Optimal Reinsurance in Relation to Ordering of Risks

Many criteria for choosing optimal reinsurance treaties have the common property that a treaty is considered preferable to another if it is lower in stop-loss order, which means that it generates uniformly lower stop-loss premiums. Such criteria include maximizing utility of wealth, minimizing ruin probability and many others. If the set of feasible reinsurance treaties contains a minimum element in stop-loss order, this element is the solution to the optimization problem. The retained loss under a stop-loss treaty is such a minimum in the set of all treaties with equal mean, implying that stop-loss reinsurance is optimal with respect to many criteria. This extends the result of Borch (1960) and Kahn (1961) on maximizing the utility of wealth.