1.2 Credibility Theory 

Credibility theory is the art of combining different collections of data to obtain an accurate 

overall estimate. It provides actuaries with techniques to determine insurance premiums for 





contracts that belong to a (more or less) heterogeneous portfolio, where there is limited or 

irregular claim experience for each contract but ample claim experience for the portfolio. 

Credibility theory can be seen as a set of quantitative tools that allows the insurers to perform 

experience rating, that is, to adjust future premiums based on past experience. In many 

cases, a compromise estimator is derived from a convex combination of a prior mean and 

the mean of the current observations. The weight given to the observed mean is called the 

credibility factor (since it fixes the extent to which the actuary may be confident in the 

data). 

3.1.3 Limited Fluctuation Theory 

There are different types of credibility mechanisms: limited fluctuations credibility and 

greatest accuracy credibility. Limited fluctuation credibility theory was developed in the early 

part of the 20th century in connection with workers compensation insurance by Mowbray 

(1914). It provides a mechanism for assigning full or partial credibility to a policyholder’s 

experience. In the former case, the policy is rated on the basis of its own claims history, 

whereas in the latter case, a weighted average of past experience and grand mean is used 

by the insurer. Although the limited fluctuation approach provides simple solutions to the 

problem, it suffers from a lack of theoretical justification. We will not consider this approach 

in this book. Instead, we will consider the greatest accuracy credibility theory formalized by 

Bühlmann (1967,1970). 

3.1.4 Greatest Accuracy Credibility 

The idea behind greatest accuracy credibility theory can be summarized as follows: Tariff 

cells include policyholders with similar underwriting characteristics; each of them is viewed 

as homogeneous with respect to the underwriting characteristics used by the insurance 

company. Of course, the risks in the cell are not truly homogeneous: there still remains some 

heterogeneity in each of the tariff cells, as explained in the preceding chapters. To reflect this 

heterogeneity, the relative risk level of each policyholder in the rating cell is characterized by 

a risk parameter , but the value of varies by policyholder. If = 50 % then the expected 

number of claims reported by this policyholder is half of the claim frequency corresponding 

to the rating cell, whereas if = 300 % then the expected number of claims for this individual 

is three times the claim frequency of the rating cell. Of course, even if assuming the existence 

of such a is reasonable, it is not observable and the actuary can never know its true value 

for a given policyholder. 

Because varies by policyholder, there is a distribution function F giving the proportion 

of policyholders in the portfolio with relative risk level less than or equal to a certain 

threshold. Stated another way, F represents the probability that a policyholder picked at 

random from the portfolio has a risk parameter that is less than or equal to . The connection 

with the random effect introduced in the statistical models of Chapter 2 to account for the 

residual heterogeneity is now clear: this random effect becomes the random risk parameter 

for a policyholder picked at random from the portfolio (the distribution function of is 

F ). Even if the risk parameter remains unknown, the distribution function F can be 

estimated from data, as explained in Chapter 2. Once estimated, the heterogeneity model can 

be used to perform prediction on longitudinal data and allows for experience rating in motor

Credibility Models for Claim Counts 123 

insurance. In an empirical Bayesian setting, the prediction is derived from the expectation 

of a random effect with respect to a posterior distribution taking into account the history of 

the individual. 

3.1.5 Linear Credibility 

Bayesian statistics offer an intellectually acceptable approach to greatest accuracy credibility 

theory. Nevertheless, practical applications involve numerical methods to perform integration 

with respect to a posteriori distribution, making more elementary approaches desirable (at 

least to get an easy-to-compute approximation of the result). In that respect, linear credibility 

formulas are especially useful. Basically, the actuary still resorts to a quadratic loss function 

but the shape of the credibility predictor is constrained ex ante to be linear. 

3.1.6 Financial Equilibrium 

When the insurer uses past claims history to reevaluate the amount of premium to be charged 

to the policyholders, the increases and decreases granted to the policyholders in the portfolio 

must exactly balance each other. The credibility mechanism indeed has little effect on the 

number of claims filed to the company. Therefore, the number of claims with and without 

credibility is very much the same. 

For this reason, we expect that the a posteriori corrections average to unity. This ensures 

that the average number of claims without credibility equals the average number of claims 

with credibility, and that the premium income will be enough to compensate the claims. 

3.1.7 Combining a Priori and a Posteriori Ratemaking 

The amount of premium paid by the policyholder depends on the rating factors of the current 

period (think for instance of the type of the car or of the occupation of the policyholder) but 

also on the claim history. The insurance premium is the product of a base premium and of a 

credibility coefficient. The base premium is a function of the current rating factors whereas 

the credibility coefficient usually depends on the history of claims at fault. Clearly, a priori 

and a posteriori ratings interact. To the extent that good drivers are rewarded in their base 

premiums (through other rating variables) the size of the bonus they require for equity is 

reduced. 

The claims history of each policyholder consists of a short integer-valued sequence of 

yearly claim counts. The basic model used for experience rating is based on the Negative 

Binomial distribution. This probability law can be seen as a Poisson mixture distribution with 

Gamma mixing. Therefore, it allows for serial dependence of claim counts, by introducing 

Gamma-distributed unobserved individual heterogeneity. The serial dependence in claim 

counts sequences is generated by integrating the unobserved factor, and by updating its 

prediction when individual information increases. Alternative models with LogNormal or 

Inverse Gaussian unobserved heterogeneity have also been considered in the actuarial 

literature (and reviewed in Chapter 2).


3.1.8 Loss Function 

Whatever the model selected for the number of claims, the a posteriori premium correction 

is derived from the application of a loss function. The standard choice is a quadratic loss. 

In this case, the credibility premium is the function of past claim numbers that minimizes 

the expected squared difference with the next year claim number. It is well known that the 

solution is given by the a posteriori expectation. 

The penalties obtained in a credibility system calling upon a quadratic loss function 

are often so severe that it is almost impossible to implement them in practice, mainly for 

commercial reasons. In order to avoid this problem, some authors have proposed resorting to 

an exponential loss function: the hope is that breaking the symmetry between the overcharges 

and the undercharges leads to reasonable penalties. This reduces the maluses and the bonuses, 

and results in a financially balanced system. 

3.1.9 Agenda 

Section 3.2 introduces the basics of credibility models taking into account a priori 

characteristics. It starts with a simple introductory example that contains all the ideas of 

credibility theory. Then, the probabilistic tools used in this context are briefly recalled. 

Section 3.3 is devoted to credibility formulas based on a quadratic loss function. The 

optimal predictor is then shown to be the conditional expectation of future claims given 

past claims history. In the particular case of Gamma distributed risk parameters, explicit 

expressions are available for a posteriori premium corrections. Discrete Poisson mixtures 

are considered in detail, providing approximate credibility formulas. Also, linear credibility 

predictions are derived. 

In Section 3.4, the quadratic loss function is replaced with an exponential one. Again, the 

general formulas simplify in the Negative Binomial case. Linear credibility predictions are 

considered as approximations to exact premium corrections. 

Section 3.5 discusses the type of dependence generated by the credibility construction. It 

is shown that the risk parameter, as well as future claim numbers, increases with the number 

of claims recorded in the past, and that future and past claim numbers are positively related. 

This confirms the intuition behind the actuarial credibility model. 

The final Section 3.6 gives the references, and discusses further issues. 

3.2 Credibility Models 

3.2.1 A Simple Introductory Example: the Good Driver / Bad Driver 

Model 

Consider an insurance portfolio where 60 % of the policyholders are good drivers. The 

probability that a good driver reports k claims during the year is given by the oi G 

distribution with G = 005. The remaining 40 % of the policyholders are bad drivers. The 

probability that they report k claims is given by the oi B distribution with B = 015.


A priori (i.e. at time t = 0 for a policyholder without any claim record), the actuary is not 

able to distinguish between good and bad drivers. The expected number of claims is then 

given by 

PrGood G + PrBad B = 009 

Considering a policyholder who reported k claims during the first year, it is nevertheless 

possible to compute the probability that he is a good driver: calling upon Bayes’ Theorem 

yields 

PrGoodk claims = 

= 

Prk claimsGood PrGood 

Prk claimsGood PrGood + Prk claimsBad PrBad 

exp− G k G PrGood 

exp− G k G PrGood + exp− B k B PrBad 

We get the values listed in Table 3.1 for increasing ks. Clearly, these probabilities are 

decreasing with the number of claims reported. If the policyholder does not report any claim 

during the first year, then the probability that he is a good driver is increased from 60 % 

a priori to 62.37 % a posteriori. As soon as one claim is reported during the first year, 

this probability decreases from 60 % a priori to 35.59 % a posteriori. The more claims are 

reported, the less likely that the policyholder is a good driver. The a posteriori probability of 

being a good driver remains nevertheless positive whatever the number of claims reported 

to the insurance company. 

A posteriori (i.e. at time t = 1), the actuary knows the number k of claims reported by the 

policyholder during the year and should incorporate this additional information in the price 

list. Specifically, if k claims have been reported, the expected number of claims for year 

two is 

PrGoodk claims G + PrBadk claims B 

The claim record of the policyholder thus modifies the weights assigned to G and B . The 

values of the expected number of claims for year two according to the number k of claims 

Table 3.1 Expected numbers of claims for year two in the good driver/bad driver 

model given the number k of claims reported during the first year. 

# of claims k 

reported during 

year 1 

PrGoodk 

claims (%) 

PrBadk 

claims (%) 

Expected 

number of 

claims for year 2 

0 6237 3763 00876 

1 3559 6441 01144 

2 1555 8445 01344 

3 578 9422 01442 

4 200 9800 01480 

5 068 9932 01493


reported during the first year are displayed in Table 3.1. This elementary example contains 

all the ingredients of experience rating. 

3.2.2 Credibility Models Incorporating a Priori Risk Classification 

This chapter aims to design merit rating plans in accordance with the a priori ratemaking 

structure of the insurance company. Specifically, let us consider a portfolio with n policies, 

each one observed during T i periods. Let N it (with mean EN it = it ) be the number of claims 

reported by policyholder i during year t, i.e. during the period t − 1t, i = 1 2n, 

t = 1 2T i . By convention, time 0 corresponds to the issuance of the policy. We thus 

face a nested structure: each policyholder generates a sequence N i = N i1 N i2 N iTi T of 

claim numbers. It is reasonable to assume independence between the series N 1 N 2 N n 

(at least in motor third party liability insurance), but we expect some positive dependence 

inside the N i s. 

The ith policy of the portfolio, i = 1 2n, is represented by a sequence 

i N i1 N i2 N i3 . At the portfolio level, the sequences i N i1 N i2 N i3 are 

assumed to be independent for i = 1 2n. The risk parameter i represents the 

risk proneness of policyholder i, i.e. unknown risk characteristics of the policyholder 

having a significant impact on the occurrence of claims; it is regarded as a random 

variable. Given i = , the random variables N i1 N i2 N i3 are assumed to be independent. 

Unconditionally, these random variables are dependent since their behaviour depends on the 

common i . 

The very basic tenets of a credibility model for claim counts are as follows: 

(i) a conditional distribution for the number of claims, that is, for the N it s given i = ; 

(ii) a distribution function F for the risk parameters 1 n to describe how the 

conditional distributions vary accross the portfolio; 

(iii) a loss function whose expectation has to be minimized in order to find the optimal 

experience premium. 

Let us briefly comment on these three aspects. In motor third party liability insurance 

portfolios, the Poisson distribution often provides a good description of the number of claims 

incurred by an individual policyholder during a given reference period (one year, say): the 

set of all Poisson assumptions should at least provide (locally in time) a good approximation 

to the accident generating mechanism. Given i = , the annual numbers of claims N it for 

policyholder i are assumed to be independent and to conform to a Poisson distribution with 

mean it . As in Chapter 2, it is a known function of the exposure-to-risk and possibly other 

covariates. 

Let us now consider the choice of F . Traditionally, actuaries have assumed that the 

distribution of values among all drivers is well approximated by a two-parameter Gamma 

distribution. The resulting probability distribution for the number of claims is Negative 

Binomial. Other classical choices for F include the Inverse-Gaussian and the LogNormal 

distributions, as explained in Chapters 1–2. 

Regarding (iii), quadratic and exponential loss functions will be considered in this chapter. 

This leads to the following model.


Definition 3.1 In the Poisson credibility model, the ith policy of the portfolio, i = 

1 2n, is represented by a sequence i N i where i is a positive random variable 

with unit mean representing the unexplained heterogeneity. Moreover, 

A1 given i = , the random variables N it , t = 1 2, are independent and conform 

to the oi it distribution; 

A2 at the portfolio level, the sequences i N i , i = 1 2n, are assumed to be 

independent. 

It is essential to understand the meaning of this classical actuarial construction. In 

Definition 3.1, dependence between annual claim numbers is a consequence of the 

heterogeneity of the portfolio (i.e. of i ); the dependence is only apparent. If we had a 

complete knowledge of policy characteristics then i would become deterministic and there 

would be no more dependence between the N it s for fixed i. The unexplained heterogeneity 

(which has been modelled through the introduction of the risk parameter i for policyholder 

i) is then revealed by the claims and premiums histories in a Bayesian way. These histories 

modify the distribution of i and hence modify the premium. 

Let 

T i 

T 

∑ 

∑ i 

N i• = N it and i• = it (3.1) 

t=1 

be the total observed and expected claim numbers for policyholder i during the T i observation 

periods; the statistic N i• is a convenient summary of past claims history. 

Let us prove that in the credibility model of Definition 3.1, N i• is an exhaustive summary 

of the past claims history. 

Property 3.1 In the Poisson credibility model of Definition 3.1, the predictive distribution 

of i only depends on N i• , i.e. the equality 

holds true whatever t ≥ 0. 

t=1 

Pr i ≤ tN i1 N i2 N iTi = Pr i ≤ tN i• 

Proof Let f ·k i1 k iTi be the conditional probability density function of i given 

that N i1 = k i1 N iTi = k iTi , and let 

T 

∑ i 

k i• = 

k it 

t=1 

be the total number of accidents reported by policyholder i to the company. We can then 

write 

f k i1 k iTi = PrN i1 = k i1 N iTi = k iTi i = f 


exp− 

= 

i• k i• f 

∫ + 

exp− 

0 i• k i• f d 

which depends only on k i• . This ends the proof.


This result has an important practical consequence: the Poisson credibility model A1–A2 

disregards the age of the claims. The penalty induced by an old claim is strictly identical 

to the one induced by a recent claim. This may of course sometimes be undesirable for 

commercial purposes. We will come back to this issue in the last section of this chapter. 

3.3 Credibility Formulas with a Quadratic Loss Function 

3.3.1 Optimal Least-Squares Predictor 

Often in applied probability, one seeks to approximate an unknown quantity by a function 

of a set of related variables, by minimizing the expected squared difference between the two 

items. This leads to the least-squares principle, and to the conditional expectation, as shown 

in the next result. 

Proposition 3.1 Let us consider a sequence of random variables X 1 X 2 X 3 and a 

risk parameter . Given , the X t s are independent. The first two moments of the X t s are 

assumed to be finite. Moreover, the conditional mean of the X t s is given by 

t = EX t t = 1 2 3 

and E t = t . 

The minimum of 

[( 

) 2 ] 

E T+1 − X 1 X 2 X T 

over all the measurable functions T → is obtained for 

⋆ X 1 X 2 X T = EX T+1 X 1 X 2 X T 

= E T+1 X 1 X 2 X T 

Proof 

An easy way to get the announced result consists in noting that 

[( 

) 2 ] 

E T+1 − X 1 X 2 X T 

[( 

= E T+1 − ⋆ X 1 X 2 X T 

) 2 ] 

+ ⋆ X 1 X 2 X T − X 1 X 2 X T 

[( 

) 2 ] 

= E T+1 − ⋆ X 1 X 2 X T 

[( 

) 2 ] 

+ E ⋆ X 1 X 2 X T − X 1 X 2 X T 

which is clearly minimal for ≡ ⋆ .


Proposition 3.1 indicates that the best approximation (with respect to the mean squared 

error) to T+1 given X 1 X 2 X T is EX T+1 X 1 X 2 X T , that is, the posterior 

expectation of X T+1 given X 1 X 2 X T (also called the predictive mean). The posterior 

distribution of X T+1 is then obtained by conditioning on past claims history. 

To calculate the predictive mean one needs a conditional distribution of losses given the 

parameter of interest (often the conditional mean) and a prior distribution of the parameter 

of interest. 

3.3.2 Predictive Distribution 

Let us now come back to the credibility model of Definition 3.1. The conditional distribution 

of N iTi +1 given N i1 = k 1 N iTi = k Ti 

is called the predictive distribution. It tells the 

actuary what the next year number of claims might be given the information contained in 

past claims history. It is the relevant distribution for risk analysis, management and decision 

making. 

In our case, we have 

PrN iTi +1 = kN i1 = k 1 N iTi = k Ti 

 

∫ ( 

∏Ti 

) 

0 t=1 PrN it = k t i = PrN iTi +1 = k i = dF 

= 

( 

∏Ti 

) 

t=1 PrN it = k t i = dF 

= 

∫ 

∫ 

0 

exp 

( 

− 

0 i• ) ( 

k • exp − iTi +1 ) iT i +1 k 

dF 

k! 

∫ 

exp ( − 

0 i• ) 

k • dF 

Now, the posterior distribution of i given past claims history N i1 = k 1 N iTi = k Ti 

given by 

( 

∏ Ti 

t=1 exp ( − it ) ) 

it k it 

dF 

k it ! 

( 

∫ ∏ Ti 

0 t=1 exp ( − it ) ) 

it k it 

dF 

k it ! 

= exp ( − i• ) k • dF 

∫ 

exp ( − 

0 i• ) k • dF 

is 

Hence, 

PrN iTi +1 = kN i1 = k 1 N iTi = k Ti 

 

∫ 

= exp ( − iTi +1 ) iT i +1 k 

dF 

0 

k! k • 

where F ·k • is the conditional distribution function of i given N i• = k. The predictive 

distribution thus appears as a Poisson mixture distribution, where the mixing is with respect


to the posterior distribution of i . Past claims history N i1 = k 1 N iTi = k Ti 

modifies the 

distribution of i , and this modified distribution is used as a new mixing law for the number 

of claims N iTi +1 for year T i + 1. 

3.3.3 Bayesian Credibility Premium 

We are looking for the function ⋆ of N i1 N iTi that is the closest to i , i.e. minimizing 

[ (i 

E − N i1 N iTi ) ] 2 

over all the measurable functions T i 

→ . Proposition 3.1 gives the solution of this 

optimization problem: 

⋆ N i1 N iTi = E [ i 

∣ ∣ N i1 N iTi 

] 

 

In general, the posterior mean of i given N i1 = k 1 N iTi = k Ti 

is given by 

E [ i 

∣ ∣ N i1 = k 1 N iTi = k Ti 

] 

= 

∫ + 

= 

( 

∏Ti 

 

0 

∫ + 

0 

) 


( 

∏Ti 

) 


∫ + 

0 

exp− i• k •+1 dF 

∫ + 

0 

exp− i• k • dF (3.2) 

This a posteriori expectation thus appears as the ratio of two Mellin transforms Mk = 

Eexp− i i k of i. It is interesting to note that the a posteriori expectation depends 

only on the total number k • of accidents caused in the past T i years of insurance, and not on 

the history of these claims. This was expected from Property 3.1. This is a characteristic of 

the credibility models with static random effects. 

The Bayesian credibility premium is simply the mean of the predictive distribution. It is 

given by 

E [ ∣ 

] ∫ 

N iTi +1 

∣N i1 = k 1 N iTi = k Ti = iTi +1dF k • 

0 

= iTi +1E [ ∣ ] 

i N i1 = k 1 N iTi = k Ti 

where the posterior expectation of i is given by (3.2). The posterior expected claim number 

EN iTi +1N i1 N iTi is obtained by multiplying iTi +1 by the correction coefficient 

E i N i1 N iTi . This approach always yields financial balance, since 

[ 

E E [ ∣ ] ] 

i N i1 = k 1 N iTi = k Ti 

= E i = 1 

so that the corrections average to unity.


3.3.4 Poisson-Gamma Credibility Model 

Gamma Distribution for the Random Effects 

Let us assume that i ∼ ama a with probability density function (1.35). The joint 

probability mass function of the random vector N i = N i1 N i2 N iTi is given by (2.19). 

The joint probability density of the random vector i N i1 N i2 N iTi is given by 

T 

∏ i 

t=1 

( ) ( ) kit 

i it 

exp − i it 

k it ! 

( 

T 

∑ i 

∝ exp − i 

t=1 

1 

a aa a−1 

i 

exp−a i 

) 

∑ Ti 

t=1 

it 

k it+a−1 

i exp−a i (3.3) 

A Posteriori Distribution of Random Effects 

In Section 2.5.1, we established that in a two-period model, the a posteriori distribution of 

i remained Gamma, with updated parameters. Let us now extend the result to a multiperiod 

setting. 

Now, the conditional distribution of i given the past claims frequencies N it = k it , t = 

1 2T i , is obtained from (2.19) and (3.3). This gives 

( 

exp 

(− i 

∫ + 

0 

exp 

( 

− 

a + ∑ T i 

t=1 it 

( 

a + ∑ T i 

t=1 it 

( )) 

T 

∑ i 

= exp 

(− i a + it 

t=1 

)) 

a+∑T i 

t=1 k it−1 

i 

)) 

a+∑T i 

t=1 k it−1 

d 

a+∑T i 

t=1 k it−1 

i 

( 

a + ∑ T i 

 

t=1 it 

) a+ 

∑ Ti 

t=1 k it 

( 

a + ∑ ) 

T i 

t=1 k it 

Coming back to (1.34), we recognize a Gamma probability density function. Specifically, 

we thus have that 

The correction coefficient is given by 

i N i1 N i2 N iTi ∼ am ( a + N i• a+ i• 

) 

(3.4) 

E i N i1 N i2 N iTi = a + N i• 

a + i• 

which clearly increases in the past claims N i• . The variance of i given past claim history 

is given by 

V i N i1 N i2 N iTi = 

a + N i• 

a + i• 2 

The expected number of claims in year T i + 1 given past claims history is given by 

a + N 

EN iTi +1N i1 N i2 N iTi = iTi +1E i N i1 N i2 N iTi = i• 

iTi +1 

a + i•


Let us briefly comment on this a posteriori correction: 

• We see that the a posteriori corrections will be more severe as the residual heterogeneity, 

measured by V i = 1/a, increases. 

• Considering two policyholders (numbered i 1 and i 2 ) in the portfolio during T i1 

= T i2 

periods 

such that i 1 is a priori a better driver than i 2 , that is, i1 • < i2 •: if these policyholders 

do not report any claim (i.e., N i1 • = N i2 • = 0) then the corrections to be applied to these 

policyholders satisfy 

a a 

> 

a + i1 • a + i2 • 

so that the a priori worse driver receives more discount. 

• If these policyholders report k ≥ 1 claims (i.e., N i1 • = N i2 • = k) then the penalties are such 

that 

a + k 

a + i1 • 

> a + k 

a + i2 • 

Hence, the penalty for the a priori bad driver is less severe than for the good one. 

3.3.5 Predictive Distribution and Bayesian Credibility Premium 

From (3.4) we know that the posterior distribution of i given past claims history is still 

Gamma, with parameters a+N i• and a+ i• . Therefore, the predictive distribution of N iTi +1 

is Negative Binomial, that is, 

( ) ( ) k ( 

) a+k• 

a + k• + k − 1 iTi +1 

a + 

PrN iTi +1 = kN i• = k • = 

i• 

 

k a + i• + iTi +1 a + i• + iTi +1 

Furthermore, the Bayesian credibility premium is given by 

a + k 

EN iTi +1N i• = k • = iTi +1E i N i• = k • = • 

iTi +1 

a + i• 

Remark 3.1 As time goes on, the Bayesian credibility premium tends to 0 for a policyholder 

reporting no claim. This can be seen as an unrealistic feature. An easy way to avoid this 

problem is to decompose N i into two parts: a component N 1 

i distributed according to a pure 

Poisson distribution that represents the claims occurring purely at random, and a component 

N 2 

i that is Negative Binomial and influenced by the driver’s abilities. This introduces a 

lower bound on the a posteriori claim frequency, which is no more allowed to vanish in the 

long term. 

It is worth mentioning that the Bayesian credibility premium can be cast into 

( a 

EN iTi +1N i• = k • = E 

a + i + 

) 

i• k • 

EN 

i• a + i• iTi +1 

i•


Recall that E i = 1 and EN iTi +1 = iTi +1. Hence, the Bayesian credibility premium is 

obtained by multiplying the a priori expected number of claims EN iTi +1 by an appropriate 

correction factor. This factor appears as a weighted average of the prior expectation of 

i , receiving weight a/a + i• , and the average claim frequency for that policy k • / i• 

receiving a weight i• /a + i• . 


Let us now compute the coefficients to apply to the pure premium according to the number 

of claims reported in the past. To this end, let us consider the Negative Binomial fit 

of Portfolio A given in Table 2.7. Table 3.2 displays the values of E i N i• = k • for 

different combinations of observed periods T i and number of past claims k • for a good 

driver (with observable characteristics: man, age 35, rural area, upfront premium, private 

use, i = 00928). Tables 3.3–3.4 are the analogues for an average driver (with observable 

characteristics: woman, age 25, urban area, upfront premium, private use, i = 01408) and 

a bad driver (with observable characteristics: man, age 22, rural area, split premium, private 

use, i = 02840), respectively. 

If the good driver does not report any accident during the first year, we see from Table 3.2 

that he will pay 92 % of the base premium to be covered during the second year. If, in 

addition, he does not file any claim during the second year, the premium decreases to 85.2 % 

of the base premium. After ten claim-free years, he will have to pay 53.4 % of the base 

premium to be covered by the insurer. 

Considering the average driver, we see from Table 3.3 that he will be awarded more 

discount than the good driver if he does not file any claim. Indeed he will have to pay 

88.3 % (instead of 92 %) of the base premium after one claim-free year, 79.1 % (instead of 

85.1 %) of the base premium after two claim-free years, and 43.1 % (instead of 53.4 %) after 

ten claim-free years. 

Table 3.2 Values of E i N i• = k • for different combinations of observed periods T i 

and number of past claims k • for a good driver from Portfolio A (average annual claim 


T i 

Number of claims k • 

0 1 2 3 4 5 

1 92.0 % 178.4 % 264.7 % 351.1 % 437.5 % 523.8 % 

2 85.2 % 165.1 % 245.1 % 325.0 % 405.0 % 485.0 % 

3 79.3 % 153.7 % 228.2 % 302.6 % 377.0 % 451.5 % 

4 74.2 % 143.8 % 213.4 % 283.0 % 352.7 % 422.3 % 

5 69.7 % 135.1 % 200.5 % 265.9 % 331.3 % 396.7 % 

6 65.7 % 127.3 % 189.0 % 250.6 % 312.3 % 374.0 % 

7 62.1 % 120.4 % 178.8 % 237.1 % 295.4 % 353.7 % 

8 58.9 % 114.3 % 169.6 % 224.9 % 280.2 % 335.6 % 

9 56.0 % 108.7 % 161.3 % 213.9 % 266.6 % 319.2 % 

10 53.4 % 103.6 % 153.8 % 204.0 % 254.1 % 304.3 %



and number of past claims k • for an average driver from Portfolio A (average annual 

claim frequency of 14.09 %). 

T i 


0 1 2 3 4 5 

1 88.3 % 171.2 % 254.2 % 337.1 % 420.0 % 503.0 % 

2 79.1 % 153.3 % 227.6 % 301.8 % 376.1 % 450.4 % 

3 71.6 % 138.8 % 206.0 % 273.3 % 340.5 % 407.7 % 

4 65.4 % 126.8 % 188.2 % 249.6 % 311.0 % 372.5 % 

5 60.2 % 116.7 % 173.2 % 229.8 % 286.3 % 342.8 % 

6 55.8 % 108.1 % 160.5 % 212.8 % 265.2 % 317.5 % 

7 51.9 % 100.7 % 149.4 % 198.2 % 247.0 % 295.7 % 

8 48.6 % 94.2 % 139.8 % 185.5 % 231.1 % 276.7 % 

9 45.7 % 88.5 % 131.4 % 174.3 % 217.1 % 260.0 % 

10 43.1 % 83.5 % 123.9 % 164.3 % 204.8 % 245.2 % 


and number of past claims k • for a bad driver from Portfolio A (average annual claim 


T i 


0 1 2 3 4 5 

1 78.9 % 1531 % 227.2 % 301.3 % 375.5 % 449.6 % 

2 65.2 % 1264 % 187.7 % 248.9 % 310.2 % 371.4 % 

3 55.5 % 1077 % 159.9 % 212.0 % 264.2 % 316.4 % 

4 48.4 % 938 % 139.2 % 184.7 % 230.1 % 275.5 % 

5 42.9 % 831 % 123.3 % 163.6 % 203.8 % 244.0 % 

6 38.5 % 746 % 110.7 % 146.8 % 182.9 % 219.0 % 

7 34.9 % 676 % 100.4 % 133.1 % 165.9 % 198.6 % 

8 31.9 % 619 % 91.8 % 121.8 % 151.8 % 181.7 % 

9 29.4 % 570 % 84.6 % 112.3 % 139.9 % 167.5 % 

10 27.3 % 529 % 78.5 % 104.1 % 129.7 % 155.3 % 

Table 3.4 shows higher discounts for the bad driver, with percentages of 78.9 % after one 

claim-free year, 65.2 % after two claim-free years and 27.3 % after ten claim-free years. 

The discounts awarded to policyholders who do not report any accident to the insurance 

company are thus increasing with the a priori annual expected claim frequency. The more 

claims are expected by the insurance company on the basis of observable characteristics, the 

higher the discount in case no claims are reported. Note that the a posteriori expected claim 

frequencies remain large for a priori bad drivers since reporting no claims happens with a 

smaller probability.


Now, considering the penalty in case one claim is reported, we see that the good driver 

who reports one claim during the first year will have to pay 178.4 % of the base premium 

to be covered by the insurance company during the second year. The average driver in the 

same situation pays 171.2 % of the base premium, and the bad driver 153.1 % of the base 

premium. The penalties in the case where an accident is reported to the company are thus 

decreasing with the a priori annual expected claim frequencies. 

The system appears rather severe. Reporting a claim entails a penalty of between 50 and 

75 %, which seems difficult to implement in practice. This is a consequence of the financial 

balance property. The weighted averages of all figures (in each infinite row) is equal to 

100 %. The modest discounts awarded to the majority of claim-free policyholders have 

then to be exactly compensated by the penalties supported by the minority of policyholders 

reporting claims to the company. This causes large penalties. The severity of the credibility 

corrections also appears in the number of claim free years needed to erase the penalty induced 

by an accident: 10 years for the good driver, 7 for the average one and 3 for the bad one. 

These periods make sense compared to the average claim number per policy. 

3.3.7 Discrete Poisson Mixture Credibility Model 

The good driver / bad driver model presented in Section 3.2.1 assumes that the portfolio is 

composed of two classes of insured drivers. This can easily be extended to several categories 

of drivers (for instance, bad, below average, average, above average, excellent). Specifically, 

let us assume that each risk class of the portfolio is made of q categories of insured drivers. 

Let us denote as p 1 p 2 p q the proportion of drivers in each of these categories. Then, 

⎧ 

1 with probability p 1 

⎪⎨ 2 with probability p 2 

i = 

(3.5) 

⎪⎩ 

q with probability p q 

with 0 < 1 < 2 < ···< q . The number N it of claims caused by policyholder i during year 

t is distributed as 

PrN it = k = 

q∑ 

exp− it j it j k 

p 

k! j k= 0 1 

j=1 

Note that the special case q = 2 gives the good driver / bad driver model. 

The expected number of claims reported by policyholder i in year T i + 1 is given by 

q∑ 

q∑ 

EN iTi +1 = EN iTi +1 = j p j = iTi +1 j p j 

j=1 

If we know that this policyholder reported k claims during the past T i years, we expect 

EN iTi +1N i• = k claims in year T i + 1. The computation of EN iTi +1N i• = k requires the 

conditional distribution of N iTi +1 given N i• = k. We get it from 

j=1


PrN iTi +1 = iN i• = k = PrN iT i +1 = i and N i• = k 

PrN i• = k 

∑ q 

j=1 

= 

PrN iT i +1 = i i = j PrN i• = k i = j p j 

∑ q 

j=1 PrN i• = k i = j p j 

= 

q∑ 

exp− iTi +1 j iT i +1 j i p j exp− i• j j 

k 

∑ 

i! q 

l=1 p 

l exp− i• l l 

k 

j=1 

Hence, given N i• = k, the law of N iTi +1 appears as a discrete Poisson mixture with modified 

weights 

˜p j k = 

p j exp− i• j j 

k 

∑ q 

l=1 p 


k 

The a posteriori expectation of N iTi +1 is then 

EN iTi +1N i• = k = iTi +1 

The posterior mean of i given N i• = k is then given by 

∑ q∑ 

q 

j=1 

j˜p j k = p j exp− i• j k+1 

j 

iTi +1 ∑ q 

l=1 p (3.6) 


k 

j=1 

E i N • = k = 

∑ q 

j=1 p j exp− i• j k+1 

j 

∑ q 

l=1 p (3.7) 


k 

Compared to (3.2), the integrals now reduce to sums over the q components of the discrete 

mixture. 

Even if the reality of the insurance portfolio is a discrete mixture (with a i specific to 

policyholder i, resulting in q = n), this model is not convenient since it involves a large 

number of parameters. The discrete Poisson mixture nevertheless deserves interest as an 

approximation of more general Poisson mixtures, as discussed below. 

3.3.8 Discrete Approximations for the Heterogeneous Component 

Moment Spaces 

Apart from the Poisson-Gamma case, the computation of the conditional expectation giving 

the credibility premium requires numerical integration. A convenient alternative is to 

approximate the mixing distribution by a suitable discrete analogue sharing the same sequence 

of moments. We are then back to the discrete Poisson mixture credibility model studied 

above, and we benefit from the easy-to-compute formulas (3.6)–(3.7) valid in this case. 

The discrete approximations to i are based on the knowledge of its support, 0b say, 

with b possibly infinite, and its first few moments 1 2 In general, let us denote by 

s 0 b the class of all the random variables X with support in 0band with prescribed 

first s −1 moments EX k = k , k = 1 2s−1. In the literature, s 0 b is referred 

to as a moment space. Properly speaking, it is a class of distribution functions rather than a 

class of random variables. Classical problems related to moment spaces are for instance the


determination of conditions on 0 b 1 2 s−1 so that s 0 b is not void, or 

the obtention of the elements in s 0 b with the minimal number of support points. 

These elements possess some extremal properties and will be used here in connection with 

credibility formulas. Henceforth, we tacitly assume that 0b and 1 2 s−1 are such 

that the associated moment space s 0 b is not void and is not a singleton (i.e., 

s 0 b consists of at least two distinct distribution functions). 

De Vylder’s Moment Problem 

De Vylder (1996, Section 8.3) investigated the following problem: within s 0 b , 

determine the random variables X s 

min 

and Xs max such that the inequalities 

EX s 

min s ≤ EX s ≤ EX s 

max s hold for all X ∈ s 0 b (3.8) 

Explicit solutions to (3.8) are available for s up to five. The supports of the extremal 

distributions are given in Tables 3.5–3.6. 

As shown in Denuit, De Vylder & Lefèvre (1999), the random variables X s 

X s 

min and 

max involved in (3.8) give bounds on EX for every function that is s − 2 times 

differentiable, with a convex s − 2th derivative (such functions are called s-convex; see 

Roberts & Varberg (1973) for details). In that context, X s 

min 

is called the s-convex 

minimum, and Xmax s is called the s-convex maximum. 

Table 3.5 

s = 1to5. 

Probability distribution of X s 

min ∈ s0 b achieving the lower bound in (3.8) for 

s Support points Probability masses 

1 0 1 

2 1 1 

3 0 

2 − 2 1 

2 

2 

1 

2 1 

2 

4 r + = 3 − 1 2 + √ 3 − 1 2 2 − 4 2 − 2 1 1 3 − 2 2 

2 2 − 2 1 

r − = 3 − 1 2 − √ 3 − 1 2 2 − 4 2 − 2 1 1 3 − 2 2 

2 2 − 2 1 

1 − r − 

r + − r − 

1 − 1 − r − 

r + − r − 

5 0 1− q + − q − 

t + = 1 4 − 2 3 + √ 1 4 − 2 3 2 − 4 1 3 − 2 2 2 4 − 2 3 

2 1 3 − 2 2 q + = 2 − t − 1 

t + t + − t − 

t − = 1 4 − 2 3 − √ 1 4 − 2 3 2 − 4 1 3 − 2 2 2 4 − 2 3 

2 1 3 − 2 2 q − = 2 − t + 1 

t − t − − t +


Table 3.6 

s = 1to5. 

Probability distribution of X s 

max ∈ s 0 b achieving the upper bound in (3.8) for 

s Support points Probability masses 

1 b 1 

b − 

2 0 

1 

 

b 

b 

1 

b 

b 

3 1 − 2 

b − 1 2 

b − 1 b − 1 2 + 2 − 2 1 

b 

2 − 2 1 

b − 1 2 + 2 − 2 1 

4 0 1− p 1 − p 2 

3 − b 2 

 

p 

2 − b 1 = 

2 − b 1 3 

1 3 − b 2 3 − 2b 2 + b 2 1 

b p 2 = 

5 z + = 1 − b 4 − b 3 − 2 − b 1 3 − b 2 + √ 

2 1 − b 3 − b 2 − 2 − b 1 2 

z − = 1 − b 4 − b 3 − 2 − b 1 3 − b 2 − √ 

2 1 − b 3 − b 2 − 2 − b 1 2 

b 

1 3 − 2 2 

b 3 − 2b 2 + b 2 1 

p + = 2 − b + z − 1 + bz − 

z + − z − z + − b 

p − = 2 − b + z + 1 + bz + 

z − − z + z − − b 

1 − p + − p − 

Where = 1 − b 4 − b 3 − 2 − b 1 3 − b 2 2 − 4 1 − b 3 − b 2 − 2 − 

b 1 2 2 − b 1 4 − b 3 − 3 − b 2 2 

Approximations Based on First Moments 

A given random variable X (think of the risk parameter i in credibility applications) with 

known moments k , k = 1 2, can be approximated either by X s 

min 

or Xs max involved in 

(3.8). An alternative approximation mixing these two extremal variables is also available. 

Let us denote as 

and 

s 

= 

s = 

min 

X∈ s 0b 

max 

X∈ s 0b 

EX s = E [ X s 

min s] 

EX s = E [ X s 

max s] 

the lower and upper bounds involved in (3.8). Explicit expressions of s 

and of s for s up 

to five are listed in Table 3.7, in the notation of Tables 3.5–3.6. The quantity s − s 

can 

be considered as the ‘width’ of s 0 b as explained in Denuit (2002).


Table 3.7 

Lower s 

and upper s bounds in (3.8) for s = 2to5. 

s s 

s 

2 2 1 

b 1 

3 

4 

2 2 

( ) b1 − 3 

2 b − 1 2 

1 b − 1 b − 1 2 + + 2 

2 b3 

b − 1 2 + 2 

( 

1 − ) 

1 − r − 

r− 4 r + − r + ( ) 

1 − r − 

3 − b 4 

r+ 4 p 2 

− r + − r 1 + p 

− 2 − b 2 b 4 

1 

5 q + t 5 + + q −t 5 − 

p + z 5 + + p −z 5 − + 1 − p + − p − b 5 

Now, let X ∈ s 0 b . The sth canonical moment of X, denoted as c s X, is given 

by 

c s X = EXs − s 

 

s − s 

Thus, c s X is simply the position of the sth moment of X relative to its possible range. 

Now, c s X gives a good indication of the ‘position’ of X in s 0 b with respect to 

the extrema X s 

min 

and Xs max. Therefore, we might expect that a satisfactory approximation 

of X ∈ s 0 b is furnished by a convex combination of the stochastic extrema in 

s−1 0 b with weights depending on the s − 1th canonical moment c s−1 X, i.e. we 

use the mixture 

˜X s = 

{ 

X 

s−1 

min 

X s−1 

max 

with probability 1 − c s−1 X 

with probability c s−1 X 

(3.9) 

in order to approximate X. In the following, we refer to ˜X s as the sth canonical approximation 

of X. It is easily seen that ˜X s ∈ s 0 b . 

The Unimodal Case 

A purely discrete approximation to the risk parameter i causes problems when an experience 

rating plan has to be designed, as shown in Walhin & Paris (1999). The a posteriori 

corrections obtained with discrete i s exhibit plateaus before and after sudden jumps, 

which is commercially unacceptable. When F only has a few support points, a ‘block’ 

structure is clearly apparent for the credibility coefficients, each block with almost constant 

a posteriori corrections corresponding to one support point of F . The policyholder is 

transferred from smaller to larger mass points as more claims are filed. In order to avoid 

this, we need a smooth risk distribution. Therefore, we would like to have simple continuous 

approximations to the risk parameter. This can be done in the unimodal case, as shown 

below. 

A situation of practical interest in actuarial sciences is when the random variables under 

consideration are known to have a unimodal distribution with a given mode m, together 

with the fixed moments 1 2 s−1 . The Gamma, LogNormal and Inverse Gaussian


mixing distributions examined in Chapter 2 are all unimodal. Henceforth, we denote by 

s 0 b m − unim the unimodal moment space of all the random variables of this type. 

In the following, we use the notation nim z, with m, z ∈ , for the Uniform 

distribution on the interval minm z maxm z: ifmz, it is the law 

with constant probability density function equal to 1/m − z on z m; ifm = z, itisthe 

law degenerated at the point m. Given some random variable Z, we denote by nim Z 

the mixed Uniform distribution with random extremal point Z as mixing parameter. 

A convenient representation of unimodal distributions is provided by Khinchine’s theorem 

(see, e.g., Theorem 1.3 in Dharmadhikari & Joag-Dev (1988)). This theorem states that a 

random variable Y has a unimodal law with a mode at 0 if, and only if, Y is distributed as 

UZ where U and Z are two independent random variables and U ∼ ni0 1. This condition 

can be rewritten as Y ∼ ni0Z for some random variable Z. 

Now, let X be any random variable valued in 0b and with a unique mode at m. By 

Khinchine’s theorem, 

X ∼ nim˜Z where ˜Z = m + Z (3.10) 

Note that Z is valued in −m b − m. Moreover, the moments j of X and ˜ j of ˜Z are 

linked by simple relations. Indeed, we have 

∫ ( 1 

∫ m 

) 

j = 

x j dx dF˜Z 

m − z 

z 

x=z 

∫ 

m j+1 − z j+1 

= 

dF˜Z z 

m − zj + 1 

= 1 ∫ 

m j + m j−1 z + m j−2 z 2 +···+z j dF˜Z 

j + 1 

z 

 

= 1 

j + 1 mj + m j−1˜ 1 + m j−2˜ 2 +···+˜ j j = 1 2 (3.11) 

From (3.11), we get 

˜ j = j + 1 j − mj j−1 j = 1 2 (3.12) 

Let us now come back to (3.8) in s 0 b m − unim . Specifically, we would like to 

determine the random variables X s⋆ 

min 

and Xs⋆ max such that the inequalities 

EX s⋆ 

min s ≤ EX s ≤ EX s⋆ 

max s hold for all X ∈ s 0 b m − unim (3.13) 

As shown in Denuit, De Vylder & Lefèvre (1999), the random variables X s⋆ 

min 

and Xs⋆ max 

involved in (3.13) give bounds on EX for every s-convex function . Finding the 

solution of (3.13) for X in s 0 b m − unim thus amounts to solving the corresponding 

problem in s 0 b ˜. 

Tables 3.8–3.9 give explicit expressions for the improved extremal distributions, also for 

values of s up to five. In these tables ∑ k 

i=1 p ini i i ,0≤ p i ≤ 1, i i ∈ , i = 1 2k, 

represents a mixture of the distributions ni i i , with respective weights p i .


Table 3.8 Probability distribution of X s⋆ 

min ∈ s0 b m − unim 

achieving the lower bound in (3.13), s = 15. 

s 

Distributions 

1 ni0m 

2 nim ˜ 1 

3 

˜ 2 −˜ 2 1 

ni0m+ ˜2 1 

nim ˜ 

˜ 2 ˜ 2 / ˜ 1 

2 

˜ 

4 

1 −˜r − 

nim ˜r 

˜r + −˜r + + ( 1 − ˜ 1 −˜r − 

) 

nim ˜r− 

− ˜r + −˜r − 

5 1 −˜q + −˜q − ni0m+˜q + nim ˜t + +˜q − nim ˜t − 

Here ˜r ± , ˜t ± and ˜q ± are those from Table 3.5, with the ˜ j s substituted for the j s. 

Table 3.9 Probability distribution of Xmax s⋆ ∈ s 0 b m − unim achieving 

the upper bound in (3.13), s = 15. 

s 

Distribution 

1 nim b 

2 

3 

b −˜ 1 

ni0m+ ˜ 1 

nim b 

b 

b 

b −˜ 1 2 

ni [ m b ˜ 1 −˜ 2 

] ˜ + 2 −˜ 2 1 

nim b 

b −˜ 1 2 +˜ 2 −˜ 2 1 

b −˜ 1 b −˜ 1 2 +˜ 2 −˜ 2 1 

4 1 −˜p 1 −˜p 2 ni0m+˜p 1 ni [ m ˜ 3 − b ˜ 2 

] 

+˜p2 nim b 

˜ 2 − b ˜ 1 

5 ˜p + nim ˜z + +˜p − nim ˜z − + 1 −˜p + −˜p − nim b 

Here ˜p 1 , ˜p 2 , ˜z ± and ˜p ± are those from Table 3.6, with the ˜ j s substituted for the j s. 

Application to Credibility Formulas 

Considering a risk parameter i with moments 1 s−1 and support 0b, b possibly 

infinite, the idea is now to approximate i by either i 

min or i 

max according to the form 

of the support of i . This amounts to replacing the general formulas (3.2) with the simpler 

(3.6)–(3.7) where the support points 1 q are those given in Tables 3.5–3.6. Note that 

(3.2) can be rewritten as 

E i N i1 = k 1 N iTi = k Ti 

= Eexp− i• i k •+1 

i 

Eexp− i• i k • 

i 

so that we need to approximate the Mellin transform 

for any integer k and real >0. 

Mk = Eexp− i k i


When the risk parameter i is known to be unimodal (with mode m, say) then the 

improved extremal distributions of Tables 3.8–3.9 can be used in lieu of those coming from 

Tables 3.5–3.6. This amounts to evaluating the numerator and denominator of (3.2), taking 

for the structure function F a discrete mixture of uniform distributions. This provides an 

easy-to-compute approximation for (3.2) based on incomplete Gamma functions, as can be 

seen from the following example. 

Example 3.1 Assume that i has support + , mode m and moments 1 and 2 . Then, we 

can use the approximation Mk ≈ M 3 

mink where 

M 3 

min k = 3 2 + 2 1 m − m 2 − 4 2 1 

1 

m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1 

m 

∫ m 

m − 2 

+ 

1 2 

1 

m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1 

c − c 

0 

exp− k d 

∫ c 

c 

exp− k d 

with c and c defined as 

and 

( 

c = min m 3 ) 

2 − 2m 1 

2 1 − m 

( 

c = max m 3 ) 

2 − 2m 1 

 

2 1 − m 

The value of M 3 

mink is easily obtained from the incomplete Gamma function. Specifically, 

M 3 

min k = 3 2 + 2 1 m − m 2 − 4 2 1 

k! 

k + 1 m 

m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1 

mk+1 m − 2 

+ 

1 2 

k! ( ) 

k + 1 c− k + 1c 

m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1 

c − c k+1 

Example 3.2 Now, if the support of i is known to be contained in 0b then we can use 

the approximation Mk ≈ Mmaxk 3 where 

M 3 

max k = 3 2 + 2 1 m − m 2 − 4 2 1 

1 

b + m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1 

b − m 

b + m − 2 

+ 

1 2 

1 

b + m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1 d − d 

∫ b 

m 

∫ d 

d 

exp− k d 

exp− k d 

with d and d defined as 

( 

d = min m 2b ) 

1 + 2 1 m − bm − 3 2 

b + m − 2 1


and 

( 

d = max m 2b ) 

1 + 2 1 m − bm − 3 2 

 

b + m − 2 1 

As above, this expression can be simplified with the help of the incomplete Gamma function 

as 

M 3 

max k = 3 2 + 2 1 m − m 2 − 4 2 1 

b + m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1 

k! ( ) 

× 

k + 1 b − k + 1 m 

b − m k+1 

b + m − 2 

+ 

1 2 

b + m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1 

k! ( ) 

× 

k + 1 d− k + 1d 

d − d k+1 

Example 3.3 If the third moment of i , 3 say, is known then we can use a mixture of the 

two preceding approximations, with weights defined by the canonical moments as suggested 

by (3.9). To this end, note that 

and 

⋆ 3 = 

⋆ 3 = 

min 

X∈ 3 0bm−unim 1 2 

EX 3 

3 

= 

2 + 2 1 m − m 2 − 4 2 1 

m 3 

m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1 

3 

m − 2 

+ 

1 2 

c 4 − c 4 

m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1 

c − c 

max 

X∈ 3 0bm−unim 1 2 

EX 3 

3 

= 

2 + 2 1 m − m 2 − 4 2 1 

b + m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1 

b + m − 2 

+ 

1 2 

b + m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1 

The approximation to the Mellin transform is then as follows 

b 4 − m 4 

b − m 

d 4 − d 4 

d − d 

Mk ≈ ⋆ 3 − 3 

⋆ 3 − M 3 

⋆ 3 

min k + 3 − ⋆ 3 

⋆ 3 − ⋆ 3 

M 3 

max k


Numerical Illustration 

Since we worked with mixing distributions with unbounded support, we cannot use the 

maximal variables described above. In practice, setting b equal to a high quantile (99.99 %, 

say) of the mixing distribution is expected to give good results. 

Let us consider the Negative Binomial model for the annual claim number in Portfolio A. 

In this case, i ∼ ama a, with estimated parameter â = 1065. The estimated moments 

of i are 

̂ 1 = 1 

̂ 2 = â + 1 = 1939 

â 

â + 1â + 2 

̂ 3 = = 5580 

̂ 4 = 

â 2 

â + 1â + 2â + 3 

â 3 

= 21299 

Applying formula (3.7) with discrete approximation given by Table 3.5, we can approximate 

i with the help of the 4-convex minimum with two support points: r + = 32883 with 

probability 01521 and r − = 05897 with probability 08479, or with the 5-convex minimum 

with three support points: t + = 45218 with probability 00474 and t − = 12341 with 

probability 06366 and 0 with probability 03160. The accuracy of these approximations 

decreases with T i and k • . Discrete approximations are useful for short claim history and 

rather good driving record. Since the Gamma distribution (playing the role of the mixing 

distribution in the Negative Binomial model) has a unimodal probability density function, 

we can also use a mixture of uniform distributions to approximate i in Portfolio A. It turns 

out that the approximation is satisfactory, except for high values of k • . Again, this is due to 

the fact that the approximation uses s-convex minima that underestimate the riskiness of the 

worse drivers. 

To get accurate approximations we need to use a mixture (3.9) of the improved 4-convex 

extrema in the unimodal case taking for b the 99.99th quantile of the Gamma distribution. 

This gives the results displayed in Table 3.10 for a good driver, in Table 3.11 for an average 

driver and in Table 3.12 for a bad driver. We can see that the approximations are now very 

satisfactory for the vast majority of the combinations T i –k • . 


Bayesian statistics offer an intellectually acceptable approach to credibility theory. Bayes 

revision E i N i• of the heterogeneity component is theoretically very satisfying but is 

often difficult to compute (except for conjugate distributions or discrete approximations). 

Practical applications involve numerical methods to perform integration with respect to a 

posteriori distributions, making more elementary approaches desirable (at least to get a first 

easy-to-compute approximation of the result). Because we have observed N i1 N iTi , one 

suggestion is to approximate E i N i1 N iTi by a linear function of the N it s. Basically, 

the actuary still resorts to a quadratic loss function but the shape of the credibility predictor


Table 3.10 Values of E i N i• = k • for different combinations of observed periods T i and 

number of past claims k • for a good driver from Portfolio A (expected annual claim frequency of 

9.28 %) and for a mixture of mixed uniform approximations of i (improved 4-convex extrema). 

T i 


0 1 2 3 4 5 

1 91.98 % 178.36 % 264.74 % 351.04 % 437.76 % 524.92 % 

2 85.16 % 165.12 % 245.13 % 325.04 % 404.57 % 486.88 % 

3 79.28 % 153.70 % 228.23 % 302.76 % 376.26 % 451.61 % 

4 74.16 % 143.77 % 213.51 % 283.40 % 352.14 % 420.74 % 

5 69.66 % 135.06 % 200.54 % 266.38 % 331.27 % 394.52 % 

6 65.67 % 127.37 % 189.00 % 251.28 % 312.83 % 372.29 % 

7 62.18 % 120.55 % 178.65 % 237.81 % 296.27 % 353.06 % 

8 58.93 % 114.47 % 169.29 % 225.73 % 281.26 % 335.95 % 

9 56.05 % 109.04 % 160.80 % 214.82 % 267.62 % 320.35 % 

10 53.43 % 104.17 % 153.05 % 204.90 % 255.23 % 305.90 % 


number of past claims k • for an average driver from Portfolio A (expected annual claim frequency 

of 14.09 %) and for a mixture of mixed uniform approximations of i (improved 4-convex 

extrema). 

T i 


0 1 2 3 4 5 

1 88.32 % 171.25 % 254.21 % 337.07 % 419.97 % 505.05 % 

2 79.09 % 153.34 % 227.69 % 302.05 % 375.37 % 450.47 % 

3 71.60 % 138.83 % 206.16 % 273.75 % 340.28 % 405.73 % 

4 65.41 % 126.87 % 188.25 % 250.30 % 311.63 % 370.88 % 

5 60.21 % 116.90 % 173.05 % 230.58 % 287.30 % 342.82 % 

6 55.76 % 108.51 % 159.96 % 213.75 % 266.28 % 318.81 % 

7 51.93 % 101.38 % 148.58 % 199.16 % 248.12 % 297.44 % 

8 48.58 % 95.27 % 138.64 % 186.29 % 232.40 % 278.28 % 

9 45.63 % 89.96 % 129.95 % 174.75 % 218.71 % 261.30 % 

10 43.01 % 85.30 % 122.35 % 164.32 % 206.58 % 246.42 % 

is constrained ex ante to be linear in past observations, i.e. the predictor ˆN iTi +1 of N iTi +1 is 

of the form 

T 

∑ i 

ˆN iTi +1 = c i0 + c it N it 

The coefficients c i0 and the c it s involved in ˆN iTi +1 are obtained from the minimization of an 

expected squared difference. 

t=1



number of past claims k • for a bad driver from Portfolio A (expected annual claim frequency 

of 28.40 %) and for a mixture of mixed uniform approximations of i (improved 4-convex 

extrema). 

