The collective risk model plays a central in non-life insurance mathematics. We present different ways to model the arrival process of claims and the claim sizes. We discuss tools for the exploratory statistical data analysis and we illustrate the theoretical methods using real data sets. The second part of the course introduces experience rating, Bayes estimation, and reserving for late claims.

(I) Collective Risk Models: Homogeneous and inhomogeneous Poisson processes, order statistics property, Poisson random measure, Cramér-Lundberg model, renewal process, mixed Poisson process, order of magnitude of the total claim amount, claim size distributions, heavy- and light-tailed distributions, exploratory statistical analysis with quantile-quantile plots and mean excess plots, regularly varying claim sizes and their aggregation, subexponential distributions, mixture distributions, space-time decomposition of a compound Poisson process, calculation of the distribution of the total claim amount using the extended Panjer recursion, approximation by the central limit theorem or Monte Carlo techniques, reinsurance treaties (II) Experience Rating: Heterogeneity model and Bayes estimation, linear Bayes estimator, credibility estimator, Bühlmann model, Bühlmann-Straub model (III) Reserving for Late Claims: Chain ladder method, grossing-up method, multiplicative model, multinomial model

Written and oral examination.

Lecture notes for this course are available. Im Sekretariat der Forschungsgruppe FAM erhältlich

• Chapters 1 to 3 as well as 5 and 6 in the book by Thomas Mikosch, Non-Life Insurance Mathematics, An Introduction with Stochastic Processes, Springer Universitext, Springer-Verlag Berlin Heidelberg 2004, ISBN 3-540-40650-6.
• Chapter 11 in the book by Klaus D. Schmidt, Versicherungsmathematik, Springer-Verlag Berlin Heidelberg 2002, ISBN 3-540-42731-7.