There are a wide range of variables for actuaries to consider when
calculating a motorist's insurance premium, such as age,
gender and type of vehicle. Further to these factors,
motorists' rates are subject to experience rating systems,
including credibility mechanisms and Bonus Malus systems (BMSs).

Actuarial Modelling of Claim Counts presents a
comprehensive treatment of the various experience rating systems
and their relationships with risk classification. The authors
summarize the most recent developments in the field, presenting
ratemaking systems, whilst taking into account exogenous
information.

The text:

* Offers the first self-contained, practical approach to a priori
and a posteriori ratemaking in motor insurance.

* Discusses the issues of claim frequency and claim severity,
multi-event systems, and the combinations of deductibles and
BMSs.

* Introduces recent developments in actuarial science and
exploits the generalised linear model and generalised linear mixed
model to achieve risk classification.

* Presents credibility mechanisms as refinements of commercial
BMSs.

* Provides practical applications with real data sets processed
with SAS software.

Actuarial Modelling of Claim Counts is essential reading
for students in actuarial science, as well as practicing and
academic actuaries. It is also ideally suited for professionals
involved in the insurance industry, applied mathematicians,
quantitative economists, financial engineers and statisticians.

Michel Denuit is Professor of Statistics and Actuarial Science at the Université Catholique de Louvain, Belgium. His major fields of research are risk theory and stochastic inequalities. He has (co-)authored numerous articles that have appeared in applied and theoretical journals and served as member of the editorial board for several journals (including Insurance: Mathematics and Economics). He is a section editor on Wiley's Encyclopedia of Actuarial Science, and is the author of two previous books, one of them with Wiley.

Xavier Maréchal Université Catholique de Louvain, Belgium & CEO of Reacfin, Belgium.

Sandra Pitrebois Université Catholique de Louvain, Belgium & Secura Belgian Re, Brussels.

Jean-François Walhin Université Catholique de Louvain, Belgium & Secura Belgian Re, Brussels

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

Sandra Pitrebois Secura, Belgium

Jean-François Walhin Fortis, Belgium

There are a wide range of variables for actuaries to consider when calculating a motorist's insurance premium, such as age, gender and type of vehicle. Further to these factors, motorists' rates are subject to experience rating systems, including credibility mechanisms and Bonus Malus systems (BMSs).

Actuarial Modelling of Claim Counts presents a comprehensive treatment of the various experience rating systems and their relationships with risk classification. The authors summarize the most recent developments in the field, presenting ratemaking systems, whilst taking into account exogenous information.

The text:

• Offers the first self-contained, practical approach to a priori and a posteriori ratemaking in motor insurance.
• Discusses the issues of claim frequency and claim severity, multi-event systems, and the combinations of deductibles and BMSs.
• Introduces recent developments in actuarial science and exploits the generalised linear model and generalised linear mixed model to achieve risk classification.
• Presents credibility mechanisms as refinements of commercial BMSs.
• Provides practical applications with real data sets processed with SAS software.

Actuarial Modelling of Claim Counts is essential reading for students in actuarial science, as well as practicing and academic actuaries. It is also ideally suited for professionals involved in the insurance industry, applied mathematicians, quantitative economists, financial engineers and statisticians.

Actuarial Modelling of Claim Counts presents a comprehensive treatment of the various experience rating systems and their relationships with risk classification. The authors summarize the most recent developments in the field, presenting ratemaking systems, whilst taking into account exogenous information.

The text:

• Offers the first self-contained, practical approach to a priori and a posteriori ratemaking in motor insurance.
• Discusses the issues of claim frequency and claim severity, multi-event systems, and the combinations of deductibles and BMSs.
• Introduces recent developments in actuarial science and exploits the generalised linear model and generalised linear mixed model to achieve risk classification.
• Presents credibility mechanisms as refinements of commercial BMSs.
• Provides practical applications with real data sets processed with SAS software.

Actuarial Modelling of Claim Counts is essential reading for students in actuarial science, as well as practicing and academic actuaries. It is also ideally suited for professionals involved in the insurance industry, applied mathematicians, quantitative economists, financial engineers and statisticians.

Foreword xiii

Preface xv

Notation 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

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 NewtonRaphson 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.…