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

Abstract

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.

Introduction

The optimal portfolio allocation problem is of great importance in finance from both theoretical and practical perspectives. The pioneering work of Markowitz (1952) first provided a mathematically elegant way to formulate the optimal portfolio allocation problem and developed the celebrated mean-variance approach for optimal portfolio allocation. He considers a single-period model and adopts variance (or standard deviation) as a measure of risk from the portfolio. The novelty of his mean-variance approach is that it reduces the optimal portfolio allocation problem to the one in which only the mean and the variance of the rate of return are involved under the normality assumption for the rates of return of the risky assets. This greatly simplifies the problem of optimal portfolio allocation and makes a great leap forward to the development of the field. The mean-variance approach by Markowitz has also laid down solid theoretical foundation to the optimal portfolio allocation problem and has opened up an important field, namely, the modern portfolio theory, which, together with risk management and asset pricing, are coined as the three pillars in modern finance. It also provides the theoretical background and motivation for the development of Capital Asset Pricing Model (CAPM) by the seminal work of Sharpe (1964), which laid down the theory of equilibrium asset pricing.

Merton, 1969, Merton, 1971 pioneered the development of the optimal portfolio allocation problem in a continuous-time framework, which provides a more realistic setting to deal with the problem. He explores the state of art of the stochastic optimal control techniques to provide an elegant solution to the optimal portfolio allocation problem. His work has opened up an important field in modern finance, namely, the continuous-time finance. Under the assumption that the returns from the risky assets are stationary (i.e. the coefficients of the dynamics of the returns are constant) and some specific forms of the utility function, Merton derives closed-form solutions to the optimal portfolio allocation in a continuous-time setting. In reality, the returns from the risky assets might not be stationary. So, it would be of practical relevance and importance to consider asset pricing models with non-constant coefficients, which can incorporate the feature of non-stationary returns. Boyle and Yang (1997) considered the optimal asset allocation problem in the presence of nonstationary asset returns and transaction costs. They considered the Duffie and Kan (1996) multi-factor stochastic interest model and adopt a viscosity solution approach to deal with the problem.

Recently, regime-switching, or Markov-modulated, models have received much attention among both researchers and market practitioners. Hamilton (1989) pioneered the econometric applications of regime-switching models by considering a discrete-time Markov-switching autoregressive time series model. Since then, regime-switching models, both discrete-time and continuous-time, have found a wide range of applications in economics and finance. Some papers on the use of regime-switching models in finance include Elliott and van der Hoek (1997) for asset allocation, Pliska (1997) and Elliott, Hunter, and Jamieson (2001), Elliott and Kopp (2004) for short rate models, Elliott and Hinz (2002) for portfolio analysis and chart analysis, Elliott, Chan, and Siu (2005) and Guo (2001) for option pricing under incomplete markets, Buffington and Elliott, 2002a, Buffington and Elliott, 2002b for pricing European and American options, Elliott, Malcolm, and Tsoi (2003) for volatility estimation, Elliott, Siu, and Chan (2006) for valuing options under Markov-switching GARCH models and Elliott, Siu, and Chan (2007) for pricing and hedging variance and volatility swaps, and others. Regime-switching models provide a natural and convenient way to describe the impact of the structural changes in (macro)-economic conditions and business cycles on the price dynamics. They provided a pertinent way to describe the non-stationary feature of returns of risky assets. More recently, Yin and Zhou (2003) and Zhou and Yin (2004) established a mean-variance portfolio selection problem under Markovian regime-switching models in a continuous-time economy. They introduced the stochastic linear-quadratic control to deal with the problem and established closed-form solutions to mean-variance efficient portfolios and efficient frontiers.

