Some Recent Developments on Pareto-optimal Reinsurance

This thesis focuses on developing Pareto-optimal reinsurance policy which considers the interests of both the insurer and the reinsurer. The optimal insurance/reinsurance design has been extensively studied in actuarial science literature, while in early years most studies were concentrated on optimizing the insurer’s interests. However, as early as 1960s, Borch argued that “an agreement which is quite attractive to one party may not be acceptable to its counterparty” and he pioneered the study on “fair” risk sharing between the insurer and the reinsurer. Quite recently, the question of how to strike a balance in risk sharing between an insurer and a reinsurer has drawn considerable attention. This thesis contributes to the existing study in terms of the following aspects: first, we derive the set of Pareto-optimal reinsurance policies within risk mimization framework; second, we obtain the set of Pareto-optimal reinsurance policies within expected utility maximization framework. In addition, we uniquely identify the policy according to classical bargaining models; third, we blend risk minimization criterion and expected utility maximization criterion and study the so called Pareto-optimal reinsurance policy with maximal synergy. The thesis is structured as follows. Chapter 1 introduces the problem and reviews the most recent advances on related topics. Chapter 2 applies a geometric approach to derive the Pareto-optimal reinsurance policy under Value-at-Risk minimization criterion. The geometric approach visualize the process of seeking the solution which greatly simplifies the mathematical proof. As a natural extention, Chapter 3 studies the design of Pareto-optimal reinsurance policy by assuming that distortion risk measures are employed to measure the risks of the insurer and the reinsurer. The optimal reinsurance policy is derived through three methods: Lagrange dual method, generalized Neyman-Pearson lemma and dynamic control approach. Chapter 4 studies the problem through maximizing the weighted expected utility and applies the results from classical bargaining models to identify the “best” policy on the Pareto efficient frontier. Chapter 5 revisits the problem by considering a mixture of risk minimization and expected utility maximization criteria. Chapter 6 gives potential directions for future research