T i 


0 1 2 3 4 5 

1 78.95 % 15307 % 227.29 % 301.52 % 374.70 % 449.61 % 

2 65.22 % 12650 % 187.69 % 249.57 % 310.74 % 369.83 % 

3 55.55 % 10812 % 159.34 % 212.96 % 265.29 % 317.66 % 

4 48.36 % 9488 % 138.01 % 185.46 % 231.41 % 277.06 % 

5 42.80 % 8493 % 121.75 % 163.48 % 205.61 % 245.24 % 

6 38.39 % 7709 % 109.33 % 145.45 % 184.61 % 220.87 % 

7 34.79 % 7065 % 99.70 % 130.79 % 166.57 % 201.26 % 

8 31.83 % 6518 % 92.05 % 119.03 % 150.92 % 184.27 % 

9 29.35 % 6042 % 85.75 % 109.64 % 137.59 % 168.95 % 

10 27.24 % 5622 % 80.40 % 102.06 % 126.49 % 155.15 % 

Specifically, we look for c i0 and c it s, t = 1T i , such that the expected square difference 

between N iTi +1 and its prediction ˆN iTi +1 is minimum, i.e. such that 

c = arg min 1 

c 

where 

⎡( 

) ⎤ 

T 2 

∑ i 

1 = E ⎣ N iTi +1 − c i0 − c it N it 

⎦ 

t=1 

Alternatively, it can be shown that c also solves 

c = arg min j j = 2 3 

c 

where 

⎡( 

) ⎤ 

T 2 

∑ i 

2 = E ⎣ EN iTi +1 i − c i0 − c it N it 

⎦ 

⎡( 

) ⎤ 

T 2 

∑ i 

3 = E ⎣ EN iTi +1N i1 N iTi − c i0 − c it N it 

⎦ 

t=1 

t=1 

Let us show for instance that arg min c 1 = arg min c 2 . To this end, let us write 

⎡ 

( 

)) ⎤ 

( ) 

T 2 

∑ i 

1 = E ⎣( 

N iTi +1 − EN iTi +1 i + EN iTi +1 i − c i0 − c it N it 

⎦ 

t=1


and let us expand the squared sum to get 

[ ( ) ] 2 

1 = E N iTi +1 − EN iTi +1 i 

[ ( 

+ 2E N iTi +1 − EN iTi +1 i ) ( )] 

T 

∑ i 

EN iTi +1 i − c i0 − c it N it + 2 

The second term in the expansion of 1 vanishes, and the first one does not depend on the 

c it s. 

Let us determine c as arg min 2 . Recall that EN iTi +1 i = iTi +1 i . Setting equal to 0 

the partial derivative of 2 with respect to c i0 allows us to write 

which gives 

T 

∑ i 

0 = iTi +1E i − c i0 − c it EN it 

t=1 

T 

∑ i 

c i0 = iTi +1 − c it it (3.14) 

Now, setting equal to 0 the partial derivatives of 2 with respect to c is for s = 1 2T i , 

yields 

Noting that 

t=1 

T 

∑ i 

0 = iTi +1EN is i − c i0 EN is − c it EN is N it (3.15) 

EN it N is = CN is N it + EN is EN it 

[ 

] [ 

] 

= E CN is N it i + C EN is i EN it i + is it 

⎧ 

⎨ is + ( ) 21 

is + Vi if s = t 

= 

⎩ 

is it 1 + V i otherwise 

t=1 

t=1 

and that 

[ 

] 

] 

EN is i = E CN is i i + C 

[EN is i i + is 

= is 1 + V i 

allows us to cast equation (3.15) into the form 

( 

) 

T 

∑ i 

c is = V i iTi +1 − c it it = V i c i0 

t=1


Hence, the value of c is does not depend on s and the same weight is given to all the past 

annual claim numbers. Now, inserting the value of c is that we just obtained in (3.14) finally 

gives 

which in turn yields. 

c i0 = 

c is = 

iTi +1 

1 + V i ∑ T i 

t=1 it 

iTi +1V i 

1 + V i ∑ T i 

t=1 

it 

The expected claim frequency for year T i + 1 given past claims history is 

̂N iTi +1 = iTi +1 

1 + V i N i• 

1 + V i i• 

(3.16) 

where N i• and i• have been defined in (3.1). Note that ̂N iTi +1 is the best linear predictor 

of each of the true unknown means iTi +1 i , of the Bayesian credibility premium 

EN iTi +1N i1 N iTi and of the number of claims N iTi +1 for year T i + 1. 

The linear predictor for year T i + 1 thus appears as the product of the a priori 

expected claim frequency, iTi +1, times an approximation of the theoretical correction 

E i N i1 N iTi . This approximation possesses a particularly simple interpretation since 

it entails a malus when N i• > i• , that is, if the policyholder reported more claims than 

expected a priori. 

Remark 3.2 Note that (3.16) agrees with the result obtained in the Poisson-Gamma case. 

This is because the Bayesian credibility premium is linear in the past observations in the 

Poisson-Gamma case. The term exact credibility is used to describe the situation where 

the linear credibility premium equals the Bayesian one. Intuitively speaking, using linear 

credibility formulas in the mixed Poisson model boils down to approximating the mixing 

distribution with the Gamma distribution (or, equivalently, the distribution of the claim 

numbers with the Negative Binomial one). 

Remark 3.3 Note that ̂N iTi +1 could have been obtained by a direct application of the 

Bühlmann–Straub formula. Let us consider a sequence of random variables X 1 X 2 X 3 

such that, given a random variable , the X t s are independent, with finite first and second 

moments 

Now, define M 2 and 2 as 

= EX t and EX t = = E 

E [ VX t ] = 2 

w t 

and V [ EX t ] = M 2


and put 

w • = 

n∑ 

w t and ˜X n = 1 n∑ 

w 

w j X j 

• 

Clearly, ˜X n is the weighted average of the X j s. The minimum of 

[ ( 

E − a − b˜X n) ] 2 

t=1 

t=1 

on all the couples a b is obtained for 

a = 

2 

2 + w • M 2 and b = w •M 2 

2 + w • M 2 

To apply this general result to the Poisson case, we define X j = N ij / ij which gives i = 

i , = 1, w j = ij , 2 = 1, M 2 = V i . The best linear approximation to i is then given 

by ̂N iTi +1/ iTi +1. 

3.4 Credibility Formulas with an Exponential Loss Function 

3.4.1 Optimal Predictor 

This section purposes to describe an alternative approach based on an exponential loss 

function. The exponential loss function is asymmetric and possesses one parameter that 

reflects the severity of the credibility correction. This allows us to soften the a posteriori 

corrections in case of claims, keeping the financial balance. 

When the new premium amount is fixed by the insurer, two kinds of errors may arise: either 

the policyholder is undercharged and the insurance company loses its money or the insured is 

overcharged and the insurer is at risk of losing the policy. In order to penalize large mistakes to 

a greater extent, it is usually assumed that the loss function is a nonnegative convex function 

of the error. The loss is zero when no error is made and strictly positive otherwise. The loss 

function is generally taken to be quadratic as in the preceding section. Among other choices 

we find also the absolute loss and the 4-degree loss; see, e.g., Lemaire & Vandermeulen 

(1983). The problem with these two last losses is that the resulting bonus-malus systems are 

unbalanced. 

We give here a technical result involving an exponential loss function. It is the analogue 

of Proposition 3.1. 

Proposition 3.2 

Under the conditions of Proposition 3.1, the minimum of 

[ ( 

E exp − c ( T+1 − X 1 X 2 X T ))] 

on all the measurable functions T → satisfying the constraint 

EX 1 X 2 X T = T+1 is obtained for


⋆⋆ X 1 X 2 X T = T+1 + 1 c 

(E [ ln E [ exp−c T+1 X 1 X 2 X T 

]] 

− ln E [ exp−c T+1 X 1 X 2 X T 

] ) 


Starting from 

[ ( 

E exp − c ( T+1 − X 1 X 2 X T ))] 

[ 

= E exp cX 1 X 2 X T E [ ] ] 

exp−c T+1 X 1 X 2 X T 

[ ( 

)] 

= E exp cX 1 X 2 X T − ⋆⋆ X 1 X 2 X T 

( [ 

× expc T+1 exp E ln E [ ] ]) 

exp−c T+1 X 1 X 2 X T 

Now, let us apply Jensen’s inequality to get 

[ ( 

E exp − c ( T+1 − X 1 X 2 X T ))] 

( [ 

]) 

≥ exp cE X 1 X 2 X T − ⋆⋆ X 1 X 2 X T 

( [ 

expc T+1 exp E ln E [ ] ]) 

exp−c T+1 X 1 X 2 X T 

Because of the constraint on the expectation of the s, the first exponential is 1, yielding 

[ ( 

E exp − c ( T+1 − X 1 X 2 X T ))] 

( [ 

≥ expc T+1 exp E ln E [ ] ]) 

exp−c T+1 X 1 X 2 X T 

[ ( 

= E exp − c ( T+1 − ⋆⋆ X 1 X 2 X T ))] 

which is the expected result. 

□ 

Remark that in Proposition 3.2 the constraint is made in order to guarantee the financial 

equilibrium. 

Let us now apply the result contained in Proposition 3.2 to the credibility problem. In this 

case, 

X t = N it and t i = it i 

so that the optimal predictor of N iTi +1 for the exponential loss function is of the form 

⋆⋆ N i1 N iTi = iTi +1 + 1 (E [ ln E [ ]] 

exp−c 

c 

iTi +1 i N i1 N iTi 

− ln E [ ] ) 

exp−c iTi +1 i N i1 N iTi


3.4.2 Poisson-Gamma Credibility Model 

Let us now apply the result contained in Proposition 3.2 to the Poisson-Gamma credibility 

model. To this end, assume that i ∼ ama a. Given N i• , we know from (3.4) that i 

follows the ama + N i• a+ i• distribution, so that we know from (1.36) that 

( 

) 

[ ] 

a+Ni• 

a + 

E exp−c iTi +1 i N i1 N iTi = 

i• 

 

a + i• + c iTi +1 

It follows that 

and 

[ ] 

ln E exp−c iTi +1 i N i1 N iTi 

( 

=−a + N i• ln 1 + c ) 

iT i +1 

a + i• 

[ 

E ln E [ ] ] 

exp−c iTi +1 i N i1 N iTi 

( 

=−a + i• ln 1 + c ) 

iT i +1 

 

a + i• 

Proposition 3.2 then gives 

⋆⋆ k i1 k iTi = iTi +1 + k i• − i• 

c 

( 

ln 1 + c ) 

iT i +1 

(3.17) 

a + i• 

Considering (3.17), we see that ⋆⋆ k i1 k iTi is equal to the a priori expectation iTi +1 = 

EN iTi +1 plus a correction term. This correction is positive, so that 

⋆⋆ k i1 k iTi > iTi +1 

if k i• > i• , that is, if the policyholder reported more claims than expected. Otherwise, the 

correction is negative. As with the quadratic loss function, the penalty is caused by an excess 

of observed claims over expected ones. 

Let us now compare the credibility formulas obtained with a quadratic and exponential 

loss function. Since for any c ≥ 0, 

( 

ln 1 + c ) 

iT i +1 

≤ c iT i +1 

 

a + i• a + i• 

it is easily seen that we have 

⋆⋆ k i1 k iTi ≤ ⋆ k i1 k iTi if k i• > i•


and 

⋆⋆ k i1 k iTi ≥ ⋆ k i1 k iTi if k i• < i• 

Let us define 

exp c = 1 ( 

c ln 1 + c ) 

iT i +1 

a + i• 

to be the weight given to k i• in the a posteriori evaluation (3.17) in the Poisson- 

Gamma model. We have that lim c→+ exp c = 0. Moreover, routine calculations show 

that d/dc exp c < 0, so that the weight given to the observed average claim number 

decreases as c increases. This provides an intuitive meaning of the parameter c: if c 

increases, then the a posteriori merit-rating scheme becomes less severe, and at the limit 

for c →+, the premium no longer depends on the incurred claims. If the asymmetry 

factor c tends to + then all the risks within the same tariff class pay the same premium: 

there is no longer an experience rating. Conversely, the weight given to past claims 

under an exponential loss function tends to the weight under a quadratic loss function 

as c → 0. 


Another possibility is to determine a linear credibility premium based on an exponential 

loss function, by considering predictors of the form b 0 + ∑ T i 

t=1 b jN it . The b j s minimize the 

Lagrangian function 

[ 

b 0 b 1 ··· b t = E exp ( − c iTi +1 i − b 1 N i1 − b 2 N i2 −···−b Ti 

N iTi − b 0 )] 

[ 

] 

− E b 0 + b 1 N i1 + b 2 N i2 +···+b Ti 

N iTi − iTi +1 

Setting to 0 the derivatives of with respect to , b 0 b 1 b Ti 

in the Poisson-Gamma case, i.e. (3.17). 

yields the same result as 


Let us now illustrate the use of the exponential loss function in credibility. To this end, let us 

consider the Negative Binomial fit to Portfolio A described in Table 2.7. Thus, i is taken 

to be ama a distributed, with estimated parameter â = 1065. 

Formula (3.17) allows us to compute the a posteriori correction as a function of the 

number T i of coverage periods and of the total number of claims k • filed to the company. 

The results obtained with c = 1 are displayed in Table 3.13 for a good driver, in Table 3.14 

for an average driver, and in Table 3.15 for a bad driver. Tables 3.16–3.18 are the analogues 

for c = 5. 

Let us first compare the a posteriori corrections listed in Table 3.13 with those of 

Table 3.2 corresponding to a quadratic loss function (i.e. to c = 0). We see that the 

application of an exponential loss function slightly reduces the penalties in case of claims


Table 3.13 Values of the a posteriori corrections obtained from (3.17) for different 

combinations of observed periods T i and number of past claims k • for a good driver 

(expected annual claim frequency equal to 9.28 %) from Portfolio A, with c = 1. 

T i 


0 1 2 3 4 5 

1 92.3 % 175.4 % 258.5 % 341.5 % 424.6 % 507.7 % 

2 85.7 % 162.8 % 240.0 % 317.1 % 394.2 % 471.4 % 

3 80.0 % 151.9 % 223.9 % 295.9 % 367.9 % 439.9 % 

4 75.0 % 142.4 % 209.9 % 277.4 % 344.8 % 412.3 % 

5 70.5 % 134.0 % 197.5 % 261.0 % 324.5 % 388.0 % 

6 66.6 % 126.6 % 186.5 % 246.5 % 306.5 % 366.4 % 

7 63.1 % 119.9 % 176.7 % 233.5 % 290.3 % 347.1 % 

8 59.9 % 113.9 % 167.9 % 221.8 % 275.8 % 329.7 % 

9 57.1 % 108.5 % 159.8 % 211.2 % 262.6 % 314.0 % 

10 54.5 % 103.5 % 152.6 % 201.6 % 250.7 % 299.7 % 


combinations of observed periods T i and number of past claims k • for an average driver 


T i 


0 1 2 3 4 5 

1 89.0 % 1674 % 245.8 % 324.3 % 402.7 % 481.1 % 

2 80.1 % 1507 % 221.4 % 292.0 % 362.6 % 433.2 % 

3 72.9 % 1371 % 201.3 % 265.5 % 329.8 % 394.0 % 

4 66.8 % 1257 % 184.6 % 243.5 % 302.4 % 361.3 % 

5 61.7 % 1161 % 170.5 % 224.9 % 279.2 % 333.6 % 

6 57.3 % 1078 % 158.3 % 208.9 % 259.4 % 309.9 % 

7 53.5 % 1007 % 147.8 % 195.0 % 242.1 % 289.3 % 

8 50.2 % 944 % 138.6 % 182.8 % 227.0 % 271.3 % 

9 47.2 % 889 % 130.5 % 172.1 % 213.7 % 255.4 % 

10 44.6 % 839 % 123.3 % 162.6 % 201.9 % 241.2 % 

(the values listed in the columns entitled k • = 1 to 5 are smaller in Table 3.13 compared 

to Table 3.2). Since the financial balance is fulfilled by the credibility premiums obtained 

with an exponential loss function, the discounts for claim-free policyholders are also 

reduced (the values in the column entitled k • = 0 are higher in Table 3.13 compared to 

Table 3.2). 

Note however that the a posteriori corrections obtained with an exponential loss function 

with c = 1 are very similar to those coming from a quadratic loss function. To see this, let 

us now increase the value of c to 5 in Table 3.16. Increasing c results in reduced discounts 

and also in reduced penalties.


Table 3.15 Values of the a posteriori corrections obtained from (3.17) for 

different combinations of observed periods T i and number of past claims k • for a 

bad driver (expected annual claim frequency equal to 28.40 %) from Portfolio A, 

with c = 1. 

T i 


0 1 2 3 4 5 

1 80.9 % 1482 % 2154 % 282.7 % 350.0 % 417.2 % 

2 67.9 % 1244 % 1808 % 237.3 % 293.8 % 350.2 % 

3 58.6 % 1072 % 1558 % 204.5 % 253.1 % 301.8 % 

4 51.5 % 942 % 1369 % 179.6 % 222.4 % 265.1 % 

5 45.9 % 840 % 1221 % 160.2 % 198.3 % 236.4 % 

6 41.4 % 758 % 1102 % 144.5 % 178.9 % 213.3 % 

7 37.7 % 691 % 1004 % 131.7 % 163.0 % 194.3 % 

8 34.7 % 634% 922 % 120.9 % 149.7 % 178.4 % 

9 32.0 % 586% 852 % 111.8 % 138.4 % 165.0 % 

10 29.8 % 545% 792 % 103.9 % 128.7 % 153.4 % 


combinations of observed periods T i and number of past claims k • for a good driver 


T i 


0 1 2 3 4 5 

1 93.3 % 165.9 % 238.5 % 311.1 % 383.8 % 456.4 % 

2 87.4 % 155.4 % 223.4 % 291.4 % 359.4 % 427.4 % 

3 82.2 % 146.1 % 210.1 % 274.0 % 338.0 % 401.9 % 

4 77.6 % 137.9 % 198.3 % 258.6 % 318.9 % 379.3 % 

5 73.5 % 130.6 % 187.7 % 244.9 % 302.0 % 359.1 % 

6 69.8 % 124.0 % 178.3 % 232.5 % 286.7 % 340.9 % 

7 66.5 % 118.1 % 169.7 % 221.3 % 272.9 % 324.6 % 

8 63.4 % 112.7 % 161.9 % 211.2 % 260.4 % 309.7 % 

9 60.7 % 107.8 % 154.8 % 201.9 % 249.0 % 296.1 % 

10 58.1 % 103.2 % 148.4 % 193.5 % 238.6 % 283.7 % 

If we compare the a posteriori corrections for the different types of drivers, we see 

that the discounts increase with the average annual claim frequency, as was the case with 

the quadratic loss function. Also, the penalties appear to decrease with the average annual 

claim frequencies. The fact that a priori bad drivers need a greater premium reduction 

when no claim is filed to the insurance company thus remains with exponential loss 

functions.



combinations of observed periods T i and number of past claims k • for an average driver 


T i 


0 1 2 3 4 5 

1 90.8 % 1561 % 221.4 % 286.7 % 352.0 % 417.4 % 

2 83.2 % 1429 % 202.6 % 262.4 % 322.1 % 381.8 % 

3 76.7 % 1318 % 186.8 % 241.9 % 296.9 % 351.9 % 

4 71.2 % 1223 % 173.3 % 224.3 % 275.4 % 326.4 % 

5 66.5 % 1141 % 161.7 % 209.2 % 256.8 % 304.4 % 

6 62.3 % 1069 % 151.5 % 196.0 % 240.6 % 285.2 % 

7 58.7 % 1006 % 142.5 % 184.4 % 226.3 % 268.2 % 

8 55.4 % 950 % 134.5 % 174.1 % 213.7 % 253.2 % 

9 52.5 % 900 % 127.4 % 164.9 % 202.4 % 239.8 % 

10 49.9 % 855 % 121.0 % 156.6 % 192.2 % 227.8 % 


combinations of observed periods T i and number of past claims k • for a bad driver 


T i 


0 1 2 3 4 5 

1 85.6 % 1363 % 186.9 % 237.5 % 288.2 % 338.8 % 

2 75.0 % 1190 % 163.1 % 207.2 % 251.2 % 295.3 % 

3 66.7 % 1058 % 144.8 % 183.8 % 222.9 % 261.9 % 

4 60.2 % 952 % 130.3 % 165.3 % 200.4 % 235.5 % 

5 54.8 % 866 % 118.5 % 150.3 % 182.1 % 213.9 % 

6 50.3 % 795 % 108.6 % 137.8 % 166.9 % 196.1 % 

7 46.5 % 734 % 100.3 % 127.2 % 154.1 % 181.0 % 

8 43.3 % 682 % 93.2 % 118.2 % 143.1 % 168.1 % 

9 40.4 % 637 % 87.0 % 110.3 % 133.6 % 156.9 % 

10 38.0 % 598 % 81.6 % 103.5 % 125.3 % 147.2 % 

3.5 Dependence in the Mixed Poisson Credibility Model 

3.5.1 Intuitive Ideas 

The main focus of this section is to formalize intuitive ideas with the help of stochastic 

orderings. Every actuary intuitively feels that the a posteriori claim frequency distribution 

must become more dangerous as more claims are reported. Here we precisely define ‘more 

dangerous’ and explain that the a posteriori premium must increase with the total claim 

number in the mixed Poisson model.


In the model A1–A2 of Definition 3.1, we intuitively feel that the following statements 

are true: 

Statement S1 

Statement S2 

Statement S3 

i ‘increases’ in the past claims N i• 

N iTi +1 ‘increases’ in the past claims N i• 

N iTi +1 and N i• are ‘positively dependent’. 

This section aims to precisely define the meaning of ‘increases’ in statements S1 and S2, as 

well as the nature of the ‘positive dependence’ involved in statement S3. The proofs will be 

omitted because of their technical nature; for a detailed study, we refer the reader to Denuit 

ET AL. (2005, Chapter 7). Note that we present the results in terms of stochastic dominance 

whereas in fact the stronger (but less intuitive) likelihood ratio order applies. 

3.5.2 Stochastic Order Relations 

In order to formalize the increasingness involved in statements S1–S2, our study will 

extensively resort to stochastic orderings. Therefore, we recall in this section the definition of 

stochastic dominance, as well as some intuitive intepretations. Given two random variables 

X and Y , X is said to be smaller than Y in the stochastic dominance, written as X ≼ ST Y ,if 

PrX>t≤ PrY>tfor all t ∈ 

We see that a ranking in the ≼ ST -sense translates the intuitive meaning of ‘being smaller 

than’ in probability models: indeed, we compare the probability that both random variables 

exceed some given threshold t, and the smallest one in the ≼ ST -sense has the smallest 

probability of exceeding the threshold. If M and N are two counting random variables then 

M ≼ ST N ⇔ 

+∑ 

j=k 

PrM = j ≤ 

+∑ 

j=k 

PrN = j for all k = 0 1 

One intuitively feels that a random variable N following the Poisson distribution with 

mean gets bigger as increases. The next implication formalizes this intuitive statement: 

≤ ′ ⇒ N ≼ ST N ′ 

The oi family thus increases in its parameter in the ≼ ST -sense. 

3.5.3 Comparisons of Predictive Distributions 

We then have the following results that formalize statements S1 and S2. First, i increases 

in the past claims N i• in the ≼ ST -sense, that is 

i N i• = k ≼ ST i N i• = k ′ for k ≤ k ′ (3.18) 

⇔ Pr i >tN i• = k ≤ Pr i >tN i• = k ′ whatever t, provided k ≤ k ′


This relation is transmitted to the number of claims, in the sense that 

N iTi +1N i• = k ≼ ST N iTi +1N i• = k ′ for k ≤ k ′ (3.19) 

⇔ PrN iTi +1 >jN i• = k ≤ N iTi +1 >jN i• = k ′ whatever j, provided k ≤ k ′ 

3.5.4 Positive Dependence Notions 

In order to formalize the positive dependence involved in statement S3, we will present 

some concepts of dependence related to ≼ ST . The study of concepts of positive dependence 

for random variables, started in the late 1960s, has yielded numerous useful results in both 

statistical theory and applications. Applications of these concepts in actuarial science recently 

received increased interest. 

Let us formalize the positive dependence existing between the two components of a 

random couple (i.e. the fact that large values of one component tend to be associated with 

large values for the other). Formally, let X = X 1 X 2 be a bivariate random vector. Then, 

X is positive regression dependent (PRD, for short) if X 2 X 1 = x 1 ≼ ST X 2 X 1 = x 

1 ′ for 

all x 1 ≤ x 

1 

′ and X 1X 2 = x 2 ≼ ST X 1 X 2 = x 

2 ′ for all x 2 ≤ x 

2 ′ . PRD imposes stochastic 

increasingness of one component of the random couple in the value assumed by the other 

component in the ≼ ST -sense. This dependence notion is thus rather intuitive. 

PRD naturally extends to higher dimension. Specifically, let X = X 1 X n be a 

n-dimensional random vector. Then, 

(i) X is conditionally increasing (CI, for short) if 

X i X j = x j j ∈ J ≼ ST X i X j = x ′ j j ∈ J 

whenever x j ≤ x 

j ′ , j ∈ J, J ⊂ 1 2nand i ∉ J. 

(ii) X is conditionally increasing in sequence (CIS, for short) if X i is stochastically increasing 

in X 1 X i−1 , for i ∈ 2ni.e. 

X i X 1 = x 1 X i−1 = x i−1 ≼ ST X i X 1 = x ′ 1 X i−1 = x ′ i−1 

whenever x j ≤ x 

j ′ j∈ 1i− 1. 

The conditional increasingness in sequence is interesting when there is a natural order in the 

components of X, induced by obervation times for instance. 

3.5.5 Dependence Between Annual Claim Numbers 

The total claim number N i• reported in the past periods and the claim number N iTi +1 for the 

next coverage period are PRD. The fact that N i• and N iTi +1 are PRD completes the statement 

(3.19). This provides a host of useful inequalities. In particular, whatever the distribution of 

i , the credibility coefficient E i N i• = k is increasing in k, which is easily deduced from 

(3.18). 

Considering the dependence existing between the components of N i , i.e. between the N it s, 

t = 1 2T i , we can prove that N i is CI.


3.5.6 Increasingness in the Linear Credibility Model 

Let ̂N iTi +1 be the predictor (3.16) of N iTi +1. It can be shown that N iTi +1 is indeed increasing 

in ̂N iTi +1, in the sense that 

N iTi +1̂N iTi +1 = p ≼ ST N iTi +1̂N iTi +1 = p ′ whenever p ≤ p ′ 

⇔ PrN iTi +1 >k̂N iTi +1 = p ≤ PrN iTi +1 >k̂N iTi +1 = p ′ whatever k, provided p ≤ p ′ 

This means that the linear credibility premium is indeed a good predictor of the future 

claim number in model A1–A2. Basically, we prove that increasing the linear credibility 

premium (i.e. degrading the claim record of the policyholder) makes the probability of 

observing more claims in the future greater. 


3.6.1 Credibility Models 

Credibility theory began with the papers by Mowbray (1914) and Whitney (1918). These 

papers purposed to derive a premium which was a balance between the experience of an 

individual risk and of a class of risks. An excellent introduction to credibility theory can be 

found, e.g., in Goovaerts & Hoogstad (1987), Herzog (1994), Dannenburg, Kaas & 

Goovaerts (1996), Klugman, Panjer & Willmot (2004, Chapter 16) and Bühlmann 

& Gisler (2005). See also Norberg (2004) for an overview with useful references and 

links to Bayesian statistics and linear estimation. The underlying assumption of credibility 

theory which sets it apart from formulas based on the individual risk alone is that the risk 

parameter is regarded as a random variable. This naturally leads to a Bayesian approach 

to credibility theory. The book by Klugman (1992) provides an in-depth treatment of the 

question. See also the review papers by Makov ET AL. (1996) and Makov (2002). The 

connection between credibility formulas and Mellin transforms in the Poisson case has been 

established by Albrecht (1984). 

In a couple of seminal papers, Dionne & Vanasse (1989, 1992) proposed a credibility 

model which integrates a priori and a posteriori information on an individual basis. The 

unexplained heterogeneity was then modelled by the introduction of a latent variable 

representing the influence of hidden policy characteristics. Taking this random effect to be 

Gamma distributed yields the Negative Binomial model for the claim number. An excellent 

summary of the statistical models that may lead to experience rating in insurance can be 

found in Pinquet (2000). The nature of serial correlation (endogeneous or exogeneous) is 

discussed there. 

There are many applications of credibility techniques to various branches of insurance. 

Let us mention a nonstandard one, by Rejesus ET AL. (2006). These authors examined 

the feasibility of implementing an experience-based premium rate discount in crop 

insurance.


3.6.2 Claim Count Distributions 

Other credibility models for claim counts can be found in the literature, going beyond 

the mixed Poisson model studied in this chapter. The model suggested by Shengwang, 

Wei & Whitmore (1999) employs the Negative Binomial distribution for the conditional 

distribution of the annual claim numbers together with a Pareto structure function. Some 

credibility models are designed for stratified portfolios. Desjardins, Dionne & Pinquet 

(2001) considered fleets of vehicles, and used individual characteristics of both the vehicles 

and the carriers. See also Angers, Desjardins, Dionne & Guertin (2006). 

An interesting alternative to the Negative Binomial model can be obtained using the 

conditional specification technique introduced by Arnold, Castillo & Sarabia (1999). 

The idea is to specify the joint distribution of N t through its conditionals. More precisely, 

the conditional distribution of N t given = is oi for some function + → + , 

and the conditional distribution of given N t = k is amk k where · and · 

are two functions mapping to + . For an application of the model to experience rating, 

see Sarabia, Gomez-Deniz & Vazquez-Polo (2004). 

3.6.3 Loss Functions 

The quadratic loss function is by far the most widely used in practice. The results with 

the exponential loss function are taken from Bermúdez, Denuit & Dhaene (2000). Early 

references about the use of this kind of loss function include Ferreira (1977) and Lemaire 

(1979). Morillo & Bermudez (2003) used an exponential loss function in connection with 

the Poisson-Inverse Gaussian model. 

Other loss functions can be envisaged. Young (1998a) uses a loss function that is a 

linear combination of a squared-error term and a second-derivative term. The squared-error 

term measures the accuracy of the estimator, while the second-derivative term constrains the 

estimator to be close to linear. See also Young & De Vylder (2000), where the loss function 

is a linear combination of a squared-error term and a term that encourages the estimator 

to be close to constant, especially in the tails of the distribution of claims, where Young 

(1997) noted the difficulty with her semiparametric approach. Young (2000) resorts to a 

loss function that can be decomposed into a squared-error term and a term that encourages 

the credibility premium to be constant. This author shows that by using this loss function, the 

problem of upward divergence noted in Young (1997) is reduced. See also Young (1998b). 

Young (2000) also provides a simple routine for minimizing the loss function, based on the 

discussion of De Vylder in Young (1998a). 

Adopting the semiparametric model proposed in Young (1997, 2000) but considering that 

the piecewise linear function has better characteristics in simplicity and intuition than the 

kernel, Huang, Song & Liang (2003) used the piecewise linear function as the estimate of 

the prior distribution and to obtain the estimates for the credibility formula. 

3.6.4 Credibility and Regression Models 

Hachemeister (1975) contributed to the credibility context by introducing a regression 

model. Since De Vylder (1985), it has been known that credibility formulas can be 

recovered from appropriate (non-)linear statistical regression models. More recently, Nelder


& Verrall (1997) recognized that credibility theory can be encompassed within the theory of 

hierarchical generalized linear models developed by Lee & Nelder (1996). This extends to 

the GLM family the pioneering work by Norberg (1986). The likelihood-based approach can 

be carried out using standard statistical packages. All the assumptions underlying the model 

can be checked (e.g., using appropriate residual analyses), avoiding dogmatic application 

of risk theory models. The mean random effects are estimated for each policyholder by 

maximizing the hierarchical likelihood. With an appropriate choice for the distribution of 

the random effects and using the canonical link function, the estimate is in the form of a 

linear credibility premium. See also Bühlmann & Bühlmann (1999) and Luo, Young & 

Frees (2004). Frees, Young & Luo (1999) developed links between credibility theory and 

statistical models for longitudinal (or panel) data, as explained below. 

Frees & Wang (2006) used a longitudinal data set-up, so that the experience from several 

risk classes are observed over time, whereas Frees ET AL. (1999) focussed on credibility 

predictors that are linear combinations of the data and/or that are minimizers of a squarederror 

loss function. In contrast, Frees & Wang (2006) considered severity distributions that 

may be long-tailed so that averaging or using squared-error loss do not yield appropriate 

prediction tools. 

Antonio & Beirlant (2007) suggested the use of the generalized linear mixed models 

(where a transformation of the mean is expressed as a linear function of both fixed and 

random effects) in credibility theory. Many actuarial credibility models appear to be particular 

cases of generalized linear mixed models. Yeo & Valdez (2006) addressed a simultaneous 

dependence of claims across individuals for a fixed time period and across time periods 

for a fixed individual. This is accomplished by introducing the notion of a common effect 

affecting all individuals and another common effect affecting a fixed individual over time. 

This construction falls within the broader framework of generalized linear mixed models. 

Lo, Fung & Zhu (2006) considered a regression credibility model with random 

regression coefficients. The variance components represented by the uncertainty about the 

regression coefficients then account for the heterogeneity in risks borne by policyholders 

across contracts. From a different perspective, the dependence between contracts has been 

introduced by treating the contract-specific regression coefficients as being generated by the 

same random mechanism such that they are random deviations from the collective mean. 

Autoregressive specifications of the error structure in the credibility context have been 

proposed by Bolancé ET AL. (2003). In Sundt (1983), the generalized Bühlmann–Straub 

model was proposed with consecutive error terms assumed to follow AR(1) dependences. 

Qian (2000) used the nonparametric regression method to establish estimators for 

credibility premiums under some principles of premium calculation. The asymptotic 

properties of the estimators are studied in this paper. 

3.6.5 Credibility and Copulas 

Copulas are a powerful tool to model dependencies between multivariate outcomes. See, 

e.g., Denuit ET AL. (2005) for an introduction. Several works have successfully applied 

copulas to solve credibility problems. Let us mention Frees & Wang (2005) who handled 

serial (time) dependence through a t-copula, and Frees & Wang (2006) who extended that 

formulation by introducing elliptical copulas for serial dependencies. Like the t-copula, the 

elliptical copulas turn out to have an analytically tractable form for predictive distributions.


Multivariate credibility models may be considered for several lines of business, or several 

types of claims. Multivariate credibility models are discussed, e.g., in Frees (2003). This 

topic will be treated in Chapter 6. 

3.6.6 Time Dependent Random Effects 

The vast majority of the papers which have appeared in the actuarial literature considered 

time-independent heterogeneous models. This chapter is restricted to the case of static random 

effects: in the classical credibility construction A1–A2 of Definition 3.1, the risk parameter 

i relating to policyholder i is assumed to be constant over time. This is of course rather 

unrealistic since driving ability may vary during the driving career (because of the learning 

effect, or modification in the risk characteristics). In automobile insurance, an unknown 

underlying random parameter that develops over time expresses the fact that the abilities 

of a driver are not constant. Moreover, the hidden exogeneous variables revealed by claims 

experience may vary with time, as do observable ones. 

Another reason to allow for random effects that vary with time relates to moral 

hazard. Indeed, individual efforts to prevent accidents are unobserved and feature temporal 

dependence. The policyholders may adjust their efforts for loss prevention according to their 

experience with past claims, the amount of premium and awareness of future consequences 

of an accident (due to experience rating schemes). The effort variable determines the moral 

hazard and is modelled by a dynamic unobserved factor. 

Of course, it is hopeless in practice to discriminate between residual heterogeneity due to 

unobservable characteristics of drivers that significantly affect the risk of accident, and their 

individual efforts to prevent such accidents. Both effects get mixed in the latent process. 

Since the observed contagion between annual claim numbers is always positive, the effect 

of omitted explanatory variables seems to dominate moral hazard. Anyway, this issue has no 

practical implication since predictions depend on observed contagion, but not on its nature. 

Hence, instead of assuming that the risk characteristics are given once and for all by a 

single risk parameter, we might suppose that the unknown risk characteristics of each policy 

are described by dynamic random effects. In the terminology of Jewell (1975), evolutionary 

credibility models allow for random effects to vary in successive periods. Now, the ith 

policy of the portfolio, i = 1 2n, is represented by a double sequence i N i where 

i is a positive random vector with unit mean representing the unexplained heterogeneity. 

Specifically, the model is based on the following assumptions: 

B1 

Given i = i , the random variables N it , t = 1 2T i , are independent and conform 

to the Poisson distribution with mean it it , i.e. 

PrN it = k it = it = exp− it it it it k 

 

k! 

k= 0 1 

B2 

with it = d it exp T˜x it . 

At the portfolio level, the i s are assumed to be independent. Moreover, i1 iTi 

is distributed as 1 Ti 

where = 1 Tmax 

T is a stationary random 

vector (with T max = max T i ). It is further assumed that E it = 1 for all i, t. The unit 

mean condition is imposed for identification (otherwise, the mean could be absorbed


B3 

into the intercept term of the Poisson regression). This condition means that the a priori 

ratemaking is correct on average. 

The sequences i N i , i = 1 2n, are assumed to be independent. 

Purcaru & Denuit (2003) studied the kind of dependence arising in these credibility models 

for claim counts. Albrecht (1985) studied such credibility models for claim counts, whereas 

Gerber & Jones (1975) and Sundt (1981,1988) dealt with general random variables. 

A fundamental difference between static A1–A3 and dynamic B1–B3 credibility models 

is that the latter incorporate the age of the claims in the risk prediction, whereas the former 

neglect this information. Since we intuitively feel that the predictive ability of a claim should 

decrease with its age, dynamic specification seems more acceptable. As pointed out by 

Pinquet, Guillén & Bolancé (2001), dynamic credibility models agree with economic 

analysis of multiperiod optimal insurance under moral hazard. In this optic, the stationarity 

of the i s implies that the predictive ability of claims depends mainly on the lag between 

the date of prediction and the date of occurrence (because of time translation invariance of 

the marginals of the i s). 

Empirical studies performed on panel data, as in Pinquet, Guillén & Bolancé (2001) 

and Bolancé, Guillén & Pinquet (2003), support time-varying (or dynamic) random 

effects. An interesting feature of credibility premium derived from stationary random effects 

with a decreasing correlogram is that the age of the claims are taken into account in the a 

posteriori correction: a recent claim will be more penalized than an old one (whereas the 

age of the claim is not taken into account with static random effects). 

This kind of a posteriori correction reconciles actuaries’ and economists’ approaches to 

experience rating. Henriet & Rochet (1986) distinguished two roles played by a posteriori 

corrections, showing that these two roles involve different rating structures. The first role 

deals with the problem of adverse selection, where the very aim is to evaluate as accurately as 

possible the true distribution of reported accidents. This is the classical actuarial perspective. 

The second role is linked to moral hazard and implies that the distribution of reported 

accidents over time must be taken into account to maintain incentives to drive carefully. 

This means that more weight must be given to recent information in order to maintain 

such incentives. This is the economic point of view. The credibility model B1–B3 with 

dynamic random effects, although theoretically more intricate, takes these two objectives into 

account. 

3.6.7 Credibility and Panel Data Models 

Frees, Young & Luo (1999) developed links between credibility theory and longitudinal 

(or panel) data models. They demonstrated how longitudinal data models can be applied 

to the credibility ratemaking problem. As pointed out by these authors, by expressing 

credibility ratemaking applications in the framework of longitudinal data models, actuaries 

can realize several benefits: (1) Longitudinal data models provide a wide variety of 

models from which to choose. (2) Standard statistical software makes analysing data 

relatively easy. (3) Actuaries have another method for explaining the ratemaking process. 

(4) Actuaries can use graphical and diagnostic tools to select a model and assess its 

usefulness.


3.6.8 Credibility and Empirical Bayes Methods 

Credibility theory has an empirical Bayes flavour, as pointed out by Norberg (1980). 

Analyses commonly used in highway safety include the widely applied empirical Bayes 

method. According to Lord (2006), this method has become increasingly popular since it 

corrects for the regression-to-the-mean bias, refines the predicted mean of an entity, and is 

relatively simple to manipulate compared to the full Bayes approach. The empirical Bayes 

method combines information obtained from a reference group having similar characteristics 

with the information specific to the individual under study. A weight factor is assigned to 

both the reference population and the individual under study, and a credibility formula is 

obtained. 

Credibility theory has thus a clear empirical Bayes flavour. We refer the interested reader 

to Quigley, Bedford & Walls (2006) for a case study involving the rate of occurrence of 

train derailments within the United Kingdom.

4 

Bonus-Malus Scales 


4.1.1 From Credibility to Bonus-Malus Scales 

One of the main tasks of the actuary is to design a tariff structure that will fairly distribute the 

burden of claims among policyholders. To this end, he often has to partition all policies into 

risk classes with all policyholders belonging to the same class paying the same premium. It 

is convenient to achieve a priori classification by resorting to generalized linear models (e.g. 

Poisson regression for claim counts), as explained in Chapter 2. However, many important 

factors cannot be taken into account at this stage. Consequently, risk classes are still quite 

heterogeneous despite the use of many a priori variables. 

Rating systems penalizing insureds responsible for one or more accidents by premium 

surcharges (or maluses), and rewarding claim-free policyholders by awarding them 

discounts (or bonuses) are now in force in many developed countries. Besides encouraging 

policyholders to drive carefully (i.e. counteracting moral hazard), they aim to better 

assess individual risks. The amount of premium is adjusted each year on the basis of 

the individual claims experience using techniques from credibility theory, as shown in 

Chapter 3. 

However, credibility formulas are difficult to implement in practice, because of their 

mathematical complexity (complexity refers here to the sphere of commercial relations, 

where customers are often reluctant in using mechanisms that they consider to be complex, 

especially in connection with insurance products). For this reason, bonus-malus scales have 

been proposed by insurance companies. Such scales have to be seen as commercial versions 

of credibility formulas. The typical customer can figure out what the premium will be for 

any given claims history. 





4.1.2 The Nature of Bonus-Malus Scales 

When a merit rating plan is in force, the amount of premium paid by the policyholder 

depends on the rating factors of the current period but also on claim history. In practice, a 

bonus-malus scale consists of a finite number of levels, each with its own relative premium. 

New policyholders have access to a specified level. After each year, the policy moves up 

or down according to transition rules and to the number of claims at fault. The premium 

charged to a policyholder is obtained by applying the relative premium associated to his 

current level in the scale to a base premium depending on his observable characteristics 

incorporated into the price list. 

4.1.3 Relativities 

The problem addressed in this chapter is the determination of the relative premiums attached 

to each of the levels of the scale when a priori classification is used by the company. The 

relativity associated to level l is denoted as r l . The meaning is that a policyholder occupying 

level l in the bonus-malus scale has to pay r l times the base premium to be covered by the 

insurance company. 

The severity of the a posteriori corrections must depend on the extent to which amounts 

of premiums vary according to observable characteristics of policyholders. The key idea 

is that both a priori classification and a posteriori corrections aim to create tariff cells 

as homogeneous as possible. The residual heterogeneity inside each of these cells being 

smaller for insurers incorporating more variables in their a priori ratemaking, the a posteriori 

corrections must be softer for those insurers. 

The framework of credibility theory, with its fundamental notion of randomly distributed 

risk parameters, was employed in analysis of bonus-malus systems by Pesonen as early as 

1963. In this chapter, we will keep the framework of Definition 3.1. According to Norberg 

(1976), once the number of classes, the starting level and the transition rules have been fixed, 

the optimal relativity associated with level l is determined by maximizing the asymptotic 

predictive accuracy. Formally, the relativities minimize the mean squared deviation between 

a policy’s expected claim frequency and its premium in the year t as t →+. The optimal 

relativity for level l is thus equal to the conditional expected risk parameter for an infinitely 

old policy, given that the policy is in level l. 

4.1.4 Bonus-Malus Scales and Markov Chains 

In most of the commercial bonus-malus systems, the knowledge of the current level and the 

number of claims during the current period suffice to determine the next level in the scale. 

So the future (the level for year t + 1) depends only on the present (the level for year t and 

the number of accidents reported during year t) and not on the past. This is closely related to 

the memoryless property of the Markov chains. If the claim numbers in different years are 

(conditionally) independent then the trajectory of a given policyholder in the bonus-malus 

scale will be a (conditional) Markov chain. Sometimes, fictitious levels have to be introduced 

to recover the memoryless property. 

The treatment of bonus-malus scales is best performed in the framework of Markov 

chains. This chapter is nevertheless self-contained, and does not require any prior knowledge

Bonus-Malus Scales 167 

of this topic. All the useful results have been derived in an elementary way (the readers 

having acquaintance with the theory of Markov chains will rapidly recognize all the classical 

machinery taught in textbooks devoted to stochastic processes). 


Exactly as for credibility mechanisms, it is important that the relativities average to 100 %, 

resulting in financial equilibrium. This fundamental property is highly desirable: it guarantees 

that the introduction of a bonus-malus system has no impact on the yearly premium collection. 

The distribution of the amounts paid by the policyholders is modified according to the 

reported claims but on the whole, the company gets the same amount of money. 

Throughout this chapter, we work with the long-run equilibrium distribution of 

policyholders in the bonus-malus levels. We will see that in the long run, the way the 

relativities are computed in this chapter ensures that the bonus-malus system is financially 

stable. Things are however more complicated in practice. Specifically, some undesirable 

phenomena can arise in a transient regime. These issues will be addressed in Chapters 8–9. 

4.1.6 Agenda 

In Section 4.2, the trajectory of the policyholder accross the bonus-malus levels is modelled 

as a Markov chain. Section 4.3 is devoted to transition probabilities, that is, the probability 

that the policyholder moves from one level to another over a given time horizon. The longterm 

behaviour of the scale is studied in Section 4.4. It is shown there that the proportions 

of policyholders in each level of the scale tend to stabilize over time. Various methods to 

compute the stationary probabilities are described. 

Section 4.5 explains how to compute the relativities using a quadratic loss function. As 

for credibility formulas, relativities that are linear in the bonus-malus level are also derived. 

Section 4.5.3 examines the interaction between the bonus-malus scale and a priori risk 

classification. It is shown there that creating several scales decreases the rating inadequacies. 

In Section 4.6, the quadratic loss function is replaced with an exponential one. A 

comparison with quadratic relativities is performed, and the influence of the severity 

parameter is carefully assessed. 

In Section 4.7, we will consider the so-called special bonus rule. According to this rule, 

a policyholder who did not report any claim for a certain number of years, and is still in the 

malus zone (i.e. in a level with a relativity above 100 %) is automatically sent to the initial 

level (i.e. to the level with relativity equal to 100 %). Many compulsory systems formerly 

imposed by governments included such a rule. Because of this special rule, the stochastic 

process describing the trajectory of the drivers accross the levels is no longer Markovian. 

The memoryless property can nevertheless be re-obtained by adding fictitious levels in the 

scale. To fix the ideas, we will study the special bonus rule in the former compulsory Belgian 

system. 

In a competitive market, it can be expected that some policyholders switch from one 

insurance company to the other. In a regulated framework, with a unique compulsory bonusmalus 

system imposed on all the insurance companies, the drivers will be subject to the 

same a posteriori corrections whatever the insurer. If a driver decides to switch to another


insurer, he must first obtain a certificate from the former insurer stating his attained bonusmalus 

level and whether pending claims could affect this level. The new insurer must then 

award the same discount or apply the same surcharges. The competition between insurance 

companies is limited to the services offered and the a priori premiums. 

Things become more complicated in a deregulated market, where each insurer is free to 

design its own bonus-malus system. Then, insurers compete also on the basis of a posteriori 

corrections. It rapidly becomes extremely difficult for policyholders to determine the optimal 

insurance provider, since companies apply different penalties when claims are reported. In 

Section 4.8, we consider a policyholder switching from insurer A to insurer B. He occupies 

the level l 1 in the bonus-malus scale used by company A, and the question is where to place 

him in the bonus-malus scale used by company B. 

Section 4.9 examines the dependence properties existing between the successive levels 

occupied by the policyholders and the random risk parameter. It is argued that contrarily to 

the results obtained with credibility models, the risk parameters do not necessarily increase 

with the level occupied in the scale. 

The final Section 4.10 gives references and addresses further issues. 

4.2 Modelling Bonus-Malus Systems 

4.2.1 Typical Bonus-Malus Scales 

Before embarking on an abstract definition of bonus-malus systems, let us discuss a couple 

of examples that will be used throughout this chapter. 

Example 4.1 (−1/Top Scale) This bonus-malus scale has 6 levels (numbered 0 to 5). 

Policyholders are classified according to the number of claim-free years since their last claim 

(0, 1, 2, 3, 4 or at least 5). After a claim all premium reductions are lost. The transition rules 

are described in Table 4.1. Specifically, the starting class is the highest level 5. Each claimfree 

year is rewarded by one bonus class. When an accident is reported, all the discounts are 

lost and the policyholder is transferred to level 5. 

Note that the philosophy behind such a bonus-malus system is different from credibility 

theory. Indeed, this bonus-malus scale only aims to counteract moral hazard: it is in fact 

more or less equivalent to a deductible which is not paid at once but smoothed over the time 

Table 4.1 Transition rules for 

the scale −1/top. 

Starting Level occupied if 

level 0 ≥ 1 

claim is reported 

0 0 5 

1 0 5 

2 1 5 

3 2 5 

4 3 5 

5 4 5


needed to go back to the lowest class. Note however that this ‘smoothed’ deductible only 

applies to the first claim: subsequent claims are ‘for free’. 

Example 4.2 (−1/+2 Scale) There are 6 levels. Level 5 is the starting level. A higher 

level number indicates a higher premium. The discount per claim-free year is one level: if no 

claims have been reported by the policyholder then he moves one level down. The penalty 

per claim is two levels. If a number of claims, n t > 0, has been reported during year t then 

the policyholder moves 2n t levels up. The transition rules are described in Table 4.2. 

In the subsequent sections, we will also make the −1/+2 scale more severe, by penalizing 

each claim by 3 levels instead of 2. This alternative bonus-malus system will be henceforth 

referred to as the −1/ + 3 system. 

4.2.2 Characteristics of Bonus-Malus Scales 

The bonus-malus scales investigated in this book are assumed to possess s + 1 levels, 

numbered from 0 to s. A specified level is assigned to a new driver. In practice, the initial 

level may depend upon the use of the vehicle (or upon another observable risk characteristic). 

Each claim-free year is rewarded by a bonus point (i.e. the driver goes one level down). 

Claims are penalized by malus points (i.e. the driver goes up a certain number of levels 

each time he files a claim). We assume that the penalty is a given number of classes per 

claim. This facilitates the mathematical treatment of the problem. More general systems can 

nevertheless be considered, with higher penalties for subsequent claims. After sufficiently 

many claim-free years, the driver enters level 0 where he enjoys the maximal bonus. 

In Chapter 3, updating premiums with credibility formulas only uses the total number 

of claims reported by the policyholder in the past. The new premium does not depend on 

the way the accidents are distributed over the years. This property is never satisfied by 

bonus-malus systems, where it would be the policyholder’s interest to concentrate all the 

claims during a single year. 

In commercial bonus-malus systems, the knowledge of the present level and of the 

number of claims of the present year suffices to determine the next level. Together with 

the (conditional) independence of annual claim numbers, this ensures that the trajectory 

accross the bonus-malus levels may be represented by a (conditional) Markov chain: the 

Table 4.2 

Transition rules for the scale −1/+2. 

Starting 

Level occupied if 

level 0 1 2 ≥3 

claim(s) is/are reported 

5 4 5 5 5 

4 3 5 5 5 

3 2 5 5 5 

2 1 4 5 5 

1 0 3 5 5 

0 0 2 4 5


future (the level for year t + 1) depends on the present (the level for year t and the number 

of accidents reported during year t) and not on the past (the complete claim history and 

the levels occupied during years 1 2t− 1). Sometimes, fictitious levels have to be 

introduced in order to meet this memoryless property. Indeed, in some bonus-malus systems, 

policyholders occupying high levels are sent to the starting class after a few years without 

claims. This issue will be addressed in Section 4.7. 

4.2.3 Trajectory 

New drivers start in level l 0 of the scale. Note that experienced drivers arriving in the 

portfolio are not necessarily placed in level l 0 , but in a level corresponding to their claim 

history or to the level occupied in the bonus-malus scale used by a competitor. This problem 

will be dealt with in Section 4.8. 

The trajectory of the policyholder in the bonus-malus scale is modelled by a sequence 

L 1 L 2 of random variables valued in 0 1s, such that L k is the level occupied 

during the k + 1th year, i.e. during the time interval k k + 1. Since movements in the 

scale occur once a year (at policy anniversary), the policyholder occupies level L k from 

time k until time k + 1. Once the number N k of claims reported by the policyholder during 

k − 1k is known, this information is used to reevaluate the position of the driver in the 

scale. We supplement the sequence of the L k s with L 0 = l 0 . 

The L k s obviously depend on the past numbers of claims N 1 N 2 N k reported by 

the policyholder. If we denote as ‘pen’ the penalty induced by each claim (expressed as a 

number of levels), then the L k s obey the recursion 

{ maxLk−1 − 1 0 if N 

L k = 

k = 0 

minL k−1 + N k × pens if N k ≥ 1 

{ 

= max min { L k−1 − 1 − I k + N k × pens } } 

0 

where 

{ 1ifNk ≥ 1 

I k = 

0 otherwise 

indicates whether at least one claim has been reported in year k. This is an example of a 

stochastic recursive equation. This representation of the L k s clearly shows that the future 

trajectory of the policyholder in the scale is independent of the levels occupied in the past, 

provided that the present level is given. This conditional independence property is at the 

heart of Markov models. 

The stochastic recursive equations given above assume that the bonus is lost in case at 

least one claim is filed with the company. In some cases (like the former compulsory Belgian 

bonus-malus scale), the bonus is granted in any case. The L k s then obey the recursion 

{ 

L k = max min { L k−1 − 1 + N k × pens } } 

0 

This means that the first claim is penalized by pen−1 levels, and the subsequent ones by 

pen levels.


4.2.4 Transition Rules 

The probability of moving from one level to another depends on the number of claims 

reported during the current year. Therefore, we can introduce more formally the transition 

rules which impose the transfer from one level to another level once the number of claims 

is known. If k claims are reported, 

{ 1 if the policy gets transferred from level i to level j, 

t ij k = 

0 otherwise. 

The t ij ks are put in matrix form Tk, i.e. 

⎛ 

t 00 k t 01 k ··· t 0s k 

t 10 k t 11 k ··· t 1s k 

Tk = ⎜ 

⎝ 

 

 

t s0 k t s1 k ··· t ss k 

⎞ 

⎟ 

⎠ 

Then, Tk is a 0-1 matrix having in each row exactly one 1. 

Example 4.3 (−1/Top Scale) 

⎛ 

T0 = 

⎜ 

⎝ 

1 0 0 0 0 0 

1 0 0 0 0 0 

0 1 0 0 0 0 

0 0 1 0 0 0 

0 0 0 1 0 0 

0 0 0 0 1 0 

and Tk = T1 for all k ≥ 2. 

In this case, we have 

⎞ ⎛ 

T1 = 

⎟ ⎜ 

⎠ ⎝ 

0 0 0 0 0 1 

0 0 0 0 0 1 

0 0 0 0 0 1 

0 0 0 0 0 1 

0 0 0 0 0 1 

0 0 0 0 0 1 

⎞ 

⎟ 

⎠ 

Example 4.4 (−1/+2 Scale) 

⎛ 

T0 = 

⎜ 

⎝ 

1 0 0 0 0 0 

1 0 0 0 0 0 

0 1 0 0 0 0 

0 0 1 0 0 0 

0 0 0 1 0 0 

0 0 0 0 1 0 

In this case, we have 

⎞ ⎛ 

T1 = 

⎟ ⎜ 

⎠ ⎝ 

0 0 1 0 0 0 

0 0 0 1 0 0 

0 0 0 0 1 0 

0 0 0 0 0 1 

0 0 0 0 0 1 

0 0 0 0 0 1 

⎞ 

⎟ 

⎠ 

⎛ 

T2 = 

⎜ 

⎝ 

0 0 0 0 1 0 

0 0 0 0 0 1 

0 0 0 0 0 1 

0 0 0 0 0 1 

0 0 0 0 0 1 

0 0 0 0 0 1 

⎞ 

⎛ 

and Tk = 

⎟ 

⎜ 

⎠ 

⎝ 

0 0 0 0 0 1 

0 0 0 0 0 1 

0 0 0 0 0 1 

0 0 0 0 0 1 

0 0 0 0 0 1 

0 0 0 0 0 1 

⎞ 

for all k ≥ 3 

⎟ 

⎠


4.3 Transition Probabilities 

4.3.1 Definition 

Let us now assume that N 1 N 2 are independent and oi distributed. The trajectory 

will be denoted as L 1 L 2 to emphasize the dependence upon the annual 

expected claim frequency . Note however that the argument in L k does not mean 

that the L k s are functions of the parameter , but only that their distribution depends 

on . 

