Optimal investment and consumption under a continuous-time cointegration model with exponential utility

Abstract

In this paper, we study the effects of cointegration on optimal investment and consumption strategies for an investor with exponential utility. A Hamilton-Jacobi-Bellman (HJB) equation is derived first and then solved analytically. Both the optimal investment and consumption strategies are expressed in closed form. A verification theorem is also established to demonstrate that the solution of the HJB equation is indeed the solution of the original optimization problem under an integrability condition. In addition, a simple and sufficient condition is proposed to ensure that the integrability condition is satisfied. Financially, the optimal investment and consumption strategies are decomposed into two parts: the myopic part and the hedging demand caused by cointegration. Discussions on the hedging demand are carried out first, based on analytical formulae. Then numerical results show that ignoring the information about cointegration results in a utility loss.

Keywords:

• Cointegration
• Hamilton-Jacobi-Bellman (HJB) equation
• Optimal investment and consumption problem
• Closed-form analytical solutions
• Verification theorem

Acknowledgements

The authors would also like to thank two anonymous referees for their valuable comments and suggestions, which have led to a substantial improvement of the readability of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Guiyuan Ma http://orcid.org/0000-0002-0257-9680

Song-Ping Zhu http://orcid.org/0000-0002-2863-0640

Notes

1 Such a constant will be given later.

2 It should be pointed out that $V_M(t,y,\mathit{\beta })$ is always negative.

3 In tables 1–3, $\begin{array}{rl}{ϵ}_1^{\ast }& =\frac{({\beta }_1+{\beta }_2)({\sigma }_2^2+{\sigma }_2^2-4r)k_2\left(t\right)}{8\left[f_1{g}_1\right(t)+f_2{g}_2(t\left)\right]}\mathrm{and}\\ {ϵ}_2^{\ast }& =-\frac{2k_3\left(t\right)}{k_2\left(t\right)({\sigma }_1^2+{\sigma }_2^2-4r)}(\frac{{\beta }_1}{{\sigma }_1^2}+\frac{{\beta }_2}{{\sigma }_2^2}).\end{array}$

Additional information

Funding

The research conducted and reported in this paper is part of an ARC (Australian Research Council)-funded three-year project, for which the authors gratefully acknowledge the financial support from the ARC under the ARC(DP) funding scheme (DP140102076, DP170101227).