A constraint-free approach to optimal reinsurance

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.

KEYWORDS:

• Optimal reinsurance
• expected utility
• convex premium principle
• Borch's theorem
• Pareto-optimal risk exchange
• constraint-free approach

Acknowledgments

The authors are grateful to Hansjoerg Albrecher, Vali Asimit, Ka Chun Cheung, Ambrose Lo, Phillip Yam and two anonymous referees for their valuable comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

We acknowledge with thanks the support from Principal Financial Group, Research Grants Council of the Hong Kong Special Administrative Region (project No. HKU 17330816), and Society of Actuaries' Centers of Actuarial Excellence Research Grants.