A constraint-free approach to optimal reinsurance. Issue 1 (2nd January 2019) - Gerber, Hans U.; Shiu, Elias S.W.; Yang, Hailiang
ABSTRACT: Reinsurance is available for a reinsurance premium that is determined according to a convex premium principle H . The first insurer selects the reinsurance coverage that maximizes its expected utility. No conditions are imposed on the reinsurer's payment. The optimality condition involves the gradient of H . For several combinations of H and the first insurer's utility function, closed-form formulas for the optimal reinsurance are given. If H is a zero utility principle (for example, an exponential principle or an expectile principle), it is shown, by means of Borch's Theorem, that the optimal reinsurer's payment is a function of the total claim amount and that this function satisfies the so-called 1-Lipschitz condition. Frequently, authors impose these two conclusions as hypotheses at the outset.