Optimal insurance contract under VaR and CVaR constraints

•We develop an optimal insurance contract under VaR and CVaR constraints.•The double deductible insurance is optimal when VaR constraint is binding.•We consider a regular insurance form for avoiding moral hazard problem.•The optimal regular form is a piecewise linear deductible insurance.•We provide intuitive comparison between our result and relevant studies.

This study endogenously develops an optimal insurance contractual form for maximizing insured expected utility under VaR and CVaR constraints. We find that CVaR constraint does not affect the contractual form, but may increase minimum insurance premium requirement. Additionally, when the VaR constraint is binding, the optimal contract is a double deductible insurance. However, if the contract is restricted to a regular form (both indemnity schedule and retained loss schedule are continuously nondecreasing) for avoiding moral hazard problem, the optimal contract is a piecewise linear deductible insurance. Finally, we provide intuitive comparison between this study result and relevant studies.