مدیریت فعال پرتفوی همراه با الگوبرداری: اضافه کردن یک محدودیت ارزش در ریسک

We examine the impact of adding a value-at-risk (VaR) constraint to the problem of an active manager who seeks to outperform a benchmark while minimizing tracking error variance (TEV) by using the model of Roll [1992. A mean/variance analysis of tracking error. Journal of Portfolio Management 18, 13–22]. We obtain three main results. First, portfolios on the constrained mean-TEV boundary still exhibit three-fund separation, but the weights of the three funds when the constraint binds differ from those in Roll's model. Second, the constraint mitigates the problem that when an active manager seeks to outperform a benchmark using the mean-TEV model, he or she selects an inefficient portfolio. Finally, when short sales are disallowed, the extent to which the constraint reduces the optimal portfolio's efficiency loss can still be notable but is smaller than when short sales are allowed.

As Roll (1992) and Cornell and Roll (2005) note, institutional investors often manage money against a benchmark. This has led to the practice by active portfolio managers (hereafter ‘managers’) of seeking to outperform the benchmark by a given percentage, subject to a limit on tracking error variance, or TEV.1 However, this practice leads such managers to select portfolios that are mean-variance inefficient and under certain conditions have systematic risk that is greater than 1 when measured against the benchmark. Not surprisingly, large losses relative to the benchmark have occurred in some cases. A recent example involved the management of Unilever's pension fund by Merrill Lynch, who in attempting to beat the FTSE All-Share Index by 1% per year, ended up as the defendant in a lawsuit due to underperforming the index by roughly 10% over a 15-month period.2 Two methods have been proposed for overcoming this tendency to invest in overly risky portfolios. Roll (1992) advocates constraining the portfolio's beta, while Jorion (2003) advocates constraining the portfolio's variance. In this paper we propose a third method that involves constraining the portfolio's Value-at-Risk, or VaR.3 A VaR constraint is of particular interest for several reasons. First, as Jorion, 2001 and Jorion, 2003 and Pearson (2002) note, the fund management industry is increasingly using VaR to: (1) allocate assets among managers, (2) set risk limits, and (3) monitor asset allocations and managers (these activities are often referred to as ‘risk budgeting’). Second, we show that Jorion's result of bringing the optimal portfolio closer to the mean-variance efficient frontier with a variance constraint can also be obtained with a VaR constraint. Third, under certain conditions, the use of VaR as a risk measure is consistent with expected utility maximization (see Alexander and Baptista, 2002). Finally, VaR can be useful to reduce the regret of losses (see Shefrin, 2000). We begin by examining the case when short sales are allowed. The set of portfolios that minimize TEV for various levels of expected return is referred to as the mean-TEV boundary, while the set of portfolios that do so given a VaR constraint is referred to as the constrained mean-TEV boundary. Like portfolios on the mean-TEV boundary, we find that portfolios on the constrained mean-TEV boundary exhibit three-fund separation, but the weights of the three funds when the constraint binds differ from those in its absence. Under certain conditions, we find that the constrained mean-TEV boundary consists of: (i) portfolios on the mean-variance boundary, (ii) portfolios on the mean-TEV boundary, and (iii) portfolios that do not belong to either of these boundaries. There are also conditions under which no portfolio on the mean-TEV boundary belongs to the constrained mean-TEV boundary. Nevertheless, there are no conditions under which the constrained mean-TEV boundary includes more than two portfolios on the mean-variance boundary. It is important to emphasize that the constrained mean-TEV boundary is related to the constant-TEV mean-variance boundary of Jorion (2003). A portfolio is on the constant-TEV mean-variance boundary if it satisfies a TEV constraint and there is no other portfolio with the same variance that satisfies the constraint and has a larger expected return. Under certain conditions, a portfolio on the constrained mean-TEV boundary with a given expected return is also on the constant-TEV mean-variance boundary for a TEV bound that depends on the required expected return and the VaR bound. Thus, the constrained mean-TEV boundary can be thought of as an extension of the constant-TEV mean-variance boundary. Next, we show that a VaR constraint mitigates the problem that when a manager seeks to outperform a benchmark using the mean-TEV model, he or she selects a portfolio that is mean-variance inefficient. First, the constrained optimal portfolio dominates the unconstrained optimal portfolio according to both mean-variance and mean-VaR criteria. Second, under certain conditions, the constrained optimal portfolio dominates the benchmark according to both criteria. Third, there are no conditions under which the former is dominated by the latter according to either criteria. Almazan et al. (2004) find that a large fraction of fund managers cannot engage in short sales. Hence, we also examine the case when short sales are disallowed. We find that when short sales are disallowed, the extent to which the constraint reduces the optimal portfolio's efficiency loss can still be notable but is smaller than when short sales are allowed. Previous papers recognize that managers may have incentives to take actions that are not in the best interest of investors. First, these adverse incentives can be induced by compensation contracts that are explicitly based on the managers’ performance relative to a benchmark. For example, Starks (1987) shows that managers spend a smaller amount of resources to produce superior returns than the amount that is optimal for investors. Admati and Pfleiderer (1997) show that when a manager has private information, he or she ends up selecting a portfolio that is suboptimal for investors. Grinblatt and Titman (1989) show that the option-like features in performance-based compensation give incentives for the manager to select a portfolio with a level of risk that may be suboptimal for investors.4Carpenter (2000) shows that compensating a manager with a call option on the assets that he or she manages may lead to (but does not necessarily imply) greater risk seeking by the manager. Elton et al. (2003) provide empirical evidence that managers with performance-based compensation select riskier portfolios than managers whose compensation is not performance-based. Second, adverse incentives can be induced implicitly by the relationship between fund inflows and performance. Sirri and Tufano (1998) document that investors flock to funds with the highest recent returns. Hence, when a manager's compensation is based on a fixed percentage of assets under management, he or she has incentives to change a fund's riskiness as a function of its performance. Chevalier and Ellison (1997) provide empirical evidence that managers respond to these incentives. Our work is related to these papers in that we explore the ability of a VaR constraint to curtail the tendency of managers who use the mean-TEV model to select overly risky portfolios. Our paper proceeds as follows. Section 2 describes the model and characterizes the constrained mean-TEV boundary when short sales are allowed. Section 3 explains how to implement the VaR constraint. Section 4 presents an example to illustrate the portfolio selection implications of the constrained mean-TEV model. Section 5 examines the effect of disallowing short sales in our results. Section 6 presents an example to illustrate this effect. Section 7 concludes.

Value-at-risk (VaR) is a popular risk management tool in the fund management industry. Moreover, active portfolio management is often conducted relative to a benchmark. We combine these two practices by examining the impact of adding a VaR constraint to the problem of a manager who seeks to outperform a benchmark by a given percentage. In doing so, we use the mean-tracking error variance (TEV) model of Roll (1992). We obtain three main results. First, portfolios on the constrained mean-TEV boundary still exhibit three-fund separation, but the weights of the three funds when the constraint binds differ from those in Roll's model. Second, the constraint mitigates the problem that when a manager seeks to outperform a benchmark using the mean-TEV model, he or she selects an inefficient portfolio. Finally, when short sales are disallowed, the extent to which the constraint reduces the optimal portfolio's efficiency loss can still be notable but is smaller than when short sales are allowed. Our results on the usefulness of a VaR constraint justify the observation of Jorion, 2001 and Jorion, 2003 and Pearson (2002) that the fund management industry is increasingly using VaR to (1) allocate assets among managers, (2) set risk limits, and (3) monitor asset allocations and managers.