Decision Support
Active allocation of systematic risk and control of risk sensitivity in portfolio optimization

https://doi.org/10.1016/j.ejor.2013.02.016Get rights and content

Abstract

Portfolio risk can be decomposed into two parts, the systematic risk and the nonsystematic risk. It is well known that the nonsystematic risk can be eliminated by diversification, while the systematic risk cannot. Thus, the portfolio risk, except for that of undiversified small portfolios, is always dominated by the systematic risk. In this paper, under the mean–variance framework, we propose a model for actively allocating the systematic risk in portfolio optimization, which can also be interpreted as a model of controlling risk sensitivity in portfolio selection. Although the resulting problem is, in general, a notorious non-convex quadratically constrained quadratic program, the problem formulation is of some special structures due to the features of the defined marginal systematic risk contribution and the way to model the systematic risk via a factor model. By exploiting such special problem characteristics, we design an efficient and globally convergent branch-and-bound solution algorithm, based on a second-order cone relaxation. While empirical study demonstrates that the proposed model is a preferred tool for active portfolio risk management, numerical experiments also show that the proposed solution method is more efficient when compared to the commercial software BARON.

Highlights

► We propose a portfolio optimization model with active control on systematic risk allocation. ► This portfolio selection model also serves a purpose of risk sensitivity control. ► By exploring its special structure, we proposed a numerically efficient solution method. ► Empirical study shows the necessity of systematical control/risk sensitivity control. ► Numerical experiments demonstrate the efficiency of the solution method.

Introduction

Modern portfolio selection theory was pioneered by Markowitz (1952) in his seminal mean–variance (MV) analysis. Although the downside risk measure, Value-at-Risk (see CreditMetrics, 1997, Philippe, 1996), has recently become another standard industrial tool for risk management, the MV model remains one of the most popular tools in portfolio selection, especially in equity portfolio management. Three possible reasons may explain why this is the case: first, equity return is (almost) symmetrically distributed; second, the effect of diversification of risk can be well modeled by the variance term; finally, an intrinsic convenience lies in the computation of the MV model.

Decomposition of the entire risk according to the risk contributors is fundamental for portfolio risk management. CreditMetrics (1997) explicitly proposed the concept of marginal risk to measure the risk contribution of a given asset, which is defined as the difference between the risk of the entire portfolio and the risk of the portfolio without this asset. Actually, in the literature, except for the marginal risk given by CreditMetrics (1997), several other forms of risk attribution were also suggested to meet the purpose of managing, monitoring and hedging portfolio risk (see, e.g., Litterman, 1996, Philippe, 1996, Grinold and Kahn, 1999, Tasche, 2000, Mina, 2002, Sharpe, 2002, Kurth and Tasche, 2003).

Although the marginal risk and its variants have been popularly used in ex post analysis, they have rarely been directly integrated as ex ante constraints in portfolio optimization. However, this type of constraints is indeed required in practice. In traditional models, this practical requirement has always been indirectly treated by simply imposing lower/upper bounds on the portfolio positions. Such a questionable simplification in practice is partly due to the intractability in dealing with a non-convex program associated with the marginal risk control problem. More specifically, the mean–variance model with marginal risk-type constraints is a non-convex quadratically constrained quadratic program (QCQP), which is NP hard by nature. Global optimal solution methods for QCQP are mainly branch-and-bound (B&B) methods (see Horst, 1986, Horst et al., 1995, Al-Khayyal et al., 1995, Audet et al., 2000, Linderoth, 2005) and many researchers have devoted to this subject in recent years. However, it still lacks efficient solution methods for general situations. In our experience, BARON (Sahinidis (1996)), the state-of-art global optimization solver, cannot solve a general non-convex QCQP with more than 50 variables within acceptable time window.

Recently, Zhu et al. (2010) proposed a portfolio selection model with marginal risk control within the mean–variance framework, which represents the first work in the literature to actively integrate the marginal risk control in portfolio selection. The problem formulation is a non-convex QCQP with a special structure. By exploiting this special structure, they developed an efficient branch-and-bound (B&B) method to solve the problem with absolute marginal risk constraints. In their study, they claimed that the relative marginal risk (ratio of the marginal risk to the entire risk) constrained problem remains hard to solve and suggested some heuristics to search for acceptable solutions. Furthermore, only the constraint of marginal risk contributed by a single asset is considered in their model.

