Asymptotic Optimality of a First-Order Approximate Strategy for an Exponential Utility Maximization Problem with a Small Coefficient of Wealth-Dependent Risk Aversion

Abstract

In Delong [8] we investigate an exponential utility maximization problem for an insurer who faces a stream of non-hedgeable claims. We assume that the insurer's risk aversion coefficient consists of a constant risk aversion and a small amount of wealth-dependent risk aversion. We apply perturbation theory and expand the equilibrium value function of the optimization problem on the parameter controlling the degree of the insurer's risk aversion depending on wealth. We derive a candidate for the first-order approximation to the equilibrium investment strategy. In this paper we formally show that the zeroth-order investment strategy π * 0 postulated by Delong (Math Methods Oper Res 89:73-113, 2019) performs better than any strategy π 0 when we compare the asymptotic expansions of the objective functions up to order O(1) as → 0, and the first-order investment strategy π * 0 + π * 1 postulated by Delong (Math Methods Oper Res 89:73-113, 2019) is the equilibrium strategy in the class of strategies π * 0 + π 1 when we compare the asymptotic expansions of the objective functions up to order O(2) as → 0, where denotes the parameter controlling the degree of the insurer's risk aversion depending on wealth.

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

“…4 and the calculations in this section. In Delong (2018b) we study an asymptotic optimality of our investment strategy and we formally show that (6.15) performs better than any strategy in the class π 0 (t) + π 1 (t) up to the second order O( 2 ) in the asymptotic expansion of the value function as → 0. We refer the reader to Delong (2018b).…”