## Effective actuarial methodsDuring the last two decades actuarial research has developed in a more applied direction. Although the original risk models generally served as convenient and sometimes tractable mathematical examples of general probabilistic and/or statistical theories, nowadays models and techniques are encountered that can be considered to be typically actuarial. Examples include ordering of risks by dangerousness, credibility theory and techniques based on IBNR models. Not only does this book present the underlying mathematics of these subjects, but it also deals with the practical application of the techniques. In order to provide results based on real insurance portfolios, use is made of three software packages, namely SLIC performing stop-loss insurance calculations for individual and collective risk models, CRAC dealing with actuarial applications of credibility theory, and LORE giving IBNR-based estimates for loss reserves. Worked-out examples illustrate the theoretical results. This book is intended for use in preparing university actuarial exams, and contains many exercises with varying levels of complexity. It is valuable as a textbook for students in actuarial sciences during their last year of study. Due to the emphasis on applications and because of the worked-out examples on real portfolio data, it is also useful for practising actuaries to guide them in interpreting their own results. |

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### Contents

Introduction to ordering of risks | 3 |

Theoretical aspects of ordering of risks | 9 |

Stochastic dominance | 16 |

Copyright | |

17 other sections not shown

### Common terms and phrases

actuarial applied approximation assumptions Bayesian inference Buhlmann Buhlmann-Straub model calculated chain ladder method claim amounts column compound Poisson distribution computed conditional distribution consider convex functions covariance credibility estimator credibility factor credibility formulae credibility model credibility premium credibility results credibility theory credibility weights d-triangle decision makers defined denotes derivative Diagram distribution function equal Esscher example Exercises expected value exponential following theorem franchise deductible Fx(x given gives heterogeneity hierarchical model IBNR identically distributed independent individual iteratively least squares linearized credibility loss function matrix mean squared error minimal number of claims observable variables obtained optimal portfolio premium principle problem Proof prove random variables regression remaining life-time risk parameter risk premium ruin probability run-off triangle Section Sector 6 Records statistical Stieltjes integrals stochastic dominance stop-loss order stop-loss premiums stop-loss transform structure parameters subsection Suppose system of equations techniques unbiased estimator utility function variability order variance Vylder Xpzw Zpsigma