Combination of traditional and parametric insurance: calibration method based on the optimization of a criterion adapted to heavy tail losses

In this paper, we consider the question of providing insurance protection against heavy tail losses, where the expectation of the loss may not even be finite. The product we study is based on a combination of traditional insurance up to some limit, and a parametric (or index-based) cover for larger losses. This second part of the cover is computed from covariates available just after the claim, allowing to reduce the claim management costs via an instant compensation. To optimize the design of this second part of the product, we use a criterion which is adapted to extreme losses (that is distribution of the losses that are of Pareto type). We support the calibration procedure by theoretical results that show its convergence rate, and empirical results from a simulation study and a real data analysis on tornados in the US. We conclude our study by empirically demonstrating that the proposed hybrid contract outperforms a traditional capped indemnity contract.