The modelling of extreme events is becoming of increased importance to actuaries. This paper outlines the various theories. It outlines the consistent theory underlying many of the differing approaches and gives examples of the analysis of models. A review of non-standard extreme events is given, and issues of public policy are outlined.

The use of classical bonus–malus systems entails very high maluses and other problems which, during recent years, have been criticised by actuaries. To avoid these problems, new bonus–malus models have been developed. For instance, it is well known that the use of an exponential loss function reduces the differences between overcharges and undercharges, solving the problem of high maluses. In order to measure the sensitivity of the exponential bonus–malus system, and according to robust Bayesian analysis, we first model the structure function by specifying a subclass of the generalised moments class. We then examine the range of relativities for each prior. Finally, we illustrate our method with a numerical example based on real data.