Continuous time mean-variance-utility portfolio problem and its equilibrium strategy

Abstract

In this paper, we propose a new class of optimization problems, which maximize the terminal wealth and accumulated consumption utility subject to a mean variance criterion controlling the final risk of the portfolio. The multiple-objective optimization problem is firstly transformed into a single-objective one by introducing the concept of overall "happiness" of an investor defined as the aggregation of the terminal wealth under the mean-variance criterion and the expected accumulated utility, and then solved under a game theoretic framework. We have managed to maintain analytical tractability; the closed-form solutions found for a set of special utility functions enable us to discuss some interesting optimal investment strategies that have not been revealed before in literature.

Mean-variance-utility portfolio selection with time and state dependent risk aversion

“…We shall solve the optimal portfolio selection problem (3) under a game theoretic framework, which was introduced in [4,5] and developed by [13,24]. The equilibrium strategy under the continuous-time game theoretic equilibrium for the problem (3) can be defined as follows.…”

“…We would like to point out that the model (3) includes several known models as special cases. In fact, if we remove the consumption component, the model degenerates into the one studied in [8,13]; if we do not consider the time and sate dependent risk aversion function, the model becomes the one reported in [24]; if δ is set to 0, the time and sate dependent risk aversion and the consumption component are not be taken into account, then the model becomes the classical mean-variance model (see [2,16,27]).…”

“…Before we are able to present the optimal solution, some preliminaries need to be outlined. As reported in studies [13,24], we can establish an extension of the HJB equation for the characterization of the optimal value function and the corresponding optimal strategy, so that the stochastic problem can be transformed into a system of deterministic differential equations and a deterministic point-wise minimization problem. We introduce the following two lemmas.…”

“…To obtain a rational consumption policy, Yang et. al [24] proposed a new portfolio selection model which simultaneously maximize the terminal wealth and accumulated consumption utility subject to a mean variance criterion controlling the final risk of the portfolio. The analytically derived policy performs the continuous influence of investors' consumption preference on the optimal consumption strategy and represents a more economically rational investment/consumption behavior.…”

“…Unfortunately, the optimal amount of moment to invest in is not dependent of wealth under the model setting [24], which means that for a given risk aversion degree a rich or poor investor optimally invest the same amount of the money in stocks. In fact, the investor will change his investment policy according to the update of her/his wealth in the case of multi-stage investment.…”