Let p l1 l 2 

be the probability of moving from level l 1 to level l 2 for a policyholder with 

annual mean claim frequency , that is, 

p l1 l 2 

= PrL k+1 = l 2 L k = l 1 

with l 1 l 2 ∈ 0 1s. Clearly, the p l1 l 2 

s satisfy 

p l1 l 2 

≥ 0 for all l 1 and l 2 , and 

s∑ 

p l1 l 2 

= 1 (4.1) 

l 2 =0 

Moreover, the transition probabilities can be expressed using the t ij ·s introduced above. 

To see this, it suffices to write 

p l1 l 2 

= 

= 

+∑ 

n=0 

∑ 

n=0 

PrL k+1 = l 2 N k+1 = n L k = l 1 PrN k+1 = nL k = l 1 

n 

n! exp −t l 1 l 2 

n 

Note that we have used the fact that N k+1 and L k are independent (since L k depends 

on N 1 N k ), so that 

PrN k+1 = nL k = l 1 = PrN k+1 = n = n 

exp − 

n! 

The transition probabilities allow the actuary to compute the probability of any trajectory 

in the scale. Specifically, since the probability that a certain policyholder with expected 

annual claim frequency is in level l 1 l n at time 1n is simply the probability 

of going from l 0 to l n via the intermediate levels l 1 l n−1 , we have 

PrL 1 = l 1 L n = l n L 0 = l 0 = p l0 l 1 

···p ln−1 l n 

(4.2) 

Furthermore, it is enough to know the current position in the scale to determine the probability 

of being transferred to any other level in the bonus-malus scale. Formally, 

PrL n = l n L n−1 = l n−1 L 0 = l 0 = p ln−1 l n 

 

whenever PrL n−1 = l n−1 L 0 = l 0 >0.


4.3.2 Transition Matrix 

Further, P is the one-step transition matrix, i.e. 

⎛ 

p 00 p 01 ··· p 0s 

p 10 p 11 ··· p 1s 

P = ⎜ 

⎝ 

 

 

p s0 p s1 ··· p ss 

⎞ 

⎟ 

⎠ 

From (4.1), we see that the matrix P is a stochastic matrix. As already mentioned, the 

future level of a policyholder is independent of its past levels and only depends on its present 

level (and also on the number of claims reported during the present year). 

In matrix form, we can write P as 

P = 

∑ 

k=0 

k 

exp −Tk 

k! 

provided the N t s are independent and oi distributed. 


system is given by 

⎛ 

P = 

⎜ 

⎝ 

The transition matrix P associated with this bonus-malus 

exp− 0 0 0 0 1− exp− 

exp− 0 0 0 0 1− exp− 

0 exp− 0 0 0 1− exp− 

0 0 exp− 0 0 1− exp− 

0 0 0 exp− 0 1− exp− 

0 0 0 0 exp− 1 − exp− 


system is given by 

⎞ 

 

⎟ 

⎠ 

The transition matrix P associated with this bonus-malus 

⎛ 

 

exp− 0 exp− 0 

2 

exp− 1 − ⎞ 

2 1 

⎜ exp− 0 0 exp− 0 1− 2 

⎜⎜⎜⎜⎝ 0 exp− 0 0 exp− 1 − P = 

3 

0 0 exp− 0 0 1− exp− 

 

⎟ 

0 0 0 exp− 0 1− exp− ⎠ 

0 0 0 0 exp− 1 − exp− 

where i represents the sum of the elements in columns 1 to 5 in row i, i = 1 2 3, that is, 

) 

1 = exp− 

(1 + + 2 

2 

2 = 3 = exp− 1 +


4.3.3 Multi-Step Transition Probabilities 

The probability 

p n 

ij = PrL k+n = jL k = i 

evaluates the likelihood of being transferred from level i to level j in n steps. Note that this 

is the probability that L k+n = j given L k = i for any k. The process describing the 

trajectory of the policyholder accross the levels is thus stationary. From 

p n 

ij = 

s∑ 

i 1 =0 i 2 =0 

s∑ 

 

s∑ 

i n−1 =0 

p ii1 p i1 i 2 

···p in−1 j 

we clearly see that it includes all the possible paths from i to j and the probability of their 

occurrence. This is the n-step transition probability p n 

ij . Therefore, the matrix 

⎛ 

P n = 

⎜ 

⎝ 

p n 

00 

p n 

10 

pn 01 

pn 11 

 

p n 

s0 

··· pn 0s 

⎞ 

··· pn 1s 

 

 

⎟ 

⎠ 

pn s1 ··· pn 

ss 

is called the n-step transition matrix corresponding to P. 

The following result shows that P n is a stochastic matrix, being the nth power of the 

one-step transition matrix P. 

Property 4.1 

For all n m = 0 1, 

P n = P n (4.3) 

and hence, 

P n+m = P n P m (4.4) 

Proof The proof is by induction on n. The result is obviously true for n = 1. Assume it 

holds for n and let us show that it is still true for n+1. Clearly, by conditioning on the level 

l occupied at time n we get 

p n+1 

ij = 

s∑ 

l=0 

p n 

il p lj (4.5) 

which corresponds to matrix multiplication. This proves (4.3), from which (4.4) readily 

follows. 

□ 

The matrix identity (4.4) is usually called the Chapman Kolmogorov equation. Taking 

the nth power of P yields the n-step transition matrix whose element l 1 l 2 , denoted


as p n 

l 1 l 2 

, is the probability of moving from level l 1 to level l 2 in n transitions. Using 

Property 4.1, we get the following representation for the distribution of the state variable L n : 

denoting as 

we have 

p k = ( PrL k = 0PrL k = s ) T 

 

p k+n = p k P n (4.6) 

Remark 4.1 (Numerical Aspects) The computation of the probability distribution of L n 

amounts to calculating the nth power of the transition matrix P. For large values of n, this 

may pose some computational difficulties. This is why we now discuss an algebraic method 

which makes use of the concept of eigenvalues and eigenvectors (that will be encountered 

further in this chapter). 

The vector v with at least one component different from 0 is a right eigenvector of P if 

Pv = v for some ∈ . In this case, is said to be an eigenvalue of P. Finding the 

eigenvalues of P amounts to solving the characteristic equation detP − I = 0. A 

nonzero vector u that is a solution of u T P = u T is called a left eigenvector corresponding 

to . 

In general, the characteristic equation possesses s + 1 solutions 0 s which can be 

complex and some of them can coincide (we assume that the eigenvalues are numbered so 

that 0 ≥ 1 ≥···≥ s ). The Perron-Froebenius theorem for regular matrices ensures 

that 

∑ 

provided the transition matrix P is regular then 0 = 1, v 0 = e, u 0 ≥ 0 with u T e = 

s 

j=0 u 0j = 1. Moreover, all the other eigenvalues of P lie inside the unit circle of the 

complex plane, that is j < 1 for j = 1s. 

Let V = v 0 v s be an s +1×s +1 matrix consisting of right column eigenvectors 

and 

⎛ 

⎜ 

U = ⎝ 

an s + 1 × s + 1 matrix consisting of left column eigenvectors. Let us assume that the 

eigenvalues 0 s are distinct. This ensures that v 0 v s are linearly independent so 

that V is invertible. Moreover, U = V −1 . In this case, P can be represented as 

⎛ 

⎞ 

0 ··· 0 

⎜ 

P = V 

⎝ 

 

 

⎟ 

s∑ 

⎠ U = j v j u T j 

j=0 

0 ··· s 

u T 0 

 

u T s+1 

This representation is useful for computing the nth power of P in that 

⎛ 

⎞ 

n 0 

··· 0 

P n ⎜ 

= V ⎝ 

 

 

⎟ 

s∑ 

⎠ U = n j v ju T j 

0 ··· n j=0 

s 

⎞ 

⎟ 

⎠


4.3.4 Ergodicity and Regular Transition Matrix 

A Markov chain with transition matrix P is said to be ergodic if P is regular, that is, if there 

exists some n 0 ≥ 1 such that all entries of P n 0 are strictly positive. This condition means 

that it is possible, with a strictly positive probability, to go from one level i to another level 

j in a finite number of transitions or, in other words, that all states of the Markov chain are 

accessible from any initial state in a finite number of steps. 

All bonus-malus scales in practical use have a ‘best’ level, with the property that a policy 

in that level remains in the same level after a claim-free period. In our framework, the best 

level is level 0 and, for any level, it is possible to reach the superbonus level 0 after a 

sufficiently large number of claim-free years, resulting in p n 

l0 

> 0 for all sufficiently 

large n. In the following, we restrict our attention to such non-periodic bonus rules. The 

transition matrix P associated with such a bonus-malus scale is regular, i.e. there exists 

some integer n 0 ≥ 1 such that all entries of the n 0 th power P n 0 of the one-step transition 

matrix are strictly positive. 

4.4 Long-Term Behaviour of Bonus-Malus Systems 

4.4.1 Stationary Distribution 

A natural question that arises concerns the long term behaviour of a bonus-malus system. 

Intuitively, we expect that the system will stabilize in the long run. Since the annual claim 

numbers have been assumed to be independent and identically distributed, each policyholder 

will ultimately stabilize around an equilibrium level correponding to the expected annual 

claim frequency , and will gravitate around this level. 

To formalize this intuitive idea, let us compute the powers of the transition matrix P 

for = 01 inthe−1/top and −1/ + 2 bonus-malus scales. This is done in the following 

examples. 


⎛ 

P01 = 

⎜ 

⎝ 

we get 

⎛ 

P 2 01 = 

⎜ 

⎝ 

Starting from 

0904837 0 0 0 0 0095163 

0904837 0 0 0 0 0095163 

0 0904837 0 0 0 0095163 

0 0 0904837 0 0 0095163 

0 0 0 0904837 0 0095163 

0 0 0 0 0904837 0095163 

0818731 0 0 0 0086107 0095163 

0818731 0 0 0 0086107 0095163 

0818731 0 0 0 0086107 0095163 

0 0818731 0 0 0086107 0095163 

0 0 0818731 0 0086107 0095163 

0 0 0 0818731 0086107 0095163 

⎞ 

⎟ 

⎠ 

⎞ 

 

⎟ 

⎠


⎛ 

P 3 01 = 

⎜ 

⎝ 

0740818 0 0 0077913 0086107 0095163 

0740818 0 0 0077913 0086107 0095163 

0740818 0 0 0077913 0086107 0095163 

0740818 0 0 0077913 0086107 0095163 

0 0740818 0 0077913 0086107 0095163 

0 0 0740818 0077913 0086107 0095163 

⎞ 

 

⎟ 

⎠ 

and 

⎛ 

P 4 01 = 

⎜ 

⎝ 

⎛ 

P 5 01 = 

⎜ 

⎝ 

067032 000000 0070498 0077913 0086107 0095163 

067032 000000 0070498 0077913 0086107 0095163 

067032 000000 0070498 0077913 0086107 0095163 

067032 000000 0070498 0077913 0086107 0095163 

067032 000000 0070498 0077913 0086107 0095163 

000000 067032 0070498 0077913 0086107 0095163 

0606531 0063789 0070498 0077913 0086107 0095163 

0606531 0063789 0070498 0077913 0086107 0095163 

0606531 0063789 0070498 0077913 0086107 0095163 

0606531 0063789 0070498 0077913 0086107 0095163 

0606531 0063789 0070498 0077913 0086107 0095163 

0606531 0063789 0070498 0077913 0086107 0095163 

where all the rows are identical. Of course, 

⎞ 

 

⎟ 

⎠ 

⎞ 

 

⎟ 

⎠ 

P k 01 = P 5 01 for any integer k ≥ 6 

This means that, whatever the initial distribution, 

p k 01 = 0606531 0063789 0070498 0077913 0086107 0095163 T 

for any k ≥ 5. The proportion of policyholders occupying each of the levels of the −1/top 

scale thus remains unchanged after 5 years. 


⎛ 

P01 = 

⎜ 

⎝ 

In this case, 

0904837 0 0090484 0 0004524 0000155 

0904837 0 0 0090484 0 0004679 

0 0904837 0 0 0090484 0004679 

0 0 0904837 0 0 0095163 

0 0 0 0904837 0 0095163 

0 0 0 0 0904837 0095163 

⎞ 

 

⎟ 

⎠


The convergence is now much slower: 

⎛ 

P 5 01 = 

⎜ 

⎝ 

0791523 0081985 0088694 0018169 0015456 0004173 

0788490 0066822 0106890 0017259 0014546 0005992 

0788490 0063789 0073531 0053651 0013637 0006902 

0606531 0245749 0070498 0020292 0050028 0006902 

0606531 0063789 0252457 0017259 0034865 0025098 

0606531 0063789 0070498 0199219 0031833 0028131 

⎞ 

⎟ 

⎠ 

⎛ 

P 10 01 = 

⎜ 

⎝ 

0784013 0081747 0090966 0022022 0016217 0005037 

0784003 0081480 0090248 0023009 0016178 0005081 

0777382 0088092 0089871 0022071 0017497 0005087 

0776278 0079263 0099795 0021694 0016890 0006080 

0776278 0078160 0090966 0031618 0016623 0006356 

0743169 0111269 0089862 0026100 0023236 0006365 

⎞ 

⎟ 

⎠ 

⎛ 

P 20 01 = 

⎜ 

⎝ 

which slowly converges to 

⎛ 

01 = 

⎜ 

⎝ 

0782907 0082338 0090996 0022276 0016387 0005096 

0782903 0082332 0091006 0022275 0016387 0005097 

0782902 0082326 0090993 0022295 0016386 0005098 

0782803 0082424 0090984 0022285 0016406 0005098 

0782776 0082352 0091082 0022278 0016403 0005108 

0782774 0082327 0091011 0022376 0016399 0005113 

0782901 0082338 0090998 0022278 0016387 0005097 

0782901 0082338 0090998 0022278 0016387 0005097 

0782901 0082338 0090998 0022278 0016387 0005097 

0782901 0082338 0090998 0022278 0016387 0005097 

0782901 0082338 0090998 0022278 0016387 0005097 

0782901 0082338 0090998 0022278 0016387 0005097 

In this case, the system is not stable after 20 years. 

Let us consider the trajectory of a policyholder with expected claim frequency 

accross the levels of the bonus-malus scale. We define the stationary distribution = 

0 1 s T as follows: l is the stationary probability for a policyholder 

with mean frequency to be in level l i.e. 

l2 

= lim 

n→+ pn l 1 l 2 

 

The term l is the limit value of the probability that the policyholder is in level l, when 

the number of periods tends to +. It is also the fraction of the time a policyholder with 

claim frequency spends in level l, once the steady state has been reached. Note that 

⎞ 

⎟ 

⎠ 

⎞ 

 

⎟ 

⎠


does not depend on the starting class. This means that the nth power P n of the one-step 

transition matrix P converges to a matrix with all the same rows T , that is 

⎛ 

lim 

n→+ Pn = = ⎜ 

⎝ 

T 

T 

 

T 

exactly as we saw in the introductory examples. 

Let us now explain how to compute the l s. Taking the limit in both sides of (4.5) 

for n →+, we see that the vector is the unique probabilistic solution to the system 

of linear equations 

j = 

⎞ 

⎟ 

⎠ 

s∑ 

l p lj j ∈ 0s (4.7) 

l=0 

In matrix notation, (4.7) can be written as 

{ T = T P 

T e = 1 

(4.8) 

where e is a column vector of 1s. This means that is the left eigenvector u 0 of P 

encountered above. Thus we see that if the initial distribution in the scale is , then the 

probability distribution remains equal to . 

4.4.2 Rolski–Schmidli–Schmidt–Teugels Formula 

Let E be the s +1×s +1 matrix all of whose entries are 1, i.e. consisting of s +1 column 

vectors e. Then, the following result provides a direct method to get . 

Property 4.2 Assume that the stochastic matrix P is regular. Then the matrix I −P+ 

E is invertible and the solution of (4.8) is given by 

T = e T I − P + E −1 (4.9) 


Let us first check that I − P + E is invertible. We must show that 

I − P + Ex = 0 ⇒ x = 0 

From (4.8), we have T I − P = 0. Thus, 

I − P + Ex = 0 ⇒ 0 = T I − P + Ex = 0 + T Ex 

⇔ T Ex = 0


On the other hand, T E = e T . Thus, e T x = 0, which implies Ex = 0. Consequently, 

This implies for any n ≥ 1 that 

I − Px = 0 ⇔ Px = x 

x = P n x → x 

i.e. x i = ∑ s 

j=0 jx j for all i = 0s. Because the right-hand side of these equations 

does not depend on i, we have x = ce for some c ∈ . Since we also have 

0 = e T x = ce T e = cs + 1 ⇒ c = 0 

Thus, I − P + E is invertible. Furthermore, since T I − P = 0, we have 

T I − P + E = T E = e T 

This proves (4.9). 

□ 

If the number s + 1 of states is small, the matrix I − P + E can easily be inverted. For 

larger s + 1, numerical methods have to be used, like the Gaussian elimination algorithm. 

Example 4.9 (−1/top scale) We have seen above that p 5 

5l = l for l = 0 15, 

so that the system needs 5 years to reach stationarity (i.e. the time needed by the best 

policyholders starting from level 5 to arrive in level 0). Formula (4.9) gives here 

T = 1 1 1 1 1 1 

⎛ 

× 

⎜ 

⎝ 

2 − exp− 1 1 1 1 exp− 

1 − exp− 2 1 1 1 exp− 

1 1− exp− 2 1 1 exp− 

1 1 1− exp− 2 1 exp− 

1 1 1 1− exp− 2 exp− 

1 1 1 1 1− exp− 1 + exp− 

Applying this formula in the particular case = 01, we get 

T 01 = 1 1 1 1 1 1 

⎛ 

× 

⎜ 

⎝ 

1095163 1 1 1 1 0904837 

0095163 2 1 1 1 0904837 

1 0095163 2 1 1 0904837 

1 1 0095163 2 1 0904837 

1 1 1 0095163 2 0904837 

1 1 1 1 0095163 1904837 

⎞ 

⎟ 

⎠ 

−1 

⎞ 

⎟ 

⎠ 

 

−1


= 1 1 1 1 1 1 

⎛ 

2305672 −0678486 −0565648 −0440943 −0303122 −0150806 

1305672 0321514 −0565648 −0440943 −0303122 −0150806 

× 

0400834 0226351 0434352 −0440943 −0303122 −0150806 

⎜ −0417897 0140245 0339189 0559057 −0303122 −0150806 

⎝ −1158715 0062332 0253083 0463895 0696878 −0150806 

−1829035 −0008166 0175170 0377788 0601716 0849194 

= 0606531 0063789 0070498 0077913 0086107 0095163. 

Coming back to Example 4.7, we see that the matrices P k 01, k ≥ 5, have all their rows 

equal to T 01. 

⎞ 

 

⎟ 

⎠ 

Example 4.10 (−1/+2 scale) 

Formula (4.9) gives here 

T = 1 1 1 1 1 1 

⎛ 

( ) 

2 − exp− 1 1− exp− 1 1− 2 

exp−exp− 1 + + 2 

2 2 

1 − exp− 2 1 1− exp− 1 exp−1 + 

× 

1 1− exp− 2 1 1− exp− exp−1 + 

⎜ 1 1 1− exp− 2 1 exp− 

⎝ 1 1 1 1− exp− 2 exp− 

1 1 1 1 1− exp− 1 + exp− 

⎞ 

−1 

 

⎟ 

⎠ 

Applying this formula in the particular case = 01, we get 

T 01 = 1 1 1 1 1 1 

⎛ 

⎞ 

1095163 1 0909516 1 0995476 0999845 

0095163 2 1 0909516 1 0995321 

× 

1 0095163 2 1 0909516 0995321 

⎜ 1 1 0095163 2 1 0904837 

⎟ 

⎝ 1 1 1 0095163 2 0904837 ⎠ 

1 1 1 1 0095163 1904837 

= 1 1 1 1 1 1 

−1 

⎛ 

2759072 −0630802 −0512948 −0658608 −0480599 −0309449 

1659534 0358730 −0524518 −0561440 −0472165 −0293474 

× 

0578107 0244995 0454957 −0475982 −0366345 −0269065 

⎜−0478699 0133850 0332122 0599117 −0272234 −0147489 

⎝−1434937 0033282 0220977 0571906 0812921 −0037482 

−2300176 −0057717 0120409 0547285 0794810 1062056 

⎞ 

⎟ 

⎠ 

= 0782901 0082338 0090998 0022278 0016387 0005097


4.4.3 Dufresne Algorithm 

Dufresne (1988,1995) proposed a simple and efficient iterative algorithm for deriving 

provided that the driver goes one level down if no claims are filed to the company, and goes 

n×pen levels up if n claims are reported to the insurer. Then, the move at the end of year k 

can be modelled as 

{ −1 if no claims 

k+1 = 

n × pen if n claims 

If the annual numbers of claims N 1 N 2 are independent and oi distributed then the 

sequence 1 2 is made up of independent and identically distributed random 

variables, with common probability mass function 

Pr k+1 =−1 = PrN k+1 = 0 = exp− 

Pr k+1 = n × pen = PrN k+1 = n = exp− n 

n! 

Pr k+1 = = 0 otherwise 

The level L k+1 can then be represented as 

for n = 1 2 

⎧ 

⎪⎨ L k + k+1 if 0 ≤ L k + k+1 ≤ s 

L k+1 = 0ifL k + k+1 =−1 

⎪⎩ 

s if L k + k+1 >s 

Let us denote as F k · the distribution function of L k , that is 

F k l = PrL k ≤ l l = 0 1s 

Furthermore, let us denote as p · the common probability mass function of the k s. We 

then have 

F k+1 l = 

l∑ 

F k l − yp y 

y=−1 

with F k+1 s = 1. The stationary distribution F · is then obtained as 

F l = lim F kl = 

k→+ 

l∑ 

F l − yp y 

with F s = 1. Obviously, the l s are then recovered from 0 = F 0 and 

y=−1 

l = F l − F l − 1 for l = 1s 

The values of F l can be computed recursively from the following algorithm:


(i) set A0 = 1 

(ii) compute for l = 0 1s− 1 

(iii) set 

Al + 1 = 

( 

1 

Al − 

p −1 

F l = Al 

As 

) 

l∑ 

Al − yp y 

y=0 

for l = 0 1s 

This recursive formula is computationally efficient, and easy to implement. 

4.4.4 Convergence to the Stationary Distribution 

Geometric Bound for the Speed of Convergence 

What matters for a unique limit to exist is that one can find a positive integer n 0 such 

that 

= 

min 

ij∈0s 

p n 0 

ij > 0 

In other words, n 0 is such that all the entries of P n 0 are positive, and thus it is possible 

to reach any level starting from any other level in n 0 periods. Various inequalities indicate 

the speed of convergence to the limit distribution . It can be shown that 

p n 

ij − j ≤ 

max 

i∈0s 

p n 

ij − 

min 

i∈0s 

p n 

ij ≤ ( 1 − )⌊ ⌋ 

n 

n0 −1 

 

This inequality provides us with a geometric bound for the rate of convergence to the limit 

distribution. Further bounds can be determined using concepts of matrix algebra. 

Total Variation Distance 

The total variation metric is often used to measure the distance to the stationary distribution 

. Recall that the total variation distance between two random variables X and Y , denoted 

as d TV X Y, is given by 

d TV X Y = 

∫ + 

− 

dF X t − dF Y t (4.10) 

For counting random variables M and N , (4.10) obviously reduces to 

d TV M N = 

+∑ 

k=0 

∣ PrM = k − PrN = k ∣ ∣ 

There is a close connection between d TV and the standard variation distance, which 

considers the supremum of the difference between the probability masses given to some


random events. Specifically, given two random variables X and Y , d TV X Y can be 

represented as 

∣ 

d TV X Y = 2 sup ∣ PrX ∈ A − PrY ∈ A∣ (4.11) 

A 

Selection of the Initial Level 

The main objective of a bonus-malus system is to correct the inadequacies of a priori rating 

by separating the good from the bad drivers. This separation process should proceed as fast 

as possible; the time needed to achieve this operation is the time needed to reach stationarity. 

A convenient way to select the initial level has been suggested by Bonsdorff (1992). The 

idea is to select it in order to minimize the time needed to reach stationarity. It relies on 

the total variation distance d TV between the nth transient distribution starting from level l 1 , 

i.e. p n 

l 1 l 2 

l 2 = 0 1s, and the stationary distribution l2 

l 2 = 0 1s, 

computed as 

d TV l 1 n= 

s∑ 

l 2 =0 

p n 

l 1 l 2 

− l2 

 

it measures the degree of convergence of the system after n transitions. Of course, 

lim 

n→+ d TV l 1 n= 0 for all l 1 and 

The convergence to is essentially controlled by the second largest eigenvalue 1 . 

For any > 1 , there exists an a


4.5.2 Bayesian Relativities 

Predictive accuracy is a useful measure of the efficiency of a bonus-malus scale. The idea 

behind this notion is as follows: A bonus-malus scale is good at discriminating among the 

good and the bad drivers if the premium they pay is close to their ‘true’ premium. According 

to Norberg (1976), once the number of classes and the transition rules have been fixed, 

the optimal relativity r l associated with level l is determined by maximizing the asymptotic 

predictive accuracy. 

Let us pick at random a policyholder from the portfolio. Both the a priori expected 

claim frequency and relative risk parameter are random in this case. Let us denote as the 

(random) a priori expected claim frequency of this randomly selected policyholder, and as 

the residual effect of the risk factors not included in the ratemaking. The actual (unknown) 

annual expected claim frequency of this policyholder is then . Since the random effect 

represents residual effects of hidden covariates, the random variables and may 

reasonably be assumed to be mutually independent. Let w k be the weight of the kth risk 

class whose annual expected claim frequency is k . Clearly, Pr = k = w k . 

Let L be the level occupied by this randomly selected policyholder once the steady state 

has been reached. The distribution of L can be written as 

PrL = l = ∑ k 

∫ + 

w k l k dF (4.12) 

0 

Here, PrL = l represents the proportion of the policyholders in level l. 

Our aim is to minimize the expected squared difference between the ‘true’ relative premium 

and the relative premium r L applicable to this policyholder (after the steady state has been 

reached), i.e. the goal is to minimize 

The solution is given by 

] 

E 

[ − r L 2 = 

= 

s∑ 

∣ ] 

E 

[ − r l 2 ∣∣L = l PrL = l 

l=0 

∫ + 

s∑ 

l=0 

= ∑ k 

0 

w k 

∫ + 

0 

− r l 2 PrL = l = dF 

s∑ 

− r l 2 l k dF 

l=0 

r l = EL = l 

[ 

] 

∣ 

= E EL = l ∣L = l 

= ∑ k 

= ∑ k 

EL = l = k Pr = k L = l 

∫ + 

0 

PrL = l = = kw k 

dF 

PrL = l = k Pr = kL= l 

PrL = l


= 

∑k w k 

∫ + 

0 

l k dF 

∑k w k 

∫ + 

0 

l k dF (4.13) 

It is easily seen that 

[ ] 

Er L = E EL = E = 1 

resulting in financial equilibrium once steady state is reached. 

Remark 4.2 If the insurance company does not enforce any a priori ratemaking system, 

all the k s are equal to E = and (4.13) reduces to the formula 

r l = 

∫ + 

0 

l dF 

∫ + 

0 

l dF 

(4.14) 

that has been derived in Norberg (1976). 

The way a priori and a posteriori ratemakings interact is described by 

EL = l = ∑ k 

k Pr = k L = l 

= ∑ k 

= 

k 

PrL = l = k w k 

PrL = l 

∑k k w k 

∫ + 

0 

l k dF 

∑k w k 

∫ + 

0 

l k dF (4.15) 

If EL = l is indeed increasing in the level l, those policyholders who have been granted 

premium discounts at policy issuance (on the basis of their observable characteristics) will 

also be rewarded a posteriori (because they occupy the lowest levels of the bonus-malus 

scale). Conversely, the policyholders who have been penalized at policy issuance (because 

of their observable characteristics) will cluster in the highest bonus-malus levels and will 

consequently be penalized again. 

Example 4.11 (−1/Top Scale, Portfolio A) The results for the bonus-malus scale −1/top 

are displayed in Table 4.3. Specifically, the values in the third column are computed with 

the help of (4.14) with â = 0889 and ̂ = 01474. Those values were obtained in Section 1.6 

by fitting a Negative Binomial distribution to the portfolio observed claim frequencies given 

in Table 1.1. Integrations have been performed numerically with the QUAD procedure of 

SAS R /IML. The fourth column is based on (4.13) with â = 1065 and the ̂ k s listed in 

Table 2.7. 

Once the steady state has been reached, the majority of the policies (58.5 % of the 

portfolio) occupy level 0 and enjoy the maximum discount. The remaining 41.5 % of the 

portfolio are distributed over levels 1–5, with about 13 % in level 5 (those policyholders who 

just claimed). Concerning the relativities, the minimum percentage of 54.7 % when the a 

priori ratemaking is not recognized becomes 61.2 % when the relativities are adapted to the


Table 4.3 Numerical characteristics for the system −1/top and for Portfolio A. 

Level l PrL = l r l = EL = l 

without a priori 

ratemaking 

r l = EL = l 

with a priori 

ratemaking 

EL = l 

5 128 % 1973 % 1812% 163% 

4 97 % 1709 % 1599% 158% 

3 77 % 1507 % 1439% 155% 

2 62 % 1348 % 1313% 152% 

1 52 % 1220 % 1209% 150% 

0 585% 547% 612% 141% 

a priori risk classification. Similarly, the relativity attached to the highest level of 197.3 % 

gets reduced to 181.2 %. The severity of the a posteriori corrections is thus weaker once the 

a priori ratemaking is taken into account in the determination of the r l s. The last column 

of Table 4.3 indicates the extent to which a priori and a posteriori ratemakings interact. 

The numbers in this column are computed as (4.15). The average a priori expected claim 

frequency clearly increases with the level l occupied by the policyholder. 

Example 4.12 (−1/Top Scale, Portfolio B) The results for the bonus-malus scale −1/top 

are displayed in Table 4.4. We only give the relativities computed by taking into account 

the a priori risk classification that is taken from Table 2.16 with ̂ = 0677. We see that 

in Portfolio B, only 46.6 % of the policyholders occupy level 0. The relativities are now 

less dispersed, ranging from 70.6 % to 146.9 % (instead of 61.2 % to 181.2 %). Again, the 

last column indicates that a priori risk classification and a posteriori premium corrections 

interact. 

Example 4.13 (−1/+2 Scale, Portfolio A) Results are displayed in Table 4.5 which is 

the analogue of Table 4.3 for the bonus-malus scale −1/ + 2. The bonus-malus system is 

perhaps too soft since the vast majority of the portfolio (about 71 %) clusters in the super 

bonus level 0. The higher levels are occupied by a very small minority of drivers. Such a 

system does not really discriminate between good and bad drivers. Consequently, only those 

Table 4.4 Numerical characteristics for the system −1/top and for portfolio B. 

Level l PrL = l r l = EL = l EL = l 

with a priori 

ratemaking 

5 163 % 1469% 195% 

4 125 % 1293% 193% 

3 99 % 1172% 191% 

2 80 % 1080% 190% 

1 66 % 1007% 189% 

0 466% 706% 184%


Table 4.5 Numerical characteristics for the system −1/ + 2 and for Portfolio A. 



ratemaking 

r l = EL = l 

with a priori 

ratemaking 

EL = l 

5 44 % 3091 % 2714% 185% 

4 47 % 2414 % 2185% 171% 

3 44 % 2077 % 1925% 164% 

2 87 % 1429 % 1388% 153% 

1 71 % 1302 % 1286% 151% 

0 706% 624% 685% 142% 

policyholders in level 0 get some discount whereas occupancy of any level 1–5 implies some 

penalty. Again, the a posteriori corrections are softened when a priori risk classification is 

taken into account in the determination of the r l s. The comments made for the scale −1/top 

still apply to this bonus-malus scale. 

Example 4.14 (−1/+2 Scale, Portfolio B) Results are displayed in Table 4.6 which is the 

analogue of Table 4.4 for the bonus-malus scale −1/ + 2. The comparison with Portfolio A 

yields the same comments as before. 

Example 4.15 (−1/+3 Scale, Portfolio A) Let us now make the −1/ + 2 bonus-malus 

scale more severe: to this end, each claim is now penalized by 3 levels (instead of 2 in the 

−1/ + 2 system). The numerical results are displayed in Table 4.7. 

We see that less policyholders occupy level 0 (64.5 % compared with 70.6 % with the 

−1/+2 system), and that the upper levels are now more populated. Drivers in level 0 deserve 

more bonus compared to the −1/+2 system (they pay 57.8 % of the base premium compared 

to 62.4% in the non-segmented case, and 64.2 % compared to 68.5 % in the segmented case). 

Also, the maximal penalties get reduced when claims are more severely penalized. 

Example 4.16 (−1/+3 Scale, Portfolio B) Let us now consider Portfolio B where each 

claim is penalized by 3 levels (instead of 2 in the −1/ + 2 system). The numerical results 

Table 4.6 Numerical characteristics for the system −1/ + 2 and for Portfolio B. 


with a priori 

ratemaking 

EL = l 

5 54 % 2232% 204% 

4 60 % 1713% 199% 

3 58 % 1489% 196% 

2 115 % 1121% 191% 

1 93 % 1050% 190% 

0 620% 747% 185%


Table 4.7 Numerical characteristics for the system −1/ + 3 and for Portfolio A. 



ratemaking 

r l = EL = l 

with a priori 

ratemaking 

EL = l 

5 73 % 2571 % 2308 % 17.4 % 

4 59 % 2194 % 2009 % 16.7 % 

3 90 % 1517 % 1452 % 15.5 % 

2 73 % 1365 % 1331 % 15.2 % 

1 60 % 1240 % 1230 % 15.0 % 

0 645% 578% 642 % 14.1 % 

Table 4.8 Numerical characteristics for the system −1/ + 3 and for Portfolio B. 


with a priori 

ratemaking 

EL = l 

5 92 % 1840% 200% 

4 78 % 1566% 197% 

3 117 % 1174% 192% 

2 95 % 1086% 190% 

1 78 % 1016% 189% 

0 540% 720% 184% 

are displayed in Table 4.8. The comparison with Portfolio A shows that the relativities are 

much less dispersed here. 

4.5.3 Interaction between Bonus-Malus Systems and a Priori Ratemaking 

Since the relativities attached to the different levels are the same whatever the risk class 

to which the policyholders belong, those scales overpenalize a priori bad risks. Let us 

explain this phenomenon, put in evidence by Taylor (1997). Over time, policyholders will 

be distributed over the levels of the bonus-malus scale. Since their trajectory is a function 

of past claims history, policyholders with low a priori expected claim frequencies will 

tend to gravitate to the lowest levels of the scale. Conversely for individuals with high a 

priori expected claim frequencies. Consider for instance a policyholder with a high a priori 

expected claim frequency, a young male driver living in a urban area, say. This driver 

is expected to report many claims (this is precisely why he has been penalized a priori) 

and so to be transferred to the highest levels of the bonus-malus scale. On the contrary, a 

policyholder with a low a priori expected claim frequency, a middle-aged lady living in 

a rural area, say, is expected to report few claims and so to gravitate to the lowest levels 

of the scale. The level occupied by the policyholders in the bonus-malus scale can thus be 

partly explained by their observable characteristics included in the price list. It is thus fair to 

isolate that part of the information contained in the level occupied by the policyholder that


does not reflect observables characteristics. A posteriori corrections should be driven only 

by this part of the bonus-malus information. 

In credibility theory, we have seen that to the extent that good drivers are rewarded in 

their base premiums (through other rating variables) the size of the bonus they require for 

equity is reduced. This can be summarized as follows: 

• when a priori segmentation is used, then the severity of the a posteriori differentiation 

has to be lowered. 

• when both a priori and a posteriori ratemakings are used, the size of the bonus is always 

smaller for good drivers than for bad ones. 

If a single bonus-malus scale is applied to the entire portfolio, even if the relativites take a 

priori risk classification into account, the resulting ratemaking is unfair to a priori bad drivers. 

The bonuses and the maluses need to be functions of the drivers’ a priori characteristics, 

used in the price list. 

We know from credibility theory that the a posteriori corrections are functions of the a 

priori characteristics; see for instance (3.16). On the contrary, when a bonus-malus system 

is in force, the same a posteriori corrections apply to all policyholders, whatever their a 

priori expected claim frequency. This of course induces unfairness in the portfolio. 

In order to reduce the unfairness of the tariff, we could propose several bonus-malus scales, 

according to the a priori characteristics. The idea is to select a few a priori characteristics 

inducing large differences in expected claim frequencies (typically, those associated with the 

largest regression coefficients), and to build separate scales according to these characteristics. 

Example 4.17 (−1/+2 System, Portfolio A, with a Dichotomy Rural–Urban) Table 4.9 

describes such a system where the company differentiates policyholders according to the 

type of district where they live (urban or rural). People living in urban areas have higher a 

priori expected claim frequencies. Thus, they should be better rewarded when they do not file 

any claim and less penalized when they report accidents compared to people living in rural 

zones. This is indeed what we observe when we compare the relative premiums obtained for 

the system −1/ + 2: the maximal discount is 66.6 % for urban policyholders, compared to 

71.3 % for rural ones. Similarly, the highest penalty is 267.2 % for urbans against 280.4 % 

for rurals. 

Table 4.9 Relativities obtained by differentiating policyholders 

according to the type of district for the system −1/ + 2 and for 

Portfolio A. 

Level l Rural Urban 

5 2804 % 2672% 

4 2264 % 2143% 

3 2008 % 1879% 

2 1444 % 1353% 

1 1343 % 1249% 

0 713% 666%


4.5.4 Linear Relativities 

As demonstrated in the numerical examples with the −1/top, −1/+2 and −1/+3 scales, the 

relativities obtained above may exhibit a rather irregular pattern, and this may be undesirable 

for commercial purposes. It may therefore be interesting to smooth this scale in order to 

obtain relativities which are regularly increasing according to the level. As suggested by 

Gilde & Sundt (1989), a linear scale of the form r lin 

l 

= + l l = 0 1s could 

then be desirable. Then, Norberg’s maximum accuracy criterion becomes a constrained 

minimization: 

[ ] ] 

min E − r lin 

L 2 = min E 

[ − − L 2 (4.16) 

Setting the derivative of the objective function with respect to equal to 0 yields 

= E − EL 

Doing the same with the derivative of the objective function with respect to gives 

[ 

] 

0 = E L − − L 

= EL − EL − EL 2 

Replacing the value of with the expression found above, we get 

0 = EL − ELE + EL 2 − EL 2 

= CL − VL 

The solution of the optimization problem (4.16) is thus given by: 

= 

CL 

VL 

and = − 

CL 

EL (4.17) 

VL 

The linear relative premium scale is thus of the form 

where 

r lin 

l 

= 1 + 

CL 

l − EL 

VL 

CL = EL − EL 

= ∑ s∑ 

w k lE l k − EL 

k l=0 

= ∑ s∑ ∫ + 

w k l l k dF − EL 

k l=0 

0


PrL = l = ∑ ∫ + 

w k l k dF 

k 

0 

s∑ 

EL = l PrL = l 

l=0 

s∑ 

VL = l − EL 2 PrL = l 

l=0 

Remark 4.3 

It is interesting to note that the optimal and solving (4.16) also minimize 

[ (EL ) ] [ 

E − r 

lin 2 (EL ) ] 2 

= E − − L 

L 

so that the linear relativities provide the best linear fit to the Bayesian ones. To check this 

assertion, note that 

[ ( ) ] 2 

E − − L 

[( ( ) ( ) ) 2] 

= E − EL + EL − − L 

[ ( ) ] 2 

= E − EL 

[ ( )( ) ] 

+ 2E − EL EL − − L 

[ (EL ) ] 2 

+ E − − L 

and the second term vanishes by definition of the conditional expectation EL, since 

− EL is orthogonal to any function of L. 

Example 4.18 (−1/+2 Scale, Portfolio A) Table 4.10 allows comparison of the relativities 

of the two scales in the segmented case. As we can see, the values are close to each other. 

The constant step between two levels in the linear scale is equal to 393%. 

The values of the expected errors Q 1 = [ − r L 2] and Q 2 = [ − rL lin2] 

are 

respectively given by 

Q 1 = 06219 and Q 2 = 06261 

The small difference between Q 1 and Q 2 indicates that the additional linear restriction does 

not really produce any deterioration in the fit. Note that the large differences in levels 1–5 

are given low weights in the computation of Q 1 and Q 2 . 

Table 4.11 displays the relativities without a priori segmentation. As can be observed, the 

scale without a priori segmentation is more elastic, which is logical since it has to take into 

account the full heterogeneity. The mean square errors are now given by 

Q 1 = 06630 and Q 2 = 06688


Table 4.10 Linear relativities with a priori risk classification for the system −1/ + 2 

and for Portfolio A. 

Level l 

Unconstrained relativities r l 

with a priori ratemaking 

Linear relativities rl 

lin 

priori ratemaking 

with a 

5 2714 % 2662% 

4 2185 % 2269% 

3 1925 % 1875% 

2 1388 % 1482% 

1 1286 % 1088% 

0 685% 695% 

Table 4.11 Linear relativities without a priori risk classification for the system 

−1/ + 2 and for Portfolio A. 

Level l Unconstrained relativities r l Linear relativities rl 

lin 

without a priori ratemaking without a priori ratemaking 

5 3091 % 2983% 

4 2414 % 2512% 

3 2077 % 2041% 

2 1429 % 1571% 

1 1302 % 1100% 

0 624% 629% 

so that again the additional linear restriction does not really produce any deterioration in the 

fit. Obviously, Q 1 and Q 2 are higher than before (when a priori risk classification was in 

force). This was expected as is now more variable. 

Example 4.19 (−1/+2 Scale, Portfolio B) 

Portfolio B. The mean square errors are 

Table 4.12 gives the linear relativities for 

Q 1 = 04134 and Q 2 = 04182 

We observe large discrepancies between r l and rl 

lin , especially in the higher levels. The 

constant penalty by step in the linear scale is 26.2 %. 

4.5.5 Approximations 

In Chapter 3, several discrete approximations to allowed us to derive simplified versions 

of the credibility formulas (replacing integrals with sums). The same idea can be applied 

here. Considering (4.13), the expression for r l when has support points 1 k with 

respective probability masses p 1 p q as in (3.5) becomes 

r l = 

∑ 

k w k 

∑ q 

j=1 j l k j p j 

∑k w k 

∑ q 

j=1 l k j p j


Table 4.12 Linear relativities with risk classification for the system −1/ + 2 

and for Portfolio B. 

Level l Unconstrained relativities r l Linear relativities rl 

lin 



5 2232 % 2047% 

4 1713 % 1785% 

3 1489 % 1522% 

2 1121 % 1260% 

1 1050% 998% 

0 747% 736% 

The discrete approximations listed in Tables 3.5–3.6 can then be used in this formula. 

In the case where has a unimodal probability density function, the mixed uniform 

approximations of Tables 3.8–3.9 can also be used. 

4.6 Relativities with an Exponential Loss Function 

4.6.1 Bayesian Relativities 

This section proposes an asymmetric loss function with one parameter that reflects the 

severity of the bonus-malus system. In order to reduce the maluses obtained with a quadratic 

loss, keeping a financially balanced system, we resort on an exponential loss function. Such 

loss functions have been applied in Section 3.4 in the classical credibility setting. Our purpose 

here is to apply exponential loss functions to determine the optimal relativities. 

When using the exponential loss function, the goal is now to minimize 

[ 

] 

Q exp = E exp−c − r L 

(4.18) 

under the financial balance constraint Er L = 1. The parameter c>0 determines the 

‘severity’ of the bonus-malus scale. The loss (4.18) puts more weight on the errors 

resulting in an overestimation of the premium (i.e. r L >), than on those coming from 

an underestimation. Consequently, the maluses are reduced, as well as the bonuses since 

financial stability has been imposed. 

Let us derive the general solution of (4.18). 

Proposition 4.1 

The solution of the constrained optimization problem (4.18) is 

r exp 

L 

= 1 + 1 c 

( [ 

] 

) 

E ln Eexp−cL − ln Eexp−cL (4.19) 


First, note that 

( [ 

]) 

exp ( ) expc exp E ln E exp−cL 

cr exp 

L = 

E [ exp−cL ]


Now, we have to minimize (4.18) that can be rewritten as 

[ 

] [ 

] 

E exp−c − r L = E expcr L − r exp 

L 

× expc exp ( E [ ln E exp−cL ]) 

Invoking Jensen’s inequality yields 

[ 

E exp−c − r L )] ( 

≥ exp cE [ ] ) 

r L − r exp 

L 

} {{ } 

=1 

× expc exp 

[ 

= E 

( [ 

E 

exp−c − r exp 

L 

]) 

ln E exp−cL 

] 

 

which ends the proof. 

□ 

Let us now compute the quantities in (4.19). Firstly, 

[ 

] 

∣ 

Eexp−cL = l =E Eexp−cL = l ∣L = l 

= ∑ k 

Eexp−cL = l = k Pr = k L = l 

= ∑ k 

∫ + 

0 

exp−c PrL = l = = kw k 

dF 

PrL = l = k 

× Pr = kL= l 

PrL = l 

∫ + ∑k w k exp−c 

0 l k dF 

= ∫ + 

(4.20) 

∑k w k 

0 l k dF 

and secondly 

[ 

] 

E ln Eexp−cL 

(4.21) 

= 

= 

s∑ 

PrL = l ln Eexp−cL = l 

l=0 

( ∑ ∫ 

s∑ 

+ 

) 

k w k exp−c 

0 l k dF 

PrL = l ln 

(4.22) 

PrL = l 

l=0


Remark 4.4 If no a priori ratemaking is in force, the expressions (4.20) and (4.22) are 

equal to those derived in Denuit & Dhaene (2001), that is 

Eexp−cL = l = 

∫ + 

0 

exp−c l dF 

PrL = l 

and 

[ 

] 

E ln Eexp−cL = 

= 

s∑ 

PrL = l ln Eexp−cL = l 

l=0 

s∑ 

PrL = l ln 

l=0 

(∫ + 

0 

exp−c l dF 

PrL = l 

) 

 

4.6.2 Fixing the Value of the Severity Parameter 

Let us briefly explain a possible criterion to fix the value of the parameter c. First, note that 

lim r exp 

l c→0 

= EL = l = r quad 

l 

 

where r quad 

l 

is the relativity obtained with a quadratic loss function. Letting c tend to 0 thus 

yields Norberg’s approach. In other words, the bonus-malus scale becomes more severe as 

c decreases. Now, the ratio of the variances of the premiums obtained with an exponential 

and a quadratic loss is given by 

Vr exp 

L 

Vr quad 

L = 1 V [ ln E exp−cL ] 

c 2 V [ ] = % ≤ 100 % 

E L 

The fact that the ratio of the variances is less than unity comes from the Jensen inequality. 

The idea is then to select the variance of the premium in the new system as a fraction of the 

corresponding variance under a quadratic loss (for instance = 25, 50 or 75 %). Of course, 

other procedures can be applied. For instance, the actuary could select the value of r 0 ,orof 

r s , and then compute c in order to match this value. 

4.6.3 Linear Relativities 

In practice, a linear scale of the form rl 

lin = + l, l = 0 1s, could be desirable. Let 

us now indicate how Gilde & Sundt’s (1989) approach can be extended using exponential 

loss functions. The aim is now to minimize the objective function 

[ 

= E exp ( − c − − L )]


under the financial balance constraint 

Er lin 

L = E = 1 

⇔ E = + EL ⇔ = E − EL 

It suffices to minimize 

[ ( ( 

))] 

˜ = E exp − c − E − L − EL 

Differentiating ˜ with respect to and equating to zero yields 

[ 

( 

E L − EL exp − c ( − E − L − EL ))] = 0 

⇔ 

∫ + 

0 

s∑ 

( ( 

)) 

l − EL exp − c − E − l − EL l dF = 0 

l=0 

which has to be solved numerically to get the value of (and hence of ). Convenient 

starting values for the numerical search are provided by (4.17). 


In this section, we give numerical examples of computation of bonus-malus scales 

when using an exponential loss function. We compare the results with those obtained 

previously. 

In order to be able to compare the results, we have computed the relativities associated 

with the same severity factor c = 1. These results are given in Table 4.13 for the −1/top, 

−1/ + 2 and −1/ + 3 systems and Portfolio A. Specifically, Table 4.13 gives the relativities 

obtained with an exponential loss function with severity parameter c = 1, with and without a 

priori risk classification, as well as the analogues for the −1/ + 2 and −1/ + 3 systems. As 

was the case with the quadratic loss function, we see that a priori risk classification reduces 

the dispersion of the relativities. 

We observe that the relativities computed when no a priori ratemaking is in force give 

bigger bonuses when the policyholders are in level 0 but also impose bigger maluses in the 

other levels. Indeed, when no a priori ratemaking is in force, the a posteriori correction 

must be more severe in order to distinguish between good and bad drivers. On the contrary, 

when an a priori ratemaking is in force, the correction applied with the a posteriori tariff 

must be softer because a greater part of the risk is already taken into account in the a priori 

ratemaking. 

Influence of the Loss Function 

We can also compare the results obtained when using the quadratic loss function and 

when using the exponential loss function (with different values of c). These are given in


Table 4.13 Numerical characteristics for the systems −1/top, 

−1/ + 2 and −1/ + 3 with a severity c = 1 and Portofolio A. 

−1/top 

Level l 

Relativity r exp 

l 


ratemaking 


l 

with a priori 

ratemaking 

5 1613 % 1541% 

4 1480 % 1423% 

3 1371 % 1329% 

2 1281 % 1252% 

1 1204 % 1187% 

0 690% 723% 

−1/ + 2 

Level l 


l 


ratemaking 


l 

with a priori 

ratemaking 

5 2534 % 2293% 

4 2044 % 1898% 

3 1834 % 1726% 

2 1328 % 1300% 

1 1249 % 1234% 

0 716% 758% 

−1/ + 3 

Level l 


l 


ratemaking 


l 

with a priori 

ratemaking 

5 2093 % 1947% 

4 1878 % 1763% 

3 1372 % 1334% 

2 1283 % 1259% 

1 1207 % 1194% 

0 693% 732% 

Tables 4.14, 4.15 and 4.16 respectively for the −1/top, −1/ + 2 and −1/ + 3 systems. The 

exponential relativities have been computed for a severity coefficient ranging from 0 to 5. 

The limit value of c = 0 provides the same result as with the quadratic loss function. We 

observe in Table 4.14 that an increasing value of c leads to less dispersed relativities. The 

maximal penalty decreases as c increases: keeping financial balance, increasing c tends to 

soften a posteriori corrections.


Table 4.14 Numerical characteristics for the system −1/top and Portofolio A. 

Level l 

Relativity r quad 

l 

with a priori 

ratemaking c = 0 


l 

with a priori 

ratemaking 

c = 1 


l 

with a priori 

ratemaking 

c = 2 


l 

with a priori 

ratemaking 

c = 5 

5 1812 % 1541 % 1418 % 1262% 

4 1599 % 1423 % 1336 % 1219% 

3 1439 % 1329 % 1270 % 1184% 

2 1312 % 1252 % 1214 % 1153% 

1 1209 % 1187 % 1166 % 1126% 

0 612% 723% 779% 854% 

Table 4.15 Numerical characteristics for the system −1/ + 2 and Portofolio A. 

Level l 


l 

with a priori 



l 

with a priori 

ratemaking 

c = 1 


l 

with a priori 

ratemaking 

c = 2 


l 

with a priori 

ratemaking 

c = 5 

5 2714 % 2293 % 2073 % 1749% 

4 2185 % 1898 % 1743 % 1511% 

3 1925 % 1726 % 1612 % 1433% 

2 1388 % 1300 % 1249 % 1172% 

1 1286 % 1234 % 1200 % 1143% 

0 685% 758% 798% 859% 

Table 4.16 Numerical characteristics for the system −1/ + 3 and Portofolio A. 

Level l 


l 

with a priori 



l 

with a priori 

ratemaking 

c = 1 


l 

with a priori 

ratemaking 

c = 2 


l 

with a priori 

ratemaking 

c = 5 

5 2308 % 1947 % 1767 % 1516% 

4 2009 % 1763 % 1632 % 1439% 

3 1452 % 1334 % 1271 % 1182% 

2 1331 % 1259 % 1216 % 1151% 

1 1230 % 1194 % 1169 % 1124% 

0 642% 732% 780% 849%


4.7 Special Bonus Rule 

4.7.1 The Former Belgian Compulsory System 

In this section we will concentrate on the former compulsory Belgian bonus-malus system, 

which all companies operating in Belgium have been obliged to use from 1992 to 2002. The 

Belgian system consists of a scale of 23 levels (numbered from 0 to 22). A new driver starts 

in class 11 if he uses his vehicle for pleasure and commuting and in class 14 if he uses his 

vehicle for business. Each claim-free year is rewarded by a bonus point. The first claim is 

penalized by four malus points and the subsequent ones by five malus points each. 

According to the special bonus rule, a policyholder with four claim-free years cannot be 

in a class above 14. This restriction is a concession to insureds with many claims in a few 

years and who suddenly improve; very few policyholders are ever able to take advantage of 

this rule. 

Actually, the Belgian bonus-malus system is not Markovian due to the special bonus rule 

(i.e. due to the fact that policyholders occupying high levels are sent to level 14 after four 

claim-free years). Fortunately, it is possible to introduce fictitious classes in order to meet 

the memoryless property by splitting the levels 16 to 21 into subclasses, depending on the 

number of consecutive years without accident. 

4.7.2 Fictitious Levels 

Splitting the levels 16 to 21 into sub-levels, depending on the number of consecutive years 

without accident, allows us to account for the special bonus rule. Let n j be the number of 

sub-levels to be associated with bonus level j. A level ji is to be understood as level j 

and i consecutive years without accidents. The transition rules are completely defined in 

Table 4.17 and the different values for n j are given in Table 4.18. We take some liberty 

with the notation by using the value 0 for the subscript i and by not using a subscript when 

n j = 1. 

4.7.3 Determination of the Relativities 

The relativities r ji are obtained by minimizing the squared difference between the true relative 

premium and the relative premium r L applicable to the policyholder when stationary state 

has been reached. 

The current situation is more complicated because some levels have to be constrained to 

have the same relativity. Indeed the artificial levels ji have the property that 

r j = r j1 =···=r jnj 

j = 0s 

We have to minimize E [ − r L 2] under these constraints. 


r j = 

∑ 

k 

∑ 

k 

∫ ∑ nj 

w k 0 

w k 

∫ 

0 

i=1 ji k dF 

∑ nj 

i=1 ji k dF (4.23)


Table 4.17 Transition rules of the Belgian bonus-malus system with fictitious levels accounting for 

the special bonus rule. 

