Optimal Private Health Insurance Contract Towards the Joint Interests of a Policyholder and an Insurer

This paper investigates the optimal private health insurance contract design problem considering the joint interests of a policyholder and an insurer. They jointly determine the premium of a private health insurance. To better reflect the reality, the illness expenditure is governed by an extended compound Poisson risk model depending on health status. Both the policyholder and the insurer aim at finding optimal time-consistent private health insurance strategies so as to maximize their respective expected terminal wealth and minimize the variance of their terminal wealth. By using the stochastic optimal control and leader-follower game techniques, we derive analytically time-consistent private health insurance strategies, form the optimal private health insurance contract, and analyze their insurance meanings. Finally, we examine the effects of model parameters on the optimal private health insurance contract and the influence of private health insurance on the policyholder through numerical experiments, which reveal some interesting phenomena and provide useful guidance for health insurance in practice