Optimal insurance design with Lambda-Value-at-Risk

This paper explores optimal insurance solutions based on the Lambda-Value-at-Risk (ΛVaR). Using the expected value premium principle, we first analyze a stop-loss indemnity and provide a closed-form expression for the deductible parameter. A necessary and sufficient condition for the existence of a positive and finite deductible is also established. We then generalize the stop-loss indemnity and show that, akin to the VaR model, a limited stop-loss indemnity remains optimal within the ΛVaR framework. Further, we examine the use of Λ′VaR as a premium principle and show that full or no insurance is optimal. We also identify that a limited loss indemnity is optimal when Λ′VaR is solely used to determine the risk-loading in the premium principle. Additionally, we investigate the impact of model uncertainty, particularly in scenarios where the loss distribution is unknown but lies within a specified uncertainty set. Our findings suggest that a limited stop-loss indemnity is optimal when the uncertainty set is defined using a likelihood ratio. Meanwhile, when only the first two moments of the loss distribution are available, we provide a closed-form expression for the optimal deductible in a stop-loss indemnity.