Class 

Class after k accidents 

k = 0 1 2 3 4 5 

22 211 22 22 22 22 22 

21.0 201 22 22 22 22 22 

21.1 202 22 22 22 22 22 

20.0 191 22 22 22 22 22 

20.1 192 22 22 22 22 22 

20.2 193 22 22 22 22 22 

19.0 181 22 22 22 22 22 

19.1 182 22 22 22 22 22 

19.2 183 22 22 22 22 22 

19.3 14 22 22 22 22 22 

18.0 17 22 22 22 22 22 

18.1 172 22 22 22 22 22 

18.2 173 22 22 22 22 22 

18.3 14 22 22 22 22 22 

17 16 210 22 22 22 22 

17.2 163 210 22 22 22 22 

17.3 14 210 22 22 22 22 

16 15 200 22 22 22 22 

16.3 14 200 22 22 22 22 

15 14 190 22 22 22 22 

14 13 180 22 22 22 22 

13 12 17 22 22 22 22 

12 11 16 210 22 22 22 

11 10 15 200 22 22 22 

10 9 14 190 22 22 22 

9 8 13 180 22 22 22 

8 7 12 17 22 22 22 

7 6 11 16 210 22 22 

6 5 10 15 200 22 22 

5 4 9 14 190 22 22 

4 3 8 13 180 22 22 

3 2 7 12 17 22 22 

2 1 6 11 16 210 22 

1 0 5 10 15 200 22 

0 0 4 9 14 190 22 

Note that it is easily seen that 

n j 

∑ PrL = ji 

r j = ∑ nj 

i=1 i=1 PrL = jir ji 

where the r ji s represent the non-constrained solution of the minimization of [ − r L 2] .


Table 4.18 Sub-levels of the 

Belgian bonus-malus system. 

j 

n j 

22 1 

21 2 

20 3 

19 4 

18 4 

17 3 

16 2 

15 1 

14 1 

13 1 

12 1 

11 1 

10 1 

9 1 

8 1 

7 1 

6 1 

5 1 

4 1 

3 1 

2 1 

1 1 

0 1 

We also immediately verify that 

s∑ 

r j PrL = j = 1 

j=0 

which ensures that the bonus-malus scale is financially balanced at stationary state. 


The numerical results for the Belgian bonus-malus system and for Portfolio A are displayed in 

Table 4.19. In this table, the special bonus rule has not been taken into account. Specifically, 

the values in the third column are computed with the help of (4.14) with â = 0889 and 

̂ = 01474. Those values were obtained by fitting a Negative Binomial distribution to the 

portfolio’s observed claims frequencies. The fourth column is based on (4.13) with â = 1065 

and the ̂ k s obtained from the a priori risk classification described in Table 2.7. The last 

column is computed with the help of (4.15). 

Once the stationary state has been reached, more or less half of the policies occupy level 

0 and enjoy the maximum discount. This is due to the fact that the transition rules of the


Table 4.19 Numerical characteristics for the Belgian bonus-malus system without the special bonus 

rule, computed on the basis of Portfolio A. 

Level l PrL = l r l = EL = l r l = EL = l EL = l 

without a priori with a priori 

ratemaking 

ratemaking 

22 54 % 3060 % 2715% 184% 

21 38 % 2739 % 2475% 177% 

20 29 % 2487 % 2291% 172% 

19 23 % 2283 % 2141% 168% 

18 19 % 2112 % 2014% 165% 

17 16 % 1965 % 1903% 162% 

16 15 % 1838 % 1803% 160% 

15 13 % 1726 % 1714% 158% 

14 13 % 1625 % 1631% 156% 

13 12 % 1528 % 1550% 155% 

12 12 % 1439 % 1472% 153% 

11 12 % 1359 % 1402% 152% 

10 12 % 1289 % 1340% 151% 

9 14 % 1198 % 1255% 149% 

8 16 % 1111 % 1173% 148% 

7 17 % 1048 % 1114% 147% 

6 18% 998 % 1067% 146% 

5 18% 956 % 1028% 146% 

4 43% 752% 826% 143% 

3 38% 727% 802% 143% 

2 34% 703% 778% 142% 

1 31% 681% 755% 142% 

0 503% 376% 451% 138% 

Belgian system are not severe enough in comparison with the average claims frequency. And 

the situation is even more serious when looking at the actual figures on the market. Indeed 

the market average claims frequency is even smaller than the one of the analysed portfolio. 

If we compute the relativities without taking into account the a priori ratemaking system, 

the relativities vary between 376 % for level 0 and 3060 % for level 22. On the other hand, 

if we adapt the relativities to the a priori risk classification, these relativities vary between 

451 % and 2715 %; the severity of the a posteriori corrections is thus weaker in this case. 

Tables 4.20, 4.21 and 4.22 display the results for the Belgian bonus-malus system when the 

special bonus rule is taken into account. Table 4.20 compares the probability mass functions 

of L when the special bonus rule is taken into account, with respect to the case without 

special bonus rule. This rule decreases the probabilities associated with upper levels. The 

decrease for level 18 is even more apparent, since policyholders in that level benefit from 

the special bonus rule. 

Table 4.21 gives the relativities computed with and without the special bonus rule. We 

see that the relativities are larger with the rule, since it boils down to softening the penalty 

in the case where claims are filed to the insurance company.


Table 4.20 Distribution of L for the Belgian bonusmalus 

system, computed on the basis of Portfolio A. 

Level l 

PrL = l 

Without special 

bonus rule 

With special 

bonus rule 

22 54% 43% 

21 38% 30% 

20 29% 22% 

19 23% 17% 

18 19% 09% 

17 16% 10% 

16 15% 10% 

15 13% 10% 

14 13% 21% 

13 12% 19% 

12 12% 18% 

11 12% 17% 

10 12% 16% 

9 14% 17% 

8 16% 19% 

7 17% 20% 

6 18% 20% 

5 18% 20% 

4 43% 45% 

3 38% 40% 

2 34% 36% 

1 31% 32% 

0 503% 509% 

We observe in Table 4.22 that the average a priori expected claim frequency EL = l 

in level l is always higher with the special bonus rule than without that rule. The effect 

is more pronounced in the highest levels of the scale and less pronounced in the lowest 

levels of the scale. This fact is obvious from the definition of the special bonus rule. The 

policyholders attaining the highest classes of the scale benefit from the special bonus rule. 

Those staying in these highest classes show therefore a higher expected frequency. Even 

below level 14 the effect remains true because the policyholders have benefitted from it 

before attaining the lowest levels. Obviously the effect is less and less pronounced at the 

bottom of the scale. 

Some insurance companies use the bonus-malus scale as an underwriting tool. For instance, 

they systematically refuse drivers with a bonus-malus level > 14. Our calculations show that 

this is unreasonable because drivers at level 15 are on average less risky than drivers at level 

14. 

We see from Table 4.19 that without the special bonus rule, the relativities are always 

increasing from level 0 to level 22. The same increasing pattern is observed for L = l. 

When looking at the results for the bonus-malus system with special bonus rule (Table 4.21), 

we observe that the relativities at levels 13–16 are not ordered any more. This can be


Table 4.21 Relativities r l = EL = l for the 

Belgian bonus-malus system computed on the basis of 

Portfolio A. 

Level l 

Relativities 

Without special With special 

bonus rule 

bonus rule 

22 2715 % 2843% 

21 2475 % 2589% 

20 2291 % 2389% 

19 2141 % 2225% 

18 2014 % 1994% 

17 1903 % 1942% 

16 1803 % 1869% 

15 1714 % 1791% 

14 1631 % 1924% 

13 1550 % 1798% 

12 1472 % 1684% 

11 1402 % 1583% 

10 1340 % 1495% 

9 1255 % 1386% 

8 1173 % 1283% 

7 1114 % 1207% 

6 1067 % 1147% 

5 1028 % 1096% 

4 826% 870% 

3 802% 841% 

2 778% 813% 

1 755% 786% 

0 451% 462% 

explained by the fact that many drivers at level 14 have benefitted from the special bonus 

rule. It is clear that such a situation is not acceptable from a commercial point of view. We 

may constrain the scale to be linear in the spirit of Gilde & Sundt (1989). However we 

propose a local adjustment to the scale in order to keep − r L 2 as small as possible. 

Let us constrain the scale to be linear between levels 13 and 16. We are looking for 

updated values for r ′ j , j = 1316. They are such that r ′ j = r ′ j−1 

+a, j = 14 15 16 where 

a = r ′ 16 − r ′ 13/3. We also want to keep the financial equilibrium of the system. Therefore 

we constrain a local equilibrium : 

∑16 

j=13 

r j j = 

∑16 

j=13 

r ′ j j 

Choosing r ′ 13 = 1798 %, we obtain r ′ 16 = 1937%.


Table 4.22 Average a priori claim frequency 

EL = l in level l for the Belgian bonus-malus 

system, computed on the basis of Portfolio A. 

Level l 

EL = l 

Without special With special 

bonus rule 

bonus rule 

22 184% 187% 

21 177% 180% 

20 172% 174% 

19 168% 170% 

18 165% 164% 

17 162% 163% 

16 160% 161% 

15 158% 160% 

14 156% 163% 

13 155% 160% 

12 153% 158% 

11 152% 156% 

10 151% 154% 

9 149% 152% 

8 148% 150% 

7 147% 149% 

6 146% 148% 

5 146% 147% 

4 143% 144% 

3 143% 143% 

2 142% 143% 

1 142% 142% 

0 138% 138% 

Now let us compare the value of the expected error Q = [ − r L 2] with the original 

model, Q 1 , and with the constrained model, Q 2 :weget 

Q 1 = 041121 and Q 2 = 041150 

This shows that the error induced by the commercial constraint is really small. So we may 

adapt the scale locally without resorting to a full linear scale constraint. 

We can perform the local minimization numerically without imposing a linear scale 

between levels 13 and 16. We use the following constraints : 

r ′ 13 ≤ r ′ 14 

r ′ 14 ≤ r ′ 15 

r ′ 15 ≤ r ′ 16


∑16 

j=13 

r j j = 

∑16 

j=13 

r ′ j ≥ 169 % 

r ′ j ≤ 194 % 

r ′ j j 

j = 1316 

j = 1316 

And we obtain r ′ 13 = 1798 % and r ′ 14 = r ′ 15 = r ′ 16 

= 1878 %. The value of Q is now 041135. 

4.7.5 Linear Relativities for the Belgian Scale 

We shall now demonstrate how to get linear relativities for the Belgian scale. In addition to 

the constraint of building a linear scale, we add a new constraint: the artificial states must 

have the same relativity. The minimization problem then becomes : 

[ ] ] 

min E − r lin 

L 2 = min E 

[ − − L 2 

Table 4.23 Relativities r l and rl 

lin for the Belgian 

bonus-malus system, taking into account the special 

bonus rule and computed on the basis of Portfolio A. 

Level l 

Relativities 

General scale Linear scale 

22 2843 % 2674% 

21 2589 % 2574% 

20 2389 % 2475% 

19 2225 % 2376% 

18 1994 % 2276% 

17 1942 % 2177% 

16 1869 % 2077% 

15 1791 % 1978% 

14 1924 % 1879% 

13 1798 % 1779% 

12 1684 % 1680% 

11 1583 % 1580% 

10 1495 % 1481% 

9 1386 % 1382% 

8 1283 % 1282% 

7 1207 % 1183% 

6 1147 % 1083% 

5 1096% 984% 

4 870% 885% 

3 841% 785% 

2 813% 686% 

1 786% 586% 

0 462% 487%


so that 

r l = r l1 = r l2 =···=r lnl 

l= 0 1s 

n 

∑ l 

The solution to this problem is the same as above. We merely have to replace l by li . 

The results are displayed in Table 4.23. The constant step between two levels of the linear 

scale is equal to 99 % and the value of the mean square error is Q = 041779. The major 

advantage of the linear scale is that it gives a system that is commercially more acceptable. 

i=1 

4.8 Change of Scale 

4.8.1 Migration from One Scale to Another 

Since the 90s insurance markets in the EU have been deregulated. More competition is 

allowed. Two related problems arise with the deregulation of the bonus-malus systems. The 

first one consists of transferring the policyholders to the new scales. The second one is more 

difficult: it consists of transferring a new policyholder to the scale of the company knowing 

his level in the scale of his previous insurer. 

The aim of the present section is to show how to develop rules allowing the transfer 

of a policyholder to a bonus-malus scale knowing his level in his previous bonus-malus 

scale. The a posteriori probability density function of given the level L occupied in the 

bonus-malus scale is given by 

f L=l = PrL = l = f 

PrL = l 

∑ 

k 

= 

k l k dF 

∫ ∑k k 0 l k dF 

Because we want to move a policyholder from one scale to the other, we should try to put 

the policyholder at a level which is as close as possible to his level in his original bonusmalus 

scale. By ‘close’, we mean here having the a posteriori random effect as close as 

possible. 

4.8.2 Kolmogorov Distance 

In addition to the total variation distance d TV used previously, we also need the Kolmogorov 

distance. The Kolmogorov (or uniform) metric based on the well-known Kolmogorov- 

Smirnov statistic (associated with the goodness-of-fit test with that name), is defined 

as follows: The Kolmogorov distance d K between the random variables X and Y is 

given by 

d K X Y = sup F X t − F Y t (4.24) 

t∈ 

Given two random variables X and Y , we have that d K X Y ≤ d TV X Y. This result is 

an immediate consequence of (4.11) since F X t = PrX ∈ t +.


4.8.3 Distances between the Random Effects 

A first measure may be to compare the expected a posteriori random effect, which actually 

is the relative premium at level l in the bonus-malus scale : 

r L=l = L = l 

So the closest level l j in scale 2 to the level l i occupied in scale 1 is given by 

argmin lj 

r L1 =l i 

− r L2 =l j 

2 

This rule simply amounts to placing the policyholder in the new scale at the level with 

the closest relativity to the one applicable in the previous scale. Because most commercial 

scales are normalized (to associate a unit relativity with the entry level), this means that the 

insurer has to compute the relativities for both scales. The implicit assumption is that the 

new entrant has the same characteristics as the policyholders in the portfolio (no adverse 

selection being allowed). 

Another measure of discrepancy consists of comparing the distribution functions of the a 

posteriori random effects. This can be done by using the Kolmogorov distance d K or the 

total variation distance d TV : 

d K L 1 = l i L 2 = l j = max Pr ≤ L 1 = l i − Pr ≤ L 2 = l j 

 

d TV L 1 = l i L 2 = l j = 

∫ 

0 

f L1 =l i 

− f L2 =l j 

d 

Summarizing, a policyholder being at level l i in scale 1 and moving to scale 2 will be put 

in the level l j of scale 2 that minimizes one of the following distances: 

(i) d E L 1 = l i L 2 = l j = r L1 =l i 

− r L2 =l j 

2 

(ii) d K L 1 = l i L 2 = l j 

(iii) d TV L 1 = l i L 2 = l j . 


In this section we use Portfolio A and its risk classification described in Table 2.7. Let us 

assume that we want to move policyholders between the following two bonus-malus scales: 

• The −1/top scale with 6 levels, numbered 0 to 5. 

• The −1/ + 2 scale of Taylor (1997) with 9 levels, having transition rules given in 

Table 4.24.


Table 4.24 Transition rules for the scale −1/ + 2 scale of 

Taylor (1997) with 9 levels. 

Starting level 

Level occupied if 

0 1 2 3 ≥ 4 

claim is/are reported 

8 7 8 8 8 8 

7 6 8 8 8 8 

6 5 8 8 8 8 

5 4 7 8 8 8 

4 3 6 8 8 8 

3 2 5 7 8 8 

2 1 4 6 8 8 

1 0 3 5 7 8 

0 0 2 4 6 8 

Table 4.25 

Distances between L 1 = l 1 and L 2 = l 2 for the three metrics. 

d K 

l 2 l 1 0 1 2 3 4 5 6 7 8 

0 0042 0377 0404 0574 0613 0701 0740 0790 0825 

1 0323 0016 0044 0245 0299 0421 0483 0562 0625 

2 0357 0061 0036 0200 0254 0376 0439 0520 0585 

3 0396 0110 0080 0153 0204 0325 0388 0471 0539 

4 0439 0168 0137 0105 0151 0267 0330 0414 0483 

5 0488 0237 0207 0061 0097 0202 0262 0344 0414 

d TV 

l 2 l 1 0 1 2 3 4 5 6 7 8 

0 0089 0754 0808 1148 1225 1402 1481 1580 1651 

1 0646 0081 0107 0490 0599 0841 0966 1125 1249 

2 0715 0123 0094 0402 0508 0751 0878 1041 1172 

3 0791 0220 0160 0320 0411 0649 0777 0943 1078 

4 0877 0337 0274 0270 0324 0535 0660 0827 0966 

5 0976 0475 0413 0281 0286 0426 0527 0688 0834 

d E 

l 2 l 1 0 1 2 3 4 5 6 7 8 

0 0001 0308 0391 1077 1415 2329 3093 4349 5908 

1 0314 0002 0001 0194 0351 0863 1350 2215 3362 

2 0440 0021 0006 0114 0239 0682 1120 1917 2993 

3 0624 0074 0041 0044 0132 0489 0868 1584 2572 

4 0902 0186 0131 0003 0041 0291 0596 1207 2085 

5 1377 0430 0343 0030 0000 0100 0301 0765 1489


Table 4.25 provides the distances between the conditional random effect for our three 

metrics. On the basis of these distances, we see that the minima are attained for the following 

transition rules. Transition rules from scale 1 to scale 2 are 

d K d TV d 

l 1 l 2 l 2 l 2 

0 0 0 0 

1 1 1 2 

2 2 2 2 

3 2 2 2 

4 3 3 3 

5 3 3 4 

and transition rules from scale 2 to scale 1 are 

d K d TV d 

l 2 l 1 l 1 l 1 

0 0 0 0 

1 1 1 1 

2 2 2 1 

3 5 4 4 

4 5 5 5 

5 5 5 5 

6 5 5 5 

7 5 5 5 

8 5 5 5 

Let us now take into account the a priori characteristics of the driver. It will be seen that 

good and bad drivers are not placed in the same way when they are transferred from one 

scale to the other. The transition rules for an a priori bad driver (specifically, a male driver 

aged more than 30 years, with premium split, having private use of the car and living in 

an urban environment, whose a priori expected frequency is 0.1794) are given in the next 

tables: from scale 1 to scale 2 

d K d TV d E 

l 1 l 2 l 2 l 2 

0 0 0 0 

1 2 1 2 

2 2 2 2 

3 2 2 3 

4 3 3 3 

5 3 4 4


and from scale 2 to scale 1 

d K d TV d E 

l 2 l 1 l 1 l 1 

0 0 0 0 

1 1 1 1 

2 1 1 1 

3 5 4 4 

4 5 4 4 

5 5 5 5 

6 5 5 5 

7 5 5 5 

8 5 5 5 

We here compare the distance between L 1 = l i = 01794 and L 2 = l j = 01794. 

We have 

 

f L=l= = l f 

∫ 

0 lf d 

The transition rules for a good policyholder (specifically, a male driver aged more than 30 

years, with upfront premium, having private use of the car and living in a rural environment, 

whose a priori expected frequency is 0.0928) are given in the next tables: from scale 1 to 

scale 2 

and from scale 2 to scale 1 

d K d TV d E 

l 1 l 2 l 2 l 2 

0 0 0 0 

1 1 1 1 

2 1 1 2 

3 2 2 2 

4 2 2 2 

5 2 2 3 

d K d TV d E 

l 2 l 1 l 1 l 1 

0 0 0 0 

1 2 2 1 

2 2 3 2 

3 5 5 5 

4 5 5 5 

5 5 5 5 

6 5 5 5 

7 5 5 5 

8 5 5 5 

We here compare the distance between L 1 = l i = 00928 and L 2 = l j = 00928.


We observe that the different metrics we have chosen do not provide very different results. 

Because the expected posterior random effect has a financial meaning (i.e. it is the multiplier 

of the average cost to get the a posteriori premium), we may be tempted to choose its 

corresponding metric to transfer a policyholder from one scale to the other. Note also that 

the a priori characteristics of the driver influence the way the policy is transferred from 

one scale to the other. This results in a number of rules according to the risk classification 

scheme applied by the insurance company. 

Remark 4.5 Transferring a policyholder from the bonus-malus scale of a given insurer 

to the bonus-malus scale of another insurer remains a more complicated task. Indeed, the 

actuary needs to know the a posteriori random effect in both situations. However the a 

priori random effect may be different due to another type of a priori tariff or due to adverse 

selection. The distance to minimize then extends to d 1 L 1 = l 1 2 L 2 = l 2 . 

4.9 Dependence in Bonus-Malus Scales 

It is clear that the sequence L 1 L 2 is CIS, but we do not have in general that given 

L = l increases with l. The reason is as follows: Because of the finite number of levels, 

it does not automatically follow that a policyholder in a higher level filed more claims in 

the past. To show this, consider the scale −1/top. A policyholder in level 3 in year 3 may 

occupy that level having filed two claims in year 1 and no claim after. On the contrary, a 

policyholder in level 4 could have filed one claim in year 2. From (3.18) we conclude that 

should be larger (in the ≼ ST -sense) for policyholder 1 than for policyholder 2 despite the 

fact that policyholder 1 is in an inferior level. 

As a consequence, we cannot be sure that the Bayesian relativities obtained with a quadratic 

loss function are increasing with the level occupied in the scale. This is why linear relativities 

are so useful in practice. 

Remark 4.6 This counterintuitive fact becomes reasonable if we allow for random effects 

to vary in time. Provided the autocorrelogram is decreasing, old claims have less predictive 

power than recent ones. Then, we have to compare two old claims to one recent claim, and 

it becomes less obvious that policyholder 1 is more dangerous than policyholder 2. 


Chapter 7 in Rolski ET AL. (1999) offers an excellent introduction to Markov chains, with 

applications to bonus-malus systems. Several parts of this chapter are directly inspired 

from this source. For the most part, this chapter is based on Pitrebois, Denuit & 

Walhin (2003b), following on from Taylor (1997) for the extension of Norberg’s (1976) 

pioneering work on segmented tariffs; on Pitrebois, Denuit & Walhin (2004) for the 

linear relativities; on Pitrebois, Walhin & Denuit (2006c) for Section 4.8; and on 

Pitrebois, Denuit & Walhin (2003a) for the Belgian bonus-malus scale and its special 

bonus rule. 

Norberg (1976), Borgan, Hoem & Norberg (1981) and Gilde & Sundt (1989) 

assumed that the bonus-malus system forms a first order Markov chain. Centeno & 

Andrade e Silva (2002) considered bonus-malus systems that are not first order Markovian


processes, but that can be made Markovian by increasing the number of states as we did in 

Section 4.7. 

The notion of distance found a second life in probability in the form of metrics in 

spaces of random variables and their probability distributions. The study of limit theorems 

(among other questions) made it necessary to introduce functionals evaluating the nearness 

of probability distributions in some probabilistic sense. In this chapter, the total variance 

and Kolmogorov distances have been used in connection with bonus-malus systems. We 

refer the interested reader to Chapter 9 of Denuit ET AL. (2005) for a detailed account of 

probability metrics and their applications in risk theory. 

The premium relativities are traditionally computed with the help of a quadratic loss 

function, in the vein of Norberg (1976). Other loss functions nevertheless also deserve 

consideration, such as the exponential loss function applied by Denuit & Dhaene (2001) 

to the computation of the relativities. For the sake of completeness, let us mention that 

the absolute value loss function has been successfully applied to the determination of the 

relativities by Heras, Vilar & Gil (2002) and Heras, Gil, Garcia-Pineda & Vilar (2004). 

If in a given market, companies start to compete on the basis of bonus-malus systems, 

many policyholders could leave the portfolio after the occurrence of an accident, in order to 

avoid the resulting penalties. Those attritions can be incorporated in the model by adding a 

supplementary level to the Markov chain (in the spirit of Centeno & Andrade e Silva 

(2001)). Transitions from a level of the bonus-malus scale to this state represent a policyholder 

leaving the portfolio whereas transitions from this state to any level of the bonus-malus scale 

mean that a new policy enters the portfolio. 

It has been assumed throughout this chapter that the unknown expected claim frequencies 

were constant and that the random effects representing hidden characteristics were timeinvariant. 

Dropping these assumptions makes the determination of the relativities much 

harder. We refer the interested reader to Brouhns, Guillén, Denuit & Pinquet (2003) 

for a thorough study of this general situation. A fundamental difference with the traditional 

approaches is that we lose the homogeneity of the chain in a dynamic segmented environment. 

Indeed, if the observable characteristics of the policyholders are allowed to vary in time, the 

claim frequencies are no longer constant and the trajectory of the policyholder in the bonusmalus 

scale is no longer described by a homogeneous Markov chain, but well described 

by a non-homogeneous one. Consequently, the classical techniques based on stationary 

distributions cannot be applied to the problem of determining the relativities. Brouhns, 

Guillén, Denuit & Pinquet (2003) propose a computer-intensive method to calibrate 

bonus-malus scales. Their paper clearly illustrates the strong complementarity of a priori 

and a posteriori ratemakings. The main originality of their approach is to compare on the 

basis of real data four different credibility models: static versus dynamic heterogeneity, with 

and without recognizing a priori risk classification. The impact of the different assumptions 

becomes clear in the numerical illustrations. 

Andrade e Silva & Centeno (2005) suggested the use of geometric relativities instead 

of linear ones. Specifically, under a quadratic loss function, the relativity associated with 

level l becomes r geo 

l 

= l , with and positive. As pointed out in Remark 4.3 for linear 

relativities, finding the geometric relativities amounts to finding the best approximation l 

to the Bayesian relativities, that is, and solve 

min E[ (EL 

− 

L ) 2 ]


No explicit expressions are available for the optimal and , which must be determined 

numerically. 

As pointed out by Subramanian (1998), the market shares of competitors in a given 

market can be strongly affected when some insurers adopt an agressive competitive behaviour 

by modifying the bonus-malus systems. In a deregulated market, insurers have an incentive 

to innovate in their pricing decisions by partitioning their portfolios (a priori ratemaking) 

and by designing new bonus-malus systems (a posteriori ratemaking). Viswanathan & 

Lemaire (2005) examined the evolution of market shares and claim frequencies in a twocompany 

market, when one insurer breaks off the existing stability by introducing a super 

discount class in its bonus-malus system. 

To end with, let us point out a final remark of primary importance. Merit-rating structures 

in automobile insurance require the insured to decide whether to file a claim for an accident 

when he is at fault. Since the penalties are independent of the claim amounts, one could 

imagine that some policyholders prefer to carry the cost of the accident themselves in order 

to avoid the premium increase in the future. Therefore, the data the actuary has at his disposal 

are contingent on the actual bonus-malus system and are ‘censored’ in a very complicated 

way. Hence, the policyholder might modify their behaviour when a new bonus-malus system 

is introduced, resulting in an increasing (or decreasing) number of reported claims. This is 

a particularly important side-effect when a bonus-malus system is modified. We will come 

back to this issue in Chapter 5. 

The numerical results presented in this chapter can be obtained using software such as 

SAS R or R. There is also a specific software package, called BM-builder, which works in 

the SAS R environment and allows for actuarial computations related to bonus-malus scales. 

It has been developed by ReacFin SA, which is a spin-off of the Université Catholique de 

Louvain, Louvain-la-Neuve, Belgium, created in January 2004 by the authors. Reacfin’s aim 

is to provide actuarial solutions to its clients. Its strong link with the university guarantees 

the use of up-to-date techniques. BM-builder aims to compute the relativities attached to 

the different levels of a bonus-malus scale, taking into account the actual structure of the 

insurance portfolio. In that respect, it properly integrates the interactions between a priori 

and a posteriori ratemakings and allows for an efficient ratemaking. For more details, see 

http://www.reacfin.com.

Part III 

Advances in 

Experience Rating 




5 

Efficiency and Bonus Hunger 


5.1.1 Pure Premium 

In nonlife business, the pure premium is the expected cost of all the claims that the 

policyholder will file during the coverage period (under the assumption of the Law of Large 

Numbers, i.e. a large portfolio comprising independent and identically distributed risks). 

The actuarial ratemaking is based on a claim frequency distribution and a loss distribution. 

The average claim frequency is defined as the number of incurred claims per unit of earned 

exposure (the exposure is usually measured in car-year for motor insurance). The average 

loss severity is the average payment per incurred claim. 

In this chapter, as well as in the next one, we need to consider the claim severities. Even if 

the premium updates induced by bonus-malus systems only depend on the number of claims 

at fault filed with the insurance company, the design of efficiency measures, as well as the 

study of the bonus-hunger phenomenon (i.e. the tendency for the policyholder to self-defray 

minor accidents to avoid premium surcharges), require an accurate modelling for the cost of 

the claims. This topic is dealt with in Section 5.2. 

5.1.2 Statistical Analysis of Claim Costs 

The computation of the pure premium relies on a statistical model incorporating all the 

available information about the risk. The technical tariff aims to evaluate as accurately as 

possible the pure premium for each policyholder via regression techniques. It is well known 

that market premiums may differ from those computed by actuaries. In that respect, the 

overall market position of the company compared to its competitors with regard to growth 

and pricing is crucial. This chapter is devoted to technical tariff only. 





The first step of any actuarial analysis consists in displaying descriptive statistics in order 

to figure out the composition of the portfolio and the marginal impact of the rating factors. 

In a second stage, available explanatory variables are incorporated to the policyholders’ 

expected claim frequencies and severities with the help of generalized regression models. 

5.1.3 Large Claims and Extreme Value Theory 

Large claims generally affect liability coverages. These major accidents require a separate 

analysis. The reason for a separate analysis of small (or moderate) and large losses is that 

no standard parametric model seems to emerge as providing an acceptable fit to both small 

and large claims. The main goal is then to determine an optimal threshold separating the 

two types of losses. 

Extreme Value Theory and Generalized Pareto distributions can be used to set the value 

of this threshold. Specifically, graphical tools including the Pareto index plot and the 

Gertensgarbe plot can be used to estimate the threshold defining the large losses. In the 

former case, the maximum likelihood estimator of the Pareto tail parameter is computed 

for increasing thresholds until it becomes approximately constant. The Gertensgarbe plot 

is based on the assumption that the optimal threshold can be found as a change point in 

the ordered series of claim costs and that the change point can be identified by means of 

a sequential version of the Mann-Kendall test as the intersection point between normalized 

progressive and retrograde rank statistics. 

5.1.4 Measuring the Efficiency of the Bonus-Malus Scales 

As explained in the preceding chapters, the basis of fair ratemaking in motor insurance is 

the fact that each policyholder is charged a premium that is proportional to the risk that 

he actually represents. The accident proneness of a policyholder being represented by the 

relative risk parameter , we expect that a relative change in will have the same impact 

on the premium paid to the insurance company. If this is the case then the system is said to 

be fully efficient. 

Section 5.3 reviews two concepts of efficiency: Loimaranta efficiency and De Pril 

efficiency. Both intend to measure how the bonus-malus system responds to a change in the 

riskiness of the driver. Loimaranta efficiency is solely based on the stationary probabilities 

whereas De Pril efficiency is a transient concept and uses the time value of money (through 

discounting). 

5.1.5 Bonus Hunger and Optimal Retention 

Since the penalty induced by the bonus-malus system is independent of the claim amount, 

a crucial issue for the policyholder is therefore to decide whether it is profitable or not to 

report small claims (in order to avoid an increase in premium). Cheap claims are likely to 

be defrayed by the policyholders themselves, and not to be reported to the company. This 

phenomenon, known as the hunger for bonus after Philipson (1960), is studied in Section 

5.4. 

Section 5.4.1 is devoted to the censorship of claim amounts and claim frequencies arising 

from bonus-malus systems. Specifically, a statistical model is specified, that takes into

Efficiency and Bonus Hunger 221 

account the fact that only ‘expensive’ claims are reported to the insurance company. We 

will consider that each policyholder has his own unknown retention limit, depending on the 

level occupied inside the bonus-malus scale as well as on observable characteristics (like 

age or gender, for instance). The policyholder reports the accident to the company only if its 

cost exceeds the retention limit. A regression model accounting for the fact that we observe 

the maximum between the accident cost and the retention limit is then fitted to the observed 

claim data. We then recover probability models for the accident costs (whereas formerly, 

we modelled claim costs). The claim frequencies can also be corrected in order to obtain 

accident frequencies (this is done in Section 5.4.2). 

Section 5.4.3 examines the optimal claiming strategy, that should be followed by rational 

policyholders. A strategy for each policyholder can be defined by a vector rl 0 rl s T 

where rl l is the retention limit for the policyholder occupying level l in the bonus-malus 

scale. This means that the cost of any accident of amount less than rl l is borne by the 

policyholder in level l. The claims causing higher costs are reported to the insurer. The 

problem is to determine optimal values for the rl l s. This can be done using the Lemaire 

algorithm. The optimal retention limits depend on the level occupied in the scale, on the 

annual expected claim frequency as well as on a discount rate. Note that the Lemaire 

algorithm gives the optimal retention limit obtained by means of dynamic programming. 

The resulting strategy should be adopted by rational policyholders, but may differ from the 

one empirically observed in insurance portfolios. The optimal retentions obtained from the 

Lemaire algorithm can also be seen as a measure of the toughness of the bonus-malus system. 

A system that induces large rl l s is more severe than another one yielding moderate retention 

limits. As such, the rl l s can also be used to measure the efficiency of the bonus-malus system. 

The claim costs play an important role in this chapter. As explained above, modelling 

claim sizes is a difficult issue because of the strong heterogeneity and of the presence of 

large claims. Nevertheless, it seems reasonable to agree that large claims will always be 

reported to the company, so that only moderate claims are subject to bonus hunger. 

In this chapter, we work within a given bonus-malus scale, whose levels are numbered 

from 0 to s, with fixed relativities r 0 r s (the r l s have been computed as explained in 

Chapter 4, or have been derived from marketing considerations, but are treated as given for 

the whole chapter). 

5.1.6 Descriptive Statistics for Portfolio C 

The numerical illustrations of this chapter are based on the observation of a Belgian motor 

third party liability insurance portfolio during the year 1997. This portfolio, henceforth 

referred to as Portfolio C, comprised 163 660 policies. 

The following variables are available for portfolio C: As far as policyholders’ 

characteristics are concerned, we know the Gender (male or female), the age (variable Ageph, 

four classes: 18–24, 25–30, 31–60 and > 60), the place of residence (variable City, rural or 

urban) and the Use of the car (private or professional). Concerning the insured vehicle, we 

know its age (variable Agev, four classes: 1–2, 3–5, 5–10 and > 10 years), the type of Fuel 

(petrol or gasoil) and its Power (three classes: < 66 kW, 66–110 kW and > 110kW). About 

the type of contract, we know whether the premium payment has been split up (variable 

Premium split, with payment once a year, or more than once a year) and the type of Coverage 

(motor third party liability only, or motor third party liability together with some more


Table 5.1 Descriptive statistics of the claim costs (only strictly 

positive values) for portfolio C. 

Statistic 

Value 

# observations 18 176 

Minimum 27.02 

Maximum 1 989 567.9 

Mean 1810.63 

Standard deviation 17 577.83 

25th percentile 145.02 

Median 598.17 




99th percentile 19 893.68 

Skewness 85.08 

optional coverages). In addition to these covariates, we know the number of claims filed by 

each policyholder during 1997, the exposure-to-risk from which these claims originated, as 

well as the resulting total claim amount. The information recorded in the data base dates 

from the end of June 1998 (6 months after the end of the observation period). Hence, most 

of the ‘small’ claims are settled and their final cost is known. However, for the large claims, 

we work here with incurred losses (payments made plus reserve). 

Descriptive statistics for claim costs are displayed in Table 5.1; we have at our disposal 

18 176 observed individual claim costs, ranging from E27.02 to almost E2 000 000, with a 

mean of E1810.63. We see in Table 5.1 that 25 % of the recorded claim costs are below 

E145.02, that half of them are smaller than E598.17, and that 90 % of them are less than 

E3021.87. The interquartile range is E1319.73. The observed claim cost distribution is highly 

asymmetric, with a skewness coefficient of about 85. 

5.2 Modelling Claim Severities 

5.2.1 Claim Severities in Motor Third Party Liability Insurance 

In nonlife business, the pure premium is the expected cost of all the claims that policyholders 

will file during the coverage period (under the assumption of the Law of Large Numbers). 

Let S i , i = 1n, be the total claim amount relating to policy number i. The S i s are 

assumed to be independent and identically distributed with common mean . The Law of 

Large Numbers ensures that 

[ 

Pr S n = 1 n 

n∑ 

i=1 

] 

S i → as n →+ = 1 

Under the conditions of the Law of Large Numbers, the pure premium is thus the expected 

claim amount.


The modelling of claim costs is much more difficult than claim frequencies. There are 

several reasons for this: In liability insurance, claims costs are often a mix of moderate 

and large claims. Usually, ‘large claim’ means exceeding some threshold, depending on the 

portfolio under study. This threshold can be selected using techniques from Extreme Value 

Theory, as described in Cebrian, Denuit & Lambert (2003). Large liability claims need 

several years to be settled. Only estimates of the final cost appear in the file until the claim 

is closed. Moreover, the statistics available to fit a model for claim severities are much more 

limited than for claim frequencies, since only 10 % of the policies in the portfolio produced 

claims. Finally, the cost of an accident is for the most part beyond the control of a policyholder 

since the payments of the insurance company are determined by third-party characteristics. 

The degree of care exercised by a driver mostly influences the number of accidents, but in 

a much lesser way the cost of these accidents. The information contained in the available 

observed covariates is usually much less relevant for claim sizes than for claim counts. 

In liability insurance, the settlement of larger claims often requires several years. Much of 

the data available for the recent accident years will therefore be incomplete, in the sense that 

the final claim cost will not be known. In this case, loss development factors can be used to 

obtain a final cost estimate. The average loss severity is then based on incurred loss data. 

In contrast to paid loss data (which are purely objective, representing the actual payments 

made by the company), incurred loss data include subjective reserve estimates. 

The total claim amount generated by policyholder i covered for motor third party liability 

can be represented as 

where 

Ni 

small 

C ik 

N large 

i 

N small N i∑ 

i∑ 

large 

S i = C ik + L ik (5.1) 

is the number of standard (or small) claims filed by policyholder i 

is the cost of the kth standard claim filed by policyholder i 

is the number of large claims filed by policyholder i 

L ik is the cost of the kth large claim filed by policyholder i. 

k=1 

All these random variables are assumed to be mutually independent. The random variables 

Ni 

small and N large 

i are analysed as explained in Chapters 1–2. Here, we explain how to model 

the C ik s and the L ik s. The first question to be addressed is to separate standard claims and 

large claims. 

k=1 

5.2.2 Determining the Large Claims with Extreme Value Theory 

Extreme Claim Amounts 

Gamma, LogNormal and Inverse Gaussian distributions (as well as other parametric models) 

have often been used by actuaries to fit claim sizes. However, when the main interest is in the 

tail of loss severity distributions, it is essential to have a good model for the largest claims. 

Distributions providing a good overall fit can be particularly bad at fitting the tails. Extreme 

Value Theory and Generalized Pareto distributions focus on the tails, being supported by 

strong theoretical arguments.


We only give hereafter a short non technical description of the fundaments of Extreme 

Value Theory; for more details, we refer the reader to Beirlant ET AL. (2004). Considering 

a sequence of independent and identically distributed random variables (claim severities, 

say) X 1 X 2 X 3 , most classical results from probability and statistics that are relevant 

for insurance are based on sums S n = ∑ n 

i=1 X i. Let us mention the Law of Large Numbers 

and the Central Limit Theorem, for instance. Another interesting yet less standard statistic 

for the actuary is M n = maxX 1 X n the maximum of the n claims. Extreme Value 

Theory mainly addresses the following question: how does M n behave in large samples (i.e. 

when n tends to infinity)? Of course, without further restriction, M n obviously diverges to 

+. Once M n is appropriately centered and normalized, however, it may converge to some 

specific limit distribution (of three different types, according to the fatness of the tails of the 

X i s). In insurance applications, heavy tailed distributions are most often encountered. Such 

distributions have survival functions that decay like a power function (in contrast to the 

Gamma, Inverse Gaussian or LogNormal survival functions, for instance, which all decay 

exponentially to zero). A prominent example of a heavy tailed distribution is the Pareto 

distribution, widely used by actuaries. 

Excess Over Threshold Approach and Generalized Pareto Distribution 

The traditional approach to Extreme Value Theory is based on extreme value limit 

distributions. Here, a model for extreme losses is based on the possible parametric form of 

the limit distribution of maxima. A more flexible model is known as the ‘Excesses Over 

Threshold’ method. This approach appears as an alternative to maxima analysis for studying 

the extreme behaviour of some random variables. Basically, given a series X 1 X n of 

independent and identically distributed random variables, the ‘Excesses Over Threshold’ 

method analyses the series X i − uX i >u, i = 1n, of the exceedances of the variable 

over a high threshold u. Mathematical theory supports the Poisson distribution for the number 

of exceedances combined with independent excesses over the threshold. 

Let F u stand for the common cumulative distribution function of the X i − uX i >us; 

F u thus represents the conditional distribution of the losses, given that they exceed the 

threshold u. The two-parameter Generalized Pareto distribution function G · provides a 

good approximation to the excess distribution F u over large thresholds. This two-parameter 

family is defined as 

G x = G 

( x 

 

) 

>0 

where 

{ 

1 − 1 + x −1/ if ≠ 0 

G x = 

1 − exp−x if = 0 

with x ≥ 0if ≥ 0 and x ∈ 0 −1/ if 0, the type II Pareto distribution when


For some appropriate function u and some Pareto index to be estimated from the 

data, the approximation 

F u x ≈ G u x x ≥ 0 (5.2) 

holds for large u. The approximation (5.2) is justified by the following formula 

lim 

sup 

u→+ x≥0 

∣ 

∣F u x − G u x ∣ = 0 (5.3) 

which is true provided that F satisfies some rather general technical conditions. These 

conditions are verified by the heavy tailed distributions. In view of (5.2) the excesses 

X i − uX i >ucan be treated as a random sample from the Generalized Pareto distribution 

provided the threshold u is large enough. 

Choice of the Threshold 

We have seen from (5.2) that if the heavy tailed character of the data is fulfilled, a high 

enough threshold is selected and enough data are available above that threshold, the use of 

Generalized Pareto distributions is justified to model large losses. The only practical problem 

in applying this result is how to determine what a ‘high enough threshold’ is; we deal with 

this problem in the present section. 

Two factors have to be taken into account in the choice of an optimal threshold u: 

• A value of u too large yields few exceedances and consequently imprecise estimates. 

• A value of u too small implies that the generalized Pareto character does not hold for 

the moderate observations and it yields biased estimates. This bias can be important as 

moderate observations usually constitute the largest proportion of the sample. 

Thus, our aim is to determine the minimum value of the threshold beyond which the 

Generalized Pareto distribution becomes a reasonable approximation to the tail of the 

distribution. 

To identify the optimal threshold value, we apply here two methods: 

Generalized Pareto Index Plot 

In virtue of the stability property of the Generalized Pareto distribution, if X is Generalized 

Pareto distributed with distribution function G , the variable X − uX>uis Generalized 

Pareto distributed with distribution function G +u , i.e. with the same index parameter , 

for any u>0. Consequently, in the plot of the index maximum likelihood estimators ˆ 

resulting from using increasing thresholds, we will observe that estimation stabilizes when 

the smallest threshold for which the Generalized Pareto behaviour holds is reached. 

Gertensgarbe Plot 

This procedure proposed by Gertensgarbe & Werner (1989) is very powerful and provides 

an estimation of the optimal threshold. Briefly, the Gertensgarbe plot aims to select a proper 

threshold, based on the determination of the starting point of the extreme value region. 

More precisely, given the series of differences i = x i − x i−1 , i = 2 3nof a sorted 

sample, x 1 ≤ x 2 ≤···≤x n , the starting point of the extreme region will be detected as a 

change point of the series i i= 2 3n. The key idea is that it may be reasonably


expected that the behaviour of the differences corresponding to the extreme observations 

will be different from the one corresponding to the non-extreme observations. This change 

of behaviour will appear as a change point of the series of differences. 

To identify the change point in a series, a sequential version of the Mann-Kendall test is 

applied. In this test, the normalized series U i is defined as 

where U ∗ 

i 

U i = U ∗ 

√ 

i 

− ii−1 

4 

ii−12i+5 

72 

= ∑ i 

k=2 n k, and n k is the number of values in 2 k lesser than k . Another 

series, denoted by U p , is calculated applying the same procedure to the series of the 

differences from the end to the start, n 2 , instead of from the start to the end. The 

intersection point between these two series, U i and U p , determines a probable change point 

that will be significant if it exceeds a high Normal percentile. 

Since, usually, these techniques can only provide approximative information about the 

threshold, simultaneous application of them is highly recommended in order to get more 

reliable results. 

Application to Claim Costs Recorded in Portfolio C 

The Generalized Pareto index plot is shown in Figure 5.1. We see that the estimates of 

tail parameter roughly stabilize after E85000. The Gertensgarbe plot gives a threshold of 

E104 397 (which corresponds to the 17th largest loss). The p-value of the Mann-Kendall 

 

0.4 

0.6 

Estimation of ξ 

0.8 

1.0 

50000 100000 150000 

Threshold 

Figure 5.1 Generalized Pareto index plot for the claim costs in Portfolio C.


test is equal to 261 %, so that the result is significant at the 5 % level. Considering the two 

analyses, the threshold for being qualified as a large claim is set to E100 000. 

5.2.3 Generalized Pareto Fit to the Costs of Large Claims 

Maximum Likelihood 

Now that the threshold defining the large losses has been determined as E100 000, we model 

the excesses over E100 000 with the help of a Generalized Pareto distribution. Descriptive 

statistics for the cost of large claims of Portfolio C are displayed in Table 5.2. The mean is 

equal to E364 6062. The limited number of large losses (17 for Portfolio C) does not allow 

for incorporating exogeneous information in these amounts. Therefore, the same parameters 

and are used for all the large losses. These parameters are estimated by maximum 

likelihood. 

Let us now fit the Generalized Pareto model to the excesses over E100 000. To this end, 

we use maximum likelihood theory, and we maximize the likelihood function given by 

= 

∏ 

ix i >100 000 

( ( 

1 

1 + − 

1 

x −1 

i − 100 000) ) 

The log-likelihood to be maximized is 

( 

L =−ln #x i x i > 100 000 − 1 + 1 ) ( ∑ 

ln 1 + ) 

 

x i − 100 000 

ix i >100 000 

where #x i x i > 100 000 = 17 in Portfolio C. This optimization problem requires numerical 

algorithms. There are different approaches to getting starting values for the parameters 

and . A natural approach consists of using moment conditions (that is, we equate sample 

mean and sample variance to their theoretical expressions involving and ). The values 

Table 5.2 Descriptive statistics of the cost of large claims 

(Portfolio C). 

Statistic 

Value 

Length 17 

Minimum 

104 3867 

Maximum 

1 989 5679 

Mean 

364 6062 

Standard deviation 

439 8827 

25th percentile 

140 0324 

Median 

252 2317 


407 4776 


499 7274 


797 8475 


1 751 2238 

Skewness 

29


of and coming from the linear fit to the empirical mean excess function could also be 

used, as shown below. 

Moment Method 

The mean and the variance of the Generalized Pareto distribution are respectively given 

by /1 − provided u 

− u = ∑ n 

i=1 x i − uIx i >u 

#x i x i >u 

where IA = 1 if the event A did occur and 0 otherwise. This means that eu is estimated by 

the sum of exceedances over the threshold u divided by the number of data points exceeding 

the threshold u. 

Usually, the mean excess function is evaluated on the observations of the sample. Denoting 

the sample observations arranged in ascending order as x 1 ≤ x 2 ≤···≤x n , we have in 

this case 

ê n x k = 1 n−k 

∑ 

x 

n − k k+j − x k 

It is easily checked that when X has a Generalized Pareto distribution function G , the 

mean excess function is a linear function in u 

eu = 

j=1 

 

1 − + 

1 − u 

provided +u > 0. Hence, the idea is to determine, on the basis of the graph of the empirical 

estimator of the excess function ê n , a region u + where ê n t becomes approximately 

linear for t ≥ u. The intercept and slope of a straight line fit to ê n determine the estimations 

of and .


Application to the Claim Costs Recorded in Portfolio C 

The initial values obtained with the the moment method applied to large claims of Portfolio 

C are ̂ 0 = 0319 and ̂ 0 = 180 1767. The maximum likelihood estimates of the Generalized 

Pareto parameters are ̂ = 04152 and ̂ = 156 7374. Different starting values have been 

used and the convergence always occurred. 

Note that the limited sample size for the large losses does not allow us to draw reliable 

conclusions about major claims (in particular, large sample properties of the maximum 

likelihood estimators cannot be invoked with a sample size as small as 17). In practice, the 

insurance company must gather the large losses together in a data base to perform detailed 

analysis. The amounts of these large losses have to be corrected for different sources of 

inflation. The assistance of a reinsurance company is useful in this respect, especially for 

insurers with small to moderate portfolios. 

5.2.4 Modelling the Number of Large Claims 

The number of large claims N large 

i for policyholder i is modelled using the oi large 

i 

distribution. This is in line with the fact that large claims occur purely at random. Poisson 

regression is then used to incorporate the information available about policyholder i, 

summarized in a vector xi 

T = x i1 x ip consisting of explanatory variables (assumed 

here to be categorical and coded by means of binary variables), in the expected frequency 

of large claims through a linear predictor by means of an exponential link function. 

The regression coefficients are estimated with Poisson regression. All the explanatory 

variables introduced in Section 5.1.6 have been excluded from the model. Only the intercept 

remained in the Poisson regression model. This is not surprising with such a small number of 

policies producing a large claim. We obtained ̂ 0 =−90555, with a standard deviation equal 

to 0.2425. The resulting frequency of large claims is 

̂ 

large 

i 

= 00117 % for all policyholders. 

Remark 5.1 (Logistic regression) In most cases, policyholders report 0 or just 1 large 

claim. Therefore, the number of large claims could also be modelled with a binary variable 

instead of a Poisson count. The probability that policyholder i reports (at least) a large claim 

can also be modelled with the help of logistic regression. Specifically, let us define the 

binary random variable J i 

{ 0 if policyholder i does not report any large claim during the observation period 

J i = 

1 if policyholder i reports at least one large claim during the observation period 

The aim is to model the probability PrJ i = 0 = q i x i that policyholder i does not report 

any large claim during the coverage period. 

Since q i x i ∈ 0 1, we resort to some distribution function F to link q i x i to the linear 

predictor 0 + ∑ p 

j=1 jx ij , that is 

( 

) 

p∑ 

p∑ 

q i x i = F 0 + j x ij ⇔ 0 + j x ij = F −1 q i x i 

j=1 

j=1


Theoretically, any distribution function F can be used; in practice, we often take for F 

the Normal or Logistic distribution function. For instance, the logistic regression model is 

specified as 

q 

ln i x i 

p∑ 

1 − q i x i = 0 + j x ij ⇔ q i = 

exp ( 0 + ∑ p 

j=1 ) 

jx ij 

1 + exp ( 0 + ∑ p 

j=1 ) 

jx ij 

j=1 

The SAS R /STAT procedure GENMOD can be used to perform this analysis. As in the 

Poisson regression case, all the explanatory variables described in Section 5.1.6 are excluded 

from the model. The estimated probability that policyholder i does not report any large 

claim is 99.99013 %. Hence, the corresponding large claim frequency is − ln 09999013 = 

000987 %, which is not too far from the estimated large 

i obtained using Poisson regression. 

5.2.5 Modelling the Costs of Moderate Claims 

Different models can be used to describe the behaviour of the moderate claims (i.e. claims 

with an incurred cost less than E100 000) as a function of the observable characteristics 

of the policyholder; including Gamma, Inverse Gaussian and LogNormal distributions. We 

briefly review these three regression models next. 

Gamma Distribution 

Here we use a new parameterization of the Gamma probability density function (1.34). 

Specifically, we use the mean as parameter, together with a parameter related to the variation 

coefficient. The probability density function with the new parameters = / and = is 

then given by 

fy = 1 

 

( ) y ( 

exp − y ) 1 

y (5.4) 

If Y has probability density function (5.4), then the first moments are given by 

EY = and VY = 2 

 

so that the variance is proportional to the square of the mean. Gamma regression assumes 

a coefficient of variation constantly equal to −1/2 . Thus it allows for heteroscedasticity 

(since the variance is proportional to the square of the mean, and is no more constant as in 

Gaussian regression models). Ideally, the Gamma regression model is best used with positive 

observations having a constant coefficient of variation. However, the model is robust to wide 

deviations from the latter assumption. 

The parameter controls the shape of the probability density function. Specifically, (i) if 

0


Let C ik be the cost of the kth claim reported by policyholder i; we assume that the 

individual claim costs C i1 C i2 are independent and identically distributed. Each C ik 

conforms to the Gamma law with mean 

( 

) 

p∑ 

i = EC ik x i = exp 0 + j x ij (5.5) 

and variance VC ik x i = 2 i 

/. Note that here we use an exponential link between the linear 

predictor 0 + ∑ p 

j=1 jx ij and the expected value i . For theoretical reasons, a reciprocal 

link function is sometimes preferable (but destroys the nice multiplicative structure of the 

resulting price list). 

Let n i be the number of claims reported by policyholder i, and let c i1 c i2 c ini be the 

corresponding claim costs. The likelihood associated with the observations is 

= ∏ 

n 

∏ i 

in i >0 k=1 

The corresponding log-likelihood is given by 

in i >0 

j=1 

( ( ) 1 

( 

cik 

exp − c ) ) 

ik 1 

 

i i c ik 

L= ln 

= ∑ 

( 

) ) 

n p∑ ∑ i 

(−n i ln + n i ln − 0 − j x ij + ln c ik − n 

∑ i 

c 

ik 

i 

+ constant 

The likelihood equations are given by 

 

L= 0 ⇔ ∑ 

j 

in i >0 

j=1 

k=1 

( 

x ij n i − c ) 

i• 

= 0 

i 

for j = 1p, where c i• = ∑ n i 

k=1 c ik is the total cost of the standard claims reported by 

policyholder i. The maximum likelihood estimators are obtained with the help of Newton- 

Raphson techniques. The estimation of can be performed by maximum likelihood as in 

Chapter 2, or it can be obtained from the Pearson- or deviance-based dispersion statistic. 

Remark 5.2 Often, only the total claim amount C i• is available, and not the individual C ik s. 

In such a case, it is convenient to work with the mean claim amount C i = C i• /n i , where 

n i is the number of claims reported by policyholder i. Considering the Gamma likelihood 

equations, this is not restrictive since only the total claim amount is needed. Specifically, 

Formula (1.36) shows that the Gamma distributions are closed under convolution in some 

particular cases. In the new parameterization of the Gamma family used in the present 

chapter, the moment generating function of C ik is 

( 

Mt = 1 − t ) − 

i 

 

k=1


so that the moment generating function of C i is 

n∏ 

j=1 

( ) ( t 

M = 1 − t ) − ′ 

i 

i 

n i i 

′ 

where ′ i = n i. Hence, the arithmetic average of the C ik s conforms to the am i n i i 

distribution. This situation is accounted for in GENMOD by specifying an appropriate 

weight n i . 

Example 5.1 (Gamma Regression for the Moderate Claim Costs in Portfolio C) The 

Gamma regression performed on the claim costs recorded in Portfolio C leads to the results 

in Table 5.3 where the following variables have been eliminated: Fuel (p-value of 9476 %), 

Gender (p-value of 8889 %), Use (p-value of 2748 %) and Power (p-value of 928 %). 

