Mean–Variance Portfolio Optimization with State‐Dependent Risk Aversion

Abstract

The objective of this paper is to study the mean–variance portfolio optimization in continuous time. Since this problem is time inconsistent we attack it by placing the problem within a game theoretic framework and look for subgame perfect Nash equilibrium strategies. This particular problem has already been studied in Basak and Chabakauri where the authors assumed a constant risk aversion parameter. This assumption leads to an equilibrium control where the dollar amount invested in the risky asset is independent of current wealth, and we argue that this result is unrealistic from an economic point of view. In order to have a more realistic model we instead study the case when the risk aversion depends dynamically on current wealth. This is a substantially more complicated problem than the one with constant risk aversion but, using the general theory of time‐inconsistent control developed in Björk and Murgoci, we provide a fairly detailed analysis on the general case. In particular, when the risk aversion is inversely proportional to wealth, we provide an analytical solution where the equilibrium dollar amount invested in the risky asset is proportional to current wealth. The equilibrium for this model thus appears more reasonable than the one for the model with constant risk aversion.

Keywords: mean–variance, time inconsistency, time‐inconsistent control, dynamic programming, stochastic control, Hamilton–Jacobi–Bellman equation