Robust insurance design with distortion risk measures

This paper studies the optimal insurance problem within the risk minimization framework and from a policyholder's perspective. We assume that the decision maker (DM) is uncertain about the underlying distribution of her loss and considers all the distributions that are close to a given (benchmark) distribution, where the "closeness" is measured by the L2 or L1 distance. Under the expected-value premium principle, the DM picks the indemnity function that minimizes her risk exposure under the worst-case loss distribution. By assuming that the DM's preferences are given by a convex distortion risk measure, we disentangle the structures of the optimal indemnity function and worst-case loss distribution in an analytical way, and provide the explicit forms for both of them under specific distortion risk measures. We also compare the results under the L2 distance and the first-order Wasserstein (L1) distance. Some numerical examples are presented at the end to illustrate the implications of our main results.