Optimal reinsurance strategy with mean-variance premium principle and relative performance concern

This paper investigates the optimal reinsurance strategies for n insurers who compete with each other within the non-zero-sum game framework, as well as the optimal reinsurance premium loadings under the Stackelberg framework. The reinsurance premium is determined in accordance with the mean-variance principle. The insurers’ objectives are to maximize their utility of relative wealth over a finite decision horizon. Firstly, utilizing the dynamic programming technique, we derive a system of coupled Hamilton–Jacobi–Bellman (HJB) equations and characterize the equilibrium reinsurance strategies. We also obtain explicit solutions in the special case where the insurers possess exponential utility functions and present numerical examples to illustrate our theoretical findings. Secondly, leveraging the outcomes from the first section, we derive the optimal premium loadings for the reinsurer. We formulate the HJB equation and, for the special case of exponential utility, we numerically and explicitly obtain optimal decisions. Furthermore, we provide numerical examples to illustrate the impact of model parameters on the optimal reinsurance premium loadings.