Optimal portfolios with regime switching and value-at-risk constraint

• Published in: Automatica (Oxford) Vol. 46; no. 6; pp. 979 - 989
• Main Authors: Yiu, Ka-Fai Cedric, Liu, Jingzhen, Siu, Tak Kuen, Ching, Wai-Ki
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 01.06.2010; Elsevier
• Subjects: Applied sciences; Constraints; Decision theory. Utility theory; Dynamic programming; Dynamical systems; Dynamics; Exact sciences and technology; Markets; Mathematical analysis; Mathematical models; Mathematical programming; Maximum value-at-risk constraints; Operational research and scientific management; Operational research. Management science; Optimal portfolio selection; Optimization; Portfolio theory; Regime-switching; Regime-switching HJB equations; Risk theory. Actuarial science; Switching; Utility maximization; Optimal portfolio selection; Utility maximization; Regime-switching HJB equations; Dynamic programming; Maximum value-at-risk constraints; Regime-switching; Markov process; Interest rate; Volatility; Financial management; Continuous time; Modeling; Optimization; Markov chain; Finite horizon; Economy; Portfolio selection; Stock exchange; Switching time; Quantile; Consumption; Sensitivity analysis; Bellman equation; Banking; Risk analysis; Brownian motion; Selection problem; Hamilton Jacobi equation; Extreme value; regime switching model; Risk management; Portfolio management
• ISSN: 0005-1098; 1873-2836
• DOI: 10.1016/j.automatica.2010.02.027

We consider the optimal portfolio selection problem subject to a maximum value-at-Risk (MVaR) constraint when the price dynamics of the risky asset are governed by a Markov-modulated geometric Brownian motion (GBM). Here, the market parameters including the market interest rate of a bank account, the appreciation rate and the volatility of the risky asset switch over time according to a continuous-time Markov chain, whose states are interpreted as the states of an economy. The MVaR is defined as the maximum value of the VaRs of the portfolio in a short time duration over different states of the chain. We formulate the problem as a constrained utility maximization problem over a finite time horizon. By utilizing the dynamic programming principle, we shall first derive a regime-switching Hamilton–Jacobi–Bellman (HJB) equation and then a system of coupled HJB equations. We shall employ an efficient numerical method to solve the system of coupled HJB equations for the optimal constrained portfolio. We shall provide numerical results for the sensitivity analysis of the optimal portfolio, the optimal consumption and the VaR level with respect to model parameters. These results are also used to investigating the effect of the switching regimes.