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 in the literature before.

Keywords:

• Merton's problem
• mean–variance portfolio problem
• equilibrium
• time-inconsistency control
• utility

Acknowledgements

The authors would like to gratefully acknowledge the technical as well as detailed comments and suggestions from all three anonymous referees. These valuable comments and suggestions have helped us to substantially improve the quality of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Although the two wealth dynamics (2(2) $\left\{\begin{array}{ll}\mathrm{d}{X}^{\pi }\left(t\right)=\left[\right(r+\pi \left(t\right)(\mu -r)\left)X^{\pi }\right(t\left)\right]\mathrm{d}t+\pi \left(t\right)\sigma X^{\pi }\left(t\right)\mathrm{d}B\left(t\right),& t\in [0,T),\\ X\left(0\right)=x_0>0.\end{array}\right.$(2) ) and (5(5) $\left\{\begin{array}{rl}\mathrm{d}{X}^{c,\pi }\left(t\right)& =\left[\right(r+\pi \left(t\right)(\mu -r)\left)X^{c,\pi }\right(t)+l(t)-c(t\left)\right]\mathrm{d}t\\ & +\pi \left(t\right)\sigma X^{c,\pi }\left(t\right)\mathrm{d}B\left(t\right),t\in [0,T),\\ X\left(0\right)& =x_0>0.\end{array}\right.$(5) ) are different, the optimal solution to the mean–variance problem (4(4) $\begin{array}{rl}\underset{\pi (\cdot )}{min}& V_1\left(\pi \right(\cdot \left)\right)+\frac{\gamma }{2}{V}_2\left(\pi \right(\cdot \left)\right)\equiv -E\left(X\right(T\left)\right)+\frac{\gamma }{2}\mathrm{V}\mathrm{a}\mathrm{r}\left(X\right(T\left)\right),\\ \mathrm{s}.\mathrm{t}.& \left\{\begin{array}{rl}& \pi (\cdot )\in L_{\mathcal{F}}^2(0,T;R),\\ & \left(X\right(\cdot ),\pi (\cdot \left)\right)\text{satisfy Equation (2)},\end{array}\right.\end{array}$(4) ) with (2(2) $\left\{\begin{array}{ll}\mathrm{d}{X}^{\pi }\left(t\right)=\left[\right(r+\pi \left(t\right)(\mu -r)\left)X^{\pi }\right(t\left)\right]\mathrm{d}t+\pi \left(t\right)\sigma X^{\pi }\left(t\right)\mathrm{d}B\left(t\right),& t\in [0,T),\\ X\left(0\right)=x_0>0.\end{array}\right.$(2) ) and that to the mean–variance problem (4(4) $\begin{array}{rl}\underset{\pi (\cdot )}{min}& V_1\left(\pi \right(\cdot \left)\right)+\frac{\gamma }{2}{V}_2\left(\pi \right(\cdot \left)\right)\equiv -E\left(X\right(T\left)\right)+\frac{\gamma }{2}\mathrm{V}\mathrm{a}\mathrm{r}\left(X\right(T\left)\right),\\ \mathrm{s}.\mathrm{t}.& \left\{\begin{array}{rl}& \pi (\cdot )\in L_{\mathcal{F}}^2(0,T;R),\\ & \left(X\right(\cdot ),\pi (\cdot \left)\right)\text{satisfy Equation (2)},\end{array}\right.\end{array}$(4) ) with (5(5) $\left\{\begin{array}{rl}\mathrm{d}{X}^{c,\pi }\left(t\right)& =\left[\right(r+\pi \left(t\right)(\mu -r)\left)X^{c,\pi }\right(t)+l(t)-c(t\left)\right]\mathrm{d}t\\ & +\pi \left(t\right)\sigma X^{c,\pi }\left(t\right)\mathrm{d}B\left(t\right),t\in [0,T),\\ X\left(0\right)& =x_0>0.\end{array}\right.$(5) ) are the same. In particular, by setting $\rho =0$ in Proposition 3.1 of [31], one can easily guarantee the optimal strategy, and the results in [31] also show that the deterministic cash flow does not bring any possible adjustment to the investment strategy under the mean–variance criterion.

2 These stochastic integrals are indeed local martingales. However, we actually adopted the idea in [27] to make the assumption that these are martingales to build and study the solution set of all admissible strategies.

3 Note that the utility function $U(\cdot )$ is a monotonically increasing and concave function, implying that its derivative and the inverse of the derivative are both decreasing functions.

4 As pointed out in [31], this seems to be economically unreasonable for a multi-period model, and a possible way to resolve this issue is to make the risk aversion be time- and wealth-dependent. We refer interested readers to [31] for a more detailed discussion.

5 The codes for numerical experiments presented in this paper are provided in the appendix of arXiv version (arXiv:2005.06782).