Mean-variance insurance design with counterparty risk and incentive compatibility

Abstract

This paper studies the optimal insurance design from the perspective of an insured when there is possibility for the insurer to default on its promised indemnity. Default of the insurer leads to limited liability, and the promised indemnity is only partially recovered in case of a default. To alleviate the potential ex post moral hazard, an incentive compatibility condition is added to restrict the permissible indemnity function. Under the actuarial premium principle and mean-variance preferences of the insured, we derive the explicit structure of the optimal indemnity function through the marginal indemnity function formulation of the problem. It is shown that the optimal indemnity function depends on the first and second order conditional expectations of the random recovery rate. The methodology and results in this paper complement the literature regarding the optimal insurance subject to the default risk and provide new insights on problems of similar types. Moreover, we also apply the techniques in this paper to the case of additive background risk, which yields an alternative proof of the main result of Chi and Tan (2021).