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Nonlife actuarial models. Theory, methods and evaluation. (English) Zbl 1207.91007

International Series on Actuarial Science. Cambridge: Cambridge University Press (ISBN 978-0-521-76465-0/hbk). xv, 524 p. (2009).
This is a book on theory, methods, and empirical implementation of nonlife actuarial models. The book includes all topics found in Exam C of the Society of Actuaries, but it also covers some topics beyond what is required by this exams and may be used by actuarial students in general.
The book is divided into four parts. Part I, concerning loss models, consists of three chapters. In Chapter 1, the author introduces some techniques for modeling nonnegative integer-valued random variables. In the next chapter, some standard continuous distributions for modeling claim-severity distributions are discussed. Chapter 3 deals with aggregate-loss models.
Part II of the book is about measuring risk and computing the likelihood of ruin and comprises two chapters. The first one deals with various measures of risk. The concept of a coherent risk measure by P. Artzner, F. Delbaen, J.-M. Eber, D. Heath and H. Ku [Ann. Oper. Res. 152, 5–22 (2007; Zbl 1132.91484)], specific measures such as Value-at-Risk, conditional tail expectation, and the distortion-function approach are discussed. In the second chapter of this part, the probability of ruin as a function of time, initial surplus, and the claim distribution is considered.
Credibility theory, which provides the basic analytic framework for pricing insurance products, is the subject of Part III of the book. The first chapter of this part deals with classical credibility, which is not based on a well-adopted statistical principle of prediction. In the next chapter, the problem is set in a rigorous statistical framework by Bühlmann’s credibility theory. The approach is further extended to the Bühlmann-Straub model, in which the loss random variables have different exposures. In the third chapter of this part, the author considers the Bayesian approach in updating the prediction for future losses. The part ends with a chapter on some empirical Bayes approaches to the estimation of the quantities necessary for the implementation of credibility prediction.
Part IV of the book deals with model construction and evaluation. It consists of six chapters. The purpose of the first one is to provide a brief review of the theory of estimation, including the properties of an estimator, and the concepts of point estimation, interval estimation, unbiasedness, consistency, and efficiency. Specific estimation methods are postponed to the next two chapters. Namely, the main focus of Chapter 11 is the estimation of distribution functions and probability (density) functions of duration and loss variables, while in the next one, various methods of estimating the parameters of a failure-time or loss distributions are discussed. Methods for evaluating a fitted model for loss distributions are reviewed in Chapter 13. They include graphical methods, misspecification tests and diagnostic checks, and information criteria. Basic Monte Carlo methods are surveyed in Chapter 14. In particular, methods for generating random numbers, and variance reduction techniques are discussed. The last chapter of this part presents some applications of Monte Carlo methods to the analysis of actuarial and financial data. The discussed topics include the use of bootstrap in model testing, and the simulation of price paths of geometric Brownian motions as well as jump diffusion processes.
The book ends with an appendix which summarizes background material found in introductory probability textbooks and develops required results for use in the main text.
The concepts and theories introduced in the book are illustrated by many practical examples. Some of these explain the theory through numerical applications, while others develop new results. Each chapter includes exercises for practice. Several chapters are provided with a section on numerical computation using Excel.
A very good balance of rigor and readability makes this book an impressive encyclopedia of results and methods for nonlife actuarial modeling.

MSC:

91-02 Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance
91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
62-02 Research exposition (monographs, survey articles) pertaining to statistics
65C05 Monte Carlo methods

Citations:

Zbl 1132.91484

Software:

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