Pareto-optimal reinsurance for both the insurer and the reinsurer under the risk-adjusted value and general premium principles

In this paper, we design the Pareto-optimal reinsurance contract for both the insurer and the reinsurer by minimizing the convex combination of the risk-adjusted value of the insurer's liability and the reinsurer's liability, where capital at risk is calculated by the value at risk (VaR) or conditional value at risk (CVaR). In order to prevent the moral hazard, we assume that both ceded and retained loss functions are increasing functions. We analyze the optimal solutions for a wide class of reinsurance premium principles. When the reinsurance premium principles satisfy three axioms: law invariance, risk loading and preserving convex order, we find that layer reinsurance is always optimal over the assumed risk measures. Then we impose an additional weak constraint on the premium principle to simplify the form of layer reinsurance which is optimal. Finally, we illustrate the applicability of our results by deriving the parameters of the optimal layer reinsurance explicitly under the expected value principle and Wang's premium principle.