Time-consistent mean-variance portfolio selection in discrete and continuous time

Using mean–variance (MV) criterion, this paper investigates a continuous-time defined contribution (DC) pension fund investment problem. The framework is constructed under a Markovian regime-switching market consisting of one bank account and multiple risky assets. The prices of the risky assets are governed by geometric Brownian motion while the accumulative contribution evolves according to a Brownian motion with drift and their correlation is considered. The market state is modeled by a Markovian chain and the random regime-switching is assumed to be independent of the underlying Brownian motions. The incorporation of the stochastic accumulative contribution and the correlations between the contribution and the prices of risky assets makes our problem harder to tackle. Luckily, based on appropriate Riccati-type equations and using the techniques of Lagrange multiplier and stochastic linear quadratic control, we derive the explicit expressions of the optimal strategy and efficient frontier. Further, two special cases with no contribution and no regime-switching, respectively, are discussed and the corresponding results are consistent with those results of Zhou & Yin [(2003) Markowitz’s mean-variance portfolio selection with regime switching: A continuous-time model, SIAM Journal on Control and Optimization 42 (4), 1466–1482] and Zhou & Li [(2000) Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization 42 (1), 19–33]. Finally, some numerical analyses based on real data from the American market are provided to illustrate the property of the optimal strategy and the effects of model parameters on the efficient frontier, which sheds light on our theoretical results.

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Markowitz's Mean-Variance Portfolio Selection With Regime Switching: From Discrete-Time Models to Their Continuous-Time Limits

IEEE Transactions on Automatic Control ◽

10.1109/tac.2004.824479 ◽

2004 ◽

Vol 49 (3) ◽

pp. 349-360 ◽

Cited By ~ 137

Author(s):

G. Yin ◽

X.Y. Zhou

Keyword(s):

Portfolio Selection ◽

Discrete Time ◽

Continuous Time ◽

Regime Switching ◽

Time Limits ◽

Mean Variance Portfolio ◽

Mean Variance ◽

Discrete Time Models

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Continuous-time mean–variance portfolio selection with value-at-risk and no-shorting constraints

International Journal of Control ◽

10.1080/00207179.2011.638326 ◽

2012 ◽

Vol 85 (1) ◽

pp. 50-57 ◽

Cited By ~ 4

Author(s):

Wei Yan

Keyword(s):

At Risk ◽

Portfolio Selection ◽

Continuous Time ◽

Value At Risk ◽

Mean Variance Portfolio ◽

Mean Variance

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Continuous-Time Mean-Variance Portfolio Selection under the CEV Process

Abstract and Applied Analysis ◽

10.1155/2014/363046 ◽

2014 ◽

Vol 2014 ◽

pp. 1-14

Author(s):

Hui-qiang Ma

Keyword(s):

Portfolio Selection ◽

Continuous Time ◽

Stock Price ◽

Efficient Frontier ◽

Optimal Portfolio ◽

Portfolio Strategy ◽

Cev Process ◽

The Mean ◽

Mean Variance Portfolio ◽

Mean Variance

We consider a continuous-time mean-variance portfolio selection model when stock price follows the constant elasticity of variance (CEV) process. The aim of this paper is to derive an optimal portfolio strategy and the efficient frontier. The mean-variance portfolio selection problem is formulated as a linearly constrained convex program problem. By employing the Lagrange multiplier method and stochastic optimal control theory, we obtain the optimal portfolio strategy and mean-variance efficient frontier analytically. The results show that the mean-variance efficient frontier is still a parabola in the mean-variance plane, and the optimal strategies depend not only on the total wealth but also on the stock price. Moreover, some numerical examples are given to analyze the sensitivity of the efficient frontier with respect to the elasticity parameter and to illustrate the results presented in this paper. The numerical results show that the price of risk decreases as the elasticity coefficient increases.

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