Moreover, for Agev, levels 6–10 and > 10 have been grouped together in a class > 5. For 

the variable Ageph, levels 31–60 and > 60 have been grouped in a class > 30. The resulting 

log-likelihood is equal to −147 62910. Type 3 analysis is presented in the following table: 


Ageph 2 6592


Remark 5.3 (Tweedie Generalized Linear Models) The Tweedie distributions are a threeparameter 

family. They allow for any power variance function and any power link. The 

Tweedie family includes the Gaussian, Poisson, Gamma and Inverse Gaussian families as 

special cases. Specifically, let i = EY i be the expectation of the ith response Y i .We 

assume that 

p∑ 

q 1 

i = 0 + j x ij and VY i = q 2 

i 

j=1 

where x i is a vector of covariates and is a vector of regression cofficients, for some , q 1 

and q 2 . A value of zero for q 1 is interpreted as ln i = 0 + ∑ p 

j=1 jx ij . The variance power 

q 2 characterizes the distribution of the responses Y . The parameter q 2 is called the index 

parameter and determines the shape of the Tweedie distribution. For various values of q 2 , 

we find the following particular cases: q 2 = 0 corresponds to the Normal distribution, q 2 = 1 

corresponds to the Poisson distribution, 1 2 corresponds to stable 

distributions for positive continuous data. 

As stated above, for 1 0 k=1 22 cik 

3 2 i 2 c ik 

)


The log-likelihood is 

L 2 = ln 2 

= ∑ 

( 

− n i 

2 ln22 − 

in i >0 

so that the likelihood equations are given by 

 

j 

L 2 = 0 ⇔ 

 

j 

⇔ ∑ 

in i >0 

∑ 

n 

∑ i 

k=1 

n 

∑ i 

in i >0 k=1 

) 

c ik − i 2 

+ constant 

2 i 2 c ik 

( 

1 − c ik 

i 

) 2 

1 

c ik 

= 0 

( 

x ij 

n 

i − c ) 

i• 

= 0 

i i 

The estimation of 2 can be performed by maximum likelihood as in Chapter 2, or it can 

be obtained from the Pearson- or deviance-based dispersion statistic. 

Remark 5.4 As pointed out for the Gamma distribution in Remark 5.2, the actuary often 

only has at his disposal the total claim amount C i• , and not the individual C ik s. This is 

not really a problem since the likelihood equations only involve C i• . Considering (1.40), 

we see that the moment generating function of the mean claim amount C i in the new 

parameterization is given by 

√ 

1 − 2 2 2 i 

( 

exp − n ( 

i 

1 − 

2 i 

)) 

t 

n i 

which corresponds to the Inverse Gaussian distribution with parameters i and 2 /n i .Asin 

the Gamma case, working with the average claim amounts is not restrictive for maximum 

likelihood estimation of the regression parameters, and this situation is accounted for in 

GENMOD by specifying an appropriate weight n i . 

Example 5.2 (Inverse Gaussian Regression for the Moderate Claim Costs in Portfolio C) 

The results of the Inverse Gaussian regression are given in Table 5.4 where the following 

variables have been eliminated: Fuel (p-value of 96.29 %), Gender (p-value of 84.58 %), 

Power (p-value of 78.32 %), Use (p-value of 56.40 %), Premium split (p-value of 23.04 %), 

City (p-value of 975 %) and Coverage (p-value of 5.37 %). Moreover, for Agev, levels 3–5, 

6–10 and > 10 have been grouped together in a class > 2. For the variable Ageph, levels 

31–60 and > 60 have been grouped in a class > 30. The resulting log-likelihood is equal to 

−150 21460. Type 3 analysis is presented in the following table: 


Ageph 2 1056 00012 

Agev 1 4187


Table 5.4 Results of the Inverse Gaussian regression on the claim costs recorded in Portfolio C. 


Intercept 71169 00282 70616 71722 636343 < .0001 

Ageph 18–24 02293 01104 00129 04457 431 0.0378 

Ageph 25–30 01699 00697 00334 03065 595 0.0147 

Ageph > 30 0 0 0 0 . . 

Agev 0–2 01609 00756 00127 03091 453 0.0334 

Agev > 2 0 0 0 0 . . 

LogNormal Distribution 

Before the generalized linear models gained popularity in the actuarial profession, claim sizes 

were often analysed using a Normal linear regression model after having been transformed 

to the log-scale. Although the results are usually quite similar for this method and Gamma 

regression, the latter approach is easier to interpret since it does not require any logarithmic 

transformation of the claim costs. 

Assume that the moderate claim sizes for policyholder i are independent and LogNormally 

distributed, with parameters 0 + ∑ p 

j=1 jx ij and 2 . Specifically, the C ik s are independent, 

and identically distributed for fixed i, with C ik ∼ Nor 0 + ∑ p 

j=1 jx ij 2 . Let n i be the 

number of claims reported by policyholder i, and let c i1 c i2 c ini be the corresponding 

claim costs. The likelihood associated with the observations is 

= ∏ 

n 

∏ i 

in i >0 k=1 

⎛ ( 

1 

√ 

2cik exp ⎝− 1 

2 2 

ln c ik − 0 − 

p∑ 

j=1 

j x ij 

) 2 

⎞ 

⎠ 

The maximum likelihood estimators are obtained with the help of Newton-Raphson 

techniques. The average cost of a standard claim for policyholder i is then obtained from 

the formula 

( 

) 

p∑ 

EC ik x i = exp 0 + j x ij + 2 

 

2 

Note that, in contrast to the Gamma and Inverse Gaussian cases, we cannot easily deal with 

the situation where only the total amount of moderate claims is available. This is due to the 

fact that the LogNormal family of distributions is not closed under convolution. Therefore, 

we fit the model to the observations made on policyholders having filed a single standard 

claim. 

Example 5.3 (LogNormal Regression for the Moderate Claim Costs in Portfolio C) The 

LogNormal regression cannot be performed with the help of SAS R /STAT procedure 

GENMOD (which does not support the LogNormal distribution). Often in practice, the 

data are first transformed on the log-scale, and a standard linear model is then fitted to 

the logarithms of the claim amounts. This ad-hoc procedure will be avoided here, and a 

maximum likelihood estimation procedure is performed on the original claim costs. 

j=1


Table 5.5 Results of the LogNormal regression analysis for the claim costs recorded in Portfolio C. 

Variable Level Coeff Std error Wald 95 % conf limit t-value Pr >t 

Intercept 61223 00268 60702 61744 23038 60 01781 00312 01169 02392 571 Chi-sq 

Ageph 2 4058 < .0001 

City 1 770 0.0055 

Agev 3 3820 < .0001 

All the three remaining variables are statistically significant, and must be kept in the model. 

5.2.6 Resulting Price List for Portfolio C 

Formula for the Pure Premium 

Let freq be the vector of the regression coefficient for the claim frequencies, that is, 

the expected annual number of standard claims is exp freq 

0 + ∑ p 

j=1 freq j x ij . Similarly, 

let cost be the regression coefficient for the moderate claim sizes. We retain here the 

LogNormal modelling for the moderate claim sizes, so that the expected moderate claim 

amount is exp cost 

0 

+ ∑ p 

j=1 cost j 

x ij + 2 /2. Neglecting the large claims, the pure premium 

for policyholder i is given by 

⎡ ⎤ 

Ni∑ 

small 

( 

p∑ ( 

E ⎣ C ik 

⎦ = EN small 

i 

EC i1 = exp freq 

0 + cost 

0 

+ 

freq 

) ) 

j + cost 

j xij + 2 

 

2 

k=1 

If all the components of x i are binary, we then get a multiplicative price list. The total 

premium is then obtained by adding the expected cost of the large losses, i.e. EN large 

i EL i1 . 

j=1


We still need to estimate freq to be able to compute the pure premium for the different 

categories of policyholders in Portfolio C. 

Risk Classification for Claim Frequencies 

Here we follow the method described in Chapter 2 for analysing the observed claim numbers. 

A Poisson regression is first performed on the claim frequencies of Portfolio C. This leads 

to the results of Table 5.6. Only the Use of the vehicle has been removed from the model 

(p-value of 3448 %). The levels 0–2, 6–10 and > 10 of the variable Agev are grouped 

together. The log-likelihood of the final model is equal to −61 5639 and the Type 3 analysis 

is presented in the following table: 


Ageph 3 89266


As explained in Chapter 2, overdispersion can be detected by using the statistics T 1 , 

T 2 or T 3 presented in Section 2.4.6. The values obtained with Portfolio C are T 1 = 2004, 

T 2 = 1456 and T 3 = 1289. All the associated p-values are less than 10 −4 leading to the 

rejection of the null hypothesis in favour of a mixed Poisson model. 

In order to take the residual heterogeneity into account, we fitted a Negative Binomial 

regression model to the Portfolio C. The results are given in Table 5.7. All the variables and 

levels are still relevant. The log-likelihood is now equal to −61 393.3 and is better than with 

Poisson regression. 

Resulting Price List 

The resulting price list is obtained thanks to the Negative Binomial model for the frequency 

(presented in Table 5.7) and to the LogNormal model for the average cost of the standard 

claims (presented in Table 5.5). 

Neglecting the large claims, the pure premium of the reference class is obtained by 

( 

) 

exp freq 

0 + cost 

0 

+ 2 

= E31676 (5.6) 

2 

This amount corresponds to the pure premium of a male policyholder living in an urban 

area, aged between 31 and 60, driving a car older than 10 years, using gasoil and with a 

power greater than 110 kW, paying his premium in several installments and having opted for 

a more extensive coverage than only the compulsory motor third party liability insurance. 

To obtain the pure premium (neglecting the large claims) for a policyholder belonging to 

another risk class, the percentages of Table 5.8 must be applied to the base pure premium 

Table 5.7 Results of the Negative Binomial regression of the claim counts recorded in Portfolio C. 


Intercept −14626 00720 −16037 −13216 41306


Table 5.8 Price list for the standard claims in Portfolio C. 

Variable Level Influence of 

frequency 

Influence of 

cost 

Total influence 

Ageph 18–24 19033 % 11908 % 22664 % 

Ageph 25–30 14339 % 10000 % 14339 % 

Ageph > 60 7997 % 11949 % 9556 % 

Ageph 31–60 10000 % 10000 % 10000 % 

Gender Female 10622 % 10000 % 10622 % 

Gender Male 10000 % 10000 % 10000 % 

Agev 0–2 10000 % 8844 % 8844 % 

Agev 3–5 9397 % 8093 % 7605 % 

Agev 6–10 10000 % 8907 % 8907 % 

Agev > 10 10000 % 10000 % 10000 % 

Fuel Petrol 8256 % 10000 % 8256 % 

Fuel Gasoil 10000 % 10000 % 10000 % 

Premium split No 7594 % 10000 % 7594 % 

Premium split Yes 10000 % 10000 % 10000 % 

Coverage MTPL only 11266 % 10000 % 11266 % 

Coverage More 10000 % 10000 % 10000 % 

City Rural 7969 % 10683 % 8514 % 

City Urban 10000 % 10000 % 10000 % 

Power < 66 kW 7445 % 10000 % 7445 % 

Power 66–110 kW 8260 % 10000 % 8260 % 

Power > 110 kW 10000 % 10000 % 10000 % 

(5.6) as a function of the characteristics of the policyholder. For example, the pure premium 

of a woman aged between 25 and 30, living in a rural region, driving a car older than 10 

years, using petrol and with a power less than 66 kW, paying her premium once a year and 

being covered only for the MTPL will be equal to 

E31676 × 10622 % (correction for being a female policyholder) 

× 14339 % (correction for being aged between 25 and 30) 

× 8514 % (correction for living in a rural area) 

× 8256 % (correction for using petrol) 

× 7445 % (correction for driving a low-power car) 

× 7594 % (correction for paying the premium once a year) 

× 11266 % (correction for buying MTPL coverage only) 

= E21601 

Finally, to obtain the total pure premium of a policyholder, the expected cost of large 

claims (which is independent of the explanatory variables) must be added. The value is equal 

to 

00117 % × E368 01880 = E4306


The total pure premium of a policyholder belonging to the reference class is then 

E31676 + E4306 = E35982 

Note that the fitted mean equal to E 368 018.80 is close to the observed mean of the 17 large 

losses recorded in Table 5.2. 

Recall that the analysis of large claims performed in this chapter is based on an estimation 

of their final cost six months after the end of the observation year 1997. We thus work with 

incurred losses (payments plus reserves). In practice, the company should maintain a data 

base recording the costs of large claims that have occurred in the past, corrected for the 

different sources of inflation (reinsurance companies can often provide valuable assistance 

to the ceding companies in this respect). The typical price for a reinsurance treaty covering 

motor third party liability insurance losses in excess of E 350 000 represents about 5 % of 

the total motor premium income of a Belgian insurance company, so that the expected cost 

of large claims computed in Portfolio C seems to be of the right order of magnitude. 

Remark 5.5 If the sum of the individual pure premiums obtained above exceeds the 

observed total loss for the insurance portfolio during the reference period, or if we expect 

larger losses in the future (because, e.g., of different sources of inflation), we can then 

keep the same relative premium amounts applied to the anticipated future total claim cost. 

This allows us to incorporate in the individual premiums observed trends in the total claim 

amount. 

5.3 Measures of Efficiency for Bonus-Malus Scales 

The elasticity of a bonus-malus system measures its response to a change in the expected 

claim frequency or expected aggregate claim amount. We expect that the premium paid by 

the policyholders subject to bonus-malus scales is increasing in the expected claim frequency 

or total claim amount. The rate of increase is related to the concept of efficiency. 

5.3.1 Loimaranta Efficiency 

Definition 

Let us denote as r the average relativity once stationarity has been reached, for a 

policyholder with annual expected claim frequency , i.e. 

r = 

s∑ 

l r l 

l=0 

The Loimaranta efficiency Eff Loi is then defined as the elasticity of the relative premium 

induced by the bonus-malus system, that is, 

Eff Loi = 

dr 

r 

d 

 

= 

d ln r 

d ln


Computation 

The computation of Eff Loi requires the determination of the derivative of r with 

respect to the annual expected claim frequency . This derivative is given by 

dr 

d 

= s∑ 

l=0 

d l 

d 

r l 

so that its computation requires the derivative of the stationary probabilities l with 

respect to the annual expected claim frequency . To get d l /d, it suffices to 

differentiate (4.8): we thus have to solve the linear system 

⎧ 

⎪⎨ 

d T 

= dT 

d d 

P + T dP 

d 

⎪⎩ 

∑ s d l 

l=0 

d = 0 

with respect to the d l /ds. 

Global Efficiency 

So far, we have defined the Loimaranta efficiency for a given value of the expected annual 

claim frequency. To get a value for the portfolio, we have to account for its composition 

with respect to rating factors as well as its residual heterogeneity. Hence, the Loimaranta 

efficiency for the portfolio is obtained by averaging over all the possible values for as 

[ ( ) ] 

Eff Loi = E Eff Loi 

Loimaranta Efficiency in Portfolio A 

Table 5.9 displays the Loimaranta efficiencies for a good driver, with annual expected claim 

frequency 9.28 %, for an average driver with annual expected claim frequency 14.09 %, and 

for a bad driver with annual expected claim frequency 28.40 %. For the −1/top bonus-malus 

scale, the efficiency is larger for the average driver than for the good and bad ones. On 

the contrary, for the −1/+ 2 and −1/+ 3 bonus-malus scales, the efficiencies appear to 

increase from the good to the average driver, and from the average driver to the bad one. 

The global efficiencies listed in Table 5.9 are rather poor, ranging from 23.23 % for the 

−1/top bonus-malus scale to 28.39 % in the −1/+ 2 bonus-malus scale. This means that 

these bonus-malus systems weakly respond to a change in the underlying claim frequency. 

Table 5.9 Loimaranta efficiency for three types of insured drivers (a good driver, with annual 

expected claim frequency 9.28 %, an average driver with annual expected claim frequency 14.09 %, 

and a bad driver with annual expected claim frequency 28.40 %) and global efficiency for Portfolio 

A, for the −1/top, −1/+ 2 and −1/+ 3 bonus-malus scales. 

Frequency Scale −1/top Scale −1/ + 2 Scale −1/ + 3 

0.0928 02865 02380 02987 

0.1409 03144 03793 04008 

0.2840 02901 06204 04733 

Portfolio A 02323 02839 02775


Figure 5.2 displays the Loimaranta efficiency as a function of the annual expected claim 

frequencies for the three bonus-malus systems. We see that the efficiency first increases 

and then decreases, reaching its maximum value at about 15 % for the −1/top bonus-malus 

system, at about 30 % for the −1/+2 bonus-malus system, and at about 25 % for the −1/+3 

bonus-malus system. 

5.3.2 De Pril Efficiency 

Definition 

Loimaranta efficiency is an asymptotic concept that does not depend on the level presently 

occupied in the scale. De Pril efficiency is a transient concept that explicitly considers time 

value of money. 

Let v


Loimaranta efficiency 

0.32 

0.30 

0.28 

0.26 

0.24 

0.22 

0.20 

0.18 

0.16 

0.14 

0.12 

0.10 

0.08 

0.06 

0.04 

0.02 

0.00 

0.7 

0.6 

Scale –1/top 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 

Claims frequency 

Scale –1/+2 


0.5 

0.4 

0.3 

0.2 

0.1 

0.0 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 



0.48 

0.46 

0.44 

0.42 

0.40 

0.38 

0.36 

0.34 

0.32 

0.30 

0.28 

0.26 

0.24 

0.22 

0.20 

0.18 

0.16 

0.14 

0.12 

0.10 

0.08 

0.06 

0.04 

0.02 

0.00 

Scale –1/+3 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 


Figure 5.2 Loimaranta efficiency as a function of the annual expected claim frequencies for the three 

bonus-malus systems and Portfolio A.


De Pril efficiency Eff DeP is thus defined analogously to Eff Loi , substituting V l for r. 

Note that Eff DeP l now depends on the starting class l. The initial class can then be 

selected so as to maximize Eff DeP l . 

Computation 

To compute Eff DeP l , we need the derivatives dV l /d of the V l satisfying (5.7). 

These derivatives can be obtained by solving the system 

dV l 

d 

= v ∑ exp− k 

k! 

k=0 

This system admits a unique solution. 

(( ) k 

− 1 V Tk l + dV ) 

T k l 

l= 0s 

d 

Global Efficiency 

At the portfolio level, the efficiency is then obtained by averaging over all the possible 

values for , that is, 

Eff DeP l = EEff DeP l 

De Pril Efficiency in Portfolio A 

Table 5.10 displays the De Pril efficiencies associated with the highest level 5 for a good 

driver, with annual expected claim frequency 9.28 %; for an average driver with annual 

expected claim frequency 14.09 %; and for a bad driver with annual expected claim frequency 

28.40 %. The discount factor is taken to be v = 1/104. De Pril efficiency behaves roughly 

as the Loimaranta efficiency displayed in Table 5.9. 

Figure 5.3 displays the De Pril efficiency as a function of the annual expected claim 

frequencies for the three bonus-malus systems. We see that the efficiency first increases and 

then decreases, reaching its maximum value at about the same frequencies as the Loimaranta 

efficiency. The shape of both efficiencies is pretty much the same. 

Let us now use the De Pril efficiency to select the optimal starting level. We have computed 

in Table 5.11 the values of Eff DeP l and Eff DeP l according to the initial level. We see 

that the optimal starting level is 0 for the three −1/top, −1/ + 2 and −1/ + 3 bonus-malus 

scales. 

Table 5.10 De Pril efficiency for three types of insured drivers (a good driver, with annual expected 

claim frequency 9.28 %, an average driver with annual expected claim frequency 14.09 %, and a bad 

driver with annual expected claim frequency 28.40 %) and global efficiency for Portfolio A, for the 

−1/top, −1/+ 2 and −1/+ 3 bonus-malus scales (starting level: level 5). 

Frequency Scale −1/top Scale −1/ + 2 Scale −1/ + 3 

0.0928 02192 01871 02252 

0.1409 02482 02880 03039 

0.2840 02414 04633 03741 

Portfolio A 01817 02186 02150


De Pril efficiency 

0.26 

0.24 

0.22 

0.20 

0.18 

0.16 

0.14 

0.12 

0.10 

0.08 

0.06 

0.04 

0.02 

0.00 

Scale –1/top 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 




0.48 

0.46 

0.44 

0.42 

0.40 

0.38 

0.36 

0.34 

0.32 

0.30 

0.28 

0.26 

0.24 

0.22 

0.20 

0.18 

0.16 

0.14 

0.12 

0.10 

0.08 

0.06 

0.04 

0.02 

0.00 

0.38 

0.36 

0.34 

0.32 

0.30 

0.28 

0.26 

0.24 

0.22 

0.20 

0.18 

0.16 

0.14 

0.12 

0.10 

0.08 

0.06 

0.04 

0.02 

0.00 

Scale –1/+2 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 


Scale –1/+3 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 


Figure 5.3 De Pril efficiency as a function of the annual expected claim frequencies for the three 

bonus-malus systems and Portfolio A.


Table 5.11 De Pril efficiency according to the starting level, for the −1/top, −1/ + 2 and −1/ + 3 

bonus-malus scales. 

Initial level 

−1/top scale 

Eff DeP l 00928 Eff DeP l 01409 Eff DeP l 02804 Eff DeP l 

0 02711379 03018164 0287755 02230948 

1 02644811 02952273 02826536 02179881 

2 02558848 02863492 02746372 02109572 

3 02454854 02755109 02648401 02025042 

4 02332928 02628293 02537603 01927769 

5 02191762 0248209 02414355 01817001 

Initial level 

−1/ + 2 scale 


0 02168178 03422846 05689437 0262244 

1 02159419 03399076 05617233 02597218 

2 02156974 03369153 05488944 02558833 

3 02120306 03274338 05247412 02474823 

4 0202593 0311189 04963046 02351387 

5 01870807 02878845 04633192 02186229 

Initial level 

−1/ + 3 scale 


0 02733728 03689638 04526559 02605588 

1 02691461 0362999 04449703 02563432 

2 02648276 03556598 04328504 02508412 

3 02572639 03446637 0417896 02429854 

4 02429893 03261114 03971923 02302266 

5 02252077 03038548 03740766 02149911 

5.4 Bonus Hunger and Optimal Retention 

5.4.1 Correcting the Estimations for Censoring 

As explained in the introduction, the policyholders subject to a bonus-malus mechanism tend 

to self-defray minor accidents to avoid premium surcharges. This means that the number of 

accidents is a censored variable: the insurer only knows the number of claims filed by the 

insured drivers, and not the number of accidents they caused. We develop here a simple 

statistical model allowing for censorship in the observed claim costs (and thus also in the 

observed numbers of claims reported to the insurer). The claiming threshold is considered 

here as a random variable, specific to each policyholder and with a distribution depending 

on the level occupied in the bonus-malus scale at the beginning of the observation period as 

well as on observable characteristics. 

Specifically, let us consider the LogNormal model for moderate claim sizes: the claim 

costs are then seen as independent and identically distributed realizations of LogNormal 

random variables in each risk class. Now, each policyholder in this class reports an accident


to the insurer if its cost exceeds a random threshold, assumed to be LogNormally distributed 

with parameters specific to the level occupied in the scale. Note that we deal here with 

moderate claim sizes only. The reason is that large claims are not subject to bonus hunger 

and are systematically reported to the insurer. The frequency of large claims will have to be 

added to the corrected frequency of moderate claims to get the actual number of accidents 

caused each year. 

Let l i be the level occupied by policyholder i in the bonus-malus scale at the beginning 

of the period, and let RL i l i ∼ N or i 2 be the random optimal retention, with a linear 

predictor of the form 

i = 0 + 

p∑ 

j x ij + fl i 

j=1 

specific to policyholder i, where the function f· expresses the effect of occupying level l i 

in the bonus-malus scale. Considering the values obtained for the optimal retention in the 

literature (reaching a maximum somewhere in the middle of the scale, and decreasing when 

approaching uppermost and lowermost levels), we will use here a quadratic effect f of l i . 

Note that other approaches are possible (see the references in the closing section for more 

details). 

This means that policyholder i will report all the accidents with a cost larger than RL i l i , 

and defray himself all those with a cost less than RL i l i . At the portfolio level, the 

RL i l i s are assumed to be independent. Now, let CA ik be the cost of the kth accident 

caused by policyholder i. We assume that for each i the random variables CA i1 CA i2 

are 

∑ 

independent and identically distributed, with CA ik ∼ N or i 2 , where i = 0 + 

p 

j=1 jx ij . Moreover, the CA ik s and RL i l i are mutually independent. We consider here 

the explanatory variables selected in the LogNormal analysis of the censored claim costs, 

presented in Table 5.5. 

Now, denoting as c i1 c ini the costs of the n i moderate claims filed by policyholder i, 

the likelihood is 

2 = ∏ n 

∏ i 

f i c ik 

f i c ik = 

in i >0 k=1 

where f i · denotes the probability density function of CA ik given CA ik >RL i l i . Each 

factor involved in the likelihood can be written is 

1 

√ exp 

(− ln c ( ) 

ik − i 2 ln cik − i 

2cik 

2 2 ) 

 

( 

1 − − ) 

i − 

√ i 

2 + 2 

 

 

The estimators of the parameters , , 2 , and are determined by maximizing the likelihood 

2 . 

This basic model could be refined in different respects. Firstly, the retention limit could 

depend on the number of claims previously filed by the policyholder during the same 

year. The retention for the second claim depends on the level to which the policyholder is


transferred after the first claim has been reported. Here, we only use the observations related 

to the policyholders having filed a single standard claim during the observation period (so 

that we have the exact cost of this claim at our disposal). Also, the estimation of f could 

be performed in a nonparametric way, allowing for an effect fl i of being in level l i (and 

imposing some smoothness in the fls if needed) before selecting an appropriate parametric 

specification. 

Let us now apply this methodology to Portfolio C. The policyholders have been subject 

to the 23-level former compulsory Belgian bonus-malus scale. The estimation of the 

regression parameters in the model containing all the explanatory variables are displayed in 

Table 5.12. Here, we take fl = l − 13 2 . The log-likelihood is −130 1542565. We see 

from this table that several covariates are not significant. Therefore, we adopt a backward 

selection procedure, and exclude the irrelevant covariates. This yields the results displayed in 

Table 5.13. The log-likelihood is now −130 1554197. The parameters and are estimated 

at ̂ = 16821 and ̂ = 10286. 

Compared to the LogNormal fit to the claim costs displayed in Table 5.5, we see that 

the intercept is now smaller, as expected. The age classes have been modified, and the 

young drivers seem to cause more expensive accidents. The effect of the covariate City 

remains approximately the same. The categories for the age of the vehicle have also been 

modified. 

Table 5.12 Fit of the model for the accident costs subject to bonus hunger in Portfolio C, containing 

all the explanatory variables. 

Variable Level Coeff Std error Wald 95 % conf limits Chi-sq Pr>Chi-sq 

Intercept 58028 00453 57122 58934 1640911 < 00001 

Ageph 18−24 02624 00640 01344 03903 1682 < 00001 

Ageph > 60 00717 00500 −00282 01716 206 01512 

Ageph 25−60 0 0 0 0 . . 

City Rural 00459 00275 −00091 01009 279 00948 

City Urban 0 0 0 0 . . 

Agev 0−2 00327 00652 −00977 01631 025 06158 

Agev 3−5 −01721 00465 −02652 −00791 1369 00002 

Agev 6−10 −01445 00357 −02159 −00732 1640 00001 

Agev > 10 0 0 0 0 . . 


Intercept 35045 00904 33238 36852 150445 < 00001 

Ageph 18−24 −03244 01637 −06519 00031 393 00476 

Ageph > 60 05419 01105 03209 07630 2405 < 00001 

Ageph 25−60 0 0 0 0 . . 

City Rural 00796 00416 −00036 01628 366 00558 

City Urban 0 0 0 0 . . 

Agev 0−2 −07825 02621 −13067 −02582 891 00028 

Agev 3−5 −03406 01107 −05620 −01193 947 00021 

Agev 6−10 00190 00449 −00708 01087 018 06725 

Agev > 10 0 0 0 0 . . 

BM level −00011 00006 −00022 00000 367 00555


Table 5.13 Fit of the final model for the accident costs subject to bonus hunger in Portfolio C. 


Intercept 58084 00402 57279 58889 2084738 < 00001 

Ageph 18–24 02522 00661 01201 03843 1457 00001 

Ageph > 24 0 0 0 0 . . 

City Rural 00692 00274 00145 01240 639 00115 

City Urban 0 0 0 0 . . 

Agev 3–5 −01866 00394 −02653 −01078 2245 < 00001 

Agev 6–10 −01582 00349 −02281 −00883 2051 < 00001 

Agev 0–2 & > 10 0 0 0 0 . 


Intercept 35269 00870 33529 37010 164272 < 00001 

Ageph 18–24 −03077 01430 −05937 −00217 463 00314 

Ageph > 60 06479 00852 04775 08183 5781 < 00001 

Ageph 25–60 0 0 0 0 . . 

Agev 0–2 −07126 01171 −09468 −04785 3704 < 00001 

Agev 3–5 −03170 00666 −04503 −01837 2263 < 00001 

Agev > 5 0 0 0 0 . . 

BM level −00011 00006 −00022 00000 370 00546 

Concerning the retention levels, we see that young drivers are more likely to defray only 

relatively cheap accidents, whereas older drivers are ready to self-defray more expensive 

accidents. The more recent the vehicle, the less accidents are self-defrayed. This may be due 

to the fact that comprehensive coverage is often bought for new vehicles, so that claims are 

filed to both third party liability and comprehensive. The effect of the level occupied in the 

scale is as follows: policyholders occupying the middle of the scale are ready to defray more 

expensive accidents than policyholders at the top or at the bottom of the scale. 

5.4.2 Number of Claims and Number of Accidents 

Let Mi 

small be the number of small accidents caused by policyholder i. The number of 

moderate claims filed by policyholder i is then given by 

By equating the expectations, we get 

EN small 

i 

= i = 

Mi∑ 

small 

N small 

i 

= ICA ik >RL i l i 

= 

+∑ 

k=0 

+∑ 

k=0 

k=1 

PrM small 

i 

PrM small 

i 

= k 

k∑ 

PrCA ij >RL i l i 

j=1 

= kk PrCA i1 >RL i l i 

= PrCA i1 >RL i l i EM small 

i


Hence, the expected number of moderate accidents is given by 

EM small 

i 

= ˜ i = 

i 

PrCA i1 >RL i l i = 

i 

( 

1 − − ) 

i − 

√ i 

2 + 2 

The expected annual number of minor accidents ˜ i caused by policyholder i in Portfolio C 

are displayed in Table 5.14. 

The total number of accidents M i caused by policyholder i is then equal to 

M i = M small 

i 

+ N large 

i 

since all the major accidents are reported to the company. The actual number of claims 

originating from the M i accidents is 

N i = N small 

i 

+ N large 

i 

Table 5.14 Expected annual claim frequency and corresponding expected annual 

accident frequency for the different risk classes in Portfolio C. 

Risk class Claim frequency Accident frequency 

18–24 + Rural + 0–2 0.3513403 03633521 

18–24 + Rural + 3–5 0.3301527 03517856 

18–24 + Rural + 6–10 0.3513403 03824574 

18–24 + Rural+>10 0.3513403 03777624 

18–24 + Urban + 0–2 0.4408723 04572087 

18–24 + Urban + 3–5 0.4142855 04435002 

18–24 + Urban + 6–10 0.4408723 04827648 

18 − 24 + Urban+>10 0.4408723 04765034 

>60 + Rural + 0–2 0.1476220 01659192 

>60 + Rural + 3–5 0.1387197 01683992 

>60 + Rural + 6–10 0.1476220 0188461 

>60 + Rural+>10 0.1476220 01831299 

>60 + Urban + 0–2 0.1852406 0209793 

>60 + Urban + 3–5 0.1740696 02137084 

>60 + Urban + 6–10 0.1852406 02396818 

>60 + Urban+>10 0.1852406 02326217 

25–60 + Rural + 0–2 0.1845933 01964037 

25–60 + Rural + 3–5 0.1734614 0193623 

25–60 + Rural + 6–10 0.1845933 02129273 

25–60 + Rural+>10 0.1845933 0208954 

25–60 + Urban + 0–2 0.2316332 02475869 

25–60 + Urban + 3–5 0.2176646 02447353 

25–60 + Urban + 6–10 0.2316332 02695796 

25–60 + Urban+>10 0.2316332 0264303


Moreover, the random variable N large 

i is independent from Mi 

small Ni small . Both M i and 

N i are mixed Poisson distributed. Specifically, given i = , M i ∼ oi˜ i + large 

i and 

N i ∼ oi i + large 

i . Note that the variable that has been analysed in the preceding chapters 

is N i , the number of accidents reported to the insurer, and not M i . 

5.4.3 Lemaire Algorithm for the Determination of Optimal Retention 

Limits 

In the preceding section, we explained how to correct the cost of claims to obtain the accident 

costs. This also allowed us to switch from claim frequencies to accident frequencies. To this 

end, we estimated the retention limits that were used by the policyholders on the basis of the 

costs of the claims they filed to the insurance company. The aim of this section is somewhat 

different. Having the distribution of the accident costs and of the accident frequencies, we 

would like to determine the optimal claiming strategy (which may differ from the observed 

claiming strategy inferred in the previous section). 

For each level of the scale, a critical claim size is determined: if the cost of the claim falls 

below this critical threshold then the rational policyholder should not report the accident to 

the company. Conversely if the cost exceeds this threshold, the rational policyholder should 

report the claim to the company. Note the close similarity with deductibles: under coherent 

behaviour, the bonus-malus scale is equivalent to a set of deductibles depending on the level 

occupied in the scale. 

Cost of Non-Reported Accidents 

Let rll be the optimal retention for a policyholder with expected annual accident 

frequency occupying level l in the bonus-malus scale. Here, rll is not a random 

variable, but an unknown constant to be determined. 

Assume that this policyholder has caused an accident with cost x at time t, 0≤ t


The expected cost of a non-reported accident for a policyholder occupying level l is 

l = 1 

p l 

∫ rll 

y=0 

yfydy 

This policyholder will pay on average l × − l per period, because of the accidents 

not reported to the company. Let us assume that the accident occurrences are uniformly 

distributed over the year (so that on average they occur in the middle of the year). The 

average annual total cost borne by a policyholder in level l is then 

CTrll = b l + v 1 2 l − l 

where b l is the premium paid at the beginning of the year, subject to the bonus-malus scale. 

Let V l be the present value of all the payments made by a policyholder with annual 

expected claim frequency occupying level l. The V l s are obtained from 

∑ 

V l = CTrll + v q l kV Tk l l = 0 1s (5.8) 

k=0 

If the policyholder reports all the accidents to the company, the system (5.8) coincides with 

(5.7). The system (5.8) admits a unique solution. For a given set of optimal retentions, the 

V l s give the cost of the strategy, according to the level occupied in the scale. 

Lemaire Algorithm 

Let us consider a policyholder in level l who just caused an accident with cost x at time t, 

0 ≤ t ≤ 1. There are two possibilities: 

(1) Either he does not claim for the accident and the expected present cost is 

v −t CTrll + x + v 1−t 

 

∑ 

k=0 

q l 

( 

k1 − t 

) 

VTk+m 

l 

where m is the number of claims that the policyholder has already filed during the year. 

(2) Or he reports the accident to the company and the expected present cost is 

v −t CTrll + v 1−t 

 

∑ 

k=0 

q l 

( 

k1 − t 

) 

VTk+m+1 

l 

The retention limit rll is the claim amount x for which the policyholder is indifferent 

between the two possibilities: the optimal retentions thus solve 

rll = v 1−t 

 

∑ 

k=0 

( ) ( ) 

q l k1 − t V Tk+m+1 l − V Tk+m l (5.9) 

for l = 0 1s. Note that (5.9) does not provide an explicit expression for the optimal 

retention since rll also appears in the q l k1 − ts. 

The optimal strategy is obtained using the following algorithm:


First Iteration 

Part A Starting from rl 0 l = 0 for l = 0s, the strategy consisting of reporting 

all the accidents to the insurer, (5.8) becomes 

∑ 

V l = b l + v exp− k 

k! V T k l 

which gives the cost V 0 corresponding to the initial strategy. 

Part B 

k=0 

An improved strategy can then be obtained from (5.9) that reduces to 

rl 1 l = v 1−t 

l = 0 1s. 

 

∑ 

k=0 

exp−1 − t 

1 − 

( 

) 

tk 

V 

k! 

Tk+m+1 l − V Tk+m l 

Second Iteration 

Part A Inserting the rl 1 l s in (5.8) gives the cost associated with this strategy. This 

cost will be smaller than the one associated with the initial strategy. 

Part B Inserting the new cost in the system (5.9), we find an improved strategy rl 2 l , 

l = 0s. 

Subsequent Iterations 

The successive insertion of updated retentions and costs in the systems (5.8)–(5.9) produces 

a sequence of strategies, with reduced costs. 

In all the cases considered in Lemaire (1995), the sequence of the rl k 

l 

s converges to 

the optimal solution with miminum cost. The optimal retention limit is thus a function of 

the level l occupied in the scale at the beginning of the insurance year, of the discount 

factor v, of the annual expected claim frequency , of the time t of occurrence of 

the accident, and of the number m of claims previously reported to the company from 

the beginning of the insurance period. The optimal strategy is an increasing function 

of t: the optimal retention increases as one approaches the end of the year (and the 

premium discount if no accidents are reported). The influence of t on the optimal 

retention limit is much weaker than the level l, the discount factor v or . Putting 

t = 0 (and so m = 0) greatly simplifies the computation but leaves the retentions almost 

unchanged. 

The optimal retentions coming from the Lemaire algorithm should not be considered as 

being the real threshold above which policyholders report the accident to the insurance 

company. Indeed, this algorithm postulates a high degree of rationality behind individual 

behaviours. It is enough to have a look at insurance statistics to see that some claims 

concern accidents bearing a cost much lower than the optimal retention, which contradicts 

the assumptions behind the Lemaire algorithm. The output of the Lemaire algorithm should 

be better understood as a measure of toughness for a particular bonus-malus system. Note 

that Walhin & Paris (2000,2001) softened this requirement by assuming that there was a 

proportion of the policyholders complying with Lemaire claiming rule, and the remainder 

reporting all the accidents, whatever their cost.


Note also that this approach requires knowledge of the uncensored distribution for claim 

costs and claim counts, which is usually not available in practice. The probability density 

function f corresponds to the cost of an accident (and not to the cost of a claim), and the 

distribution of the number of accidents is actually needed (not only the distribution of the 

number of claims that has been studied in the preceding chapters). Hence, the methodology 

described in Sections 5.4.1–5.4.2 has first to be applied to obtain the uncensored accident 

distribution. 

Application of the Lemaire Algorithm to Portfolio C 

The Lemaire algorithm can be applied with the distribution obtained for the cost of accidents 

(i.e. with the help of the LogNormal model with corrected regression coefficients) after 

having transformed the claim frequencies for Portfolio C into the accident frequencies. 

Let us consider the −1/+2 bonus-malus scale, with relativities 62.4 % for level 0, 130.2 % 

for level 1, 142.9 % for level 2, 207.7 % for level 3, 241.4 % for level 4, and 309.1 % for 

level 5. We consider here a discount rate of 4 %. 

Let us consider an individual aged between 25 and 60, living in a rural area, and driving a 

vehicle between 6 and 10 years old. His claim frequency is 18.46 %. His accident frequency 

is 21.29 %. The base pure premium is taken as the product of the claim frequency times the 

grand mean of all the claim sizes (large and moderate ones), that is, 01846 × E 181063. 

The optimal retentions are as follows: 

Level l 

Optimal retention 

0 E57475 

1 E105039 

2 E134116 

3 E176077 

4 E126069 

5 E69370 

We see that this policyholder should defray accidents with a cost up to E 1760.77 if he 

occupied level 3. 

Let us now consider an individual aged over 60, living in a urban area, and driving a 

vehicle between 3 and 5 years old. His claim frequency is 17.41 %. His accident frequency 

is 21.37 %. The base premium amounts to 01741 × E 181063. The optimal retentions are 

as follows: 

Level l 

Optimal retention 

0 E53989 

1 E98774 

2 E126284 

3 E166053 

4 E118990 

5 E65534 

The optimal retentions are now slightly smaller than before.



5.5.1 Modelling Claim Amounts in Related Coverages 

In this chapter, only techniques for motor third party liability have been described. Besides 

the compulsory motor third party liability insurance, a number of related coverages are 

proposed to the drivers (like medical benefits, uninsured or underinsured motorist coverage, 

theft and collision and other than collision insurance). The problem caused by the late 

settlement of large claims generally disappears when optional coverages are considered. The 

annual claim amount S i produced by policyholder i is then represented as 

N 

∑ i 

S i = 

C ik 

k=1 

where N i is the number of claims, and the C ik s are the corresponding claim sizes. The S i s are 

assumed to be mutually independent, the C ik s to be independent and identically distributed 

for fixed i, and independent of N i . 

The analysis of the N i s usually starts with a Poisson regression model. Then, the 

residual heterogeneity is taken into account by the inclusion of a random effect. The 

impact of the deductibles has to be carefully assessed for optional coverages. In case 

deductibles are specified in the insurance policies, the actuary has to keep in mind that the 

statistical summaries relate to conditional distributions (given that the claim costs exceed the 

corresponding deductibles). 

According to the type of coverage, different models can be used for the C ik s. The claim 

size is usually expressed as a percentage of the sum insured. For collision insurance, the 

C ik s can be decomposed as 

C ik = ( J ik + 1 − J ik P ik 

) 

vi 

where v i is the sum insured for policy i (the value of the vehicle fixed according to the rules 

contained in the policy); J ik = 1ifthekth claim generates a total loss (that is, a loss that 

exhausts the sum insured, i.e. C ik = v i ), and 0 otherwise; and 0


distributions as the Normal and Gamma, the purely discrete scaled Poisson distribution, as 

well as the class of mixed compound Poisson distribution with Gamma summands. The 

name Tweedie has been associated with this family by Jorgensen (1987,1997) in honour 

of the pioneering works by Tweedie (1984). 

In nonlife ratemaking, the Tweedie model is very convenient for risk classification. 

However, it does not allow the actuary to isolate the frequency part of the pure premium, 

and thus does not provide the actuary with the input for the design of bonus-malus scales. 

This is why in this book (which is mainly devoted to motor insurance pricing) we favoured 

the separate analysis of claim costs and claim sizes. For other insurance products, where 

only the total amount of claims is available for actuarial analysis, the Tweedie distribution 

is an excellent candidate for loss modelling. 

In the actuarial literature, Jorgensen & Paes de Souza (1994) assumed Poisson arrival 

of claims and Gamma distributed costs for individual claims. These authors directly modelled 

the risk or expected cost of claims per insured unit using the Tweedie Generalized Linear 

Model. Smyth & Jorgensen (2002) observed that, when modelling the cost of insurance 

claims, it is generally necessary to model the dispersion of the costs as well as their mean. 

In order to model the dispersion, these authors used the framework of double generalized 

linear models. Modelling the dispersion increases the precision of the estimated tariffs. The 

use of double generalized linear models also allows the actuary to handle the case where 

only the total cost of claims and not the number of claims has been recorded. 

5.5.3 Large Claims 

The analysis of large losses performed in this chapter is based on Cebrian, Denuit & 

Lambert (2003). Large losses are modelled using the Generalized Pareto distribution, and 

the main concern is to determine the threshold between small and large losses. An alternative 

has been developed by Buch-Kromann (2006) based on Buch-Larsen, Nielsen, Guillén 

& Bolanće (2005). This approach is based on a Champernowne distribution, corrected 

with a nonparametric estimator (that is obtained by transforming the data set with the 

estimated modified Champernowne distribution function and then estimating the density of 

the transformed data set using the classical kernel density estimator). Based on the analysis of 

a Danish data set, Buch-Kromann (2006) concluded that the Generalized Pareto approach 

performs better than the Champernowne one in terms of goodness-of-fit, whereas both 

methods are comparable in terms of predicting future claims. 

Another approach is proposed by Cooray & Ananda (2005) who combined a LogNormal 

probability density function together with a Pareto one. Specifically, these authors introduced 

a two-parameter smooth continuous composite LogNormal-Pareto model that is a twoparameter 

LogNormal density up to an unknown threshold value and a two-parameter Pareto 

density for the remainder. Continuity and differentiability are imposed at the unknown 

threshold to ensure that the resulting probability density function is smooth, reducing the 

number of parameters from four to two. The resulting two-parameter probability density 

function is similar in shape to the LogNormal density, yet its upper tail is thicker than the 

LogNormal density (and accomodates to the large losses observed in liability insurance). 

This approach clearly outperforms the one proposed in this chapter, in that all the parameters 

(including the threshold) are estimated in the same model. The approaches obtained with 

the methodology developed in this book can be used as starting values in the maximum


likelihood maximization. Note however that Cooray & Ananda (2005) did not consider the 

case where explanatory variables were available, so that their approach has to be extended 

to this more realistic situation. 

5.5.4 Alternative Approaches to Risk Classification 

There are numerous techniques applied to the modelling of insurance losses. Early references 

in actuarial science include ter Berg (1980a,b). Various mathematical and statistical models 

for estimation of automobile insurance pricing are reviewed in Weisberg, Tomberlin & 

Chatterjee (1984). The methods are compared on their predictive ability based on two sets 

of automobile insurance data for two different states collected over two different periods. 

The issue of model complexity versus data availability is resolved through a comparison of 

the accuracy of prediction. The models reviewed range from the use of simple cell means 

to various multiplicative-additive schemes to the empirical Bayes approach. The empirical 

Bayes approach, with prediction based on both model-based and individual cell estimates, 

seems to yield the best forecast. See also Jee (1989). 

Williams & Huang (1996) applied KDD (for Knowledge Discovery in Databases) 

techniques for insurance risk assessment. Daengdej, Lukose & Murison (1999) considered 

CBR (for Case-Based Reasoning) techniques for claim predictions. 

Classification techniques are also often used for risk classification. Retzlaff-Roberts & 

Puelz (1996) adopted an efficiency approach to the two-group linear programming method 

of discriminant analysis, using principles taken from data envelopment analysis, to predict 

group membership in an insurance underwriting scheme. Yeo ET AL. (2001) applied clustering 

techniques before modelling insurance losses. 

5.5.5 Efficiency 

The efficiency (or elasticity) of a bonus-malus system was first studied by Loimaranta 

(1972) and De Pril (1978). Note however that in these papers, the unknown individual risk 

factors are not viewed as random variables. These authors work in the fixed effects model 

that is close in spirit to the limited fluctuations credibility theory. Their efficiency concepts 

are therefore entirely different from the notion of efficiency proposed in Norberg (1976). 

Other efficiency measures have been proposed in the literature. For instance, Heras ET AL. 

(2002) evaluated the asymptotic fairness of bonus-malus systems (i.e., their ability to assess 

the individual risk in the long run) assuming the simplest case when there is no hunger for 

bonus. 

5.5.6 Optimal Retention Limits and Bonus Hunger 

The problem of determining the optimal claim size has been the topic of several papers. 

De Leve & Weeda (1968) considered a −1/top bonus-malus scale, so that the decision to 

file or not has to be made only if no claim has been made during the same period. Lemaire 

(1976,1977) studied the hunger for bonus and proposed a dynamic programming algorithm 

to determine the optimal claiming behaviour. De Pril (1979) considered that the claims 

were generated by a Poisson process and adapted a continuous-time approach. Specifically,


De Pril (1979) defined L n l k t as the amount that the actual accident must exceed in 

order to justify the filing of a claim, if the policyholder is at time t of period n in level l and 

has already filed k claims. The optimal value of L n l k t is determined to minimize the 

discounted expectation of the total future costs (premiums and self-defrayed accidents) for 

the policyholder. After De Pril (1979), De Pril & Goovaerts (1983) determined bounds 

for the optimal critical claim size when only incomplete information about the claim amount 

distribution is available. They considered −1/top bonus-malus scales. 

Dellaert ET AL. (1990) proved that under mild conditions the optimal decision rule is 

to claim for damages with amount above a certain limit. In some instances, policyholders 

are allowed to decide at the end of an insurance year which damages occurred during the 

year should be claimed; see, e.g., Martin-Lof (1973). This means that the policyholder 

has perfect information about the number of accidents and the corresponding damages at 

the moment he/she decides which damages to claim. This situation has been investigated 

by Dellaert ET AL. (1991). Let us also mention that Dellaert ET AL. (1993) considered 

damage insurance (where, in addition to the bonus hunger phenomenon, the optimal stopping 

rule to terminate the insurance has to be determined). 

Holtan (2001) envisaged the loss of bonus after a claim as a rate of interest paid from 

the customer to the insurer, and studied the hunger for bonus from this viewpoint. 

Optimal claiming rules have also been considered in Operational Research, using Markov 

decision processes. When a driver is involved in a motor accident decisions have to be made 

as to whether or not a claim should be made. Hastings (1976) considered this problem as 

a Markov decision process with the expected cost over a finite horizon (where the relevant 

costs are repair costs and premium costs) as objective function. See also Haehling von 

Lanzenauer (1974) and Hey (1985). Norman & Shearn (1980) proposed the following 

simple rule of thumb that is shown to work well in their case: irrespective of when the 

accident occurs, claim only if the amount of the claim exceeds the difference over the next 

4 years between the total premiums payable if a claim is made and those payable if it is not, 

assuming that no further claims will be made. Chappell & Norman (1989) demonstrated 

that this simple rule was less efficient in the case of protected bonus. Kolderman & 

Volgenant (1985) examined the same problem under the assumption that only one claim 

is needed to change the insurance premium category, and any extra claims in the year have 

no effect on the current repair estimate for an accident. 

Walhin & Paris (2000) derived the actual claim amount and frequency distributions 

within a bonus-malus system. As explained above, policyholders should defray the small 

claims to avoid the penalties induced by the bonus-malus system. Consequently, there are 

more accidents than the number of claims filed by the insurer: the insurance data are censored. 

The kind of censorship is nevertheless very particular, and much more complicated than the 

phenomena encountered in classical nonlife problems (where losses are censored because 

they exceed some policy limit, or fall below a given deductible). The procedure described 

in Section 5.4.1 is taken from Denuit, Maréchal, Pitrebois & Walhin (2007a), where 

alternative approaches to obtain uncensored accident distributions can be found. 

Even if the bonus-hunger phenomenon has been extensively studied in connection with 

bonus-malus scales, the same idea applies to credibility systems. See, e.g., Norberg (1975) 

and Sundt (1988) for an illustration.

6 

Multi-Event Systems 


The majority of bonus-malus systems in force throughout the world penalize the number of 

reported claims, without taking the cost of these claims into account. This can be considered 

as a shortcoming, since large claims should intuitively be more severely penalized. The first 

part of this chapter aims to develop credibility models that allow us to subdivide the claims 

into two categories, small and large losses (the extension to more than two categories is 

easy). Instead of determining a limiting amount to decide whether a loss should be qualified 

as large instead of small (such a criterion would lead to substantial practical problems, due 

to the time needed to evaluate the cost of the claim), we distinguish the accidents that caused 

property damage only from those that caused bodily injuries. Since the latter cost much more 

on average, this approach implicitly integrates the cost of the claim in a posteriori premium 

corrections. The Bayesian credibility approach turns out to lead to numerical integration. This 

is why we favour the linear credibility approach. Linear credibility formulas are developed 

to update the expected claim frequencies given past claim histories. 

The second part of this chapter is devoted to bonus-malus scales with several types of 

events (assuming a Multinomial partitioning scheme). As mentioned above, all the classical 

bonus-malus systems are based on a single type of event: the occurrence of claims at fault, 

regardless of their severity or whether the policyholders are only partially liable for them. 

This over-simplification can be regarded as problematic for commercial purposes: it seems 

desirable to integrate the severity of the claims and to recognize the partial liability of the 

policyholder. For example, the bonus-malus system in force in France (studied in Chapter 9) 

entails a reduced penalty if the policyholder is only partially liable for the claim. 

Prominent examples of a posteriori ratemaking mechanisms based on several types of 

events are provided by the experience rating systems in force in North America. These 

systems not only incorporate accidents at fault but also elements of the policyholders’ driving 

record. For instance, the Massachusetts safe driver insurance plan encourages safe driving 





by rewarding drivers who do not cause an accident, or incur a traffic law violation. It is 

based on several types of events (major and minor at-fault accidents and traffic violations). 

Specifically, each policyholder is assigned a level between 9 and 35, based on his driving 

record during the previous six years. A new driver begins at level 15 (relativity of 100 %). 

Occupying any level below 15 entails a premium discount, while above level 15 the driver 

pays a surcharge. For each incident-free year of driving, the policyholder goes down one 

level. The driver will move up a certain number of levels based on the type of incident: 

two levels for a minor traffic violation, three levels for a minor at-fault accident, four levels 

for a major at-fault accident and five levels for a major traffic violation. The Massachusetts 

system ‘forgets’ all incidents after six years. 

This chapter addresses the actuarial modelling of such systems, with penalties depending 

on different types of events. As before, the modelling uses the concept of Markov Chains. 

We will see that under mild assumptions, the trajectory of each policyholder in the scale 

can be modelled with the aid of discrete-time Markov processes. The relativities associated 

with each level will then be computed using the maximum accuracy principle discussed in 

Chapter 4. 

6.2 Multi-Event Credibility Models 

6.2.1 Dichotomy 

Let Nit 

mat be the number of claims with material damage only, reported by policyholder i 

during period t. Similarly, let Nit 

bod be the number of claims with bodily injuries, and let 

N tot 

it 

= N mat 

it 

+ N bod 

it 

be the total number of claims. Policyholder i, i = 1n, is assumed to have been observed 

during T i periods. In the previous chapters, Nit 

tot was the variable of interest. Here, a dichotomy 

is operated, and we study Nit 

mat and Nit 

bod separately. 

6.2.2 Multivariate Claim Count Model 

Overdispersion and possible dependence between N mat 

it 

correlated random effects i 

mat 

i 

mat = i 

mat , we assume that 

where mat 

it 

and bod 

i 

N mat 

it 

such that E mat 

i 

∼ oi ( mat 

it 

= d it expscore mat 

it 

, and given i 

bod 

N bod 

it 

∼ oi ( bod 

it 

mat 

i 

and Nit 

bod 

= E bod 

) 

= i 

bod , we assume that 

) 

bod 

i 

are introduced via possibly 

= 1. Specifically, given 

where bod 

it 

= d it expscore bod 

it 

. Both scores are linear combinations of explanatory variables 

specific to policyholder i and year t (summarized in a vector x it ). Specifically, 

score mat 

it 

= mat 

p∑ 

0 

+ 

j=1 

mat 

j 

x itj 

i

Multi-Event Systems 261 

score bod 

it 

= bod 

p∑ 

0 

+ 

j=1 

bod 

j 

x itj 

We make the following assumptions about the dependence structure of the random variables: 

(i) Given bod 

i 

, the Nit 

bod s, t = 1 2T i , are independent. 

, the Nit 

mat s, t = 1 2T i , are independent. 

i 

bod , the sequences Nit 

bod t = 1 2T i and Nit 

mat t = 1 2T i 

are independent. 

(ii) Given mat 

i 

(iii) given mat 

i 

As explained in the previous chapter, overdispersion and serial correlation are induced by 

missing explanatory variables, whose effect is modelled with the help of the random effects 

and bod 

mat 

i 

i 

. 

