Catastrophic Risk Management in Non-life Insurance

The paper deals with some aspects of modelling catastrophic risk and with its application to non- -life insurance claims. First, we formulate the problem of generalization of classical Cramér-Lundberg collective risk model. Then using some well-known extreme value results we study two methods for extremal claims registration. Finally, we apply the theoretical results for real insurance data. As suitable mathematical models for large insurance claims are used heavy-tailed distributions (subexponential, stable and max-stable distributions).The main reason why we are interested in stable distributions is, that for the extreme value distributions the classical central limit theorem (CLT) condition (finite mean and variance) doesn‘t hold. Instead of CLT we use the Fisher-Tippett theorem which specifies the limit laws for maximum of independent identically distributed (iid) random variables as Generalised Extreme Value (GEV) distribution. For recording extreme insurance claims we use two approaches. The first one is based on modelling maximum of the sample and called method of block-maxima. This method is based just on the Fisher-Tippett theorem and in non-life insurance we can use it for non-proportional Largest Claim Reinsurance (LCR). The second approach is based on modelling excess values over the chosen threshold. This approach is called Peaks Over Threshold method and is based on the Picands theorem which specifies the limit law for the exceedances as Generalised Pareto Distribution (GPD). This method is used in non-proportional Excess-of-Loss Reinsurance (XL). In the end, we apply these methods for modelling real fire insurance claims. We find an optimal exceedance level for reinsurance and identify lognormal distribution for all data and Pareto distribution for the tail. The empirical data are compared with considered theoretical distribution using chi-squared and Kolmogorov-Smirnov goodness-of-fit tests. For detailed statistical analysis of data we use STATGRAPHICS and its procedure Distribution fitting.