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 - Professor, Institute of Actuarial Science, UCL, Belgium.

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.

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 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 Introduction 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

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…