Optimal Reinsurance Design under Ambiguity and Value-at-Risk Preference with Wasserstein and Lk Distance Metrics

This article delves into the optimal reinsurance problem from the perspective of a decision maker (DM) who exhibits a preference for value-at-risk (VaR) and experiences ambiguity regarding the underlying loss distribution. The uncertainty set considered in this study encompasses distributions that closely envelop a reference distribution, with the proximity quantified using the Wasserstein metric. Through a rigorous analysis, we present a thorough characterization of both the optimal indemnity function and the worst-case VaR for our proposed problem. Furthermore, we extend our examination to a pertinent problem wherein the Lk distance metric substitutes for the Wasserstein metric. Numerical examples are provided to demonstrate the implications of our main findings, and a comparative analysis is conducted with relevant literature to further enhance our understanding of the outcomes. Explicit comparative analysis demonstrates that Wasserstein ambiguity sets minimize worst-case VaR through aggregate tail mass shifting penalties, while Lk distance ambiguity sets prioritize local distributional shifts near the VaR quantile, generating higher worst-case VaR estimates particularly sensitive to extreme-loss scenarios. The insights and analytical tools elucidated in this article hold consequential implications for insurers and reinsurers in proficiently navigating risk management within an uncertain environment.