6.2.3 Bayesian Credibility Approach 

The Bayesian approach requires numerical integration, which sometimes prevents the 

practical implementation of the resulting formulas (even if, nowadays, numerical integration 

has become straightforward with modern computers). It consists of deriving the conditional 

distribution of the random effects mat 

i 

and bod 

i 

conditional distribution then drives a posteriori premium corrections. 


∑ 

T i 

k mat 

i• 

= k mat 

it 

t=1 

, given the past claims history. This 

the total number of claims with material damage only filed by policyholder i during the T i 

coverage periods, and as 

∑ 

T i 

k bod 

i• 

= k bod 

it 

t=1 

the total number of claims with bodily injuries filed by this policyholder. The corresponding 

expected claim frequencies are 

The joint distribution of the N mat 

it 

PrN mat 

it 

∫ 

= 

0 

∑ 

T i 

mat 

i• 

= mat 

it 

t=1 

and bod 

∑ 

T i 

i• 

= 

t=1 

s and N bod s is given by 

= k mat 

it 

N bod 

it 

= k bod 

it 

for t = 1T i 

∫ 

0 

exp− mat 

i 

× f mat 

i 

mat 

i• 

− bod i 

it 

bod 

i 

d mat 

i 

d bod 

i 

bod 

i• 

mat i 

kmat i• 

bod 

it 

 

 

bod 

i 

kbod i• 

∏ Ti 

t=1 mat it 

kmat it 

∏ Ti 

t=1 kmat it !kit bod ! 

bod 

it 

kbod it


where f · · is the bivariate probability density function of the couple of random effects 

i 

mat i 

bod . The joint probability density function of i 

mat , i 

bod , Nit 

mat , Nit 

bod , t = 1T i 

is given by 

exp− mat 

i 

mat 

i• 

− bod i 

bod 

i• 

mat i 

kmat i• 

∏ Ti 

 

bod 

i 

kbod i• 

so that the conditional probability density function of mat 

i 

∫ 

0 

exp− mat 

∫ 

i 

mat 

i• 

− bod i 

bod 

i• 

mat i 

kmat i• 

bod 

i 

exp− 0 1 mat 

i• 

The posterior distribution of i 

mat 

in Chapter 3. 

− 2 bod 

i• 

i 

bod 

t=1 mat it 

kmat it bod 

it 

kbod it 

∏ Ti 

t=1 kmat it !kit bod ! 

i 

bod 

f mat 

i 

bod 

i 

given past claims history is 

kbod i• f i 

mat i 

bod 

(6.1) 

1 kmat i• 2 kbod i• f 1 2 d 1 d 2 

then allows for a posteriori corrections, as explained 

 

6.2.4 Summary of Past Claims Histories 

Denote as 

∑ 

T i 

N bod 

i• 

= N bod 

it 

t=1 

and N mat 

∑ 

T i 

i• 

= N mat 

it 

t=1 

the total claim numbers of each category caused by policyholder i during the observation 

period. Since the random effects i 

mat and i 

bod do not vary with time, Ni• 

mat and Ni• 

bod are 

sufficient summaries of past claims histories (in the sense that the posterior distributions of 

i 

mat and i 

bod as well as the predictive distributions of NiT mat 

bod 

i +1 

and NiT i +1 

only depend on 

Ni• 

mat and Ni• 

bod ; see (6.1) where past claims histories enter through kmat 

i• 

and ki• bod). 

Clearly, 

mat 

i• 

It is then easy to see that given mat 

i 

and that given bod 

i 

= bod 

i 

= EN mat and bod 

i• 

= mat 

i 

N mat 

i• 

N bod 

i• 

i• 

∼ oi ( mat 

i• 

∼ oi ( bod 

i• 

= EN bod 

i• 

 

mat i 

bod i 

) 

) 

 

invoking the conditional independence of the annual claim numbers of each category and the 

stability of the Poisson family under convolution. Therefore, Ni• 

mat and Ni• 

bod are both mixed 

Poisson distributed.


6.2.5 Variance-Covariance Structure of the Random Effects 

For deriving linear credibility formulas, we only need the moment structure of the risk 

variables. Let us introduce the variance-covariance matrix of i 

bod i 

mat that is denoted as 

( ) 

 

2 

= bod 

bm 

bm mat 

2 

In words, 2 bod and 2 mat 

are the variances of bod 

i 

covariance between bod 

i 

and mat 

i 

and i 

mat 

. Note that the following inequalities 

, respectively, and bm is the 

2 bod ≥ 0 2 mat ≥ 0 and bm≤ bod mat 

must be fulfilled to ensure that is positive definite. The estimated variances and covariance 

have to fulfill the same constraints. If not, this rules out the linear credibility model. 

6.2.6 Variance-Covariance Structure of the Annual Claim Numbers 

Let us now compute the variance and covariance of Ni• mat and Ni• bod. 

Since 

Ni• 

mat ∼ oi mat 

i• 

mat i 

, we have 

Similarly, from N bod 

i• 

VN mat 

i• 

∼ oi bod 

i• 

bod i 

VN bod 

i• 

= mat 

i• 

we get 

= bod 

i• 

+ ( ) 2 

mat 2 

i• mat 

+ 

( ) 2 

bod 2 

i• bod 

To have an idea about the dependence existing between the numbers of claims with material 

damage only and with bodily injuries, let us now compute the covariance between Ni• 

mat 

and Ni• bod: 

[ 

] 

CN mat 

i• 

Nbod i• = E N mat 

i• 

− mat bod 

i• 

Ni• 

− bod 

i• 

[ 

= E E [ N mat 

i• 

− mat 

i• 

[ 

= E E [ ∣ 

N mat 

i• 

− mat 

i• 

[ 

= E 

= mat 

i• 

mat 

i• 

( 

 

mat 

i 

bod i• 

bm 

As expected, the covariance bm between mat 

i 

bod 

Ni• 

∣ mat 

i 

− bod 

i• 

∣ ∣ mat 

] [ ∣ 

E N 

bod 

− bod 

i• 

− 1 ) ( 

bod 

i• 

bod 

i 

− 1 )] 

and bod 

i 

i 

i• 

bod 

i 

∣ bod 

i 

] ] 

] ] 

drives the covariance of Ni• 

mat and Ni• bod.


6.2.7 Estimation of the Variances and Covariances 

The formulas derived above for VNi• 

mat bod 

mat 

, VNi• 

and CNi• 

estimates for the parameters mat 2 , 2 bod and bm: 

∑ 

( 

n 

(k ) ) 

mat mat 2 

̂ mat 2 i=1 i• 

− ̂ i• − k 

mat 

i• 

= 

̂ 2 bod = ∑ n 

i=1 

̂ bm = 

∑ n 

i=1 

∑ n 

i=1 (̂mat i• 

( (k 

bod 

i• 

( 

) 2 

) ) 

bod 2 

− ̂ − k 

bod 

i• 

∑ n 

i=1 (̂bod i• 

ki• 

mat 

∑ n 

i=1 

) 2 

)( 

mat 

− ̂ 

i• 

̂ 

mat 

i• 

k bod 

i• 

bod ̂ i• 

i• 

Nbod i• 

) 

bod 

− ̂ 

i• 

suggest the following 

that are consistent in the random effects model. 

The parameters mat 2 , 2 bod and bm have been estimated above on aggregate data, giving 

the estimators ̂ mat, 2 ̂ bod 2 and ̂ bm. Alternatively, these parameters could be estimated from 

individual data as follows: 

∑ n ∑ 

( 

Ti 

(k ) ) 

mat mat 2 

i=1 t=1 it 

− ̂ it − k 

mat 

it 

˜ 2 mat = 

∑ n ∑ Ti 

˜ 

bod 2 = i=1 t=1 

˜ bm = 

∑ n 

i=1 

∑ Ti 

t=1 

∑ n 

i=1 

∑ Ti 

t=1 

( (k 

bod 

it 

− 

∑ n 

i=1 

∑ Ti 

t=1 

( 

k 

mat 

it 

− 

∑ n 

i=1 

∑ Ti 

t=1 

(̂mat it 

̂ 

bod 

it 

(̂bod it 

̂ 

mat 

it 

̂ 

mat 

it 

) 2 

) ) 

2 

− k 

bod 

) 2 

it 

)( 

k 

bod 

− 

it 

̂ bod 

it 

) 

bod ̂ it 

The estimators ̂ 2 mat, ̂ 2 bod and ̂ bm are preferred over ˜ 2 mat, ˜ 2 bod and ˜ bm, respectively, since 

the variances of the former are smaller. As shown by Pinquet ET AL. (2001), the condition 

0 < ̂ 2 mat < ˜ 2 mat is necessary for the introduction of dynamic random effects. 

 

6.2.8 Linear Credibility Premiums 

Denote as 

mat 

iT i +1 = d iT i +1 expscore mat 

iT i +1 and as bod 

iT i +1 = d iT i +1 expscore bod 

iT i +1 

the expected claim frequencies for policyholder i in period T i + 1. The best linear predictor 

∑ 

T i 

c mat 

i0 

+ 

t=1 

c mat/mat 

it 

N mat 

∑ 

T i 

it 

+ 

t=1 

c bod/mat 

it 

N bod 

it


of the true expected claim frequency mat 

iT i +1 mat i 

minimizes 

⎡( 

mat = E ⎣ 

mat 

iT i +1 mat i 

Similarly, the best linear predictor 

of bod 

iT i +1 bod i 

minimizes 

⎡( 

bod = E ⎣ 

∑ 

T i 

c bod 

i0 

+ 

t=1 

bod 

iT i +1 bod i 

− c mat 

i0 

T 

∑ i 

− 

t=1 

c mat/bod 

it 

N mat 

− c bod 

i0 

c mat/mat 

it 

∑ 

T i 

it 

+ 

t=1 

T 

∑ i 

− 

t=1 

c mat/bod 

it 

N mat 

it 

T 

∑ i 

− 

t=1 

c bod/bod 

it 

N bod 

it 

N mat 

it 

T 

∑ i 

− 

t=1 

c bod/mat 

it 

N bod 

it 

c bod/bod 

it 

N bod 

it 

) ⎤ 2 

⎦ 

) ⎤ 2 

⎦ 

The optima are obtained by setting to zero the derivatives of mat with respect to ci0 

mat 

to cis 

mat/mat , that is, 

c mat 

i0 

c mat/mat 

is 

T i 

= mat 

iT i +1 − ∑ 

= mat 

t=1 

c mat/mat 

it 

mat 

it 

T 

∑ i 

− 

t=1 

T 

iT i +1 2 mat − ∑ i 

2 mat 

c mat/mat 

it 

mat 

it 

t=1 

c bod/mat 

it 

bod 

it 

T 

∑ i 

− bm 

t=1 

c bod/mat 

it 

bod 

it 

The last relation shows that cis 

mat/mat does not depend on s. Similarly, one can check that 

cis 

mat/bod , cis 

bod/mat and cis 

bod/bod do not depend on s. This justifies the approach based on aggregate 

data Ni• 

mat and Ni• 

bod (that are an exhaustive summary of past claims histories in the credibility 

model with static random effects). 

Denoting as ci 

mat/mat (ci 

bod/mat , ci 

mat/bod and ci 

bod/bod , respectively) the common values of the 

cis 

mat/mat (cis 

bod/mat , cis 

mat/bod and cis 

bod/bod , respectively), the best linear predictors are thus of the form 

for mat 

for bod 

follows: 

iT i +1 mat i 

iT i +1 bod i 

, and 

c mat 

i 

c bod 

i 

+ c mat/mat 

i 

+ c mat/bod 

i 

N mat 

i• 

N mat 

i• 

+ c bod/mat 

i 

N bod 

i• 

+ c bod/bod 

i 

N bod 

i• 

. The meaning of the coefficients involved in these linear predictors is as 

 

and 

c mat/mat 

i 

c bod/mat 

i 

evaluates the information contained in past material claims on the occurrence of 

future material claims; 

evaluates the information contained in past claims with bodily injuries on the 

occurrence of future material claims;


c mat/bod 

i 

c bod/bod 

i 

evaluates the information contained in past material claims on the occurrence of 

future claims with bodily injuries; 

evaluates the information contained in past claims with bodily injuries on the 

occurrence of future claims with bodily injuries. 

The values of these coefficients are determined by minimizing simultaneously mat and bod 

that may be rewritten as 

and as 

mat = E [( mat 

bod = E [( bod 

iT i +1 mat i 

iT i +1 bod i 

− c mat 

i 

− c bod 

i 

− c mat/mat 

i 

− c mat/bod 

i 

N mat 

i• 

N mat 

i• 

− c bod/mat 

i 

− c bod/bod 

i 

N bod 

i• 

N bod 

i• 

) 2 ] 

) 2 ] 

 

Setting to zero the partial derivatives of mat and bod with respect to the six parameters gives: 

c mat 

i 

c bod 

i 

= mat 

= bod 

0 = mat 

iT i +1 − cmat/mat i 

mat 

i• 

iT i +1 − cmat/bod i 

mat 

i• 

iT i +1 Emat i 

N mat 

i• 

− c bod/mat 

i 

0 = mat 

EN mat 

i• 

N bod 

iT i +1 Emat i 

N bod 

i• 

− cbod/mat i 

bod 

i• 

(6.2) 

− cbod/bod i 

bod 

i• 

(6.3) 

− cmat 

i 

mat 

i• 

− cmat/mat i 

E [ N mat 

i• 2] 

i• (6.4) 

− cmat 

i 

bod 

i• 

− cmat/mat i 

EN mat 

i• 

N bod 

i• 

− c bod/mat 

i 

E [ N bod 

i• 2] (6.5) 

0 = bod 

iT i +1 Ebod i 

N mat 

i• 

− c bod/bod 

i 

0 = bod 

EN mat 

i• 

N bod 

iT i +1 Ebod i 

N bod 

i• 

− cbod 

i 

mat 

i• 

− cmat/bod i 

E [ N mat 

i• 2] 

i• (6.6) 

− cbod 

i 

bod 

i• 

− cmat/bod i 

EN mat 

i• 

N bod 

i• 

− c bod/bod 

i 

E [ N bod 

i• 2] (6.7) 

The expectancies involved in this system are given by 

E mat 

i 

E bod 

i 

E mat 

i 

E bod 

i 

N mat 

i• 

N bod 

i• 

N bod 

i• 

N mat 

i• 

= mat 

i• 

= bod 

i• 

2 mat + mat i• 

2 bod + bod i• 

= bod 

bm + bod 

i• 

i• 

= mat 

bm + mat 

i• 

i•


E [ N mat 

i• 

E [ N bod 

i• 

EN mat 

i• 

N bod 

i• 

2] = mat 

i• 

2] = bod 

i• 

= mat 

i• 

+ ( ) 2 

mat 2 

i• mat + 1 

+ 

( ) 2 

bod 2 

i• bod + 1 

bod i• 

Inserting these expressions in (6.4)–(6.7), we get 

and finally 

mat 

iT i +1 2 mat = cmat/mat i 

mat 

iT i +1 bm = c mat/mat 

i 

bod 

bm + 1 

1 + mat 2 mat + cbod/mat 

mat 

i• 

i• 

bm + c bod/mat 1 + bod 

iT i +1 bm = c mat/bod 

i 

1 + mat 

i• 

2 mat + cbod/bod i 

bod 

iT i +1 2 bod = cmat/bod i 

mat 

i• 

c bod/mat 

i 

= 

c mat/bod 

i 

= 

c mat/mat 

i 

c bod/bod 

i 

1 + mat 

i• 

1 + mat 

i• 

= mat 

iT i +1 

= bod 

iT i +1 

i 

i 

bod 

i• 

bm 

i• 

bm + c bod/bod 1 + bod 

mat 

iT i +1 bm 

bod 2 − mat 

2 mat1 + bod 

i• 

i 

i• 

bod 

iT i +1 bm 

bod 2 − mat 

2 mat1 + bod 

i• 

i• 

2 bod 

bod 

i• 

bm 

i• 

2 bod 

bod 

i• bm 

2 

bod 

i• bm 

2 

mat 2 + bod i• 

2 mat 2 bod − 2 bm 

1 + mat 

i• mat1 2 + bod 

i• bod 2 − mat i• 

bod 2 + mat i• 

2 mat 2 bod − 2 bm 

1 + mat 



bod 

i• bm 

2 

bod 

i• bm 

2 

We see that ci 

bod/mat and ci mat/bod are increasing with bm , and decreasing with mat 2 and 

bod 2 . This is intuitively acceptable: the more the random effects are correlated, the more 

information is contained in the claims with material damage only about the claims with 

bodily injuries, and vice-versa. Inserting these solutions in the equations (6.2)–(6.3) gives 

mat 

iT i +1 

c mat 

i 

c bod 

i 

= mat 

iT i +1 

= bod 

iT i +1 

( ) 

1 + bod 

i• 

2 

bod 

− bm 

1 + mat 



( ) 

1 + mat 

i• 

2 

mat 

− bm 

1 + mat 



bod 

i• bm 

2 

bod 

i• bm 

2 

To sum up, the best linear predictor for the expected frequency of claims with material 

damage only occurring in year T i + 1is 

( ) 

1 + bod 

i• 

2 

bod 

− bm + bm Ni• 

bod + ( mat 2 + bod i• 

2 mat 2 bod − 2 bm ) Ni• 

mat 

1 + mat 

i• 

mat1 2 + bod bod 2 − mat 

i• 

i• 

bod 

i• bm 

2


and the best linear predictor for the expected frequency of claims with bodily injuries in 

year T i + 1is 

bod 

iT i +1 

1 + mat 

i• 

( ) 

 

2 

mat 

− bm + bm Ni• 

mat + ( bod 2 + mat i• 

2 mat 2 bod − 2 bm ) Ni• 

bod 

1 + mat 


i• bod 2 − 

mat i• bod 

i• 

2 bm 

6.2.9 Numerical Illustration for Portfolio A 

A Priori Ratemaking 

The observed annual frequency for claims with bodily injuries is 16 %. The observed annual 

frequency for claims with material damage only is 130%. 

In this portfolio, the two types of claims we consider are positively correlated. This can 

be seen from Table 6.1, where the conditional expectation of the number of claims of one 

type is computed given the number of claims of the other type. The more claims of one type 

reported, the higher this conditional expectation, resulting in positive dependence. 

A Posteriori Corrections 

Let us now update the claim frequencies with the help of the formulas obtained with linear 

credibility. To this end, we first estimate . This gives 

Let us now consider two types of drivers: 

̂ 2 mat = 08458 

̂ 

bod 2 = 11188 

̂ bm = 06255 

• a good driver with mat 

it 

= 0083 and bod 

it 

= 0010, and 

• a bad driver with mat 

it 

= 0246 and bod 

it 

= 0030. 

Table 6.1 Observed annual frequency of claims with bodily injuries, given the number 

of claims with material damage only; and observed annual frequency of claims with 

material damage only, given the number of claims with bodily injuries; for portfolio A. 

Conditional expectation of the 

number of claims with bodily 

injuries 

Conditional expectation of the 

number of claims with 

material damage only 

Given the number of claims 

with material damage only 

Given the number of claims 

with bodily injuries 

= 0 is 1.5 % = 0 is 12.9 % 

= 1 is 2.7 % = 1 is 22.8 % 

= 2 is 3.6 % = 2 is 37.7 %


Table 6.2 Evolution of relativities and pure premiums (taking the average cost of a claim with 

material damage only (mat) as the monetary unit, and assuming that claims with bodily injuries (bod) 

are on average ten times more expensive) if no claim has been reported. 

Time Good driver Bad driver 

Relativity 

mat 

Relativity 

bod 

Pure 

premium 

Relativity 

mat 

Relativity 

bod 

Pure 

premium 

1 92.9 % 94.2 % 16.8 % 81.5 % 84.6 % 45.1 % 

2 86.7 % 89.1 % 15.8 % 68.7 % 73.9 % 38.8 % 

3 81.3 % 84.6 % 15.0 % 59.3 % 66.0 % 34.1 % 

4 76.6 % 80.6 % 14.2 % 52.1 % 59.9 % 30.6 % 

5 72.3 % 77.1 % 13.5 % 46.5 % 55.1 % 27.7 % 

6 68.5 % 73.9 % 12.8 % 41.9 % 51.1 % 25.4 % 

7 65.0 % 71.0 % 12.3 % 38.2 % 47.8 % 23.5 % 

8 61.9 % 68.4 % 11.8 % 35.0 % 45.0 % 21.9 % 

9 59.1 % 66.0 % 11.3 % 32.3 % 42.5 % 20.5 % 

10 56.5 % 63.8 % 10.9 % 30.0 % 40.4 % 19.3 % 

Table 6.2 displays the results for the case where no claim is reported for 10 years. The 

first column gives the coefficient to be applied on mat 

it 

, the second column gives the 

coefficient to be applied on bod 

it 

and the third column gives the premium to be charged 

if the average cost of a material damage claim is 1 and the average cost of a bodily 

injury claim is 10 (the monetary unit is thus the average cost of a claim with material 

damage only, and claims with bodily injuries are assumed to be on average ten times 

more expensive than claims with material damage only). The first three columns are for 

the good driver and the next three are for the bad driver. We see that the correction 

coefficients are always smaller for the claims with material damage only than for the claims 

with bodily injuries. This is due to the fact that the former claims occur more frequently 

than the latter ones, so that not reporting any claim with material damage only entails 

more premium discount. As explained previously, the discounts are always larger for a 

bad driver than for a good one. However, the premiums always stay higher for the bad 

drivers. 

Table 6.3 considers the case where the policyholder reported a single claim with 

material damage only, for 10 years. Finally, Table 6.4 considers the case where the 

policyholder reported a single claim with bodily injuries, for 10 years. Comparing these 

two tables, we see that the premium amount is larger if a claim with bodily injuries has 

been reported, compared to the case where a claim with material damage only has been 

reported. The cost of the claim is thus taken into account in the premium correction. 

The correction coefficients are always larger for the good driver than for the bad one, as 

explained previously. It is also interesting to note that reporting a claim of one type always 

increases the probability of reporting a claim of the other type. There is thus a double 

effect when updating the premium: the frequency of claims of the same type as the one 

that has been reported is increased, but the frequency of claims of the other type gets 

inflated, too.




are on average ten times more expensive) if a single claim with material damage only has been reported 

during the first year. 


Relativity 

mat 

Relativity 

bod 

Pure 

premium 

Relativity 

mat 

Relativity 

bod 

Pure 

premium 

1 171.6 % 152.0 % 29.0 % 150.7 % 134.9 % 77.0 % 

2 160.3 % 142.8 % 27.2 % 127.3 % 115.7 % 65.6 % 

3 150.4 % 134.7 % 25.6 % 110.1 % 101.6 % 57.2 % 

4 141.7 % 127.6 % 24.1 % 97.0 % 90.7 % 50.8 % 

5 133.9 % 121.2 % 22.9 % 86.7 % 82.2 % 45.7 % 

6 126.9 % 115.5 % 21.7 % 78.3 % 75.2 % 41.6 % 

7 120.6 % 110.3 % 20.7 % 71.4 % 69.4 % 38.1 % 

8 114.9 % 105.7 % 19.8 % 65.6 % 64.6 % 35.3 % 

9 109.7 % 101.4 % 18.9 % 60.7 % 60.4 % 32.8 % 

10 105.0 % 97.5 % 18.2 % 56.5 % 56.8 % 30.7 % 



are on average ten times more expensive) if a single claim with bodily injuries has been reported 

during the first year. 


Relativity 

mat 

Relativity 

bod 

Pure 

premium 

Relativity 

mat 

Relativity 

bod 

Pure 

premium 

1 150.7 % 201.9 % 32.1 % 131.7 % 185.5 % 87.3 % 

2 140.5 % 193.1 % 30.4 % 110.4 % 166.8 % 76.6 % 

3 131.5 % 185.4 % 28.9 % 94.8 % 152.9 % 68.6 % 

4 123.5 % 178.5 % 27.6 % 82.9 % 142.0 % 62.4 % 

5 116.5 % 172.3 % 26.4 % 73.6 % 133.2 % 57.5 % 

6 101.1 % 166.7 % 25.3 % 66.0 % 125.8 % 53.5 % 

7 104.4 % 161.7 % 24.3 % 59.8 % 119.6 % 50.1 % 

8 99.2 % 157.0 % 23.5 % 54.6 % 114.3 % 47.3 % 

9 94.5 % 152.8 % 22.7 % 50.2 % 109.6 % 44.8 % 

10 90.2 % 148.9 % 21.9 % 46.4 % 105.5 % 42.6 % 

6.3 Multi-Event Bonus-Malus Scales 

6.3.1 Types of Claims 

Here we adopt the same assumptions as in Chapter 4. Let us pick a policyholder at random 

from the portfolio and let us denote as N the number of claims reported during the year. 

Furthermore, let be the (unknown) a priori expected claim frequency, with Pr = k = w k .


Denoting as the (unknown) accident proneness of this policyholder, the conditional 

probability mass function of N is given by 

PrN = j = = k = exp− k − k j 

j = 0 1 2 

j! 

The risk profile of the portfolio is described by the distribution function F of and we 

assume that E = 1. Since represents the residual effect of unobserved characteristics, it 

seems reasonable to assume that and are mutually independent. Hence, the unconditional 

probability mass function of N is given by 

PrN = j = ∑ k 

∫ + 

w k PrN = j = = k dF 

0 

j = 0 1 

We distinguish among m different types of claim reported by the policyholder. Each type 

of claim induces a specific penalty for the policyholder. For instance, one could think of 

• claims with bodily injuries and claims with material damage only (m = 2) 

• claims with partial liability and claims with full liability (m = 2) 

• introducing claim severities (for instance, claims with amount less than E1000, between 

E1000 and E10 000, and claims above E10 000, so that m = 3). In this case, we have to 

assume that claim severities and claim frequencies are mutually independent. 

Here, we will assume that the claims are classified according to a multinomial scheme. 

Specifically, each time a claim is reported, it is classified in one of the m possible categories, 

with probabilities q 1 q m . Let us denote as N i the number of claims of type i. Then, the 

random vector N 1 N m is Multinomially distributed, with probability mass function 

PrN 1 = k 1 N m = k m = 

{ 

n! 

k 1 !···k m ! qk 1 

1 ···qk m m 

0 otherwise 

if k 1 +···+k m = n 

where n is the total number of claims. 

Each of the m components separately has a Binomial distribution with parameters n and 

q i , for the appropriate value of the subscript i, that is, N i ∼ inn q i . Because of the 

constraint that the sum of the components is n, that is, N 1 +···+N m = n, they are negatively 

correlated. 

The expected value is EN i = nq i . The covariance matrix is as follows: Each diagonal 

entry is the variance of a Binomially distributed random variable, and is therefore 

VN i = nq i 1 − q i . The off-diagonal entries are the covariances. These are 

CN i N j =−nq i q j for i, j distinct. This is a m × m nonnegative-definite matrix of rank 

m − 1. 

We will use the following result. 

Property 6.1 Let us assume that the total number of claims N is oi distributed. 

Assume that the N claims may be classified into m categories, according to a multinomial 

partitioning scheme with probabilities q 1 q m . Let N i represent the number of claims of


type i i = 1m. Then the random variables N 1 N m are independent and Poisson 

distributed with respective parameters q 1 q m . 


Since given N = n, N i ∼ inn q i , we can write 

PrN i = k = 

∑ 

PrN i = kN = n PrN = n 

n=k 

( ∑ n 

= 

)q k i 

n=k 

k 

1 − q i n−k exp− n 

n! 

= exp− q i k ∑ 1 − q i n 

k! 

n=0 

n! 

= exp−q i q i k 

k! 

which proves that N i ∼ oiq i . 

We now prove the independence property of the N i s. To this end, let us show that the 

joint probability mass function factors in the product of marginal probability mass functions: 

PrN 1 = n 1 N m = n m 

n 1+···+n m 

= PrN 1 = n 1 N m = n m N = n 1 +···+n m exp− 

n 1 +···+n m ! 

= n 1 +···+n m ! 

q n 1 

1 

n 1 !···n m ! ···qn m 

m exp− n 1+···+n m 

n 1 +···+n m ! 

m∏ 

= exp−q j q j n j 

j=1 

n j ! 

m∏ 

= PrN j = n j 

j=1 

which completes the proof. 

□ 

Let us now denote as q k1 q k2 q km the probability that the claim is of type 1 2m, 

respectively, for a policyholder with = k . The identity q k1 +q k2 +···+q km = 1 obviously 

holds true. Now, let N 1 N 2 N m be the number of claims of type 1 2m, 

respectively. Considering Property 6.1, given and , the random variables N 1 N 2 N m 

are mutually independent, with respective conditional probability mass function 

PrN l = j = = k = exp− k q kl − kq kl j 

 

j! 

j = 0 1 

for l = 1m.


6.3.2 Markov Modelling for the Multi-Event Bonus-Malus Scale 

The scale is assumed to have s +1 levels, numbered from 0 to s. A specified level is assigned 

to a new driver. Each claim free year is rewarded by a bonus point (i.e. the driver goes one 

level down). Each type of claim entails a specific penalty, expressed as a fixed number of 

levels per claim. 

We assume that the scale possesses the following memoryless property: the knowledge 

of the present level and of the number of claims of each type filed during the present 

year suffices to determine the level to which the policy is transferred. This ensures that the 

bonus-malus system may be represented by a Markov chain (at least conditionally on the 

observable characteristics and random effects). 

Let p l1 l 2 

q be the probability of moving from level l 1 to level l 2 for a policyholder 

with annual mean claim frequency and vector probability q = q 1 q m T ; here q j is the 

probability that the claim be of type j. Further, P q is the one-step transition matrix, i.e. 

⎛ 

⎞ 

p 00 q ··· p 0s q 

P q = 

⎜ 

 

⎝ 

⎟ 

⎠ 

p s0 q ··· p ss q 

Taking the nth power of P q yields the n-step transition matrix whose element l 1 l 2 , 

denoted as p n 

l 1 l 2 

q, is the probability of moving from level l 1 to level l 2 in n transitions. 

The transition matrix P q associated with such a bonus-malus system is assumed 

to be regular, i.e. there exists some integer 0 ≥ 1 such that all entries of ( P q ) 0 

are 

strictly positive. Consequently, the Markov chain describing the trajectory of a policyholder 

with expected claim frequency and vector probability q is ergodic and thus possesses a 

stationary distribution 

q = 0 q 1 q s q T 

Here, l q is the stationary probability for a policyholder with mean frequency to be 

in level l i.e. 

l2 

q = lim 

n→+ pn l 1 l 2 

q 

The stationary probabilities are directly obtained from formula (4.9). 

Let L q be valued in 0 1sand conform to the distribution q i.e. 

PrL q = l = l q 

l = 0 1s 

The variable L q thus represents the level occupied by a policyholder with annual expected 

claim frequency and probability vector q once the steady state has been reached. 

Now, let L be the level occupied in the scale by a randomly selected policyholder once 

the steady state has been reached. The distribution of L can be written as 

PrL = l = ∑ k 

∫ + 

w k l k q k dF l = 0 1s (6.8) 

0


6.3.3 Determination of the relativities 

The relativity associated with level l is denoted as r l , as before. The meaning is that an 

insured occupying that level pays an amount of premium equal to r l % of the reference 

premium determined on the basis of his observable characteristics. 

As in Chapter 4, our aim is to minimize the expected squared difference between the 

‘true’ relative premium and the relative premium r L applicable to this policyholder (after 

the stationary state has been reached), i.e. the goal is to minimize 


] 

E 

[ − r L 2 = 

s∑ 

l=0 

= ∑ k 

∣ ] 

E 

[ − r l 2 ∣∣L = l PrL = l 

w k 

∫ + 

0 

s∑ 

− r l 2 l k q k dF 

l=0 

∫ + ∑k w k 

0 l k q k dF 

r l = EL = l = ∫ + ∑k w k 

0 l k q k dF (6.9) 

It is easily seen that Er L = 1, resulting in financial equilibrium once steady state is reached. 

6.3.4 Numerical Illustrations 

− 1/ + 2/ + 3 Bonus-Malus Scale 

Let us now consider the scale with six levels (numbered from 0 to 5) already used in the 

previous chapters. But now, instead of considering one type of claim, we penalize differently 

claims with bodily injuries and claims with material damage only. If no claims have been 

reported then the policyholder moves one level down. Claims with material damage only 

are penalized by two levels whereas claims with bodily injuries entail a penalty of three 

levels. If n 1 claims with bodily injuries and n 2 claims with material damage only are reported 

during the year then the policyholder moves 3n 1 + 2n 2 levels up. This system is abbreviated 

as −1/ + 2/ + 3, in obvious notations. 

The transition matrix for a policyholder with annual mean claim frequency and vector 

probability q = q 1 q 2 T is given by 

⎛ 

⎞ 

P 0 0 P 1 P 2 P 3 1 − 1 

P 0 0 0 P 1 P 2 1 − 2 

P q = 

0 P 0 0 0 P 1 1 − 3 

⎜ 0 0 P 0 0 0 1− 4 

⎟ 

⎝ 0 0 0 P 0 0 1− 4 

⎠ 

0 0 0 0 P 0 1 − 4 

where 

P 0 = exp− 

P 1 = q 2 exp−q 2 exp−q 1


P 2 = exp−q 2 q 1 exp−q 1 

P 3 = q 2 2 

exp−q 

2 

2 exp−q 1 

and i represents the sum of all the elements in row i. Specifically, 

1 = P 0 + P 1 + P 2 + P 3 

2 = P 0 + P 1 + P 2 

3 = P 0 + P 1 

4 = P 0 

Computation of the Relativities 

Portfolio A Table 6.5 gives for each of the six levels of the bonus-malus scale the proportion 

of the portfolio in that level (column 2) and the relativity attached to that level (column 3) 

for the system −1/ + 2/ + 3. About 70 % of the portfolio is in level 0 and enjoys a discount 

of about 30 %. The rest of the portfolio is spread out among levels 1–5. The r l s range from 

68.0 % to 262.6 %. 

In order to compare the results with those of traditional bonus-malus scales, we have 

also recalled the results given by the three other scales already considered in the previous 

chapters. For all of them, each claim-free year is rewarded by one level down in the scale. 

The first bonus-malus system penalizes each claim (with or without bodily injuries) by 

two levels up in the scale (−1/ + 2 bonus-malus scale), the second one by three levels up 

(−1/ + 3 bonus-malus scale) and the third one sends the policyholder to the maximal level 

after one claim (−1/top bonus-malus scale). 

Clearly, the more the claims are penalized, the more the policyholders occupying the 

lowest levels are awarded discounts, and the less the policyholders in the upper part of the 

scale are penalized. The scale −1/ + 2/ + 3 is closer to the scale −1/ + 2. This is because 

the majority of the claims only induce material damage. Nevertheless, r 5 is reduced from 

271.4 % to 262.6 % when the claims with bodily injuries are more severely penalized. 

Table 6.5 Results for the bonus-malus systems −1/ + 2/ + 3, −1/ + 2, −1/ + 3 and −1/top for 

Portfolio A. 

Level l −1/ + 2/ + 3 −1/ + 2 −1/ + 3 −1/top 

l r l l r l l r l l r l 

5 47 % 2626% 44 % 2714% 73 % 2308% 128 % 1812% 

4 47 % 2158% 47 % 2185% 59 % 2009% 97 % 1599% 

3 49 % 1821% 44 % 1925% 90 % 1452% 77 % 1439% 

2 86 % 1380% 87 % 1388% 73 % 1331% 62 % 1313% 

1 71 % 1278% 71 % 1286% 60 % 1230% 52 % 1209% 

0 700% 680% 706% 685% 645% 642% 585% 612%


Table 6.6 Results for the bonus-malus systems −1/ + 2/ + 3, −1/ + 2, −1/ + 3 and −1/top for 

Portfolio B. 

Level l −1/ + 2/ + 3 −1/ + 2 −1/ + 3 −1/top 

l r l l r l l r l l r l 

5 59 % 2159% 54 % 2232% 92 % 1840% 163 % 1469% 

4 62 % 1695% 60 % 1713% 78 % 1566% 125 % 1293% 

3 64 % 1434% 58 % 1489% 117 % 1174% 99 % 1172% 

2 112 % 1118% 115 % 1121% 95 % 1086% 80 % 1080% 

1 92 % 1046% 93 % 1050% 78 % 1016% 66 % 1007% 

0 611% 745% 620% 747% 540% 720% 466% 706% 

Portfolio B Table 6.6 gives for each of the six levels of the bonus-malus scale the proportion 

of the portfolio in that level (column 2) and the relativity attached to that level (column 3) 

for the system −1/ + 2/ + 3. About 60 % of the portfolio is in level 0 and enjoys a discount 

of about 25 %. The rest of the portfolio is spread out among levels 1–5. The r l s range from 

74.5 % to 215.9 %. 

In order to compare the results with those of traditional bonus-malus scales, we have also 

recalled the results given by the three bonus-malus scales −1/ + 2, −1/ + 3 and −1/top 

already considered in the previous chapters. The scale −1/ + 2/ + 3 is close to the scale 

−1/ + 2 (again, this is because the majority of the claims only induce material damage). 

Nevertheless, r 5 is reduced from 223.2 % to 215.9 % when the claims with bodily injuries 

are more severely penalized. 


The second part of this chapter is based on Pitrebois, Denuit & Walhin (2006a). 

Lemaire (1995, Chapter 13) applied a model proposed by Picard (1976) to Belgian data, 

distinguishing the accidents that caused property damage only, from those that caused bodily 

injuries. The credibility model proposed by Lemaire (1995) is based on a Poisson-Gamma 

mixture, and assumes that given the expected annual claim frequency of the policyholder, 

the frequency of claims with bodily injuries conforms to a Beta distribution. This approach 

can be extended to several categories of claims using a Dirichlet distribution (that is, using 

a suitable multivariate Beta distribution). 

With the aid of multi-equation Poisson models with random effects, Pinquet (1998) 

designed an optimal credibility model for different types of claims. As an example, claims 

are separated into two groups according to fault with respect to a third party. See also 

Pinquet (1997) on allowing for the costs of the claims. 

This chapter gives only basic methods to deal with different types of claims in a posteriori 

ratemaking. Advanced statistical and econometrics models could certainly improve the 

actuarial analysis. For instance, Wedel, Böckenholtb & Kamakurac (2003) developed a 

general class of factor-analytic models for the analysis of multivariate (truncated) count data. 

These models provide a parsimonious and easy-to-interpret representation of multivariate 

dependencies in counts that extend the general linear latent variable model.

7 

Bonus-Malus Systems with 

Varying Deductibles 


In this chapter, we compare bonus-malus systems to deductibles. Specifically, we design a 

system in which the policyholders in the malus zone are allowed to choose at each renewal 

between a premium surcharge (induced by relativities associated with the bonus-malus scale) 

or a deductible in case of a claim during the forthcoming year. If the deductible is selected, 

this induces a strong incentive to careful driving. According to signal theory, drivers opting 

for the deductible are expected to be better drivers (on average) than those paying the 

premium surcharge induced by the upward mode in the bonus-malus scale. 

Bonus-malus systems do not take claim amounts into account so that a posteriori 

corrections only rely on the number of claims. Holtan (1994) suggested the use of very high 

deductibles that may be borrowed by the policyholder from the insurance company. Although 

technically acceptable, this approach obviously causes considerable practical problems. While 

Holtan (1994) assumes a high deductible which is constant for all policyholders, and thus 

independent of the level they occupy in the bonus-malus scale at the claim occurrence time, 

the present chapter lets the deductible vary between the levels of the bonus-malus scales, and 

also considers a mixed case setup of both premium and deductible surcharge after a claim. 

Specifically, the a posteriori premium correction induced by the bonus-malus scale is 

replaced with a deductible (in whole or in part). To each level of the bonus-malus scale is 

attached an amount of deductible, applied to the claims filed during the coverage period. 

Combining bonus-malus scales with varying deductibles presents a number of advantages: 

(1) according to signal theory, policyholders choosing varying deductible should be good 

drivers; 





(2) even if the policyholder leaves the company after a claim, he has to pay for the deductible 

(but in motor third party liability insurance, there may be difficulties linked to the 

collection of the deductible); 

(3) in the mixed system, the r l s and the severity of the deductibles may be tuned in an 

optimal way in order to attract the policyholders. 

The numerical study conducted in this chapter shows, provided appropriate values for the 

parameters are selected, that the amounts of deductible are moderate in the mixed case 

(reduced relativities combined with deductibles per claim). 

Although deductibles can be difficult to implement in motor third party liability insurance 

(companies indemnify the third parties directly and so cannot simply reduce the amount they 

pay), the system is nevertheless easy to implement for first party coverages (for which the 

payments are made directly to the policyholders), like material damage for instance. In the 

European Union, the premium for the material damage cover is either subject to the bonusmalus 

system applying to motor third party liability or to a specific bonus-malus system. 

In the latter case, using the techniques suggested in this chapter, one could replace the 

bonus-malus scale for material damage with deductibles determined by past claim history. 

This chapter will focus on optional coverages. 

7.2 Distribution of the Annual Aggregate Claims 

7.2.1 Modelling Claim Costs 

As explained in the introduction, the strategy designed in this chapter is difficult to apply in 

motor third party liability insurance. To fix the ideas, we consider here first party coverages 

for material damages, so that the actuary is not faced with possible large losses. The total 

claim cost can thus be modelled using a compound mixed Poisson model. Specifically, let 

us denote as C 1 C 2 C N the amounts of the N claims reported by the policyholder. The 

total claim amount for this policy is 

S = 

N∑ 

C k (7.1) 

k=1 

with the convention that the empty sum equals 0. The severities C 1 C 2 , are assumed 

to be independent and identically distributed, and independent of the claim frequency N . 

As in the preceding chapters, N is taken to be oi distributed. This construction 

essentially states that the cost of an accident is for the most part beyond the control of 

a policyholder. The degree of care exercised by a driver mostly influences the number of 

accidents, but in a much lesser way the cost of these accidents. Nevertheless, this assumption 

seems acceptable in practice, at least as an approximation. Note that the severities C 1 C 2 

are also independent of . 

Considering Expression (7.1) for the total cost of claims S, the pure premium for a 

policyholder without claim history is EC 1 . This amount will be corrected by a specific 

relativity according to the level occupied in the bonus-malus scale, that is, according to the 

claims reported to the company in the past.

Bonus-Malus Systems with Varying Deductibles 279 

The distribution function of S is then given by 

PrS ≤ x = = = 

where F is the common distribution function of the C k s and 

∑ 

n=0 

exp− n F ⋆n x x ≥ 0 (7.2) 

n! 

F ⋆n x = PrC 1 +···+C n ≤ x 

is the n-fold convolution of F. Computing F ⋆n x amounts to performing an n-dimensional 

integration of the probability density function corresponding to F. Together with the sum 

over n in (7.2), this of course makes the formula to get PrS ≤ x = = very time 

consuming. Furthermore, we still have to integrate over the possible values of and to 

get PrS ≤ x, that is, 

PrS ≤ x = ∑ k 

∫ + 

w k PrS ≤ x = k = dF x ≥ 0 

0 

Fortunately, the Poisson distribution belongs to the Panjer’s class of counting distributions 

for which there exists a recursive algorithm. 

7.2.2 Discretization 

The computation of the distribution function of S with the help of the Panjer algorithm 

requires the discretization of the individual claim amounts C k . This amounts to replacing 

each C k with a discrete claim cost Ck 

multiple of some fixed monetary unit . Usually, 

actuaries use one of the following two methods to discretize the claim amounts: 

Rounding up 

In this case, a step is selected and the discretized distribution function F is defined as 

F x = Fk for k ≤ xEC k).


Mass Dispersion 

It is also possible to discretize the C k s keeping the expected claim cost unchanged. This is 

done by spreading the probability mass of each i i + 1 on the two extremities of the 

interval (instead of placing all the mass on i + 1 as before). Define for i = 0 1 2, 

f + 

i 

Using integration by parts, f + 

i 

f + 

i 

= 1 

∫ i+1 

x=i 

can be cast into 

( 

i + 1 − x 

) 

dFx 

= 1 ( [(i ) ] i+1 

+ 1 − x Fx + 

 

x=i 

= 1 ( ∫ i+1 

) 

−Fi + Fxdx 

 

x=i 

= 1 

∫ i+1 

x=i 

Similarly, for i = 0 1 2, define 

f − 

i+1 = 1 

= 1 

( 

Fx − Fi 

) 

dx 

∫ i+1 

x=i 

∫ i+1 

x=i 

( 

x − i 

) 

dFx 

∫ i+1 

x=i 

( 

Fi + 1 − Fx 

) 

dx 

) 

Fxdx 

Let us now spread the probability mass Fi + 1 − Fi of the interval i i + 1 on 

its extremities i (which receives f + 

i ) and i + 1 (which receives fi+1 − ). Then, 

PrC 1 = 0 = f 0 = F0 + f + 0 

PrC 1 = i = f i = f − 

i 

+ f + 

i 

for i ≥ 1 (7.4) 

When the discretization is performed according to (7.4), C1 

 

It is easy to see that for k ≤ x


7.2.3 Panjer Algorithm 

Direct Convolution Approach 

The Panjer algorithm then allows us to compute the distribution function of the discretized 

version of S, defined as 

S = 

N∑ 

C k 

k=1 

To realize the merit of the Panjer approach, let us first write the formulas in the direct 

approach. Let us denote as f i = PrC1 

= i the probability mass function of C 1 . The 

probability mass function of C1 +···+C k 

is 

[ ] 

k∑ 

f ⋆k 

i = Pr C j 

= i 

j=1 

In the applications we have in mind, the distribution function for the claim costs satisfies 

F0 = 0. Clearly, with Discretization Method (7.3), we then have f 0 = 0 so that f ⋆k 

i = 0if 

i ≤ k − 1 (since C1 ⋆k 

≥ with probability 1). The fi 

s satisfy 

f ⋆k 

i−k+1 

∑ 

i = f ⋆k−1 

i−j f j if i ≥ k (7.5) 

Then, the probability mass function g i = PrS = i of S satisfies 

g i = 

i∑ 

k=0 

j=1 

PrN = kf ⋆k 

i i∈ (7.6) 

This direct computation of the probability mass function of S requires a lot of computation 

time. Things are even worse in the Discretization (7.4) for which f 0 > 0. 

Panjer Family 

The Panjer formula holds for a class of probability distributions referred to as the Katz 

family in statistical circles, and as the Panjer family in the actuarial literature. This family 

contains all the counting distributions such that the relation 

( 

p k = a + b ) 

p 

k k−1 k= 1 2 (7.7) 

is fulfilled for some a and b. The Panjer family contains three elements. Specifically, the 

probability distributions satisfying (7.7) are 

(i) the Poisson distribution, obtained with a = 0 and b>0; 

(ii) the Negative Binomial distribution, obtained with 0


Panjer Algorithm with f 0 = 0 

The Panjer algorithm is easily established with probability generating functions. Here, we 

use another approach with conditional expectations. The proof of the Panjer formula is based 

on the following technical lemma. 

Lemma 7.1 

The relation 

f ⋆n 

j 

= n j 

j∑ 

i=1 

if i f ⋆n−1 

j−i 

is valid for j ≥ n. 


The proof consists of noting that, on the one hand, for any j ≥ n 

[ ∣ ] [ ∣ ] 

∣∣∣∣ n∑ 

E X 1 X k = j = 1 n∑ ∣∣∣∣ n∑ 

k=1 

n E X k X k = j = j 

k=1 k=1 

n 

and on the other hand, if f ⋆k 

j > 0, 

[ ∣ ] [ 

] 

∣∣∣∣ n∑ 

j∑ 

n∑ 

E X 1 X k = j = i Pr X 1 = i 

X 

∣ k = j 

k=1 

i=1 

k=1 

∑ j 

i=1 

= 

i Pr X 1 = i and ∑ n 

k=2 X k = j − i 

Pr ∑ n 

k=1 X k = j 

∑ j 

i=1 

= 

if if ⋆n−1 

j−i 

 

f ⋆n 

j 

Equating these two formulas yields the expected result. 

□ 

We are now ready to derive the Panjer algorithm if f 0 = 0 (as is the case with Discretization 

Method (7.3)). 

Proposition 7.1 If the probability distribution of N belongs to the Panjer family and if 

f 0 = 0, then the g i s are obtained recursively from 

starting with g 0 = p 0 . 

g j = 

j∑ 

i=1 

( 

a + i b ) 

f 

j i g j−i (7.8) 


Since C k ≥ almost surely, we clearly have 

PrS = 0 = PrN = 0 = p 0


For j ≥ 1, Lemma 7.1 allows us to write 

g j = 

= 

= 

= 

+∑ 

( 

a + b ) 

p 

n n−1 f ⋆n 

j 

n=1 

+∑ 

j∑ 

ap n−1 

n=1 i=1 

j∑ 

i=1 

j∑ 

i=1 

( 

a + i b j 

+∑ 

j−i + 

n=1 

f i f ⋆n−1 

) 

∑+ f i 

n=1 

( 

a + i b ) 

f 

j i g j−i 

p n−1 f ⋆n−1 

j−i 

b 

j∑ 

j p n−1 

i=1 


j−i 


□ 

Corollary 7.1 (Compound Poisson Case) 

If N ∼ oi (a = 0 and b = ), we get 

{ exp− if i = 0 

g i = 

 

∑ i 

i j=1 jf jg i−j if i ≥ 1 

Panjer Algorithm with f 0 > 0 

Let us now consider that f 0 > 0 (resulting from Discretization Method (7.4)). 

Proposition 7.2 If the probability distribution of N belongs to the Panjer family and if 

f 0 ≠ 0, then the g i s are obtained recursively from 

g j = 

starting from g 0 = N f 0 . 

Proof Since f ⋆k 

0 = f 0 k , k ∈ , we easily see that 

( 

1 

j∑ 

a + i b ) 

f 

1 − af 0 j i g j−i (7.9) 

g 0 = 

+∑ 

k=0 

i=1 

p k f 0 k = N f 0 

Following the reasoning in the proof of Proposition 7.1, we get for j>0 

g j = 

+∑ 

ap n−1 

n=1 i=0 

j∑ 

f i f ⋆n−1 

∑+ = af 0 p n−1 f ⋆n−1 

n=1 

= af 0 g j + 

j∑ 

i=1 

+∑ 

j−i + 

n=1 

j∑ 

j + 

i=1 

( 

a + i b j 

b 

j∑ 

j p n−1 

i=1 

( 

a + i b ) 

∑+ f 

j i 

) 

f i g j−i 


j−i 

n=1 

p n−1 f ⋆n−1 

j−i 


□


Computation of the Distribution Function of S 

Clearly, 

i∑ 

PrS > i = 1 − g j 

j=0 

i∈ where g j = PrS = j 

The recursive formula 

{ 1 if i =−1 

PrS > i = 

PrS >i− 1 − g i if 

i ∈ 

gives the distribution of S. 

7.3 Introducing a Deductible within a Posteriori Ratemaking 

Often, the r l s for high levels l are so large that the system has to be softened before a 

possible commercial implementation, resulting in financial instability (since the company 

then faces a progressive decrease of the average premium level because of a clustering 

of the policyholders in the high-discount classes). To avoid this deficiency, the premium 

increase that the policyholder has to pay when he goes up in the scale could be (at least 

partly) replaced by a deductible that would be applied on claims filed by the policyholder 

during the following year. The company compensates for the reduced penalties in the 

malus zone with the deductibles paid by policyholders who report claims whilst in the 

malus zone. This can be commercially attractive since the policyholders are penalized 

only if they file claims in the future. The amount of these deductibles depends on 

the level attained by the policyholder and can be applied either annually or claim by 

claim. 

7.3.1 Annual Deductible 

Let us consider a policyholder occupying level l in the scale. If this policyholder is subject 

to the a posteriori premium corrections induced by the bonus-malus scale, he will have to 

pay r l EC 1 to be covered by the company. If the policyholder is subject to the annual 

deductible instead, he will have to pay the pure premium EC 1 as well as minS d l . The 

indifference principle is now expressed for level l by the equation 

r l EC 1 = EC 1 + ESS


Equation (7.10) is to be solved for all levels l such that r l > 100 % (that is, for all levels 

in the malus zone). So, the premium surcharge r l − 1EC 1 is replaced with an annual 

deductible d l . The relation (7.10) ensures that the substitution is actuarially fair. 

In practice, equation (7.10) does not possess an explicit solution so that numerical 

techniques have to be used. Panjer’s algorithm is employed to derive the distribution of 

S. The claim amounts are discretized according to Method (7.4) and the probability mass 

is concentrated on 500 points. Then, the equation can be solved using appropriate routines 

available in the IML package of SAS R . 

7.3.2 Per Claim Deductible 

Of course, the deductible could also be applied to each claim filed by the policyholder. 

The indifference principle invoked above will again be used to determine the amount of 

the deductible. Considering a policyholder in level l, he will have to pay r l EC 1 if he is 

subject to the a posteriori corrections induced by the bonus-malus scale. If, on the contrary, 

a fixed deductible d l is applied per claim, he will have to pay EC 1 as well as minC k d l 

for each of the claims C k reported to the company. Note that the expected number of claims 

is now r l because past claims history is used to update the claim frequency distribution. 

According to the indifference principle, the amount of deductible d l for a policyholder in 

level l is the solution to the equation 

) 

r l EC 1 = EC 1 + r l 

(EC 1 C 1


Let us now give the equations providing the d l s. In the case of an annual deductible, the 

indifference principle allows us to write 

r l EC 1 = 1 − r l EC 1 + EminS d l 

= 1 − r l EC 1 + ESS


monetary units. The distribution of S is then determined by Panjer’s algorithm. Claim 

amounts are discretized according to Method (7.4). 

7.4.3 Annual Deductible 

The relativities computed for the scale −1/top are displayed in the second column of Table 7.1 

and for the scale −1/ + 2 in Table 7.2. In the pure bonus-malus case, it is thus clear that the 

r l s associated with the upper levels are considerable (more than 300 % for level 5 in case of 

the scale −1/ + 2). Now, let us replace the r l s in the malus zone (i.e. levels 1 to 5) with an 

annual deductible d l . In order to obtain the d l s from equation (7.10), we first discretize the 

claim sizes using Method (7.4). Finally, (7.10) is solved numerically using routines available 

from the SAS R /IML package. 

The third column of each table displays the new relativities. In this case, the maluses 

disappear (r l = 100 % for l = 15) and are compensated for by the deductibles listed 

in the two last columns. The fourth column shows the deductible to be applied if the loss 

amounts are Negative Exponentially distributed and the last column shows the deductible to 

be applied if the loss amounts are LogNormally distributed. Since the LogNormal distribution 

has a thicker tail than the Negative Exponential one, we expect larger amounts of deductible 

for the former. This is indeed the case, as can be seen from Tables 7.1 and 7.2. 

The very high r l s in the second column induce high amounts of deductible, even in the 

Negative Exponential case. Therefore, this solution seems to be difficult (if not impossible) 

to implement in practice. 

Table 7.1 Results for an annual deductible varying according to the level occupied in the malus zone 

and for the scale −1/top. 

Level l r l r l with deductible d l if C 1 ∼ xp d l if C 1 ∼ or 

5 1973 % 100 % 18 367 35 253 

4 1709 % 100 % 14 028 29 303 

3 1507 % 100 % 10 472 23 544 

2 1348 % 100 % 7470 17 934 

1 1220 % 100 % 4897 12 452 

0 547 % 54.7 % 0 0 

Table 7.2 Results for an annual deductible varying according to the level occupied in the malus zone 

and for the scale −1/ + 2. 


