Optimal insurance contract under mean-variance preference with value at risk constraint

In this paper, we investigate the optimal insurance arrangement for an agent who exhibits a mean-variance preference. For the purpose of risk management, the agent's terminal wealth is constrained via a Value at Risk condition. As for the admissible indemnity functions, we suppose that they are subjected to principle of indemnity, incentive compatibility condition, and a so-called Vajda condition as well. The Vajda condition stipulates that within an insurance contract, the proportion of the loss borne by the insurance company should be non-decreasing as the total loss amount increases. By employing a non-decreasing rearrangement technique and a modification approach, our results show that the optimal insurance is either a pure deductible insurance or a mixed proportional insurance with a deductible under expected value premium principle. As by-products, we also obtain the optimal insurance policies under preferences of mean-variance, mean-VaR, and mean-variance with a portfolio insurance constraint, respectively. Finally, we present numerical studies to provide economic insights into these findings.