Optimal Reinsurance in a Market of Multiple Reinsurers Under Law-Invariant Convex Risk Measures

Abstract

It is natural to connect reinsurance problems with risk measures since a reinsurance contract is an efficient risk management tool for an insurer and the reinsurance premium can also be viewed as a measure of a reinsurer's risk. In this paper, we assume that the insurer uses a law-invariant convex risk measure, while reinsurers use a Wang's premium principle to determine their premiums. We study an optimal reinsurance policy design from an insurer's perspective in a market of multiple reinsurers. Both the insurer's risk measure and the reinsurer's premium principle represent broad families of risk measures with considerable generality. We provide a general formula for the optimal solution which recovers existing results if particular law-invariant convex measures, such as the AVaR, and particular premium principles are assigned.