5 3091 % 100 % 34 576 50 794 

4 2414 % 100 % 25 076 42 704 

3 2077 % 100 % 20 004 37 241 

2 1429 % 100 % 9027 20 933 

1 1302 % 100 % 6563 16 079 

0 624 % 62.4 % 0 0


7.4.4 Per Claim Deductible 

In this case, there is an analytical solution when the claims are Negative Exponentially 

distributed: the deductible d l involved in (7.11) is simply given by ln r l times the 

expected claim cost. No explicit solution is available when claim amounts are LogNormally 

distributed, and numerical procedures have to be used in this case to find the 

deductibles d l . 

Table 7.3 displays the results obtained for a deductible per claim for the scale −1/top 

and Table 7.4 for the scale −1/ + 2, when the premium paid by the policyholder is 

held constant whatever the claim history. As was the case for the annual deductible, 

the second column gives the relativities associated with each level of the scale in the 

case of a classical bonus-malus system. The third column gives the relative premium 

in the case of a scale with the deductible system. The fourth column shows the 

deductible to be applied if the loss amounts are Negative Exponentially distributed and 

the last column shows the deductible to be applied if the loss amounts are LogNormally 

distributed. 

Again, the amounts of deductible displayed in the last two columns are very high compared 

to the annual premium, especially for the scale −1/ + 2. This results from the severe r l s 

listed in column 2. In order to get acceptable amounts of deductible, keeping the financial 

stability of the system, in the next section we will combine softened penalties in the malus 

zone with moderate deductibles. 

Table 7.3 Results for a deductible per claim varying according to the level occupied in the malus 

zone for the scale −1/top. 


5 1973 % 100 % 14 041 16 254 

4 1709 % 100 % 11 073 12 191 

3 1507 % 100 % 8474 8941 

2 1348 % 100 % 6170 6288 

1 1220 % 100 % 4108 4080 

0 547 % 54.7 % 0 0 


zone for the scale −1/ + 2. 


5 3091 % 100 % 23 317 31 592 

4 2414 % 100 % 18 209 22 633 

3 2077 % 100 % 15 101 17 803 

2 1429 % 100 % 7376 7651 

1 1302 % 100 % 5451 5502 

0 624 % 62.4 % 0 0


7.4.5 Annual Deductible in the Mixed Case 

Another solution would be to keep the system of maluses but to reduce the penalties r l 

applied to the policyholders in the malus zone, for example by choosing = 20 %. In 

exchange for these reduced r l s, the policyholders are subject to an annual deductible on the 

claims they will eventually file. Equation (7.12) used to compute the annual deductible then 

becomes 

r l EC 1 = 80 %r l EC 1 + ESS 100 %. 

Tables 7.5 and 7.6 display the numerical results. The r l s displayed in column 3 are equal 

to 80 % of those in column 2, except for the bonus level 0. Note that the reduced level 1 for 

the scale −1/top enters the bonus zone. The annual deductibles in the Negative Exponential 

case (column 4) are now reasonable (about twice the annual premium for most levels) but 

those in the LogNormal case remain considerable (up to five times the pure premium). 

7.4.6 Per Claim Deductible in the Mixed Case 

If we combine softened penalties for the bonus-malus system with deductibles per claim, 

Equation (7.13) becomes 

20 %EC 1 = EC 1 C 1



zone, combined with reduced relativities r l , for the scale −1/top. 


5 197.3 % 157.8 % 4610 4604 

4 170.9 % 136.7 % 4610 4604 

3 150.7 % 120.6 % 4610 4604 

2 134.8 % 107.8 % 4610 4604 

1 122.0 % 97.6 % 4610 4604 

0 54.7 % 54.7 % 0 0 


zone, combined with reduced relativities r l , for the scale −1/ + 2. 


5 309.1 % 247.3 % 4610 4604 

4 241.4 % 193.1 % 4610 4604 

3 207.7 % 166.2 % 4610 4604 

2 142.9 % 114.3 % 4610 4604 

1 130.2 % 104.2 % 4610 4604 

0 62.4 % 62.4 % 0 0 

As already mentioned, since this equation does not depend on the level l, the amount of 

deductible will be the same for each level of the scale. 

Tables 7.7 and 7.8 display the numerical results. The third column gathers the relativities: 

those in the malus zone have been reduced by 20 % compared to column 2. The last two 

columns display the amounts of deductible. In this case, the amounts of deductible are 

reasonable and can be implemented in practice (about 150 % of the annual pure premium). 

Quite surprisingly, the LogNormal distribution now produces smaller deductibles than its 

Negative Exponential counterpart. 


This chapter is based on Pitrebois, Denuit & Walhin (2005) for the most part. 

The Panjer family of counting distributions is known as the Katz family in applied 

probability. This family attracted a lot of attention in the literature, due to the fact that it 

contains underdispersed (Binomial), equidispersed (Poisson), and overdispersed (Negative 

Binomial) distributions. As a result of Panjer’s (1981) publication, a lot of other articles 

have appeared in the actuarial literature covering similar recursion relations. Multivariate 

versions of the Panjer algorithm will be used in Chapter 9. 

Claim amounts have also been taken into account by Bonsdorff (2005), who studied 

bonus-malus systems where the transitions between bonus levels in the entire interval a b


are determined by the number of claims of the previous year and the total amount of claims 

of the previous year. 

Vandebroek (1993) analysed the efficiency of bonus-malus systems and partial coverages 

in preventing moral hazard problems, by means of stochastic dynamic programming. See 

also Holtan (1994) and Lemaire & Zi (1994a) for the trade-off between bonus-malus 

systems and deductibles.

8 

Transient Maximum Accuracy 

Criterion 


8.1.1 From Stationary to Transient Distributions 

All developments so far have been based on the stationary distribution of the Markov process 

describing the trajectory of the policyholder in the bonus-malus scale. As Borgan, Hoem 

& Norberg (1981) objected, an asymptotic criterion is moderately relevant for bonusmalus 

systems needing relatively long periods to reach their steady state, since policies 

are in force only during a limited number of insurance periods. These authors modified 

the criterion in order to take into account the rating error for new and young policies. 

As Norberg (1976), Borgan ET AL. (1981) measured the performances of a bonus-malus 

system by a weighted average of the expected squared rating errors for selected insurance 

periods. 

In Chapter 4, the relativities associated with the levels of the bonus-malus scale were 

computed on the basis of an asymptotic criterion. The implicit assumption behind the results 

in the preceding chapters is thus that the Markov process reaches its steady state after a 

relatively short period, as is the case for the −1/top bonus-malus scale for instance. If a 

majority of the policies are far from the steady state, it seems desirable to modify the criterion 

so as to take into account the rating error for new policies and for policies of a moderate 

age as well. 

8.1.2 A Practical Example: Creating a Special Scale for New Entrants 

Before entering into the mathematical developments, let us describe a concrete situation 

where the asymptotic criterion used in the previous chapters is no longer relevant, and 





where the transient regime must be considered. This concerns new entrants in a bonusmalus 

system (especially the young drivers). Often, age is included in the a priori 

ratemaking, raising the premium for young drivers (especially young males). The large 

premium surcharges imposed on young drivers pose social problems in many countries. 

As shown by Boucher & Denuit (2006), the heterogeneity is huge inside classes of 

young drivers. Once individual factors have been accounted for (on the basis of a fixed 

effect model for panel data), young drivers even became less risky on average than 

mature and old ones in the empirical study conducted by Boucher & Denuit (2006). In 

fact, the vast majoriy of claims reported by young drivers is concentrated on just a few 

policies. 

In addition to the severe explicit penalties contained in the a priori tariff, young drivers 

enter the bonus-malus scale far above the average level they should occupy given their 

annual expected claim frequency. There is thus an implicit penalty for new drivers (added to 

the explicit penalty found in most commercial price lists), since the relativity corresponding 

to the access level of all bonus-malus systems is in every case substantially higher than the 

average stationary relativity. The implicit surcharge paid by newcomers can be evaluated by 

comparing the access level to the stationary level for the sub-population of the policyholders 

insured for a period of 20 years, say. 

Young inexperienced drivers generally cause many more accidents than the other 

categories of the policyholders. At the same time, classes composed of young 

drivers are more heterogeneous than the other ones: the numerous claims are filed 

by a minority of insured drivers (causing several claims per year). There are 

basically two ways to take this phenomenon into account when designing bonus-malus 

systems: 

• Either the more important residual heterogeneity is recognized (by a larger variance of the 

random effect) and particular transition rules (i.e. heavier penalties when a claim is filed) 

are imposed on young drivers during the first few years. 

• Or young drivers are first placed in a special −1/top scale, and once the bottom level is 

attained they are sent to the regular bonus-malus scale (entering the scale at their average 

stationary level). 

Let us follow the second approach. The bonus-malus scale is as follows: young drivers 

are first placed in the highest level of a −1/top scale (with six levels, say). Careful young 

drivers then reach level 0 in five years, and enter the regular scale at that time (the regular 

bonus-malus scale can be of the −1/+2 type, for instance). 

In such a case, the levels of the initial −1/top scale form a transient class in the 

Markov chain describing the trajectory of the policyholders accross the bonus-malus 

scales. The policyholders will all leave the initial scale sooner or later and never 

come back to it, so that the associated stationary probabilities are all equal to 0. 

Applying the asymptotic criterion to compute the relativities of such a hybrid bonusmalus 

system does not account for the initial scale. The transient distribution will be 

influenced by the −1/top scale, and should therefore be used in this case to determine 

the relativities associated with the −1/top initial scale and with the regular −1/+2 

scale.

Transient Maximum Accuracy Criterion 295 

8.1.3 Agenda 

The transient regime is discussed in Section 8.2, where the convergence of bonus-malus 

systems is analysed. The modified criterion with a quadratic loss function is presented in 

Section 8.3. The exponential loss function is briefly discussed in Section 8.4. 

In Section 8.5, we give the results obtained on the examples studied in the preceding 

chapters (former compulsory Belgian bonus-malus scale, −1/top scale and −1/+2 scale) 

when using the transient maximum accuracy criterion. All the results have been computed 

with the formulas taking the a priori ratemaking into account. We first examine the 

convergence to the steady state. Then we give the transient probability distributions and 

the transient relativities obtained when using a uniform initial distribution and a uniform 

distribution of the age of policy. We also compare the relativities computed using the 

transient maximum accuracy criterion to the relativities obtained with the help of the 

asymptotic maximum accuracy criterion. Finally, we give the evolution of the expected 

financial income. 

Many EU insurers recently started to compete on the basis of bonus-malus systems. 

Because of marketing and competition for market shares, several insurers now offer the best 

level ‘for life’: provided the insured drivers reach level 0, they are allowed to stay in that 

level whatever the claims reported to the company. Note however that insurance companies 

remain free to cancel the policy after each claim. There is thus a super bonus level: the 

driver reaching level 0 of the scale is then allowed to ‘claim for free’. These gifts to the 

best drivers are in contradiction to the actuarial and economic purposes of the a posteriori 

ratemaking systems. They are nevertheless very efficient from the marketing point of view, 

to keep the best drivers in the portfolio. There is thus an absorbing state in the Markov 

model describing the trajectory of the driver in the bonus-malus scale. Consequently, the 

stationary distribution is degenerated, placing a unit probability mass in level 0. Making 

level 0 absorbing for the Markov chain thus forbids the use of the stationary distribution. 

This particular case will be considered in Section 8.6. 

All the numerical illustrations of this chapter are based on Portfolio A. 

8.2 Transient Behaviour and Convergence of Bonus-Malus Scales 

A method of computation of the convergence rate based on the eigenvalues of the transition 

matrix has been discussed in Chapter 4. Here, we examine the evolution of the total variation 

distance between the transient distribution p n 

l 

l = 0 1s and the stationary 

distribution l l = 0 1s: 

d TV p n = 

s∑ 

l=0 

p n 

l 

− l n= 0 1 2 

for some given expected annual claim frequency . Considering a policyholder, picked at 

random in the portfolio, let us denote as L n the level occupied by this policyholder in the 

bonus-malus scale after n years, and as L the level occupied once the stationary regime has 

been reached. The convergence can thus be assessed with 

d TV p n = 

s∑ 

∣ PrL n = l − PrL = l∣ 

l=0


= 

s∑ 

∣ ∑ k 

l=0 

∫ + ( 

w k 

0 

p n 

l 

) 

∣ 

k − l k dF ∣ (8.1) 

for n = 0 1 2 which is the sum on each level l of the absolute difference between the 

probability for a policyholder to be in the level l after n periods and the probability for this 

policyholder to be in level l when the stationary state is reached. 

We can assume that the convergence to the steady state is acceptable when we reach n 0 

such that 

d TV p n 0 ≤ (8.2) 

for some fixed >0. The convergence could then be checked by computing and analysing 

the evolution of (8.1) with n. 

Example 8.1 (Former compulsory Belgian bonus-malus scale) The convergence of the 

former compulsory Belgian bonus-malus scale is assessed using the evolution of 

C n = d TV p n 

displayed in Figure 8.1. We notice a fast convergence during the first 20 years (C n decreasing 

from 0.9 to 0.2) and then a very slow convergence to reach C n


8.3 Quadratic Loss Function 

8.3.1 Transient Maximum Accuracy Criterion 

In this chapter, a policyholder picked at random from the portfolio is now characterized by 

three random variables: , and A. As before, is the expected claim frequency derived 

from the a priori ratemaking and the residual effect due to the heterogeneity remaining 

inside each risk class. The integer-valued random variable A then represents the age of the 

policy in the portfolio and will enable us to take the transient behaviour of a bonus-malus 

system into account. The probability mass function of A is denoted as 

PrA = n = a n n= 1 2 

In words, this means that a proportion a n of the policies in the portfolio have been in force 

for n years. 

We have already seen that the assumption of the independence between and is 

reasonable. As for and A, it seems that they are likely to be correlated. Indeed, it seems 

probable that the age of the policyholder, which is included in the a priori ratemaking, is 

correlated with the age of the policy. However, in order to simplify the computation, we will 

further assume that and A are independent. We also assume the independence between 

and A. 

Recall from Chapter 4 that the sequence of levels occupied by the policyholder in the 

bonus-malus scale is denoted as L 0 L 1 L 2 . Here, we denote as L A the level occupied 

by a policyholder subject to the bonus-malus system for A years. Let us assume that the 

relativity applied to a policyholder picked at random from the portfolio is r A 

L A 

. For a 

policyholder with age of policy n, this relativity becomes r n 

L n 

. As a result, for each group of 

policyholders with age of policy n, the goal is then to minimize the expected squared rating 

error 

[ ∣ ] [ ] 

Q n = E − r A 

L A 

2 ∣∣A = n = E − r n 

L n 

2 


= ∑ k 

w k 

∫ + 

0 

s∑ 

− r n 

l 

2 p n 

l kdF (8.3) 

l=0 

r n 

l 

= EL A = l A = n 

∫ + ∑k w k p n 

0 l 

= 

kdF 

∫ + ∑k w k p n 

0 l kdF (8.4) 

We see that (4.13) is a limiting form of equation (8.4) when n tends to infinity. 

The asymptotic criterion Q seems reasonable when a majority of the risks are close to 

the steady state. In practice, however, real portfolios will often have a substantial fraction of 

comparatively young policies. Then it is desirable to obtain a solution for the relativities not


only by means of Q but also by means of Q n for various finite n. This is why we introduce 

a new criterion based on a weighted average of the form 

[ ] ∑ 

¯Q = E − r A 

L A 

2 = a n Q n 

n=1 

The solution of (8.5) is then given by 

k 

0 

n=1 

∑ ∑ ∫ + s∑ 

= a n w k − r n 

l 

2 p n 

l kdF (8.5) 

l=0 

¯r l = EL A = l 

∑ 

= r n 

l 

PrA = nL A = l 

n=1 

∑ 

n=0 

= 

a ∫ + 

n 

∑k w k p n 

0 l 

kdF 

∑ 

n=1 a ∫ + 

n 

∑k w k p n 

0 l kdF (8.6) 

Remark 8.1 Let us mention that if the insurance company does not enforce any a priori 

ratemaking system, all the k s are equal to and (8.6) reduces to the formula 

∑ 

n=1 

¯r l = 

a ∫ + 

n 

∑ 

n=1 a ∫ + 

n 

p n 

0 l 

p n 

0 l 

dF 

dF 

that has been derived in Borgan, Hoem & Norberg (1981). 

Example 8.2 (Former Compulsory Belgian Bonus-Malus Scale) We saw from Figure 8.1 

that the steady state was approached at a slow rate for the 23-level Belgian bonus-malus scale. 

This is why we selected 50 years in the distribution of policy age. The transient relativities 

presented in Table 8.2 correspond to the transient distributions displayed in Table 8.1. 

They have all been computed with a uniform initial distribution (PrL 0 = l = 1/23 for 

l = 0 122). 

We observe that the transient relativities slowly converge to the steady state relativities 

given in the last column of Table 8.2 but also that they generally underestimate these 

stationary relativities. This agrees with the fact that the transient probabilities of being in 

level 0 underestimate the steady state probability of being in this level. Therefore, the bonuses 

of levels 0 1 2, 3 and 4 must be greater and the maluses of levels 5 to 22 must be smaller 

to ensure financial balance. Moreover, we clearly see that, in the first steps of the transient 

behaviour, there are more levels granting a bonus in order to balance the lack (with respect 

to the stationary state) of policyholders in the lowest levels. 

In addition to the uniform initial distribution, let us now consider a ‘top’ distribution (78 % 

of the policyholders are concentrated in level 22 at time 0 and each other level contains 1 % 

of the policyholders) and a ‘bottom’ distribution (78 % of the policyholders are placed in 

level 0 at time 0 and the others are spread uniformly over levels 1 to 22). Let us mention 

that all the policyholders could not be placed in level 0 or 22 because Equation (8.4) is not 

defined when p n 

l 

equals 0. The results are displayed in Tables 8.3 and 8.4. We can see there


Table 8.1 Evolution of the uniform initial distribution for the former compulsory Belgian bonusmalus 

scale. 

Level l PrL 0 = l PrL 10 = l PrL 20 = l PrL 30 = l PrL 40 = l PrL 50 = l PrL = l 

22 43% 46% 51% 53% 53% 53% 54% 

21 43% 34% 36% 37% 38% 38% 38% 

20 43% 28% 28% 29% 29% 29% 29% 

19 43% 24% 23% 23% 23% 23% 23% 

18 43% 23% 21% 20% 19% 19% 19% 

17 43% 25% 19% 18% 17% 17% 16% 

16 43% 26% 18% 16% 15% 15% 15% 

15 43% 26% 18% 15% 14% 14% 13% 

14 43% 27% 17% 15% 14% 13% 13% 

13 43% 28% 18% 15% 13% 13% 12% 

12 43% 41% 20% 15% 14% 13% 12% 

11 43% 40% 20% 15% 14% 13% 12% 

10 43% 40% 20% 16% 14% 13% 12% 

9 43% 40% 22% 17% 15% 14% 14% 

8 43% 40% 23% 18% 17% 16% 16% 

7 43% 40% 28% 20% 18% 18% 17% 

6 43% 39% 28% 21% 19% 18% 18% 

5 43% 37% 28% 21% 19% 18% 18% 

4 43% 47% 45% 42% 42% 42% 43% 

3 43% 42% 40% 38% 38% 38% 38% 

2 43% 38% 44% 35% 34% 34% 34% 

1 43% 34% 40% 32% 31% 31% 31% 

0 43% 236% 393% 469% 490% 497% 503% 

Table 8.2 Evolution of the transient relativities for the uniform initial distribution for the former 

compulsory Belgian bonus-malus scale. 

Level l 

r 0 

l 

r 10 

l 

r 20 

l 

r 30 

l 

r 40 

l 

r 50 

l 

r l 

22 00 % 2641 % 2700 % 2709 % 2712 % 2714 % 2715% 

21 00 % 2290 % 2419 % 2450 % 2462 % 2468 % 2474% 

20 00 % 2011 % 2195 % 2248 % 2269 % 2279 % 2291% 

19 00 % 1792 % 2008 % 2079 % 2110 % 2124 % 2141% 

18 00 % 1640 % 1854 % 1937 % 1974 % 1992 % 2014% 

17 00 % 1413 % 1704 % 1807 % 1853 % 1876 % 1903% 

16 00 % 1321 % 1594 % 1700 % 1750 % 1775 % 1803% 

15 00 % 1239 % 1499 % 1608 % 1659 % 1685 % 1714% 

14 00 % 1179 % 1422 % 1528 % 1578 % 1603 % 1631% 

13 00 % 1149 % 1364 % 1459 % 1504 % 1526 % 1549% 

12 00% 869 % 1253 % 1376 % 1427 % 1450 % 1472% 

11 00% 857 % 1201 % 1319 % 1365 % 1385 % 1402%


Table 8.2 

(Continued). 

Level l 

r 0 

l 

r 10 

l 

r 20 

l 

r 30 

l 

r 40 

l 

r 50 

l 

r l 

10 00% 831 % 1151 % 1266 % 1310 % 1327 % 1340% 

9 00% 821 % 1107 % 1206 % 1238 % 1249 % 1255% 

8 00% 826 % 1073 % 1150 % 1170 % 1174 % 1173% 

7 00% 815% 947 % 1078 % 1109 % 1115 % 1114% 

6 00% 791% 913 % 1038 % 1065 % 1070 % 1067% 

5 00% 755% 876 % 1000 % 1026 % 1030 % 1028% 

4 00% 774% 809% 833% 831% 829% 826% 

3 00% 741% 782% 807% 806% 804% 802% 

2 00% 704% 648% 767% 779% 779% 778% 

1 00% 664% 618% 742% 754% 755% 755% 

0 00% 515% 438% 435% 442% 446% 451% 

Table 8.3 Evolution of the transient relativities for the top initial distribution for the former 


Level l 

r 0 

l 

r 10 

l 

r 20 

l 

r 30 

l 

r 40 

l 

r 50 

l 

r l 

22 00 % 2227 % 2488 % 2587 % 2639 % 2668 % 2715% 

21 00 % 1952 % 2217 % 2328 % 2387 % 2421 % 2474% 

20 00 % 1749 % 2012 % 2131 % 2195 % 2233 % 2291% 

19 00 % 1592 % 1850 % 1973 % 2041 % 2081 % 2141% 

18 00 % 1470 % 1719 % 1843 % 1913 % 1954 % 2014% 

17 00% 946 % 1330 % 1546 % 1678 % 1763 % 1903% 

16 00 % 1093 % 1431 % 1594 % 1684 % 1735 % 1803% 

15 00 % 1031 % 1354 % 1515 % 1603 % 1652 % 1714% 

14 00% 980 % 1288 % 1446 % 1531 % 1577 % 1631% 

13 00% 942 % 1234 % 1385 % 1464 % 1504 % 1549% 

12 00% 460% 890 % 1116 % 1256 % 1344 % 1472% 

11 00% 857 % 1026 % 1222 % 1317 % 1361 % 1402% 

10 00% 831% 987 % 1177 % 1266 % 1305 % 1340% 

9 00% 821% 954 % 1126 % 1199 % 1229 % 1255% 

8 00% 826% 926 % 1074 % 1130 % 1152 % 1173% 

7 00% 815% 589% 858 % 1001 % 1069 % 1114% 

6 00% 791% 746% 944 % 1013 % 1039 % 1067% 

5 00% 755% 722% 911% 975% 999 % 1028% 

4 00% 774% 711% 743% 774% 794% 826% 

3 00% 741% 685% 719% 751% 770% 802% 

2 00% 704% 311% 632% 748% 776% 778% 

1 00% 664% 618% 651% 691% 717% 755% 

0 00% 515% 438% 361% 404% 425% 451%


Table 8.4 Evolution of the transient relativities for the bottom initial distribution for the former 


Level l 

r 0 

l 

r 10 

l 

r 20 

l 

r 30 

l 

r 40 

l 

r 50 

l 

r l 

22 00 % 3049 % 2902 % 2818 % 2774 % 2750 % 2715% 

21 00 % 2681 % 2629 % 2566 % 2529 % 2508 % 2474% 

20 00 % 2429 % 2402 % 2361 % 2334 % 2318 % 2291% 

19 00 % 2075 % 2240 % 2215 % 2189 % 2172 % 2141% 

18 00 % 1901 % 2071 % 2066 % 2050 % 2038 % 2014% 

17 00 % 1776 % 1929 % 1933 % 1926 % 1919 % 1903% 

16 00 % 1813 % 1841 % 1827 % 1819 % 1814 % 1803% 

15 00 % 1866 % 1788 % 1745 % 1729 % 1722 % 1714% 

14 00 % 1408 % 1607 % 1633 % 1637 % 1636 % 1631% 

13 00 % 1482 % 1566 % 1562 % 1557 % 1554 % 1549% 

12 00 % 1341 % 1513 % 1499 % 1486 % 1480 % 1472% 

11 00 % 1426 % 1493 % 1455 % 1430 % 1417 % 1402% 

10 00 % 1486 % 1474 % 1419 % 1384 % 1364 % 1340% 

9 00 % 1153 % 1293 % 1285 % 1272 % 1264 % 1255% 

8 00 % 1223 % 1263 % 1231 % 1206 % 1192 % 1173% 

7 00 % 1255 % 1203 % 1185 % 1158 % 1140 % 1114% 

6 00 % 1268 % 1181 % 1152 % 1120 % 1099 % 1067% 

5 00 % 1270 % 1159 % 1122 % 1087 % 1064 % 1028% 

4 00% 994% 920% 884% 861% 847% 826% 

3 00% 981% 901% 863% 839% 824% 802% 

2 00% 966% 848% 838% 816% 801% 778% 

1 00% 950% 827% 816% 794% 779% 755% 

0 00% 566% 498% 475% 465% 460% 451% 

that even if the ultimate distribution of the policyholders in the scale is the same whatever 

the initial distributions, the transient relativities are affected by these distributions. Starting 

with a uniform distribution of the policyholders in the scale (Table 8.2), the r n 

l 

s increase 

to the r l s for levels 1 to 22, and they decrease to the limit for l = 0. The same phenomenon 

arises when starting with the top distribution (Table 8.3), but the difference between the 

r 10 

l 

s and the asymptotic r l s is now larger. On the contrary, if the bottom distribution is used 

(Table 8.4) then the r n 

l 

s decrease to their limit r l . 

The relativities ¯r l computed using (8.6) are given in Table 8.5 according to the initial 

distribution of the policyholders in the scale and for a uniform distribution of age of policy 

over 50 years, that is, a n = 1/50 for n = 150. For the sake of completeness, the steady 

state relativities are displayed in the last column. When a uniform initial distribution of the 

policyholders is used, the relativities based on the transient maximum accuracy criterion are 

smaller than the steady state relativities, except for level 0. This is the case for all levels if 

the top distribution is assumed. On the contrary, if the bottom distribution is used then the 

¯r l s are larger than the corresponding r l s. 

In order to figure out the impact of the maturity of the portfolio on the relativities, 

we considered two alternative age structures to the uniform age of policy distribution 

used so far. The three distributions (henceforth referred to as mature, young and old


Table 8.5 Relativities computed on the basis of the transient maximum accuracy criterion for the 

former compulsory Belgian bonus-malus scale, and for different initial distributions. 

Level l Uniform Top Bottom Steady state 

distribution ¯r l distribution ¯r l distribution¯r l distribution r l 

22 2661 % 2411 % 2835 % 2715% 

21 2344 % 1854 % 2552 % 2474% 

20 2086 % 1640 % 2326 % 2291% 

19 1874 % 1475 % 2137 % 2141% 

18 1701 % 1345 % 1983 % 2014% 

17 1561 % 1242 % 1856 % 1903% 

16 1448 % 1157 % 1750 % 1803% 

15 1356 % 1088 % 1660 % 1714% 

14 1281 % 1029 % 1589 % 1631% 

13 1219% 980 % 1537 % 1549% 

12 1166% 937 % 1483 % 1472% 

11 1118% 898 % 1426 % 1402% 

10 1075% 864 % 1371 % 1340% 

9 1038% 835 % 1337 % 1255% 

8 1001% 808 % 1287 % 1173% 

7 963% 781 % 1228 % 1114% 

6 927% 753 % 1172 % 1067% 

5 892% 727 % 1120 % 1028% 

4 818% 689% 997% 826% 

3 780% 658% 941% 802% 

2 745% 630% 893% 778% 

1 713% 602% 850% 755% 

0 458% 401% 525% 451% 

portfolios, respectively) are summarized in Table 8.6. Table 8.7 displays the bonus-malus 

relativities obtained with these different age structures and a uniform initial distribution of 

the policyholders in the bonus-malus scale. We see that the older the portfolio, the closer 

the ¯r l s to the corresponding r l s. 

8.3.2 Linear Scales 

Sometimes, it is desirable to have the same relative penalty associated with each level. To this 

end, the actuary can linearize the ¯r l s, as suggested by Gilde & Sundt (1989). The optimal 

linear relativity ¯r 

l 

lin = +l, l = 0 1s, in the transient case is thus the solution of the 

minimization of 

E [ − r A 

L A 

2] = E [ − − L A 2] 

It is easy to check that the solution of this optimization problem is 

= CL A 

VL A 

and = E − CL A 

EL 

VL A A


Table 8.6 

Values of the a n s for three portfolios with different maturities. 

Age of policy Mature portfolio Young portfolio Old portfolio 

1 2 % 10 % 0.5 % 

2 2 % 10 % 0.5 % 

3 2 % 5 % 0.5 % 

4 2 % 5 % 0.5 % 

5 2 % 5 % 0.5 % 

6 2 % 5 % 0.5 % 

7 2% 5% 1% 

8 2 % 2.5 % 1 % 

9 2 % 2.5 % 1 % 

10 2 % 2.5 % 1 % 

11 2 % 2.5 % 1 % 

12 2 % 2.5 % 1 % 

13 2 % 2.5 % 1 % 

14 2 % 2.5 % 1 % 

15 2 % 2.5 % 1 % 

16 2 % 2.5 % 1 % 

17 2 % 2.5 % 1 % 

18 2% 1% 1% 

19 2% 1% 1% 

20 2% 1% 1% 

21 2% 1% 1% 

22 2% 1% 1% 

23 2% 1% 1% 

24 2% 1% 1% 

25 2% 1% 1% 

26 2% 1% 1% 

27 2% 1% 1% 

28 2% 1% 1% 

29 2% 1% 1% 

30 2% 1% 1% 

31 2% 1% 1% 

32 2% 1% 1% 

33 2% 1% 1% 

34 2 % 1 % 2.5 % 

35 2 % 1 % 2.5 % 

36 2 % 1 % 2.5 % 

37 2 % 1 % 2.5 % 

38 2 % 1 % 2.5 % 

39 2 % 1 % 2.5 % 

40 2 % 1 % 2.5 % 

41 2 % 1 % 2.5 % 

42 2 % 1 % 2.5 % 

43 2 % 1 % 2.5 % 

44 2% 1% 5% 

45 2 % 0.5 % 5 % 

46 2 % 0.5 % 5 %


Table 8.6 

(Continued). 

Age of policy Mature portfolio Young portfolio Old portfolio 

47 2 % 05% 5% 

48 2 % 05% 5% 

49 2 % 05 % 10 % 

50 2 % 05 % 10 % 

Table 8.7 Relativities computed on the basis of the transient maximum accuracy criterion for the 

former compulsory Belgian bonus-malus scale, and for different maturities of the portfolio. 

Level l Mature Young Old Steady state 

portfolio ¯r l portfolio ¯r l portfolio ¯r l distribution r l 

22 2661 % 2536 % 2697 % 2715% 

21 2344 % 2080 % 2426 % 2474% 

20 2086 % 1738 % 2210 % 2291% 

19 1874 % 1529 % 2030 % 2141% 

18 1701 % 1382 % 1876 % 2014% 

17 1561 % 1275 % 1744 % 1903% 

16 1448 % 1197 % 1631 % 1803% 

15 1356 % 1136 % 1529 % 1714% 

14 1281 % 1097 % 1442 % 1631% 

13 1219 % 1064 % 1365 % 1549% 

12 1166 % 1035 % 1297 % 1472% 

11 1118 % 1008 % 1238 % 1402% 

10 1075% 982 % 1186 % 1340% 

9 1038% 965 % 1131 % 1255% 

8 1001% 942 % 1078 % 1173% 

7 963% 914 % 1032 % 1114% 

6 927% 888% 991 % 1067% 

5 892% 863% 954 % 1028% 

4 818% 844% 822% 826% 

3 780% 792% 790% 802% 

2 745% 754% 761% 778% 

1 713% 725% 733% 755% 

0 458% 505% 448% 451% 

The linear premium scale is thus of the form 

where 

r lin 

l 

= E + CL A 

l − EL 

VL A 

A 

CL A = 

+∑ 

a n 

n=1 

s∑ 

l=0 

l ∑ k 

w k 

∫ + 

0 

p n 

l 

kdF − EL A E


EL A = 

VL A = 

∑ 

a n 

n=1 l=0 k 

∑ 

8.3.3 Financial Balance 

a n 

n=1 l=0 k 

s∑ ∑ ∫ + 

w k 

0 

lp n 

l 

kdF 

s∑ ∑ ∫ + 

w k l − EL A 2 p n kdF 

0 

We know from Chapter 4 that when the relativities are derived from the asymptotic criterion, 

the bonus-malus system is financially balanced when the steady state has been reached. The 

only way of keeping this financial balance during the transient behaviour of the bonusmalus 

scale is to change the relativities applied to the policyholders so that they are equal 

to the r n 

l 

s in each period n. For commercial reasons, it seems difficult to adopt such a 

strategy. 

But, when the number of levels and the transistion rules of the bonus-malus system have 

been fixed, it is possible to check how the expected financial income evolves with respect to 

the financial balance for the different periods n = 1 2 until the steady state is reached. 

If the relativities in force are the r l s computed on the basis of the asymptotic maximum 

accuracy criterion, then the expected financial income for policies with age n is 

s∑ 

I n = r l PrL n = l 

= 

l=0 

l=0 

k 

0 

l 

s∑ ∑ ∫ + 

w k r l p n 

l kdF (8.7) 

where the p n 

l 

·s are computed using (4.6). 

Alternatively, if the relativities are the ¯r l s computed on the basis of the transient maximum 

accuracy criterion, then the expected financial income for policies with age n is 

Ī n = 

= 

s∑ 

¯r l PrL n = l 

l=0 

s∑ ∑ ∫ + 

w k 

l=0 

k 

0 

¯r l p n 

l kdF (8.8) 

The evolution of the expected income, until steady state has been reached, is probably 

one of the most important parameters to take into account. 

Example 8.3 (Former Belgian Compulsory Bonus-Malus Scale) Table 8.8 displays the 

evolution of the expected financial income I n (computed with the steady state relativities) 

according to the initial distribution of the policyholders. It slowly converges to 100 %. The 

choice of the uniform initial distribution or the top initial distribution leads to an expected 

financial income greater than 100 %. Indeed, too many policyholders (with respect to the 

steady state situation) are in the malus levels, thus providing a greater income to the company. 

Conversely, with a bottom initial distribution, the expected financial income is smaller than 

100 % as too many policyholders are in the bonus levels.


Table 8.8 Evolution of the expected financial income I n based on r l (influence of 

the initial distribution). 

Age of policy Uniform Top Bottom 

distribution I n distribution I n distribution I n 

0 1465 % 2428% 684% 

1 1431 % 2258% 720% 

2 1401 % 2141% 753% 

3 1374 % 2055% 784% 

4 1349 % 1987% 812% 

5 1326 % 1932% 826% 

6 1304 % 1878% 837% 

7 1282 % 1828% 848% 

8 1261 % 1783% 860% 

9 1241 % 1740% 872% 

10 1222 % 1701% 880% 

11 1203 % 1663% 886% 

12 1186 % 1629% 892% 

13 1169 % 1589% 899% 

14 1153 % 1551% 906% 

15 1139 % 1521% 910% 

16 1125 % 1493% 914% 

17 1112 % 1468% 918% 

18 1099 % 1407% 922% 

19 1089 % 1383% 926% 

20 1079 % 1363% 929% 

21 1070 % 1345% 932% 

22 1061 % 1275% 935% 

23 1055 % 1243% 938% 

24 1050 % 1231% 942% 

25 1046 % 1220% 944% 

26 1041 % 1210% 947% 

27 1037 % 1176% 949% 

28 1034 % 1158% 952% 

29 1031 % 1150% 954% 

30 1028 % 1144% 957% 

31 1026 % 1137% 959% 

32 1023 % 1119% 960% 

33 1022 % 1108% 962% 

34 1020 % 1103% 965% 

35 1019 % 1099% 966% 

36 1017 % 1095% 968% 

37 1016 % 1084% 969% 

38 1015 % 1076% 971% 

39 1014 % 1073% 972% 

40 1013 % 1070% 974% 

41 1012 % 1067% 975% 

42 1011 % 1060% 976% 

43 1010 % 1055% 977% 

44 1009 % 1053% 978% 

45 1009 % 1051% 980%


46 1008 % 1049% 980% 

47 1008 % 1044% 981% 

48 1007 % 1041% 982% 

49 1007 % 1039% 983% 

50 1006 % 1038% 984% 

8.3.4 Choice of an Initial Level 

As the number of years increases, the influence of the initial level diminishes for the 

individual policy, and it vanishes in the limit. Therefore, the choice of the initial level cannot 

be made part of the optimizing procedure based on the asymptotic criterion. 

When the transient behaviour is taken into account, it is convenvient to select the optimal 

starting level by maximizing a measure of efficiency for the bonus-malus scale. In the 

transient case, the Q n -efficiency is defined as 

s∑ 

e n = r n 2 PrL n = l 

= 

l=0 

l 

s∑ ∑ ∫ + 

w k 

l=0 

k 

0 

r n 

l 

2 p n 

l kdF (8.9) 

The Q n -efficiency is equal to the variance Vr n 

L n 

up to a constant term, since 

[ ] ( [ ]) 2 

Vr n 

L n 

= E r n 

L n 

2 − E r n 

L n = en − 1 (8.10) 

Looking for an optimum of e n , it is then equivalent to use (8.9) or (8.10). The Q-efficiency, 

which is closely related to the variance Vr L ,is 

e = 

= 

s∑ 

r l 2 PrL = l 

l=0 

s∑ ∑ ∫ + 

w k r l 2 l k dF = Vr L + 1 

l=0 

k 

Finally, the ¯Q-efficiency, which is equivalent to V¯r LA 

,is 

ē = 

= 

0 

s∑ 

¯r l 2 PrL A = l 

l=0 

∑ 

a n 

n=1 l=0 k 

= V¯r LA 

+ 1 

s∑ ∑ ∫ + 

w k ¯r l 2 p n kdF 

0 

To choose the initial level, it suffices to compute the relativities ¯r l , with the help of (8.6), 

for each initial distribution e k , where e k is the vector with 1 in the kth entry and 0 elsewhere. 

After having repeated the process for each e k , k = 0 1s, we compute the ¯Q-efficiency 

of each solution. The optimal initial level is then the one that maximizes the ¯Q-efficiency. 

l


8.4 Exponential Loss Function 

Under an exponential loss function, the aim is to minimize the objective function 

[ 

n = E exp ( − c − r n 

L n 

)] 

under the constraint Er n 

L n 

= E. This yields 

r n 

l 

= E + 1 c 

( [ 

] 

) 

E ln Eexp−cL n − ln Eexp−cL n = l 

for l = 0 1s. 

Of course, there is no reason to focus on the particular nth period. Then, nonnegative 

weights a 1 a 2 a 3 summing to 1 representing the age distribution of the policies in the 

portfolio are introduced and the aim of the actuary is to minimize 

= 

+∑ 

n=1 

a n n 

Minimizing means minimizing the expected squared rating error for a randomly chosen 

policy. 

8.5 Numerical Illustrations 

8.5.1 Scale −1/Top 

The transition rules of the −1/top scale are given in Table 4.1. 

Initial Distribution 

We have tested three different initial distributions PrL 0 = l. In the first case (uniform 

distribution), the policyholders are uniformly spread in the scale (16.67 % of the policyholders 

in each of the six levels). In the second case (top distribution), 95 % of the policyholders 

are concentrated in the top of the scale (and the remaining 5 % are evenly spread over 

levels 0 to 4). In the last case (bottom distribution), 95 % of the policyholders start 

from the bottom of the scale (and the remaining 5 % are evenly spread over levels 

1to5). 

Convergence of the −1/Top Scale 

Figure 8.2 represents the evolution of C n with n for a uniform initial distribution. It gives an 

idea of the speed of convergence of the −1/top scale. We clearly see that C n = 0 for n ≥ 5, 

i.e. that the stationary state is reached after 5 years. This was known from Chapter 4. 

Transient Relativities 

Table 8.9 gives the evolution with n of the corresponding transient relativities for the three 

starting distributions. We see that the transient relativities do not depend on the initial


Cn 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0.0 

0 1 2 3 4 5 

Age of policy 

Figure 8.2 Convergence of the transient policyholders’ distributions to the steady state distribution 

for the system −1/top. 

distribution of the policyholders inside the scale. This comes from the fact that the transient 

distributions do not depend on the initial distribution for the −1/top scale. 

This somewhat surprising situation can be explained as follows. The distribution of the 

policyholders in the −1/top scale after one year is given by 

⎛ 

⎜ 

⎝ 

p 1 

0 

⎞ 

p 1 

1 

p 1 

2 

p 1 

3 

p 1 

4 

⎟ 

⎠ 

p 1 

5 

⎛ 

= 

⎜ 

⎝ 

p 0 

0 

p 0 

1 

p 0 

2 

p 0 

3 

p 0 

4 

p 0 

5 

⎞T ⎛ 

⎞ 

exp− 0 0 0 0 1− exp− 

exp− 0 0 0 0 1− exp− 

0 exp− 0 0 0 1− exp− 

0 0 exp− 0 0 1− exp− 

⎟ ⎜ 

⎠ ⎝ 0 0 0 exp− 0 1− exp−⎟ 

⎠ 

0 0 0 0 exp− 1 − exp− 

⎛ 

= 

⎜ 

⎝ 

p 0 

0 + p 0 

1 exp− ⎞ 

⎟ 

⎠ 

p 0 

2 exp− 

p 0 

3 exp− 

p 0 

4 exp− 

p 0 

5 

exp− 

1 − exp−


Table 8.9 

Transient relativities for the different initial distributions (−1/top scale). 

Level l 

r 0 

l 

r 1 

l 

Uniform initial distribution 

r 2 

l 

r 3 

l 

r 4 

l 

r 5 

l 

r l 

5 00 % 1812 % 1812 % 1812 % 1812 % 1812 % 1812% 

4 00% 882 % 1599 % 1599 % 1599 % 1599 % 1599% 

3 00% 882% 792 % 1439 % 1439 % 1439 % 1439% 

2 00% 882% 792% 720 % 1312 % 1312 % 1312% 

1 00% 882% 792% 720% 661 % 1209 % 1209% 

0 00% 882% 792% 720% 661% 612% 612% 

Level l 

r 0 

l 

r 1 

l 

r 2 

l 

Top initial distribution 

r 3 

l 

r 4 

l 

r 5 

l 

r l 

5 00 % 1812 % 1812 % 1812 % 1812 % 1812 % 1812% 

4 00% 882 % 1599 % 1599 % 1599 % 1599 % 1599% 

3 00% 882% 792 % 1439 % 1439 % 1439 % 1439% 

2 00% 882% 792% 720 % 1312 % 1312 % 1312% 

1 00% 882% 792% 720% 661 % 1209 % 1209% 

0 00% 882% 792% 720% 661% 612% 612% 

Level l 

r 0 

l 

r 1 

l 

Bottom initial distribution 

r 2 

l 

r 3 

l 

r 4 

l 

r 5 

l 

r l 

5 00 % 1812 % 1812 % 1812 % 1812 % 1812 % 1812% 

4 00% 882 % 1599 % 1599 % 1599 % 1599 % 1599% 

3 00% 882% 792 % 1439 % 1439 % 1439 % 1439% 

2 00% 882% 792% 720 % 1312 % 1312 % 1312% 

1 00% 882% 792% 720% 661 % 1209 % 1209% 

0 00% 882% 792% 720% 661% 612% 612% 

The corresponding relativities are obtained by (8.4). They are given by 

⎛ 

⎜ 

⎝ 

r 1 

0 

r 1 

1 

r 1 

2 

r 1 

3 

r 1 

4 

r 1 

5 

⎛ ∫ 

⎞ 

+ ∑k w k p 0 

0 0 + p 0 

1 exp− ⎞ 

kdF 

∫ + 

∑k w k p 0 

0 0 + p 0 

1 exp− kdF 

∫ + 

∑k w k p 0 

0 2 exp− k dF 

∫ + 

∑k w k p 0 


∫ + 

∑k w k p 0 


∫ + 

∑k w k p 0 


= 

∫ + 

∑k w k p 0 


∫ + 

∑k w k p 0 


∫ + 

∑k w k p 0 

0 5 

exp− k dF 

∫ + 

∑k w k p 0 

0 5 

exp− k dF 

⎟ 

⎠ ⎜ ∫ + 

⎟ 

⎝ 

∑k w k 1 − exp− 

0 k dF ⎠ 

∫ + ∑k w k 1 − exp− 

0 k dF


⎛ 

∑k w k 

∫ + 

0 

exp− k dF 

∑k w k 

∫ + 

0 

exp− k dF 

= 

⎜ 

∫ + 

⎝ 

∑k w k 

∑k w k 

∫ + 

0 

exp− k dF 

∑k w k 

∫ + 

0 

exp− k dF 

∑k w k 

∫ + 

0 

exp− k dF 

∑k w k 

∫ + 

0 

exp− k dF 

∑k w k 

∫ + 

0 

exp− k dF 

∑k w k 

∫ + 

0 

exp− k dF 

∑k w k 

∫ + 

0 

exp− k dF 

∑k w k 

∫ + 

0 

exp− k dF 

0 

1 − exp− k dF 

∑k w k 

∫ + 

0 


⎞ 

⎟ 

⎠ 

We see that the relativities at time 1 do not depend on the initial distribution p 0 

l l = 

0 15. Moreover, we observe that the values r 1 

0 to r 1 

4 are all equal. Only the value 

of r 1 

5 

is different from the others and is equal to the steady-state value r 5 . 

The distribution of the policyholders in the −1/top scale after two years is then given by 

⎛ 

⎜ 

⎝ 

p 2 

0 

⎞ 

p 2 

1 

p 2 

2 

p 2 

3 

p 2 

4 

⎟ 

⎠ 

p 2 

5 

⎛ 

= 

⎜ 

⎝ 

p 0 

0 

p 0 

1 

p 0 

2 

p 0 

3 

p 0 

4 

p 0 

5 

⎞T ⎛ 

⎞ 

exp−2 0 0 0 exp−1 − exp− 1 − exp− 

exp−2 0 0 0 exp−1 − exp− 1 − exp− 

exp−2 0 0 0 exp−1 − exp− 1 − exp− 

0 exp−2 0 0 exp−1 − exp− 1 − exp− 

⎟ ⎜ 

⎠ ⎝ 0 0 exp−2 0 exp−1 − exp− 1 − exp− ⎟ 

⎠ 

0 0 0 exp−2 exp−1 − exp− 1 − exp− 

⎛ 

p 0 

0 

+ p 0 

1 

+ p 0 

2 exp−2 ⎞ 

p 0 

3 

exp−2 

p 0 

4 

exp−2 

= 

 

p 0 

5 

exp−2 

⎜ 

⎝ exp−1 − exp− ⎟ 

⎠ 

1 − exp−


The corresponding relativities are obtained by 

⎛ 

⎜ 

⎝ 

r 2 

0 

r 2 

1 

r 2 

2 

r 2 

3 

r 2 

4 

r 2 

5 

⎛ ∫ + 

⎞ 

∑k w k p 0 

0 0 

+ p 0 

1 

+ p 0 

2 exp−2 ⎞ 

kdF 

∫ + 

∑k w k p 0 

0 0 

+ p 0 

1 

+ p 0 

2 exp−2 kdF 

∫ + 

∑k w k p 0 

0 3 

exp−2 k dF 

∫ + 

∑k w k p 0 

0 3 

exp−2 k dF 

∫ + 

∑k w k p 0 

0 4 

exp−2 k dF 

∫ + 

∑k w = 

k p 0 

0 4 

exp−2 k dF 

∫ + 

∑k w k p 0 

0 5 

exp−2 k dF 

∫ + 

∑k w k p 0 

0 5 

exp−2 k dF 

∫ + 

∑k w k exp− 

0 k 1 − exp− k dF 

∫ + 

⎟ 

∑k w 

⎠ 

k exp− 

0 k 1 − exp− k dF 

⎜ 

∫ + 

⎟ 

⎝ 

∑k w k 1 − exp− 

0 k dF ⎠ 

∫ + ∑k w k 1 − exp− 

0 k dF 

⎛ 

= 

∫ + 

∑k w k ⎜ 

⎝ 

∑k w k 

∫ + 

0 

exp−2 k dF 

∑k w k 

∫ + 

0 

exp−2 k dF 

∑k w k 

∫ + 

0 

exp−2 k dF 

∑k w k 

∫ + 

0 

exp−2 k dF 

∑k w k 

∫ + 

0 

exp−2 k dF 

∑k w k 

∫ + 

0 

exp−2 k dF 

∑k w k 

∫ + 

0 

exp−2 k dF 

∑k w k 

∫ + 

0 

exp−2 k dF 

0 

exp− k 1 − exp− k dF 

∑k w k 

∫ + 

0 

exp− k 1 − exp− k dF 

∑k w k 

∫ + 

0 


∑k w k 

∫ + 

0 


⎞ 

 

⎟ 

⎠ 

Once again, we can see that the relativities at time 2 do not depend on the initial distribution. 

Now, r 2 

4 and r 2 

5 

are equal to the stationary relativities r 4 and r 5 , respectively, and the 

values r 2 

0 to r 2 

3 are equal. Similar expressions can be computed for time 3, 4 and 5 to 

show that the transient relativities do not depend on the initial distribution. 

The relativities ¯r l are displayed in Table 8.10 for the three initial distributions assuming 

a uniform distribution of age of policy a n = 1/5, for n = 1 to 5 . The relativies computed 

from the bottom initial distribution are close to the steady state relativities given in the last 

column (except for level 0) whereas the relativities computed from a uniform or a top initial 

distribution are weaker than the stationary relativities for levels 1 to 5. So we see that the 

initial distribution can have a great influence on the resulting ¯r l s.


Table 8.10 Relativies computed on the basis of the transient maximum accuracy criterion 

for the different initial distributions (−1/top scale). 


distribution ¯r l distribution ¯r l distribution ¯r l distribution r l 

5 1812 % 1812 % 1812 % 1812% 

4 1403 % 1111 % 1583 % 1599% 

3 1114% 946 % 1399 % 1439% 

2 928% 815 % 1233 % 1312% 

1 815% 709 % 1051 % 1209% 

0 711% 632% 746% 612% 

Influence of the Maturity of the Portfolio 

Let us now examine the influence of the maturity of the portfolio. To this end, let us consider 

the three different distributions of the age of the policies PrA = n = a n that are displayed 

in the following table: 

Age of policy Mature Young Old 

portfolio portfolio portfolio 

1 20% 30% 10% 

2 20% 25% 15% 

3 20% 20% 20% 

4 20% 15% 25% 

5 20% 10% 30% 

The first one is a uniform distribution which represents a mature portfolio. The second 

distribution represents a relatively young portfolio or a portfolio which is growing, i.e. 

PrA = n decreases with n. Finally, the third distribution represents an old portfolio or a 

portfolio which is declining, i.e. PrA = n increases with n. 

Asssuming a uniform initial distribution of the policyholders in the scale, we get the ¯r l s 

displayed in Table 8.11. 

Table 8.11 Relativies computed on the basis of the transient maximum accuracy criterion 

for different maturities of the portfolio (-1/top scale). 



5 1812 % 1812 % 1812 % 1812% 

4 1403 % 1318 % 1496 % 1599% 

3 1114 % 1030 % 1214 % 1439% 

2 928% 883% 984 % 1312% 

1 815% 811% 819 % 1209% 

0 711% 744% 683% 612%


Financial Balance 

Finally, we compare the evolution of the expected financial income in two different cases. 

First, Table 8.12 presents the evolution of I n (computed using (8.7)) when three different 

initial distributions are used. We see that, in each situation, the financial balance is reached 

after 5 years (the time needed to reach the steady state). We notice that the uniform and the 

top initial distribution ensure profit in the first years whereas the bottom initial distribution 

causes losses in the first years. Too many policyholders (with respect to the steady state 

situation) are in the malus levels of the scale in the first two cases whereas too many 

policyholders are in the bonus level in the last case. 

Table 8.12 also gives the evolution of the expected financial income Ī n computed using 

(8.8). The varying parameter is the distribution of the age of the policies. We see that this 

parameter has a little influence on the results. The most interesting point to notice is that the 

expected financial income does not converge to 100 % but goes down under 100 %. This is 

the result of the use of the ¯r l s. 

Choice of the Initial Level 

On the basis of the concepts presented in Section 8.3.4, we now try to find the most 

efficient initial level for the −1/top scale. The procedure can be summarized as follows: 

for each initial distribution p 0 = e k (where e k is the vector with 095 in the kth entry 

and 001 elsewhere), we compute the relativities ¯r l with the help of (8.6), as well as the 

¯Q-efficiency of each solution. The optimal initial level is then the one which maximises the 

¯Q-efficiency. 

Table 8.12 Evolution of the expected financial income I n 

based on the r l s and Ī n based on the ¯r l s. 

n Uniform Top Bottom 

distribution I n distribution I n distribution I n 

0 1330 % 1783% 655% 

1 1217 % 1601% 792% 

2 1135 % 1480% 877% 

3 1075 % 1393% 934% 

4 1032 % 1329% 972% 

5 1000 % 1000 % 1000% 

n Mature Young Old 

portfolio Ī n portfolio Ī n portfolio Ī n 

0 1131 % 1100 % 1168% 

1 1057 % 1035 % 1086% 

2 1012 % 1000 % 1031% 

3 987% 982% 998% 

4 975% 974% 981% 

5 969% 970% 973%


Table 8.13 

Choice of the initial class for the −1/top scale. 

Starting level Level of the resulting scale Efficiency ē 

5 4 3 2 1 0 

5 1812% 111% 946% 815% 709% 632% 11231 

4 1812 % 1583 % 1002% 867% 757% 645% 1155 

3 1812 % 1583 % 1399% 936% 818% 671% 11671 

2 1812 % 1583 % 1399 % 1233% 897% 704% 11692 

1 1812 % 1583 % 1399 % 1233 % 1051% 746% 11657 

0 1812 % 1583 % 1399 % 1233 % 1051% 746% 11657 

The results are given in Table 8.13. We conclude that level l = 2 is the ¯Q-optimal initial 

level whereas level 5 is the usual starting level of this scale. The evolution of the expected 

financial income I n when starting in level 2 is as follows: 

I 0 = 1314% 

I 1 = 1282% 

I 2 = 877% 

I 3 = 934% 

I 4 = 972% 

I 5 = 100 % 

8.5.2 −1/+2 Scale 

The transition rules of the −1/ + 2 bonus-malus scale are given in Table 4.2. 

Initial Distribution 

As for the −1/top scale, we have tested three different initial distributions of the policyholders 

in the scale. In the first case (uniform distribution), the policyholders are uniformly spread 

in the scale (16.67 % of the policyholders in each of the six levels). In the second case (top 

distribution), 95 % of the policyholders are concentrated in the top of the scale (and the 

remaining 5 % are spread over levels 0 to 4). In the last case (bottom distribution), 95 % of 

the policyholders start from the bottom of the scale. 

