Mean-variance insurance design under heterogeneous beliefs

In this paper we study an optimal insurance problem within the mean-variance framework for the case when the insured and insurer hold heterogeneous beliefs about the loss distribution. The implicit characterization of the optimal ceded loss function is obtained first, and we then parameterize the optimal ceded loss function in an explicit way for a general setup in which the insurer's belief Q either is or is not absolutely continuous with respect to the insured's belief P. We also show that the stop-loss function is optimal for the insured when the likelihood ratio function of the two parties' beliefs is decreasing. The connection between our framework and the expected utility framework is discussed. The situation where the insured can reduce both the mean and the variance of its loss through purchasing insurance is also investigated. Some analytical and numerical examples are presented to illustrate our results.