Optimal reinsurance programs: An optimal combination of several reinsurance protections on a heterogeneous insurance portfolio

Abstract

In most practical cases, the reinsurance protection of an insurance portfolio is not limited to one reinsurance type such as quota share, surplus, excess of loss or stop loss, but is organised through a combination of several methods of protection, a so-called reinsurance program. In this paper, we will analyse optimal reinsurance programs for a given portfolio based on the “mean–variance” optimisation criterion. Special attention is given to a description of a “surplus reinsurance” in combination with an “excess of loss per risk protection” for a heterogeneous insurance portfolio. We derive the equations one needs to solve for finding the optimal solution for the following combinations: excess of loss after surplus, excess of loss after quota share, stop loss after quota share, quota share after stop loss, quota share after excess of loss, quota share before surplus and quota share after surplus. It turns out that the application order of the reinsurance protections has its importance.