It is well known that the portfolio risk can be divided into nonsystematic risk and systematic risk. As the nonsystematic risk represents the risk that is specific to a firm, it could be eliminated via diversification. On the other side, as the systematic risk is due to the risk factors that affect the entire market, it cannot be reduced through diversification. In general, factor models are used to decompose the portfolio risk into systematic risk and nonsystematic risk, where the systematic risk is associated with the uncertainty of common factors that drive the financial market. In the literature, there are essentially three types of factor models, namely, fundamental, macroeconomic and statistic factor models. Fundamental factor models use the observed asset attributes such as price/earnings ratios, company size, book-to-price and financial leverage as factors. A representative of this type of models is the three-factor model proposed by Fama and French (1995). Macroeconomic factor models adopt macroeconomic variables correlated with asset returns as factors. The five-factor model proposed by Burmeister et al. (2003) is a typical macroeconomic factor model. The statistic factor models extract factors directly from the data of asset returns using some statistical approaches, such as the principal-component-based factor analysis procedure (Bryant and Yarnold (1995)). Nowadays, factor models have been widely used in portfolio risk management (see http://www.mscibarra.com and http://www.birr.com). The famous capital asset pricing model (CAPM) (Sharpe (1964)) and the arbitrage pricing theory (APT) (Ross (1976)) are both within the framework of factor models.

Although the systematic risk cannot be eliminated, it can be actively managed. For example, when an investor feels that a specific industrial sector will be affected by the common market factors positively, he/she can attribute more systematic risk to it, otherwise he/she can act conversely. For two sufficiently diversified portfolios with almost the same total risk, the decomposition of the entire (systematic) risk according to industrial sectors may exhibit significant differences in their performance. Thus, for sufficiently diversified portfolios, risk management is essential to allocate the systematic risk among different risk contributors. In this paper, different from the model proposed by Zhu et al. (2010), where the systematic risk and the nonsystematic risk are not distinguished, we use a factor model to capture the systematic risk, and integrate it into the MV formulation to construct a portfolio optimization model with constraints on the ratios of systematic risks attributed to risk contributors to the entire systematic risk. Furthermore, unlike Zhu et al. (2010), where only the marginal risk of a single asset is considered, a risk contributor in this new model is defined in a more flexible way. For example, a risk contributor could be a bundle of assets drawn from some specific industrial sector. We believe that this new model can serve better as a more flexible tool for active portfolio risk management.

The necessity of introducing constraints on systematic risk contributions can be interpreted from some other points of view, for example, from the angle of risk sensitivity control. In the current literature, the overall return-risk tradeoff has been the only main consideration in portfolio selection. However, in practice, the response to the market recession is another critical criterion which always worries investors. In plain language, the portfolio decision would be better if it is not so sensitive to the recession of the market. Controlling the systematic risk contributions is a direct way to restrict the risk sensitivity, thus weakening the response to the market recession.

The paper is organized as follows. In the next section, we give a detailed description of the problem formulation for controlling systematic risk contributions (or controlling risk sensitivity), resulting in a formulation of non-convex QCQP. In Section 3, we propose a B&B method for identifying the global optimal solution to the problem by investigating and utilizing the noticeable special structure of the non-convex QCQP formulation. We present some empirical analysis and test the efficiency of this new approach in Sections 4 Empirical study, 5 Efficiency test of. We finally draw concluding remarks in Section 6.

Section snippets

Portfolio selection with control on allocation of systematic risk/risk sensitivity

In this section, we propose a portfolio selection model with constraints on allocation of systematic risk/risk sensitivity by integrating the factor model into the MV model.

Global optimization method

In this section, we develop a solution method for our proposed portfolio selection problem (PSMR). The proposed method can be viewed as a specific implementation of B&B method within the framework described by Horst, 1986, Horst et al., 1995. Simply speaking, our B&B method finds the global optimal solution by solving a series of subproblems over rectangles which are selected according to some specific branch-and-bound rules. In this section, we first derive outer and inner convex

Empirical study

In this section, we present the results of empirical analysis of the proposed portfolio selection model (PSMR) and compare the performance of (PSMR) with (PMV). Algorithm 1 is coded with Matlab 7.1 and is run on a PC with 1.6GHZ CPU processor to achieve an ϵ-optimal solution. We terminate Algorithm 1 when v  min{vi}  10−5 or (v  min{vi})/∣min{vi}—  1%. In our implementation, the second-order cone subproblems, (PSMRl) and (PSMRu), are solved by SeDumi 1.10 (Sturm (1999)) with accuracy parameter

Efficiency test of Algorithm 1

In this section, we report the computational comparison results between Algorithm 1 and the state-of-the-art global optimization software BARON for (PSMR). The parameters of Algorithm 1 are set as the same as those in the empirical analysis stated in the previous section. We set the upper bound of CPU time of BARON at 7200 seconds. The absolute and relative precision parameters for stopping in BARON are set, respectively, as 10−5 and 1% in consistent with those for Algorithm 1.

Algorithm 1 and

Concluding remarks

As nonsystematic risk can be eliminated through diversification, portfolio risk management should primarily be concerned about allocation of systematic risk. In this paper, we have proposed a model for portfolio optimization with active control on the allocation of systematic risk (or control on risk sensitivity) under the framework of mean–variance analysis. Although the proposed portfolio model is a non-convex QCQP, by exploiting the special features of the problem, we have developed a

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    This research work was supported by NSFC, under projects Nos. 71071036 and 70933003 and by Hong Kong Research Grants Council, under Grant CUHK414808.

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