Regret-based optimal insurance design

In this paper, we investigate an optimal insurance design problem in which the insured's risk preference is of the regret-theoretical expected utility type and admissible insurance contracts are set to satisfy both the principle of indemnity and the incentive-compatible condition. When the insurance premium is calculated by the expected value principle, we show that the optimal solution for a regret-averse insured can be in the form of no insurance or partial insurance above a deductible. We obtain a necessary and sufficient condition for the optimality of no insurance and provide a numerical algorithm to derive the optimal partial insurance above a deductible. Notably, the deductible disappears if and only if the insurance premium is actuarially fair. Finally, we carry out a comparative statics analysis to examine the effect of the insured's regret coefficient on the insured's final wealth and insurance demand. When the insurance pricing is actuarially fair, we find that the increase of the regret coefficient incentivizes the insured to purchase less insurance coverage and leads to an increase in risk of the insured's final wealth. However, our numerical analysis indicates that this finding is not always true for unfair insurance pricing.