Michel Denuit & Xavier Marechal 
Actuarial Modelling of Claim Counts 
Risk Classification, Credibility and Bonus-Malus Systems

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.

Table of Content

Foreword.

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-Log Normal 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.

About the author

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