Distributionally robust insurance under the Wasserstein distance

Abstract

This paper studies the optimal insurance contracting from the perspective of a decision maker (DM) who has an ambiguous understanding of the loss distribution. The ambiguity set of loss distributions is represented as a p-Wasserstein ball, with $p\in Z^{+}$, centered around a specific benchmark distribution. The DM selects the indemnity function that minimizes the worst-case risk within the risk-minimization framework, considering the constraints of the Wasserstein ball. Assuming that the DM is endowed with a convex distortion risk measure and that insurance pricing follows the expected-value premium principle, we derive the explicit structures of both the indemnity function and the worst-case distribution using a novel survival-function-based representation of the Wasserstein distance. We examine a specific example where the DM employs the GlueVaR and provide numerical results to demonstrate the sensitivity of the worst-case distribution concerning the model parameters.