Optimal investment for insurance company with exponential utility and wealth-dependent risk aversion coefficient

Abstract

We investigate an exponential utility maximization problem for an insurer who faces a stream of non-hedgeable claims. The insurer’s risk aversion coefficient changes in time and depends on the current insurer’s net asset value (the excess of assets over liabilities). We use the notion of an equilibrium strategy and derive the HJB equation for our time-inconsistent optimization problem. We assume that the insurer’s risk aversion coefficient consists of a constant risk aversion and a small amount of a wealth-dependent risk aversion. Using perturbation theory, the equilibrium value function, which solves the HJB equation, is expanded on the parameter controlling the degree of risk aversion depending on wealth. We find the first-order approximations to the equilibrium value function and the equilibrium investment strategy. Some new results for exponential utility maximization problem with constant risk aversion are derived in order to approximate the solution to our exponential utility maximization problem with wealth-dependent risk aversion.