Optimal reinsurance design under convex premium principles and distortion risk measures

This paper studies an optimal reinsurance problem from an insurer’s perspective under convex premium principles. The insurer’s preference is assumed to be dictated by the distortion risk measure. When doing business with only one reinsurer, the general form of the optimal indemnity function for the insurer is derived by jointly applying the calculation of variation and marginal indemnification function approaches. We demonstrate that the optimal indemnity function for the insurer takes the form of a limited stop-loss when the insurer adopts a Range Value-at-Risk preference. In contrast, when the insurer applies strictly convex distortion risk measures, we show that, under mild conditions, the optimal indemnity function may include a co-insurance component. We also extend the results to the case of multiple reinsurers through a representative reinsurer lens, and present a sufficient condition under which the representative reinsurer’s premium principle is of the same mathematical form of the convex premium principle studied in this paper. We also show the connection between the optimal reinsurance problems under the certainty-equivalent premium principle and under the convex premium principle. Some interesting results are presented for the problem between one insurer and multiple reinsurers when each reinsurer applies an ith-moment premium principle.