Extreme severity modeling using a GLM-GPD combination: application to an excess of loss reinsurance treaty

This article studies the model proposed by Laudagé et al. (Insur Math Econ 88:77–92, 2019) and examines whether the combination between a Generalized Linear Model (GLM) and a Generalized Pareto Distribution (GPD) is valid for modeling claims severity in a practical framework. For this, we consider a real fire insurance dataset and fit the proposed model to these data. In this modeling, the threshold is of great importance since it separates the data into two parts and represents the point from which the observations become extremes. Therefore, in order to guarantee the correct choice of this threshold, one extra method is adopted in addition to that used by Laudagé et al. (2019). Furthermore, we build on the authors' results and extend them by fitting the attritional data to three well-known distributions. The results of this study show that the GLM-GPD combination outperforms the benchmark model (classical GLM) in terms of predictive power. In addition, the application of an excess of loss reinsurance treaty to these two models proves that it is more interesting for an insurer to adopt a GLM-GPD combination so as not to underestimate the risk and go bankrupt. This justifies that the combined modeling is reasonably good to describe insurance claim costs.