From the authors’ introduction: This paper examines the optimal choice of a reinsurance contract. Applying the stop-loss ordering of risks to the retained risks after reinsurance leads to short and elegant derivations.
Several types of reinsurance contracts and degrees of coverage are possible. The optimal form and amount, from the insurer’s point of view, depends on his optimization criterion. In case of maximizing the expected utility of the insurer using a concave utility function a classical result says that if the reinsurance premium is proportional to the expected value of the risk, then for every fixed reinsurance premium the optimal contract must be of stop-loss form.
This important result is extended to several other optimization criteria: a partial preference ordering of risks after reinsurance is shown to imply the same preferences for a number of criteria a risk-averse decision maker might use. So if there is a retained risk that is minimal in stop-loss order, then it will be optimal from the insurer’s point of view when optimizing any of these criteria. The results found for a single (total) risk can be extended to compound models for portfolios of risks, as well as to the case where the set of reinsurance contracts to choose from is restricted.