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**Insurance premiums. Theory and applications.**
*(English)*
Zbl 0532.62082

Amsterdam - New York - Oxford: North-Holland. XI, 406 p. $ 63.75; Dfl. 150.00 (1984).

Premium calculation is one of the most important topics of current mathematical risk theory, and many substantial contributions to this subject have been made by the authors of the present monograph which is essentially a compilation of some twenty papers published by them in the past five years.

The headings of the six chapters are: Introduction, Premium calculation principles, Properties of premium calculation principles, Ordering among risks, Bounds on stop-loss premiums, Applications. In Chapter 2, most of the premium calculation principles that have been developed theoretically are introduced. In the following chapter these principles are examined with respect to certain desirable properties. In Chapter 4, orderings among random variables are considered with special emphasis on notions that have an application or interpretation in risk theory. Chapter 5 contains applications of convex analysis to the problem of calculating bounds on stop-loss premiums. In Chapter 6, applications of the general theory to actuarial problems such as the critical claim size of a bonus system and bounds on ruin probabilities are given.

This is not a cookery-book-like handbook providing actuaries with recipes how to calculate premiums in practical situations. Instead, it gives an elegant and rigorous presentation of the mathematical foundations of essential parts of modern risk theory. It is sure to be a standard work for years.

The headings of the six chapters are: Introduction, Premium calculation principles, Properties of premium calculation principles, Ordering among risks, Bounds on stop-loss premiums, Applications. In Chapter 2, most of the premium calculation principles that have been developed theoretically are introduced. In the following chapter these principles are examined with respect to certain desirable properties. In Chapter 4, orderings among random variables are considered with special emphasis on notions that have an application or interpretation in risk theory. Chapter 5 contains applications of convex analysis to the problem of calculating bounds on stop-loss premiums. In Chapter 6, applications of the general theory to actuarial problems such as the critical claim size of a bonus system and bounds on ruin probabilities are given.

This is not a cookery-book-like handbook providing actuaries with recipes how to calculate premiums in practical situations. Instead, it gives an elegant and rigorous presentation of the mathematical foundations of essential parts of modern risk theory. It is sure to be a standard work for years.

Reviewer: W.-R.Heilmann

### MSC:

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

91-02 | Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance |

91B30 | Risk theory, insurance (MSC2010) |

90C25 | Convex programming |

60E15 | Inequalities; stochastic orderings |