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.

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

“…From mathematical point of view, the results complete the results from Delong [8] and give a more rigorous justification for the strategy derived in Delong [8]. From economic point of view, the proof that the candidate strategy from Delong [8] is optimal (in some sense) is crucial for applications and conclusions derived from the model.…”

“…Delong [8] proves a lot of results which are essential to characterize the firstorder approximation to the equilibrium investment strategy and justify the choice of his investment strategy as the first-order approximation. However, the order of the error of approximating the true equilibrium investment strategy with the candidate first-order approximate solution has not been proved.…”

“…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 changes in time and depends on the current insurer's net asset value (the excess of assets over liabilities).…”

“…In this paper we formally study an asymptotic optimality of the investment strategy postulated by Delong [8]. More precisely, we show that the zeroth-order investment strategy π * 0 postulated by Delong [8] 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 [8] 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. From mathematical point of view, the results complete the results from Delong [8] and give a more rigorous justification for the strategy derived in Delong [8].…”

“…To the best of our knowledge, there are only two papers by Dong and Sircar [9] and Delong [8] which study exponential utility maximization problems for investors with wealth-dependent risk aversion coefficients. Moreover, the first-order approximation to the equilibrium investment strategy postulated by Delong [8] is a new investment strategy and its properties are worth investigating.…”