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 epsilon 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 pi _0^* postulated by Delong (Math Methods Oper Res 89:73–113, 2019) performs better than any strategy pi _0 when we compare the asymptotic expansions of the objective functions up to order {mathcal {O}}(1) as epsilon rightarrow 0, and the first-order investment strategy pi _0^*+pi _1^*epsilon postulated by Delong (Math Methods Oper Res 89:73–113, 2019) is the equilibrium strategy in the class of strategies pi ^*_0+pi _1epsilon when we compare the asymptotic expansions of the objective functions up to order {mathcal {O}}(epsilon ^2) as epsilon rightarrow 0, where epsilon denotes the parameter controlling the degree of the insurer’s risk aversion depending on wealth.