Principle of equivalent utility and universal variable life insurance pricing

In this paper we study the pricing problem for a class of universal variable life (UVL) insurance products, using the idea of principle of equivalent utility. As the main features of UVL products we allow the (death) benefit to depend on certain indices or assets that are not necessarily tradable (e.g., pension plans), and we also consider the “multiple decrement” cases in which various status of the insured are allowed and the benefit varies in accordance with the status. Following the general theory of indifference pricing, we formulate the pricing problem as stochastic control problems, and derive the corresponding HJB equations for the value functions. In the case of exponential utilities, we show that the prices can be expressed explicitly in terms of the global, bounded solutions of a class of semilinear parabolic PDEs with exponential growth. In the case of general insurance models where multiple decrements and random time benefit payments are all allowed, we show that the price should be determined by the solutions to a system of HJB equations, each component corresponds to the value function of an optimization problem with the particular status of the insurer.