Optimal insurance design with Lambda-Value-at-Risk

TL;DR

This paper investigates optimal insurance strategies under Lambda-Value-at-Risk, revealing conditions under which truncated stop-loss or full insurance are optimal, and analyzing the impact of model uncertainty on these solutions.

Contribution

It extends the Lambda-Value-at-Risk framework to derive explicit optimal insurance indemnities and examines the effects of different uncertainty sets on optimal solutions.

Findings

Truncated stop-loss is optimal under certain Lambda-VaR conditions.

Full or no insurance is optimal when using Lambda'-VaR as premium principle.

Optimal deductible formulas are provided under model uncertainty.

Abstract

This paper explores optimal insurance solutions based on the Lambda-Value-at-Risk ($\Lambda\VaR$). If the expected value premium principle is used, our findings confirm that, similar to the VaR model, a truncated stop-loss indemnity is optimal in the $\Lambda\VaR$ model. We further provide a closed-form expression of the deductible parameter under certain conditions. Moreover, we study the use of a $\Lambda'\VaR$ as premium principle as well, and show that full or no insurance is optimal. Dual stop-loss is shown to be optimal if we use a $\Lambda'\VaR$ only to determine the risk-loading in the premium principle. Moreover, we study the impact of model uncertainty, considering situations where the loss distribution is unknown but falls within a defined uncertainty set. Our findings indicate that a truncated stop-loss indemnity is optimal when the uncertainty set is based on a likelihood ratio. However, when uncertainty arises from the first two moments of the loss variable, we provide the closed-form optimal deductible in a stop-loss indemnity.