Optimal reinsurance with concave ceded loss functions under VaR and CTE risk measures

Most of the studies on optimal reinsurance are from the viewpoint of the insurer and the optimal ceded functions always turn out to be convex. However reinsurance contracts always involve a limit on the ceded loss function in practice, thus it may not be enough to confine the analysis to the class of convex functions only. In this paper, we study the problem of optimal reinsurance under VaR and CTE optimization criteria when the ceded loss functions are in the class of increasing concave functions. By using a simple geometric approach, we prove that under the VaR optimization criterion, the quota-share reinsurance with a policy limit is always optimal, while the full reinsurance with a policy limit is optimal under the CTE optimization criterion. Some illustrative examples are presented.