Optimal Insurance with Background Risk and Incentive Compatibility

Abstract

This paper reexamines the design of an optimal insurance contract with background risk from the perspective of an insured by imposing an incentive compatible constraint. The incentive compatibility means that both parties in an insurance contract are obligated to pay more for a larger realization of loss. As standard in the literature, it is assumed that the insurer is risk-neutral, that is, the insurance premium is calculated based only upon the expected indemnity. When the insured has a general mean-variance preference, we derive explicitly an optimal insurance form, which heavily relies on conditional expectation function (CEF) of background risk with respect to the insurable risk and is often significantly different from one that is obtained in the absence of incentive compatibility. It suggests that both the incentive compatible constraint and the dependence between background risk and the insurable risk play very important roles in the insured's risk transfer decision. Finally, optimal insurance is discussed in detail for some special CEFs the derivative of which has less than one change point.