Two Optimal Reinsurance Issues

As the reinsurance proposed, the study of optimal reinsurance policies has received remarkable attention and the system of reinsurance has been basically perfected. In real life, reinsurance is proposed when the expect of loss beyond the underwriting capacity. The reinsurance is the behavior that insurers transfer the part or whole insurance services to the other insurers. It is significant that how to share risks and premiums to maximize their profits. And any changes of them can lead to the changes of the terminal profits. In this paper, two optimal reinsurance problems are considered to study under two premium principles. One is to maximize the value function. The other is to minimize the total risk exposure of reinsurer. The tools are used contain cooperation game, stochastic optimal control, VaR and CVaR risk measures and so on. According to the problems we studied, this paper is divided into two chapters:(1) Stochastic Pareto-optimal reinsurance policies.In this chapter, we model reinsurance as a stochastic cooperation game in a continuous-time framework. We generalize the classical Cramer-Lundberg model in Zeng and Luo. We proposed a new model with new stochastic process{B(t)} Employing dynamic programming technologies and stochastic control theory, we study Pareto-optimal reinsurance and drive the corresponding Hamilton-Jacobi-Bellman equation. We study the HJB equations under standard deviation principle and modified variance principle. After analyzing the HJB equation, we find that the Pareto-optimal reinsurance policies is proportional functions based on two different premium share principles. At last, we give the explicit solutions in example to illustrate our results.(2) Optimal reinsurance with Wang’s premium principles.In this chapter, we study two classes of optimal reinsurance models in terms of minimize the total risk exposure of reinsurance under the VaR and CVaR risk measures. We assume that the premium principles satisfy three basic axioms:risk loading, distribution invariance and stop-loss ordering preserving. The total risk exposure of the reinsurer is given by the difference between the transfer loss from the insurer and the reinsurance premium paid by the insurer,i.e. Tf(X)=f(X)-π(f(X)).We generalize the results in Chi and Tan. We suppose that both the insurer and reinsurer are obligated to pay more for larger loss. If the optimal reinsurance policy is relatively stable with respect to the changes of either the risk measures or the premium principles or both, the optimal reinsurance is said to be robust. Under the above assumption, the layer reinsur-ance is robust since it is always optimal under the two risk measures and Wang’s premium principle. We choose Wang’s premium principle as it satisfies the above three basic axioms and follows the property of increasing relative risk loading. We give an example to illustrate our result at last.