Worst-case reinsurance strategy with likelihood ratio uncertainty

Abstract

In this paper, we explore a non-cooperative optimal reinsurance problem incorporating likelihood ratio uncertainty, aiming to minimize the worst-case risk of the total retained loss for the insurer. We establish a general relation between the optimal reinsurance strategy under the reference probability measure and the optimal reinsurance strategy in the worst-case scenario. This relation can be generalized to insurance design problems quantified by tail risk measures. We characterize distortion risk measures such that the insurer's optimal strategy remains the same in the worst-case scenario. As an application of this relation, optimal policies for the worst-case scenario quantified by an expectile risk measure are determined. Additionally, we propose and explore a cooperative problem, which can be viewed as a general risk sharing problem between two agents in a comonotonic market. We determine the risk measure value and the optimal reinsurance strategy in the worst-case scenario for the insurer, and  the  comparison between the non-cooperative and the cooperative models is presented.

Keywords: Likelihood ratio uncertainty set, distortion risk measure, tail risk measures, expectile risk measure, optimal reinsurance design, robust reinsurance

JEL Classification: J60, G22