Combination of traditional and parametric insurance: calibration method based on the optimization of a criterion adapted to heavy tail losses[Combinaison d'assurance traditionnelle et paramétrique : méthode de calibration basée sur l'optimisation d'un critère adapté aux pertes à queue lourde]

In this paper, we address the problem of providing insurance protection against heavy-tailed losses, for which the expected loss may not even be finite. The product we study is based on a combination of traditional insurance up to a given limit and a parametric (or index-based) cover for larger losses. This second component of the coverage is computed from covariates available immediately after the loss occurs, allowing claim management costs to be reduced through rapid compensation. To optimize the design of this second component, we use a criterion adapted to extreme losses, that is, to loss distributions of Pareto type. We support the calibration procedure with theoretical results establishing its convergence rate, as well as empirical evidence from both a simulation study and a real-data analysis on tornado losses in the United States. We also propose a two-step optimization procedure as a potential solution to the issue of data scarcity in the tails of loss distributions. We conclude by empirically demonstrating that the proposed hybrid contract outperforms a traditional capped indemnity contract.