Optimal Reinsurance Under VaR and CTE Risk Measures When Ceded Loss Function is Concave

Cai et al. (2008) explored the optimal reinsurance designs among the class of increasing convex reinsurance treaties under VaR and CTE risk measures. However, reinsurance contracts always involve a limit on the ceded loss function in practice, and thus it may not be enough to confine the analysis to the class of convex functions only. The object of this article is to present an optimal reinsurance policy under VaR and CTE optimization criteria when the ceded loss function is in a class of increasing concave functions and the reinsurance premium is determined by the expected value principle. The outcomes reveal that the optimal form and amount of reinsurance depend on the confidence level p for the risk measure and the safety loading θ for the reinsurance premium. It is shown that under the VaR optimization criterion, the quota-share reinsurance with a policy limit is optima, while the full reinsurance with a policy limit is optima under CTE optimization criterion. Some illustrative examples are provided.