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

Preface.

Notation.

Part I Modelling Claim Counts.

1 Mixed Poisson Models for Claim Numbers.

1.1 Introduction.

1.2 Probabilistic Tools.

1.3 Poisson Distribution.

1.4 Mixed Poisson Distributions.

1.5 Statistical Inference for Discrete Distributions.

1.6 Numerical Illustration.

1.7 Further Reading and Bibliographic Notes.

2 Risk Classification.

2.1 Introduction.

2.2 Descriptive Statistics for Portfolio A.

2.3 Poisson Regression Model.

2.4 Overdispersion.

2.5 Negative Binomial Regression Model.

2.6 Poisson-Inverse Gaussian Regression Model.

2.7 Poisson-LogNormal Regression Model.

2.8 Risk Classification for Portfolio A.

2.9 Ratemaking using Panel Data.

2.10 Further Reading and Bibliographic Notes.

Part II Basics of Experience Rating.

3 Credibility Models for Claim Counts.

3.1 Introduction.

3.2 Credibility Models.

3.3 Credibility Formulas with a Quadratic Loss Function.

3.4 Credibility Formulas with an Exponential Loss Function.

3.5 Dependence in the Mixed Poisson Credibility Model.

3.6 Further Reading and Bibliographic Notes.

4 Bonus-Malus Scales.

4.1 Introduction.

4.2 Modelling Bonus-Malus Systems.

4.3 Transition Probabilities.

4.4 Long-Term Behaviour of Bonus-Malus Systems.

4.5 Relativities with a Quadratic Loss Function.

4.6 Relativities with an Exponential Loss Function.

4.7 Special Bonus Rule.

4.8 Change of Scale.

4.9 Dependence in Bonus-Malus Scales.

4.10 Further Reading and Bibliographic Notes.

Part III Advances in Experience Rating.

5 Efficiency and Bonus Hunger.

5.1 Introduction.

5.2 Modelling Claim Severities.

5.3 Measures of Efficiency for Bonus-Malus Scales.

5.4 Bonus Hunger and Optimal Retention.

5.5 Further Reading and Bibliographic Notes.

6 Multi-Event Systems.

6.1 Introduction.

6.2 Multi-Event Credibility Models.

6.3 Multi-Event Bonus-Malus Scales.

6.4 Further Reading and Bibliographic Notes.

7 Bonus-Malus Systems with Varying Deductibles.

7.1 Introduction.

7.2 Distribution of the Annual Aggregate Claims.

7.3 Introducing a Deductible within a Posteriori Ratemaking.

7.4 Numerical Illustrations.

7.5 Further Reading and Bibliographic Notes.

8 Transient Maximum Accuracy Criterion.

8.1 Introduction.

8.2 Transient Behaviour and Convergence of Bonus-Malus Scales.

8.3 Quadratic Loss Function.

8.4 Exponential Loss Function.

8.5 Numerical Illustrations.

8.6 Super Bonus Level.

8.7 Further Reading and Bibliographic Notes.

9 Actuarial Analysis of the French Bonus-Malus System.

9.1 Introduction.

9.2 French Bonus-Malus System.

9.3 Partial Liability.

9.4 Further Reading and Bibliographic Notes.

Bibliography.

Index.