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**Actuarial modelling of claim counts. Risk classification, credibility and bonus-malus systems.**
*(English)*
Zbl 1168.91001

Chichester: John Wiley & Sons (ISBN 978-0-470-02677-9/hbk; 978-0-470-51742-0/ebook). xxvii, 356 p. (2007).

This book presents a comprehensive treatment of various rating systems and their relationships with risk classification. It treats the most recent developments in this field. It offers the first self-contained practical approach to a priori and a posteriori rate making in motor insurance and discusses the issues of claim frequency and claim severity, and multi-event-systems. It introduces recent developments in actuarial science and exploits the generalized linear model and generalized mixed models to achieve risk classification. It presents credibility mechanisms of commercial Bonus-Malus-Systems, and provides practical applications with real data sets.

For understanding of this book, a preknowledge in statistics and probability theory is necessary. This book is essential 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 mathematics, quantitative economy, financial engineers and statisticans.

This book has three parts. Part I models claim counts. Part II treats basics of experience rating. Part III describes advances in experience rating.

Part I (chapter 1 and 2) presents basic notations of risk and risk characteristics and their theoretical representation in stochastic models. Chapter 1 treats mixed Poisson models for claim numbers. It describes probabilistic tools, the Poisson distributions, the mixed Poisson distributions, and statistical inference for discrete distributions. Chapter 2 regards risk classifications. After an introduction, it treats different regression models, as for example the Poisson or the negative binominal regression model. Further, it regards risk classification and rate making using panel data.

Part II (chapter 3 and 4) describes the basics of experience data. Chapter 3 treats credibility modells for claim counts as credibility modells with different loss functions. Chapter 4 presents Bonus-Malus scales. After modelling Bonus-Malus-Systems, it describes transition probabilities, long-term behaviour of Bonus-Malus-Systems and relativities with different loss functions.

Part III (chapter 5-9) presents advances in experience rating. In chapter 5, efficiency and Bonus hunger, modelling claim severities, measures of efficiency for Bonus-Malus-Systems, Bonus hunger and optimal retention are treated. Chapter 6 describes multi-event credibility modells and multi-event Bonus-Malus scales. Chapter 7 regards Bonus-Malus Systems with varying deductibles by introducing a deductible within a posteriori ratemaking. Chapter 8 treats the transient maximum accuracy criterion. Chapter 9 ends the book with an actuarial analysis of the french Bonus-Malus-System.

For understanding of this book, a preknowledge in statistics and probability theory is necessary. This book is essential 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 mathematics, quantitative economy, financial engineers and statisticans.

This book has three parts. Part I models claim counts. Part II treats basics of experience rating. Part III describes advances in experience rating.

Part I (chapter 1 and 2) presents basic notations of risk and risk characteristics and their theoretical representation in stochastic models. Chapter 1 treats mixed Poisson models for claim numbers. It describes probabilistic tools, the Poisson distributions, the mixed Poisson distributions, and statistical inference for discrete distributions. Chapter 2 regards risk classifications. After an introduction, it treats different regression models, as for example the Poisson or the negative binominal regression model. Further, it regards risk classification and rate making using panel data.

Part II (chapter 3 and 4) describes the basics of experience data. Chapter 3 treats credibility modells for claim counts as credibility modells with different loss functions. Chapter 4 presents Bonus-Malus scales. After modelling Bonus-Malus-Systems, it describes transition probabilities, long-term behaviour of Bonus-Malus-Systems and relativities with different loss functions.

Part III (chapter 5-9) presents advances in experience rating. In chapter 5, efficiency and Bonus hunger, modelling claim severities, measures of efficiency for Bonus-Malus-Systems, Bonus hunger and optimal retention are treated. Chapter 6 describes multi-event credibility modells and multi-event Bonus-Malus scales. Chapter 7 regards Bonus-Malus Systems with varying deductibles by introducing a deductible within a posteriori ratemaking. Chapter 8 treats the transient maximum accuracy criterion. Chapter 9 ends the book with an actuarial analysis of the french Bonus-Malus-System.

Reviewer: Klaus Ehemann (Karlsruhe)

### MSC:

91-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to game theory, economics, and finance |

91B30 | Risk theory, insurance (MSC2010) |