Abstract
This paper studies the optimal insurance policy that maximizes the decision maker (DM)'s expected utility under distortion risk constraints. To alleviate the ex post moral hazard issues arising from the discontinuity of the indemnity functions in Huang (Geneva Risk Insur Rev 31(2):91-110, 2006) and Bernard and Tian (Geneva Risk Insur Rev 35(1):47-80, 2010) we re-visit their problems under the so called incentive compatibility condition, which requires that both the ceded and retained loss functions are non-decreasing. In addition, we generalize the value-at-risk (VaR) constraints used in the literature to the distortion-risk-measure-based constraints. We first implicitly characterize the optimal indemnity function when the risk constraints are defined in terms of the general distortion risk measure and then provide explicit solutions for the VaR and tail value-at-risk (TVaR) cases. The effect of the risk constraints on the optimal indemnity function are analyzed in great detail. Our results show that under the VaR risk constraints, the DM chooses to ignore the risk which does not contribute to its VaR value and only manages the risk that influences its VaR value. This problem is alleviated under the TVaR risk constraints.