Maximizing profits by choosing the optimal portfolio allocation rule is an important issue. Another equally important issue is to control the risk we are taking. This raises the issues of risk measurement and management. Some recent financial crises, such as the Asian financial crisis, the collapse of Long-Term Capital Management (LTCM), and the turmoil at Barings and Orange Country, point out the importance of appropriate practice of risk measurement and management. Various methods and techniques for measuring, managing and controlling risk have been proposed in the literature that cater for the practical needs of regulators, central bankers and market traders. Value at Risk (VaR) has emerged as an important and popular tool for risk measurement. It has widely been adopted in the finance industries. VaR describes the extreme loss from a portfolio at a certain (small) probability level over a fixed time horizon, say 1 day or 7 days. Technically, it can be defined as a quantile of the profit/loss distribution of a portfolio. For example, if the VaR of the portfolio at a 1% probability level over the next trading day is £2 millions, it is expected that the actual loss from the portfolio over the next trading day is at least £2 millions with 1% probability. Duffie and Pan (1997) and Jorion (2001) provided excellent introduction and survey to VaR. In the original work of Merton, the expected utility of wealth or consumption was maximized over a fixed time horizon without imposing any risk limits or constraints. However, maximizing profits is not the only objective that needs to be taken into account in the optimal portfolio allocation problem. It is also important to control or limit the amount of risk being taken. Several approaches have been introduced in the literature to investigate the mean-VaR optimization, in which VaR is used as a measure of risk and a VaR constraint is imposed in the problem of maximizing the mean rate of return of the portfolio. Some of the works along this direction include Alexander and Baptista (1999), Kast, Luciano, and Peccati (1999) and Kluppelberg and Korn (1997). These studies were conducted in a static setting. Recently, the formulation of the problem in a continuous-time was introduced by Basak and Shapiro (2001) and Luciano (1998). Both of them considered the optimal portfolio allocation problem by maximizing the utility function of an economic agent with the VaR constraint. Luciano (1998) provided analysis on derivations from the VaR constraint instead of explicitly applying the constraint to the optimal portfolio allocation problem while Basak and Shapiro (2001) imposed the VaR constraint at one point in time to investigate trading between recalculated VaRs. Cuoco, He, and Issaenko (2001) employed the martingale approach to study optimal dynamic trading strategies with VaR constraints. They considered the case that the price dynamics of the risky asset are governed by a geometric Brownian motion. Yiu (2004) imposed the VaR as a dynamic constraint. To make the calculations tractable, he calculated the constraints abstracting from within-interval trading and from considerations of backtesting. His approach applied the VaR constraint over time and stresses the repeated recalculations of the VaRs. It also described how the VaR affects the investment decision dynamically. However, in his framework, the returns from the risky assets are assumed to be stationary. The empirical studies show that the traditional log normal model can not catch the extreme stock movement and stock variability in the variance model. The switching behavior of the economic states can be due to the structural changes in economic conditions and business cycles. There can be substantial fluctuations in economic variables, which affect the dynamics of the market values of the assets, over a long period of time. Hence, it is of practical importance and relevance to incorporate the switching behavior of the economic states in modelling the dynamics of the market values of the assets for investment and risk management. Gabih, Sass, and Wunderlich (2005) considered the utility maximization problem with shortfall risk constraints when the dynamics of the stock returns are modulated by a continuous-time, finite-state hidden Markov chain. They employed the separation principle to separate the control problem or the utility maximization problem and the filtering problem of the hidden Markov chain. Gundel and Weber (2008) obtained closed-form solution to an utility maximization problem under a joint budget and downside risk constraint, where the risk constraint is specified by a class of convex risk measures proposed in Föllmer and Schied (2002) and Frittelli and Rosazza Gianin (2002). They considered a general semi-martingale framework for the asset price dynamics and developed the closed-form solution based on the martingale approach for constrained maximization problems. Sotomayor and Cadenillas (2009) considered an optimal consumption and investment problem with the bankruptcy constraint under a Markovian regime-switching model for the asset price dynamics. They determined a consumption-investment policy so as to maximize the expected total discounted utility of consumption until bankruptcy and employed techniques of classical stochastic optimal control to derive the regime-switching Hamilton–Jacobi–Bellman equation. They were able to obtain explicit solutions to the problem for some HARA utility functions.

In this paper, we consider the optimal portfolio selection problem subject to the MVaR constraint when the price dynamics of the risky asset are governed by a Markov-modulated geometric Brownian motion (GBM). In particular, 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 optimal control theory (see, for example, Ahmed and Teo (1981) and Teo, Reid, and Boyd (1980)) and the dynamic programming principle, we shall derive a regime-switching Hamilton–Jacobi–Bellman (HJB) equation. Our work is different from those in the existing literature in four major aspects. Firstly, we consider the use of the dynamic programming approach together with the method of static constrained optimization approach to deal with the constrained optimal portfolio problem while most of the literature considered the use of the martingale approach for the constrained optimization problem. Secondly, we consider the case that both the regime-switching effect and the VaR constraint are present while most of the aforementioned literature, except Gabih et al. (2005), considered the case that the regime-switching effect is absent. Thirdly, most of the literature mentioned above did not consider the consumption process while we consider the optimal consumption and investment problem. Indeed, the proposed approach seems to be a convenient method to deal with the optimal consumption and investment problem with risk constraints. Lastly, the constraint we considered here is the maximum VaR over different states of an economy. This means that the optimal consumption and investment results developed here are uniformly optimal over different states of the economy described by the chain. In other words, our method here can provide a conservative and prudent approach to determine the optimal consumption and investment with risk constraints. Most of the literature considered the case that the VaR does not depend on the states of the economy. Our work is different from Gabih et al. (2005). Firstly, we consider an observable Markov chain while Gabih et al. (2005) assumed that the Markov chain is hidden. Secondly, the approach to deal with the risk constraint here is different from that employed in Gabih et al. (2005). Thirdly, we consider the maximum VaR constraint here, which is different from the constraint considered in Gabih et al. (2005). The work in this paper is also different from that of Sotomayor and Cadenillas (2009). The methodology and risk constraints used in the two papers are different. We shall propose an efficient numerical method to solve the regime-switching HJB equation and 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 investigate the effect of the switching regimes.

The introduction of the regime-switching effect to the optimal consumption and investment problem with VaR constraints is important for at least three reasons. From a statistical perspective, the Markovian regime-switching model can describe and explain some important “stylized” features of financial time series, namely, time-varying conditional volatility, the heavy-tailedness of unconditional distribution of returns, regime switchings and nonlinearity. So, it provides a more realistic description to the behavior of asset returns than its constant-coefficient counterpart. Neglecting the regime-switching effect may lead to underestimation of the risk of a portfolio and sub-optimal results for consumption and investment. It is especially important to incorporate the regime-switching effect when the decision horizon of the optimal consumption and investment problem is long, say 30–40 years, since there could be substantial changes in the economic condition over a long period of time. For example, one may consider the asset allocation problem of a pension fund. From an economic perspective, the Markovian regime-switching model can describe the stochastic evolution of investment opportunity sets due to structural changes in the state of the economy. This important economic feature cannot be captured by a constant-coefficient model.

This paper is structured as follows. In Section 2, we shall describe the price dynamics of the model and formulate the constrained optimal portfolio problem. We shall derive the regime-switching HJB equation using the dynamic programming principle and present the method of the Lagrange multiplier to deal with the MVaR constraint in Section 3. The results of the numerical experiments will be presented and discussed in Section 4. The final section summarizes the paper.