Optimal reinsurance of dependent risks

The talk will focus on the optimal reinsurance problem for two dependent risks, from the point of view of the ceding insurance company. We aim at maximizing the expected utility or the adjustment coefficient of the insurer wealth. The insurer buys reinsurance on each risk separately. By risk we mean a line of business, a portfolio of policies or a policy. We assume a generic known dependence structure, so that the optimal solution depends on the joint distribution. Due to dependencies, the optimal level of reinsurance for each risk involves a trade-off between the reinsurance premia of both risks. We study the shape of this trade-off and characterize the optimal treaties. We show that an optimal solution exists and provide an optimality condition. Unfortunately, explicit optimal treaties are not easy to compute from this condition. We discuss some strategies to obtain numerical approximations for the optimal treaties and discuss some aspects of the structure of the optimal strategy. Numerical results are presented assuming that the two risks are dependent by means of a copula structure and that the reinsurance treaty consists of a combination of quota-share and stop-loss. Sensitivity of the optimal reinsurance strategy is analyzed numerically to several factors, including the dependence structure, through the copula chosen, and the dependence strength, by means of the dependence parameter, corresponding to different values of the Kendall’s tau. A variety of reinsurance premium calculation principles are also considered.