Severity modeling of extreme insurance claims for tariffication

Generalized linear models are common instruments for the pricing of non-life insurance contracts. They are used to estimate the expected frequency and severity of insurance claims. However, these models do not work adequately for extreme claim sizes. To accommodate for these extreme claim sizes, we develop the threshold severity model, that splits the claim size distribution in areas below and above a given threshold. More specifically, the extreme insurance claims above the threshold are modeled in the sense of the peaks-over-threshold methodology from extreme value theory using the generalized Pareto distribution for the excess distribution, and the claims below the threshold are captured by a generalized linear model based on the truncated gamma distribution. Subsequently, we develop th e corresponding concrete log-likelihood functions above and below the threshold. Moreover, in the presence of simulated extreme claim sizes following a log-normal as well as Burr Type XII distribution, we demonstrate the superiority of the threshold severity model compared to the commonly used generalized linear model based on the gamma distribution.