Optimal insurance design under Vajda condition and exclusion clauses

In this paper, we explore the optimal insurance problem where the exclusion clause is taken into account. Assume that the insurable loss is mutually exclusive from another loss that is denied in the insurance coverage. Our objective is to characterize the optimal insurance strategy by minimizing the risk-adjusted value of a policyholder's liability, where the unexpected loss is calculated by either the value at risk (VaR) or the tail value at risk (TVaR). To prevent moral hazard and to reflect the spirit of insurance, we analyze the optimal solutions over the class of ceded loss functions such that the policyholder's retained loss and the proportion paid by an insurer are both increasing. We show that every admissible insurance contract is suboptimal to a ceded loss function composed of three interconnected line segments if the insurance premium principles satisfy risk loading and convex order preserving. The form of optimal insurance can be further simplified if the premium principles satisfies an additional weak property. Finally, we derive the optimal insurance explicitly for the expected value principle and Wang's principle.