Optimal insurance design with Lambda-Value-at-Risk

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.