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

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 combines traditional insurance coverage up to a given limit with parametric (or index-based) coverage for larger losses. This second component is computed using covariates available immediately after a loss occurs, thereby reducing claims management costs through rapid compensation. To optimize its design, we use a criterion tailored to extreme losses, that is, to Pareto-type loss distributions. 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 of tornado losses in the United States. We also propose a two-step optimization procedure as a potential solution to the scarcity of data in the tails of loss distributions. Finally, we empirically demonstrate that the proposed hybrid contract outperforms a traditional capped indemnity contract.