Convergence of the −1/+2 Scale 

Figure 8.3 shows the convergence of the −1/ + 2 scale by plotting the evolution of C n . The 

convergence is rather fast during the first five years and then slows down. The level = 005 

is reached after about eight years (C 8


Cn 

1.1 

1.0 

0.9 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0.0 

0 1 2 3 4 5 6 7 8 

Age of policy 

Figure 8.3 Convergence of the transient policyholders’ distributions to the steady state distribution 

for the system −1/ + 2. 

The corresponding relativies ¯r l are presented in Table 8.15 for a uniform distribution of 

age of policy. The last column gives the steady state relativities. We notice that the relativities 

are more severe when the initial distribution is closer to the bottom initial distribution. 

This seems reasonable since, with the bottom initial distribution, many policyholders are in 

the bonus level from the beginning. Therefore, the bonus must be weaker and the maluses 

stronger to ensure the financial balance. As for the −1/top bonus-malus scale, the relativities 

obtained starting from the bottom initial distribution are the closest to the stationary 

relativities. 

Influence of the Maturity of the Portfolio 

Let us consider the following three distributions of age of policy: 

Age of Mature Young Old 

policy portfolio portfolio portfolio 

1 125 % 25 % 5 % 

2 125 % 20 % 5 % 

3 125% 15% 10% 

4 125% 10% 10% 

5 125% 10% 10% 

6 125% 10% 15% 

7 125% 5% 20% 

8 125% 5% 25%


Table 8.14 

Evolution of the transient relativities for the different initial distributions (−1/+2 scale). 

Level l 

r 0 

l 

r 1 

l 

Uniform initial distribution 

r 2 

l 

r 5 

l 

r 7 

l 

r 8 

l 

r l 

5 00 % 1886 % 2048 % 2503 % 2631 % 2652 % 2714% 

4 00% 994 % 1673 % 2031 % 2088 % 2142 % 2185% 

3 00% 972% 935 % 1586 % 1821 % 1838 % 1925% 

2 00% 972% 971 % 1323 % 1379 % 1378 % 1388% 

1 00% 882% 863 % 1222 % 1210 % 1276 % 1286% 

0 00% 882% 792% 674% 678% 680% 685% 

Level l 

r 0 

l 

r 1 

l 

r 2 

l 

Top initial distribution 

r 5 

l 

r 7 

l 

r 8 

l 

r l 

5 00 % 1815 % 1820 % 2278 % 2546 % 2547 % 2714% 

4 00% 883 % 1602 % 1987 % 1888 % 2126 % 2185% 

3 00% 972% 794 % 1292 % 1772 % 1661 % 1925% 

2 00% 972% 971 % 1313 % 1358 % 1358 % 1388% 

1 00% 882% 863 % 1210 % 1082 % 1255 % 1286% 

0 00% 882% 792% 616% 616% 660% 685% 

Level l 

r 0 

l 

r 1 

l 

Bottom initial distribution 

r 2 

l 

r 5 

l 

r 7 

l 

r 8 

l 

r l 

5 00 % 2420 % 2817 % 2764 % 2735 % 2734 % 2714% 

4 00 % 1850 % 2154 % 2178 % 2180 % 2183 % 2185% 

3 00% 972 % 1554 % 1887 % 1910 % 1915 % 1925% 

2 00 % 1627 % 1448 % 1408 % 1415 % 1394 % 1388% 

1 00% 882 % 1447 % 1349 % 1281 % 1307 % 1286% 

0 00% 882% 792% 724% 708% 699% 685% 

Table 8.15 Relativities computed on the basis of the transient maximum accuracy criterion 

(influence of the initial distribution). 


distribution ¯r l distribution ¯r l distribution ¯r l distribution r l 

5 2326 % 2071 % 2749 % 2714% 

4 1643 % 1231 % 2140 % 2185% 

3 1318 % 1074 % 1847 % 1925% 

2 1145% 952 % 1447 % 1388% 

1 994% 839 % 1317 % 1286% 

0 710% 634% 759% 685%


Table 8.16 Relativities computed on the basis of the transient maximum accuracy 

criterion (influence of the maturity of the portfolio). 



5 2326 % 2186 % 2459 % 2714% 

4 1643 % 1440 % 1846 % 2185% 

3 1318 % 1164 % 1521 % 1925% 

2 1145 % 1066 % 1227 % 1388% 

1 994% 938 % 1070 % 1286% 

0 710% 738% 693% 685% 

The influence of the maturity of the portfolio is illustrated in Table 8.16. We notice that the 

relativities are more severe when the age of the portfolio is greater. 

Choice of the Initial Level 

We have computed the efficiency with different starting levels. Table 8.17 indicates that level 

2isthe ¯Q-optimal level ( ¯Q-efficiency of 1.2330). The evolution of the expected financial 

income when using an initial level l = 2 is as follows: 

I 0 = 1406% 

I 1 = 1413% 

I 2 = 1019% 

I 3 = 1009% 

I 4 = 1044% 

I 5 = 1000% 

I 6 = 1000% 

I 7 = 1009% 

I 8 = 999% 

Table 8.17 

Choice of the initial class for the −1/ + 2 scale. 

Starting level Level of the resulting scale Efficiency ē 

5 4 3 2 1 0 

5 2071 % 1231 % 1074% 952% 839% 634% 11532 

4 2167 % 1774 % 1118 % 1002% 878% 651% 11942 

3 2313 % 1851 % 1582 % 1059% 941% 675% 12187 

2 2582 % 1969 % 1675 % 1349 % 1009% 711% 12330 

1 2675 % 2102 % 1777 % 1404 % 1276% 749% 12321 

0 2749 % 2140 % 1847 % 1447 % 1317% 759% 12251


8.6 Super Bonus Level 

8.6.1 Mechanism 

As explained above, the companies operating in Belgium continue to use the former 

compulsory bonus-malus scale, despite the deregulation of the a posteriori corrections. Slight 

modifications have nevertheless been brought to the system. For marketing purposes, and 

to keep the best drivers in the portfolio, many insurance companies operating in Belgium 

have allowed the insured drivers reaching level 0 to ‘claim for free’: whatever the number of 

accidents reported to the company, they stay in level 0. This makes the stationary distribution 

degenerate: all its probability mass is concentrated at 0. Therefore, computations must be 

based on the transient distribution. 

The transition probabilities for the Belgian bonus-malus scale with a super bonus level 0 

are the same as before, except for the line corresponding to level 0, which is replaced with 

a line of 0s and a single 1 on the diagonal. 

8.6.2 Initial Distributions 

We use the following initial distributions: the uniform distribution (1/23 of the portfolio in 

each level), the top distribution (78 % of the policyholders are concentrated in level 22 and 

the remaining 22 % are spread over levels 0 to 21) and the steady-state distribution displayed 

in Table 8.1. Starting with the stationary distribution gives an idea of the influence of the 

introduction of a super bonus level in an existing bonus-malus scale. 

8.6.3 Transient Relativities 

The results for the uniform initial distribution are given in Tables 8.18 (transient distributions) 

and 8.19 (corresponding relativities). We note that after 50 years, the majority of the 

policyholders (76.0 %) are in the super bonus level. Most of the other policyholders are in 

the upper levels. They correspond to the very bad drivers. 

Concerning the transient relativities, we see that the super bonus level has a greater 

relativity than level 1 during the first 20 years. This side-effect of the introduction of a super 

bonus level is undesirable. 

Table 8.20 presents the transient relativities for the top initial distribution. The relativities 

also show particular patterns as policyholders principally move by clusters in the scale. For 

example, the relativities of levels 0 to 2 are not ordered. 

The transient relativities for the steady-state initial distribution are given in Table 8.21. 

Once again, the transient relativities are not always in ascending order. 

Table 8.22 shows the ¯r l s for the three initial distributions. The results for the steady-state 

initial distribution are not in ascending order. Indeed, the relativity associated with the super 

bonus level is larger than the relativities associated with levels 1 to 5. 

Therefore, we see that the introduction of a super bonus level in an actual scale leads to 

some undesirable side-effects: concentration of the majority of the policyholders in the super 

bonus level after a few years and optimal relativities no longer ordered for the lowest levels 

(these side-effects could be overcome with the introduction of a linear scale).


Table 8.18 Evolution of the uniform initial distribution for the former Belgian compulsory bonusmalus 

scale with a super bonus level 0. 

Level l PrL 0 = l PrL 10 = l PrL 20 = l PrL 30 = l PrL 40 = l PrL 50 = l 

22 435 % 443 % 465 % 462 % 454 % 446 % 

21 435 % 328 % 334 % 327 % 318 % 309 % 

20 435 % 268 % 259 % 248 % 238 % 230 % 

19 435 % 239 % 215 % 200 % 189 % 180 % 

18 435 % 225 % 188 % 169 % 157 % 147 % 

17 435 % 245 % 175 % 150 % 135 % 124 % 

16 435 % 248 % 164 % 135 % 119 % 107 % 

15 435 % 251 % 157 % 124 % 106 % 095 % 

14 435 % 259 % 155 % 118 % 098 % 085 % 

13 435 % 265 % 154 % 113 % 091 % 078 % 

12 435 % 392 % 168 % 113 % 088 % 073 % 

11 435 % 385 % 168 % 110 % 083 % 068 % 

10 435 % 376 % 168 % 107 % 079 % 063 % 

9 435 % 372 % 169 % 105 % 076 % 059 % 

8 435 % 365 % 169 % 102 % 072 % 055 % 

7 435 % 355 % 207 % 109 % 072 % 053 % 

6 435 % 344 % 205 % 106 % 068 % 049 % 

5 435 % 330 % 201 % 103 % 064 % 046 % 

4 435 % 308 % 186 % 093 % 057 % 040 % 

3 435 % 287 % 172 % 084 % 050 % 034 % 

2 435 % 267 % 249 % 093 % 049 % 032 % 

1 435 % 249 % 233 % 085 % 044 % 028 % 

0 435 % 3198 % 5439 % 6744 % 7293 % 7600 % 

Table 8.19 Evolution of the transient relativities for the uniform initial distribution for the former 

Belgian compulsory bonus-malus scale with a super bonus level 0. 

Level l 

r 0 

l 

r 10 

l 

r 20 

l 

r 30 

l 

r 40 

l 

r 50 

l 

22 000 % 26207 % 26976 % 27344 % 27638 % 27889 % 

21 000 % 22708 % 24106 % 24676 % 25073 % 25393 % 

20 000 % 19913 % 21822 % 22594 % 23096 % 23483 % 

19 000 % 17797 % 19938 % 20875 % 21476 % 21929 % 

18 000 % 16238 % 18385 % 19432 % 20115 % 20628 % 

17 000 % 13881 % 16834 % 18109 % 18909 % 19495 % 

16 000 % 12885 % 15685 % 17026 % 17891 % 18529 % 

15 000 % 12052 % 14693 % 16081 % 17003 % 17685 % 

14 000 % 11568 % 13938 % 15289 % 16230 % 16941 % 

13 000 % 11155 % 13299 % 14606 % 15555 % 16288 %


12 000 % 8273 % 12051 % 13714 % 14811 % 15627 % 

11 000 % 8093 % 11488 % 13132 % 14248 % 15089 % 

10 000 % 7845 % 10951 % 12589 % 13730 % 14597 % 

9 000 % 7757 % 10570 % 12157 % 13289 % 14164 % 

8 000 % 7612 % 10198 % 11757 % 12888 % 13771 % 

7 000 % 7413 % 8695 % 10867 % 12245 % 13253 % 

6 000 % 7163 % 8334 % 10471 % 11864 % 12891 % 

5 000 % 6863 % 7971 % 10083 % 11495 % 12544 % 

4 000 % 6599 % 7672 % 9777 % 11196 % 12253 % 

3 000 % 6320 % 7369 % 9474 % 10904 % 11972 % 

2 000 % 6026 % 5355 % 8400 % 10212 % 11465 % 

1 000 % 5718 % 5108 % 8088 % 9906 % 11176 % 

0 000 % 7163 % 6314 % 6293 % 6476 % 6630 % 

Table 8.20 Evolution of the transient relativities for the top initial distribution for the former Belgian 

compulsory bonus-malus scale with a super bonus level 0. 

Level l 

r 0 

l 

r 10 

l 

r 20 

l 

r 30 

l 

r 40 

l 

r 50 

l 

22 000 % 22221 % 24839 % 25892 % 26490 % 26902 % 

21 000 % 19477 % 22125 % 23280 % 23952 % 24420 % 

20 000 % 17451 % 20074 % 21298 % 22029 % 22544 % 

19 000 % 15898 % 18455 % 19719 % 20495 % 21049 % 

18 000 % 14672 % 17143 % 18426 % 19236 % 19821 % 

17 000 % 9441 % 13250 % 15414 % 16827 % 17847 % 

16 000 % 10865 % 14242 % 15920 % 16947 % 17675 % 

15 000 % 10237 % 13464 % 15130 % 16170 % 16914 % 

14 000 % 9745 % 12808 % 14450 % 15498 % 16256 % 

13 000 % 9322 % 12235 % 13856 % 14911 % 15681 % 

12 000 % 4584 % 8839 % 11105 % 12623 % 13779 % 

11 000 % 8093 % 10089 % 12177 % 13474 % 14391 % 

10 000 % 7845 % 9679 % 11741 % 13047 % 13976 % 

9 000 % 7757 % 9331 % 11357 % 12667 % 13605 % 

8 000 % 7612 % 9005 % 11005 % 12321 % 13270 % 

7 000 % 7413 % 5811 % 8528 % 10242 % 11495 % 

6 000 % 7163 % 7005 % 9500 % 11125 % 12256 % 

5 000 % 6863 % 6753 % 9207 % 10838 % 11980 % 

4 000 % 6599 % 6507 % 8952 % 10588 % 11739 % 

3 000 % 6320 % 6270 % 8707 % 10348 % 11508 % 

2 000 % 6026 % 3003 % 6363 % 8404 % 9831 % 

1 000 % 5718 % 5108 % 7154 % 9175 % 10560 % 

0 000 % 7163 % 6314 % 4542 % 5095 % 5487 %


Table 8.21 Evolution of the transient relativities for the steady-state initial distribution for the former 

Belgian compulsory bonus-malus scale with a super bonus level 0. 

Level l 

r 0 

l 

r 10 

l 

r 20 

l 

r 30 

l 

r 40 

l 

r 50 

l 

22 000 % 25793 % 26951 % 27415 % 27744 % 28011 % 

21 000 % 22081 % 23992 % 24691 % 25134 % 25473 % 

20 000 % 19252 % 21680 % 22592 % 23139 % 23543 % 

19 000 % 17466 % 19788 % 20813 % 21460 % 21940 % 

18 000 % 16052 % 18299 % 19397 % 20107 % 20636 % 

17 000 % 12472 % 16258 % 17829 % 18756 % 19409 % 

16 000 % 11795 % 15198 % 16793 % 17767 % 18458 % 

15 000 % 11134 % 14253 % 15879 % 16904 % 17634 % 

14 000 % 11543 % 13902 % 15224 % 16158 % 16876 % 

13 000 % 11181 % 13287 % 14580 % 15521 % 16252 % 

12 000 % 6621 % 11164 % 13200 % 14479 % 15400 % 

11 000 % 6849 % 10789 % 12722 % 13989 % 14915 % 

10 000 % 6899 % 10357 % 12236 % 13514 % 14458 % 

9 000 % 8004 % 10588 % 12130 % 13235 % 14095 % 

8 000 % 7991 % 10209 % 11734 % 12854 % 13729 % 

7 000 % 7824 % 7619 % 10195 % 11793 % 12933 % 

6 000 % 7525 % 7520 % 9948 % 11511 % 12643 % 

5 000 % 7118 % 7330 % 9651 % 11202 % 12341 % 

4 000 % 7833 % 7660 % 9728 % 11137 % 12184 % 

3 000 % 7315 % 7369 % 9424 % 10854 % 11921 % 

2 000 % 6774 % 4242 % 7629 % 9672 % 11074 % 

1 000 % 6228 % 4283 % 7507 % 9493 % 10874 % 

0 000 % 9241 % 8794 % 8493 % 8485 % 8514 % 

Table 8.22 Relativies computed on the basis of the transient maximum 

accuracy criterion for the former Belgian compulsory bonus-malus scale 

with a super bonus level 0. 

Level l Uniform Top Steady state 

distribution ¯r l distribution ¯r l distribution ¯r l 

22 2675 % 2410 % 2652% 

21 2346 % 1848 % 2278% 

20 2079 % 1633 % 1990% 

19 1859 % 1468 % 1768% 

18 1680 % 1338 % 1595% 

17 1635 % 1234 % 1461% 

16 1418 % 1149 % 1356% 

15 1323 % 1078 % 1272% 

14 1244 % 1018 % 1204% 

13 1177% 966 % 1146% 

12 1119% 921 % 1097% 

11 1069% 880 % 1055% 

10 1024% 844 % 1017%


9 981% 810% 979% 

8 940% 778% 944% 

7 901% 748% 912% 

6 865% 718% 881% 

5 832% 691% 852% 

4 783% 656% 798% 

3 740% 624% 759% 

2 703% 595% 723% 

1 671% 566% 692% 

0 655% 514% 876% 


There are relatively few papers dealing with the study of bonus-malus scales using transient 

distributions. The study of bonus-malus scales with transient distributions started with the 

seminal paper by Borgan, Hoem & Norberg (1981). Gilde & Sundt (1989) studied linear 

scales in a transient regime. The examples mentioned in this chapter (special scale for new 

entrants, and absorbing level 0) are extensively treated in Denuit, Maréchal, Pitrebois 

& Walhin (2007b).

9 

Actuarial Analysis of the French 

Bonus-Malus System 


As discussed in the preceding chapters, bonus-malus systems usually take the form of a scale 

comprising a number of levels. The policyholders move inside the scale according to the 

number of claims they report to the insurance company. To each level of the scale is attached 

a relativity (that is, a percentage, or relative premium). These relativities are applied to a 

base premium. Usually, bonus-malus systems may be modelled through a Markov chain, 

which makes mathematics easy for the actuary. 

France is an exception. The French law imposes on the insurers operating in France 

a unique bonus-malus system that is not based on a scale. Instead the French bonusmalus 

system uses the concept of an increase-decrease coefficient (coefficient de réductionmajoration 

in French, henceforth abbreviated as CRM). More precisely, the French bonusmalus 

system implies a malus of 25 % per claim and a bonus of 5 % per claim-free year. So 

each policyholder is assigned a base premium and this base premium is adapted according 

to the number of claims reported to the insurer, multiplying the premium by 1.25 each time 

an accident at fault is reported to the company, and by 0.95 per claim-free year. In the case 

of shared responsibility, the increase is reduced by half (12.5 % instead of 25 %). Note that 

these increases are applied to the previous relativity: the first claim causes the premium to 

pass from 100 to 125, the second increases the premium to 156, the third to 195, and so 

on (all the numbers are rounded down). The penalties are thus convex in the number of 

claims reported by the driver, ensuring that the more claims are reported, the heavier they are 

penalized. The highest percentage is 350, and the lowest is 50 (attained after 13 consecutive 

claim-free years). According to the French special bonus rule, after two consecutive years 

without a claim, the driver goes back to the initial level 100 %. This special bonus rule is 

particularly generous. 





In 1994, the European Union decreed that all its member countries had to drop their 

mandatory bonus-malus systems, claiming that such systems reduced competition between 

insurers and were in contradiction to the total rating freedom implemented by the Third 

Directive. However, the mandatory French system is still in force. Quite surprisingly, the 

European Court of Justice decided in 2004 that the mandatory bonus-malus systems of 

France and the Grand Duchy of Luxembourg were not in contradiction to the rating freedom 

imposed by the European legislation. These two countries were thus allowed to stick to their 

respective uniform bonus-malus mechanisms. 

In this chapter, we show that the framework of credibility theory can be used to analyse 

the French bonus-malus system. Specifically, the greatest accuracy credibility approach 

presented in Chapter 3 is adapted to fit the CRM coefficients: the actuary resorts to a 

quadratic loss function but the shape of the credibility predictor is constrained ex ante to 

the form imposed by the French law. Let us mention that the approach developed in this 

chapter is not the only possible method to deal with CRM coefficients. It has been shown 

in Kelle (2000) that the French bonus-malus system corresponds to a scale comprising 

several hundreds of levels (530 levels, precisely), that can be analysed in the Markovian 

setting of Chapter 4. The large number of states needed is due to the malus reduction 

in the case of claims with shared responsibility, forcing the author to consider the pair 

(number of claims with whole responsibility, number of claims with partial liability) to 

make the computation. The form of the transition matrix is somewhat intricate and we 

believe that the alternative developed in this chapter offers an appropriate treatment of 

the CRMs. 

Let us now detail the contents of this chapter. In Section 9.2, we model the CRMs and 

we compute the parameters involved in the French bonus-malus system. We also examine 

whether the bonus-malus system is financially balanced or not. Some numerical applications 

illustrate the methodological results. Section 9.3 discusses a special rule associated with the 

French bonus-malus system: claims for which the policyholder is only partially liable entail 

a reduced penalty. The impact of this reduction is evaluated, and numerical illustrations are 

discussed. The final Section 9.4 concludes with bibliographic notes. 

9.2 French Bonus-Malus System 

9.2.1 Modelling Claim Frequencies 

We adopt here the framework of the preceding chapters. Let us pick at random a policyholder 

from the portfolio. We denote as N t the number of claims reported by this policyholder in 

period t. We assume that N t is Poisson distributed with parameter where is a random 

effect accounting for the heterogeneity present in the portfolio. By assumption, is a positive 

random variable that represents the annual mean frequency in the portfolio (or in the risk class 

in the case of a segmented tariff). Given = , the conditional probability mass function of 

N t is oi. We further assume that E = 1, so that EN = . The heterogeneity present 

in the portfolio is described by a structure function. Formally, the structure function is the 

probability density function f of . Therefore, the unconditional probability mass function 

of N t is Poi . Furthermore, the random variables N 1 N 2 N 3 are assumed to be 

independent and identically distributed given the risk proneness of the policyholder. Since 

is unknown to the insurer, this induces serial dependence among the N t s.

Actuarial Analysis of the French Bonus-Malus System 327 

9.2.2 Probability Generating Functions of Random Vectors 

In this chapter we will use multivariate models for counting random vectors. Specifically, let 

us consider random vectors M = M 1 M n T valued in n . The multivariate probability 

mass function of M is 

p M k 1 k n = PrM 1 = k 1 M n = k n 

Throughout the chapter we will extensively use the multivariate extension of the probability 

generating function introduced in Chapter 1, which is defined as 

M z = z M 1 

1 ···z M n 

n 

 

∑ ∑ 

= z k 1 

1 ···zk n 

n p Mk 1 k n 

k 1 =0 

k n =0 

Let us now point out several interesting properties of the multivariate probability generating 

functions. If any function that is known to be a multivariate probability generating function 

for a random vector M is expanded as a power series in z, then the coefficient of z k 1 

1 ···zk n 

n 

must be p M k 1 k n . Furthermore, 

• z ↦→ M zzz is the probability generating function of M 1 +···+M n ; 

• z ↦→ M z 00 is the probability generating function of M 1 ; 

• M z 1 z n = M1 

z 1 ··· Mn 

z n when the random variables M 1 M n 

independent. 

are 

9.2.3 CRM Coefficients 

We will assume that the CRM coefficients only depend on the observed number of reported 

claims and not on their severity. Therefore the base premium is simply multiplied by a 

constant (essentially the expected cost of a claim). 

Let t be the ‘reduction’ coefficient and t be the ‘majoration’ coefficient applying to a 

policyholder who has been covered for t years. The CRM coefficient for years 1 to t then 

becomes 

with 

where I j is defined as 

r t t 

N • I • t= 1 + t N • 

1 − t I • 

N • = 

t∑ 

N j and I • = 

j=1 

t∑ 

I j (9.1) 

j=1 

{ 

1ifNj = 0 

I j = 

0ifN j ≥ 1


In words, N • is the total number of claims reported by the policyholder during the period 

0t and I • is the number of years without any claim reported to the company. Note that 

the CRM coefficients depend on t, so that the penalties and discounts may change for every 

0t period. 

To obtain the parameters t and t , we minimize the expected squared difference between 

the ‘true’ relative premium and the relative premium r t t 

applicable to the policyholder 

according to the French-type bonus-malus system. More specifically, for a policyholder 

observed during t years, and having filed N 1 N 2 N t claims, we aim to determine t and 

t so as to minimize the objective function 

t = − r N • I • t 2 

with respect to the arguments and . We therefore have to solve the first order conditions 

 

t = 0 

and 

 

t = 0 

which rewrites as 

⎧ 

⎨ [ ] 

N • 1 + N •−1 1 − I • − 1 + 

N •1 − 

I • = 0 

⎩ 

[ ] 

I • 1 + N • 1 − 

I • −1 − 1 + N • 1 − 

I • = 0 

⎧ 

⎨ [ [ ] 

N • 1 + N •−1 1 − •] I = N• 1 + 2N •−1 1 − 2I • 

⇐⇒ 

⎩ 

[ I • 1 + ] N • 1 − 

I • −1 

= [ I • 1 + ] 2N • 1 − 

2I • −1 

 

(9.2) 

9.2.4 Computation of the CRMs at Time t 

Let us define the conditional probability generating function of the random couple N • I • 

given = as 

1 2 = N • 

1 I • 

2 

= 

The conditional independence assumption of N 1 N 2 N t allows us to write 

1 2 = 

= 

t∏ 

j=1 

We can then rewrite the system (9.2) as 

⎧ 

⎨ 

⎩ 

[ 

N j 

1 I j 

2 

] 

∣ 

∣ = 

( 

e − 2 − 1 + e −1− 1) t 

2 10 1 + 1 − = 10 

2 1 + 1 − 

2 01 1 + 1 − = 01 

2 1 + 1 −


where 

xy a b = x y ∣ 

s x t s t ∣∣s=at=b 

y 

xy 

2 a b = x y 

s x t y s2 t 2 ∣ 

 

for x y ∈ 0 1. 

We can rewrite the first order conditions as 

∫ 

∣ 

s=at=b 

) t−1e 

 

(e 2 − e − f d 

0 

∫ ( 

t−1e 

= 1 + e 2+2 + e − 2 2+ 

− 2) 2 f d 

0 

and 

∫ 

( ) t−1e 

e − e − − f d 

0 

∫ ( 

t−1e 

= 1 − e 2+2 + e − 2 − 2) − f d 

0 

These equations do not possess a closed form solution as is the case for the Markovian 

systems studied in Chapter 3. Nevertheless, they can be solved numerically using on the one 

hand a numerical integration algorithm or on the other hand either an algorithm allowing us 

to numerically solve a system of nonlinear equations or an optimisation algorithm, depending 

on what type of procedure is available. In the numerical illustrations proposed in this chapter, 

we have used an optimisation algorithm from the SAS/IML package, trying to minimize the 

sum of the squared differences between the left-hand side and the right-hand side of the two 

equations displayed above. 

9.2.5 Global CRM 

Note that we have obtained so far a numerical solution for each t: minimizing t with 

respect to and gives the optimal solution t t for the period 0t. However we want 

to obtain a unique set of CRM coefficients. These may be obtained in the transient setting 

developed in the preceding chapter. To this end, let us introduce the age structure of the 

portfolio. Specifically, we denote as A the number of years the driver is covered by the 

company, and as N 1 N 2 N A the annual numbers of claims reported by this policyholder. 

Note that A is a random variable since we work with a policyholder picked at random from 

the portfolio. The idea is then to determine and so as to minimize E A . The 

objective function then becomes 

= E A = 

to be minimized with respect to the parameters and . 

∑ 

a t t where a t = PrA = t 

t=1


Some algebra immediately leads to the following system of equations to solve: 

∑ 

a t t 

t=1 

∫ 

∑ 

= 1 + a t t 

0 

t=1 

2 (e − e − ) t−1e f d 

∫ 

0 

( 

t−1e 

e 2+2 + e − 2 2+ 

− 2) 2 f d 

and 

∑ 

a t t 

t=1 

∫ 

∑ 

= 1 − a t t 

0 

t=1 

( ) t−1e 

e − e − − f d 

∫ 

0 

( 

e 2+2 + e − 2 − 2) t−1e − f d 

Again, this system does not admit any closed-form solution, but can be solved numerically 

(using an appropriate SAS/IML optimization algorithm). 

Remark 9.1 Note that here, we have made an averaging with respect to the age structure 

of the portfolio. In the case where the portfolio is partitioned into a series of risk classes, an 

average with respect to the composition of the portfolio (in terms of classification variables) 

could also be performed. If some explanatory variables are correlated with A, care must be 

taken in the second averaging. 

9.2.6 Multivariate Panjer and De Pril Recursive Formulas 

Notations 

In Sections 9.2.7 and 9.3.3, we will need the bivariate and trivariate extensions of the 

Panjer algorithm that was described in Section 7.2. The present section is devoted to the 

presentation of this method as well as a particular case for the sum of independent and 

identically distributed random vectors, known as multivariate De Pril’s recursive formula. 

Assume independent and identically distributed realizations of possibly dependent losses 

X i = X i1 X ik T affected by a common event, denoted by the counting variable N . For 

example N may count the number of hurricanes hitting the United States and the X j s may 

represent the cost of the hurricane in state numbered j, j = 1k. It is natural to try to 

obtain the distribution of the aggregate claim: 

( ) T 

∑ N N∑ 

S = S 1 S k T = X i1 X ik (9.3) 

Even when the components of X are independent, the components of S will have some 

positive dependence due to the common counter N . 

When N belongs to the Panjer family of counting random variables, let us show that a 

multivariate version of Panjer’s recursive formula emerges. To this end, we will use the 

following notations: 

i=1 

i=1


• the probability mass function of S is denoted as 

gs = PrS 1 = s 1 S k = s k 

• the probability mass function of X is denoted as 

fx = PrX 1 = x 1 X k = x k 

• the difference between vectors has to be understood componentwise, that is, 

s − x = s 1 − x 1 s k − x k T 

• and, finally, 

( 

s∑ ∑ s1 

fx = ··· 

s k 

∑ 

x≠0 

x 1 =0 x k =0 

) 

fx 1 x k − f00 

Multivariate Panjer Algorithm 

We are now ready to state and prove the following result. 

Property 9.1 Let S be as in (9.3) with X i = X i1 X ik T , i = 1 2, independent and 

identically distributed, arithmetic and independent of N . Furthermore, we assume that N 

belongs to Panjer’s class, i.e. its probability mass function satisfies (7.7). Then, if N denotes 

the probability generating function of N , we have 

g0 = N f0 (9.4) 

( 

1 

s∑ 

gs = 

a + b x ) 

i 

gs − xfx 

1 − af0 s i 

s i ≥ 1 i= 1k (9.5) 

x≠0 


From 

Let X · and S · be the probability generating functions of X and S, respectively. 

∑ 

gs = PrN = nf ⋆n s 

n=0 

we get 

∑ 

S u = PrN = n ( X u ) n 

from which (9.4) follows immediately. By hypothesis, one has 

n=0 

n PrN = n = an − 1 PrN = n − 1 + a + b PrN = n − 1 

n ≥ 1


Multiplying on both sides of the equality by n−1 

X uu i/u i X u and summing over n = 1 

to +, weget 

u i 

 

 

 

 

u S u = a X uu i 

i u S u + a + bu i 

i u X u S u 

i 

Comparing identical powers of u on both sides of the equality gives: 

s∑ 

s∑ 

s i gs = a fxs i − x i gs − x + a + b fxx i gs − x 

x=0 

⇔ s i gs = as i f0gs + 

⇔ gs = 

1 

1 − af0 

x≠0 

x=0 

s∑ 

fxgs − x ( ) 

as i + bx i 

x≠0 

( 

s∑ 

fxgs − x a + b x ) 

i 

 

s i 


□ 

The Panjer formula in dimension 1 immediately follows from Property 9.1 by putting 

k = 1. 

Multivariate De Pril Algorithm 

Another interesting problem is to compute the multivariate convolution of a random vector. 

This is actually the result we will need in the following sections. Let X i = X i1 X ik T , 

i = 1 2n, be independent and identically distributed realizations of the random vector 

X = X 1 X k T . We want to obtain the distribution of the random vector 

( ) n∑ T 

n∑ 

S = S 1 S k T = X i1 X ik (9.6) 

i=1 

i=1 

The probabiliy mass function of S is the n-fold convolution f ⋆n x. The multivariate version 

of De Pril’s formula provides a recursive formula to derive the distribution of S. 

Property 9.2 Let X i = X i1 X ik T be independent and identically distributed 

realizations of the random vector X defined on the nonnegative integers and with probability 

function such that f0>0. Then, the following recursion holds : 

f ⋆n 0 = f n 0 

f ⋆n s = 1 ( ) 

s∑ n + 1 

x 

f0 s i − 1 f ⋆n s − xfx 

i 

s i ≥ 1 i= 1k 

x≠0 

Proof Let us introduce an auxilliary random vector W with probability mass function 

h0 = 0 and 

hx = 

fx 

1 − f0 x > 0


and the auxilliary random variable N ∼ inn 1 − f0. 

The probability generating function of W 11 +···+W 1N W k1 +···+W kN T is given by 

( 

n ( 

1 − 1 − f01 − W u) 

= X u ) n 

 

from which we conclude that S = W 11 +···+W 1N W k1 +···+W kN T . 

Applying Property 9.1 with 

a = f0 − 1 

f0 

and b = 1 − f0 n + 1 

f0 

we get the desired result. 

□ 

9.2.7 Analysis of the Financial Equilibrium of the French Bonus-Malus 

System 

An interesting property of the relativities associated with Markovian bonus-malus systems 

and obtained through Norberg’s least-squares criterion is that they make the bonus-malus 

system financially balanced, i.e. the premium income of the insurer does not increase nor 

decrease over time (on average). In this section, we would like to check whether or not the 

French-type bonus-malus system enjoys this property. 

More precisely, once t and t have been obtained, we would like to verify whether 

Er t t 

N • I • t is equal to 1, where N • and I • are as defined in (9.1). The computation of 

Er t t 

N • I • t requires knowledge of the joint distribution of the random couple N • I • . 


fx y = PrN 1 = x I 1 = y = 

the joint discrete mass function of the random couple N 1 I 1 , conditional on = , and as 

f ⋆t x y = PrN • = x I • = y = 

the joint discrete mass function of the random couple N • I • defined in (9.1), conditional 

on = . We then have the following result. 

Property 9.3 

For fixed , the following recursive formulas 

g ⋆t x y = f ⋆t x t − y for 0 ≤ y ≤ t and x>0 

g ⋆t 0 0 = e −t 

− 

x 

fx 0 = e for x>0 

x! 

f0 1 = e − 

( ) 

x∑ t + 1 

g ⋆t x y = e − 1 g ⋆t x − u y − 1gu 1 for y ≥ 1 and x ≥ 1 

y 

u=1


hold true, with the convention that the functions take the value 0 where they have not been 

defined. 


It is trivial that for t = 1 we have 

− 

x 

fx 0 = e x>0 

x! 

f0 1 = e − 

As f ⋆t is the t-fold convolution of a lattice random vector, we are in a position to apply the 

bivariate extension of De Pril’s algorithm given in Property 9.2. As that algorithm needs a 

mass at the origin, we define an auxilliary probability mass function 

We find 

g ⋆t 0 0 = e −t 

g ⋆t x y = e ( x∑ 

u=0 

+ 

x∑ 

u=1 

g ⋆t x y = f ⋆t x t − y 

( t + 1 

x u − 1 ) 

g ⋆t x − u y − 1gu 1 

( t + 1 

x u − 1 ) 

g ⋆t x − u ygu 0 

) 

x≥ 1 

( x∑ ( ) t + 1 

g ⋆t x y = e − 1 g ⋆t x − u y − 1gu 1 

u=0 

y 

) 

x∑ 

+ −1g ⋆t x − u ygu 0 y≥ 1 

u=1 

Because g0 1 = 0 and gx 0 = 0 for x>0, we obtain 

g ⋆t 0 0 = e −t 

( ) 

x∑ t + 1 

g ⋆t x y = e 

u=1 

x u − 1 g ⋆t x − u y − 1gu 1 x ≥ 1 

( ) 

x∑ t + 1 

g ⋆t x y = e − 1 g ⋆t x − u y − 1gu 1 y ≥ 1 

y 

u=1 

Because g ⋆t x 0 = 0 for x>0, only the second recursive formula has to be used. This 

formula is numerically stable because t + 1/y − 1 > 0. 

□ 

In order to obtain the unconditional probability mass function 

f ⋆t x y = PrN • = x I • = y


of N • and I • , it suffices to integrate the conditional mass function f ⋆t x y with respect 

to the structure function f , that is, 

f ⋆t x y = 

∫ 

0 

f ⋆t x yf d x > 0 0 ≤ y ≤ t 

These quantities can then be used to evaluate Er t t 

N • I • t. 


We will assume that is Gamma distributed with probability density function given by 

(1.35). The parameters a and are estimated on the basis of Portfolio A, that is, ̂ = 01474 

and â = 0889. 

Table 9.1 displays, for different values of t, the coefficients t and t obtained by solving 

the system given in Section 9.2.4. We observe a dramatic decrease of the values of t and 

t over time. The last column of the table allows us to verify the financial equilibrium 

of the system. The total premium income first decreases to 97.61 % and then increases to 

107.29 % after 30 years. The discount per claim-free year decreases from 14.23 % to about 

1 %. Similarly, the penalty induced by each reported claim decreases from 61.45 % to 6.09 %. 

The a posteriori corrections are therefore considerably softened with time. 

The decrease of t and t with time t that is apparent from Table 9.1 can be explained 

as follows: The aim is that r t t 

be as close as possible to the unknown risk parameter . 

Since does not depend on t whereas N • and I • are almost surely nondecreasing with t, 

the optimal parameters t and t must decrease to compensate for the increase in N • and I • . 

This is why averaging over time is needed. 

Table 9.2 gives the CRM coefficient r t t 

x y = 1 + t x 1 − t y for different periods 

of length t and for different values of the total number of claims x. The index ty means that 

we have y claimsfree years during the period 0t. For the sake of comparison, Table 9.3 

gives the CRM coefficients obtained from classical Bayesian credibility. In this case, the 

a priori annual expected claim frequency is multiplied by a + x/a + t as discussed in 

Chapter 3. We observe some large discrepancies between the values listed in Tables 9.2 

and 9.3. 

We see from Tables 9.2 and 9.3 that the discounts awarded to the policyholders who 

did not report any claim (column x = 0 in Tables 9.2 and 9.3) are larger with Bayesian 

Table 9.1 

values of t. 

Parameters t and t and financial equilibrium for different 

t t t Financial equilibrium 

1 0.6145 0.1423 0.9761 

2 0.4595 0.0955 0.9985 

3 0.3690 0.0727 1.0001 

4 0.3092 0.0589 1.0099 

10 0.1585 0.0279 1.0431 

20 0.0880 0.0149 1.0638 

30 0.0609 0.0102 1.0729


Table 9.2 CRM coefficients r t t 

x y for different values of t, 

x and y. 

ty 

x 

0 1 2 3 4 5 6 

1 86 % 161 % 261 % 421 % 680 % 1097 % 1771 % 

2.≥1 82 % 132 % 193 % 281 % 410 % 599 % 874 % 

2.0 213 % 311 % 454 % 662 % 967 % 

3.≥2 80 % 118 % 161 % 221 % 302 % 414 % 566 % 

3.1 174 % 238 % 326 % 446 % 610 % 

3.0 257 % 351 % 481 % 658 % 

Table 9.3 Premium update coefficients derived from Bayesian 

credibility in the Poisson-Gamma model. 

t 

x 

0 1 2 3 4 5 6 

1 86 % 182 % 279 % 375 % 472 % 568 % 665 % 

2 75 % 160 % 244 % 329 % 413 % 498 % 582 % 

3 67 % 142 % 217 % 292 % 367 % 442 % 518 % 

credibility than with CRM coefficients. From the approximate financial stability evidenced 

in Table 9.1, the penalties induced by CRM coefficients must therefore be softer compared 

to Bayesian credibility. For instance, policyholders who reported a single claim (column 

x = 1 in Tables 9.2 and 9.3) have a premium surcharge ranging from 118 to 161 % with 

CRM coefficients, and ranging from 142 to 182 % with Bayesian credibility. However, 

policyholders reporting many claims are more heavily penalized with CRM coefficients 

than with Bayesian credibility. This comes from the convex behaviour of the CRM 

coefficients, whereas Bayesian credibility corrections are linear in the past number of claims. 

In the Poisson-Gamma (or Negative Binomial) case, the penalties corresponding to CRM 

coefficients are thus convex functions of the number of claims reported in the past, whereas 

corrections induced by credibility mechanisms are linear in this number. Compared with 

credibility, the bonus-malus system grants less discounts, penalizes policyholders reporting 

a single claim to a lesser extent but induces more severe premium corrections for those 

reporting at least two claims. 

To obtain unique values for the CRM coefficients, we have to decide about an age 

structure of the policies comprised in the portfolio. Here, we take the following hypothetical 

distribution of the portfolio: 

a 1 = 10 % 

a 5 = 20 % 

a 12 = 30 % 

a 20 = 20 %


a 25 = 10 % 

a 30 = 10 % 

a t = 0 for all other t 

The minimization of 

∑ 

E A = a t t 

t=1 

with respect to and then gives the optimal solutions 

= 00710 and = 00133 

The financial equilibrium is achieved, as the total premium income tends to 104.29 % of the 

initial one. When working with a weighted average of the t s, the values associated 

with large t play the prominent role, resulting in values for and similar to those obtained 

for t>20 in Table 9.1. 

With optimal CRM coefficients, the discount for claim-free policyholders is rather modest 

(1.33 % per claim-free year), but the penalty in case of a claim is also moderate (7.1 %). The 

large differences compared with the official values of today’s bonus-malus system in France 

(5 % of discount per claim-free year, and 25 % increase per claim) can be explained by the 

fact that all the penalties are suppressed after two claim-free years according to the terms of 

the French law, which is particularly generous. 

We have also tested two different sets of a t s, to study the influence of the age structure of 

the portfolio on the optimal CRM coefficients. With the age structure of an ‘old’ portfolio, 

that is, 

the minimization of E A gives 

a 1 = 10 % 

a 5 = 10 % 

a 12 = 10 % 

a 20 = 20 % 

a 25 = 20 % 

a 30 = 30 % 


= 00658 and = 00115 

With the age structure of a ‘young’ portfolio, that is, 

a 1 = 30 % 

a 5 = 20 %


a 12 = 20 % 

a 20 = 10 % 

a 25 = 10 % 

a 30 = 10 % 


the minimization of E A gives 

= 00694 and = 00132 

The influence of the age structure on the optimal CRM coefficients is thus rather moderate. 

9.3 Partial Liability 

9.3.1 Reduced Penalty and Modelling Claim Frequencies 

The French bonus-malus system possesses many particular rules. This section is devoted 

to the study of one of them. Specifically, according to the terms of the French law, if the 

policyholder is partially liable for the claim then the premium is multiplied by 1.125 instead 

of 1.25. To take such a rule into account, we have to model the random couple N 1t N 2t 

where N 1t counts the number of full liability claims filed during year t and N 2t counts the 

number of partial liability claims filed during the same year. Clearly, N 1t + N 2t is the total 

number of claims N t used in the preceding section. 

Let q be the probability that the policyholder is only partially liable for the claim he 

files. Further, let us assume a Bernoulli scheme for the claim types. This ensures that, 

conditionally on , N 1t and N 2t are independent and both conform to the Poisson distribution 

(see Property 6.1). Specifically, we have now 

−1−q 

1 − qk 

PrN 1t = k = = e k= 0 1 2 

k! 

−q 

qk 

PrN 2t = k = = e k= 0 1 2 

k! 

The random variables N 1t and N 2t are obviously dependent if the risk proneness is 

unknown. The joint probability mass for the random couple N 1t N 2t is given by 

PrN 1t = k 1 N 2t = k 2 = 

∫ + 

This is a mixed bivariate Poisson model. 

0 

PrN 1t = k 1 = PrN 2t = k 2 = f d 

9.3.2 Computations of the CRMs at Time t 

Let us consider a policyholder covered for t years. In addition to the parameter t giving the 

magnitude of the penalty in case of a full-liability claim, we introduce the new parameter t


giving the reduced penalty in case of a partial liability claim. Now the CRM coefficient for 

the time period 0t is 

with 

r t t t 

N 1• N 2• I 12• t= 1 + t N 1• 

1 + t N 2• 

1 − t I 12• 

N 1• = 

N 2• = 

I 12• = 

t∑ 

N 1j 

j=1 

t∑ 

N 2j 

j=1 

t∑ 

j=1 

{ 1ifN1j = N 

I j with I j = 

2j = 0 

0 otherwise 

We will assume that t = t with fixed by the actuary. The value of describes the way 

a claim with full liability is penalized, compared to a claim with partial liability. Then the 

CRM coefficient becomes 

r t t 

N 1• N 2• I 12• t= 1 + t N 1• 

1 + t N 2• 

1 − t I 12• 

 

In order to obtain t and t we have now to minimize the objective function 

t = [( − r N 1• N 2• I 12• t ) 2] 

with respect to the parameters and . The first order conditions are 

⎧ 

[ ( 

1 − I 12• N1• 1 + N1•−1 1 + N 2• + N2• 1 + N2•−1 1 + 1•)] 

N 

⎪⎨ 

= [ ( 

1 − 2I 12• N1• 1 + 2N1•−1 1 + 2N 2• + N2• 1 + 2N2•−1 1 + 1•)] 

2N 

[ 1 + N 1• 1 + 

N 2•I12• 1 − I 12•−1 ] 

⎪⎩ 

= [ 1 + 2N 1• 1 + 

2N 2•I12• 1 − ] 2I 12•−1 

 

Let us define 

[ 

1 2 3 = N 1• 

1 N 2• 

2 I 12• 

3 

] 

∣ 

∣ = 

to be the conditional probability generating function of the random vector N 1• N 2• I 12• 

given = . We clearly have that 

( )) 

1 2 3 = 

(e − 3 − 1 + e 1−q 1 +q 2 −1 t


It can be verified that the first order conditions are as follows: 

⎧ 

2 

⎪⎨ 

100 1 + 1 + 1 − + 010 1 + 1 + 1 − 

= 100 

2 1 + 1 + 1 − + 010 

2 1 + 1 + 1 − 

⎪⎩ 

2 001 1 + 1 + 1 − = 001 

2 1 + 1 + 1 − 

where 

xyz a b c = 

x y z 

s x t y u z s t u ∣ 

∣∣s=at=bu=c 

xyz 

2 a b c = x y z 

s x t y u z s2 t 2 u 2 

∣ 

∣ 

s=at=bu=c 

for x y z ∈ 0 1. Again, numerical procedures are needed to find the solution of this 

optimization problem. 


Analyzing the financial equilibrium of the system now amounts to checking whether 

r t t 

N 1• N 2• I 12• t is equal to 1 with the optimal values t and t . To this end, we 

need the joint distribution of the random vector N 1• N 2• I 12• . The joint probability mass 

function of this vector is given in the following result that extends Property 9.3 in the present 

setting. 

Property 9.4 

For fixed , the following recursive formulas 

g ⋆t x y z = f ⋆t x y t − z for 0 ≤ z ≤ t x y ≥ 0 and x + y>0 

g ⋆t 0 0 0 = e −t 

fx y 0 = e − x+y 1 − q x q y 

x!y! 

for x y ≥ 0 and x + y>0 

f0 0 1 = e − 

( ) 

x∑ y∑ t + 1 

g ⋆t x y z = e 

− 1 g ⋆t x − u y − v z − 1gu v 1 

u=0 v=0 

z 

for 1 ≤ z ≤ t x y ≥ 0 and x + y>z− 1 

hold true with the convention that the defined functions take the value 0 where they have 

not been defined. 


It is trivial that for t = 1 we have 

fx 0z = e −1+ x z 

 

x!z! 

f0 1 0 = e −1+ 

xz>0


As f ⋆t is the t-fold convolution of a lattice random vector, we will apply the trivariate 

extension of De Pril’s algorithm described in Property 9.2. As this algorithm needs a mass 

at the origin, we define an auxilliary density function 

g ⋆t x y z = f ⋆t x t − y z 

Using similar arguments as before we obtain the following recursion 

( ) 

x∑ z∑ t + 1 

g ⋆t x y z = e 1+ 

− 1 g ⋆t x − u y − 1z− wgu 1w 

y 

u=0 w=0 

□ 

9.3.4 Numerical Illustrations 

To illustrate this special case, we use the same parameters as in Section 9.2.8. We assume 

that 20 % of the claims concern partial liability, that is q = 02. We numerically solve the 

system of two equations for different values of . Table 9.4 gives the results. As before, t 

and t decrease with t and an averaging is needed to get a unique set of parameters. 

The total income of the company is not much influenced by the value of , and is quite 

close to the values listed in Table 9.1. 

To obtain unique values for the CRM coefficients, we choose the first age distribution of 

Section 9.2.8. The minimization of 

E A = 

∑ 

a t t 

then gives the values displayed in Table 9.5. The same comments apply to this case. 

Specifically, the t s with large values of t play the prominent role, giving optimal 

CRM coefficients close to the values obtained with t>20 in Table 9.1. 

The values of the optimal CRM coefficients displayed in Table 9.5 are again much smaller 

than those implemented by the French law. As before, this is due to the special bonus rule 

of the French system (after two consecutive years without claim, the driver goes back to the 

initial level of 100 %). 

t=1 

Table 9.4 Parameters t and t and financial equilibrium for different values of t and . 

t = 15 = 20 = 25 

t t Financial 

equilibrium 

t t Financial 

equilibrium 

t t Financial 

equilibrium 

1 0.4336 0.1423 09747 0.3316 0.1423 0.9729 0.2674 0.1423 09714 

2 0.3253 0.0954 09870 0.2498 0.0953 0.9850 0.2022 0.0953 09832 

3 0.2616 0.0727 09984 0.2014 0.0726 0.9965 0.1633 0.0726 09946 

4 0.2194 0.0589 10083 0.1692 0.0589 1.0062 0.1374 0.0588 10047 

10 0.1128 0.0279 10419 0.0873 0.0279 1.0402 0.0711 0.0279 10386 

20 0.0627 0.0149 10634 0.0486 0.0149 1.0624 0.0397 0.0149 10617 

30 0.0435 0.0102 10725 0.0337 0.0102 1.0707 0.0276 0.0102 10712


Table 9.5 Parameters and and financial equilibrium for different values of . 

= 15 = 20 = 25 

Financial 

equilibrium 

Financial 

equilibrium 

Financial 

equilibrium 

0.0507 0.0133 1.0419 0.0393 0.0133 1.0403 0.0321 0.0133 1.0393 


This chapter is based on Pitrebois, Denuit & Walhin (2006b). Despite its apparent 

difference with bonus-malus scales, the French bonus-malus system can be treated as a scale 

with many levels. Kelle (2000) followed this route, and used a Markov chain with 530 

states to analyse the French system. The large amount of states needed is due to the malus 

reduction in the case of claims with shared responsibility, forcing the author to consider the 

pair (number of claims with whole responsibility, number of claims with partial liability) to 

make the computation. 

In this chapter, we did not consider all the characteristics of the bonus-malus system 

in force in France. We have disregarded the special bonus rule (which suppresses all 

the penalties after two claim-free years). The French law imposes other specific rules on 

insurance companies. For instance, the French bonus-malus system is such that drivers never 

pay more than 350 % of the base premium nor less than 50 % of the base premium. Therefore 

the minimization process has to be carried with an adapted CRM coefficient of the form 

r ∗ = max05 min35r 

Several simplifying assumptions can be considered to ease the numerical computations. 

For instance, we could work with binary annual claim numbers: either the policyholder 

does not report any claim or he reports a single claim to the company. Such an assumption 

replacing N t by minN t 1 leads to smaller discounts and higher penalties, which is a prudent 

strategy for the insurer. 

Even if the vast majority of bonus-malus systems appear as scales in which policyholders 

move according to their claims history, there are some exceptions (such as the system 

in force in France). We refer the reader to Neuhaus (1988) for another example, where 

the malus after a claim is expressed by a fixed monetary amount (instead of a relativity). 

This interesting mechanism restores some fairness in case of differentiated a priori 

price lists. 

The multivariate version of Panjer’s recursive formula has been derived by Sundt 

(1999) and Ambagaspitya (1999). Sundt (1999) provided a proof based on conditional 

expectations whereas Ambagaspitya (1999) used a proof based on generating functions. 

Sundt (1999) also showed that the following recursive formula can be used : 

f S s = 

1 

1 − af X 0 

s∑ 

x≠0 

( 

a + b s ) 

1 +···+s k 

f 

x 1 +···+x X xf S s − x 

k 

x > 0


The multivariate version of De Pril’s recursive formula for the convolution of independent 

and identically distributed random vectors has been derived by Sundt (1999) as a particular 

case of the multivariate Panjer algorithm, and by Walhin (2001) as a particular case of 

the multivariate version of Dhaene & Vandebroek’s (1995) recursive formula for the 

multivariate individual risk model. It can also be deduced from the multivariate extension 

of De Pril’s methodology as shown in Dickson & Waters (1999).

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Index 

Binomial distribution 13–15, 271–2, 281, 

333 

Bonus hunger 51–2, 220–1, 246–54, 257–8 

Chapman-Kolmogorov equations 174 

De Pril algorithm 330–3, 334, 341 

Deductible 168–9, 251, 276–91 

Deviance 71, 72, 75 

Deviance residuals 72, 77 

Efficiency 220, 257, 307, 314, 318 

De Pril 242–6 

Loimaranta 240–3 

Eigenvalue 175, 184, 295 

Eigenvector 175, 179 

Ergodicity 176, 273 

Exposure-to-risk 20, 44–5, 53, 56, 73, 93 

Financial equilibrium 123, 130, 135, 167, 186, 

197, 202, 205, 305, 314, 318, 333–5, 340–1, 

342 

Fisher 

information matrix 38, 40, 69, 73, 104 

score 36, 67–8 

Gamma 

distribution 28, 29, 230 

regression 231–2 

GEE 101–5 

Generalized Pareto 

distribution 224–9 

index 224–6 

Gertensgarbe plot 225–6 

Information criteria 

AIC 43, 110 

BIC 43, 110 

Interaction 58–9, 62–3, 74 

Inverse Gaussian 

distribution 31–2, 233 

regression 233–4 

Kolmogorov distance 208–13 

Large claim 223–30, 256–7 

Lemaire algorithm 251–4 

Likelihood ratio 

confidence interval 69 

test 41–2, 45, 72, 76, 82 

LogNormal 

distribution 34 

regression 235–6, 246–9 

Mixed Poisson 

credibility model 126–8, 155–8 

discrete credibility model 135–6, 193–4 

distribution 24–8, 46, 270–2, 278, 326 

process 25 






Negative Binomial 

credibility model 131–5, 144, 148, 151–2 

distribution 28–31, 45–6, 281, 284, 336 

regression 83–6, 105–6, 238 

Offset 73, 101 

Optimal retention 220–1, 247–9, 251–4, 257–8 

Panjer algorithm 279–84, 287, 330–2 

Perron-Froebenius theorem 175 

Poisson 

distribution 15–17, 45–6, 54–6, 172–3, 

229–30, 281, 283 

process 17–21 

regression 64–79, 94–101, 229–30, 237–8 

Poisson-Inverse Gaussian 

distribution 32–3, 45–6 


Poisson-LogNormal 

distribution 34, 45–6 


Regularity 175, 176, 179, 273 

Score test 43, 82–3 

Special bonus rule 200–8 

Specification error 72–3, 80 

Total variation distance 183–4, 208–13, 295–6, 

308, 315 

Vuong test 43–4, 46, 89, 109–10 

Wald 

confidence interval 69 

test 42–3

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