Optimal Reciprocal Reinsurance Strategy Under The VaR Risk Measure

A reinsurance contract involves two parties, the insurer and the reinsurer. The two parties have conflicting interests. However, most existing optimal reinsurance treaties only consider the interest of the insurer. In fact, an optimal reinsurance contract for an insurer may not be optimal for a reinsurer and it might be unac-ceptable for a reinsurer. An interesting question about optimal reinsurance is to design a reinsurance so that it considers the interests of both the insurer and the reinsurer and it is fair, in some sense, to both of the parties.In this paper, we study the optimal reinsurance strategies from the perspective both the insurer and the reinsurer. We define the loss function with the VaR risk measure such that the sum of squares of the insurer's loss of VaR and the reinsurer's loss of VaR is minimal. Different forms of optimal reinsurance strategy can be obtained over different ceded loss functions under the criterion of minimizing the loss function, assuming that the premium calculation principle of reinsurance is monotone, (1)the change-loss reinsurance is optimal among the class of increasing convex ceded loss functions; (2)when ceded loss functions loss functions are Lipschitz continuous, the layer reinsurance is shown to be optimal; (3)and finally when the ceded loss functions are in the class of increasing concave functions, the quota-share reinsurance with a policy limit is always optimal. Finally, we derives the optimal parameters and the conditions under expectation premium principle and Dutch premium principle.