Portfolio Optimization with Constraints on Tracking Error

Abstract

This paper explores the risk and return relationship of active portfolios subject to a constraint on tracking error volatility (TEV), which can also be interpreted in terms of Value At Risk (VAR). This is the typical setup for active managers who are given the task of beating a benchmark. The problem with this setup is that the portfolio manager pays no attention to the total portfolio risk, resulting in seriously inefficient portfolios, unless some additional constraints are imposed. This paper shows that TEV-constrained portfolios are described by an ellipse in the traditional mean-variance plane. This yields a number of new insights. We show that, due to the sat shape of this ellipse, adding a constraint on total portfolio volatility can substantially improve the performance of the active portfolio. This constraint is most beneÞcial in situations with low values of the admissible TEV, or when the benchmark is relatively inefficient. P.Jorion-Portfolio Optimization with TEV Constraints 1 In a typical portfolio delegation problem, the investor assigns the management of assets to a portfolio manager who is given the task of beating a benchmark. When outperformance is observed for the active portfolio, the issue is whether the added value is in line with the risks undertaken. This is particularly important with performance fees, which induce an option-like pattern in the compensation of the agent, who may have an incentive to take on more risk to increase the value of the option. To control for this, institutional investors commonly impose a limit on the volatility of the deviation of the active portfolio from the benchmark, also known as Tracking Error Volatility (TEV). The problem with this setup, however, is that it induces the manager to optimize in an excess return space only, totally ignoring the investor’s overall portfolio risk. In an insightful paper, Roll (1992) noted that excess return optimization leads to the unpalatable result that the active portfolio has systematically higher risk than the benchmark and is not optimal. Jorion (2002) examines a sample of enhanced index funds, which are more likely to go through a formal excess return optimization, and Þnds that such funds have systematically greater risk than the benchmark. Thus, the problem is real. It is not clear why the industry maintains this widespread emphasis on tracking error risk control. Roll conjectured that diversifying across managers could mitigate the inherent saw in TEV optimization but, as we will show later, this is not the case. Instead, this paper investigates whether this problem can be corrected with additional restrictions on the active portfolio without eliminating the usual TEV constraint. Thus, this paper takes the TEV constraint as given, even if this restriction is not optimal, given its widespread practice. The paper derives the constant TEV This is the case, for instance, with hedge funds, which typically charge a variable fee of 20 percent of proÞts. For a good introduction to the major issues with performance fees, see Davanzo and Nesbitt (1987), Grinold and Rudd (1987), and Kritzman (1987). Going one step further, Admati and Pseiderer (1997) examine the rationale for benchmarkadjusted compensation schemes. They argue that such schemes are generally inconsistent with optimal risk-sharing and do not help in solving potential contracting problems with the portfolio manager. P.Jorion-Portfolio Optimization with TEV Constraints 2 frontier in the original mean-variance space. Traditionally, TEV has been checked after the fact, i.e. from the volatility of historical excess returns. More recently, the industry has witnessed the advent of forward-looking measures of risk such Value at Risk (VAR). The essence of VAR is to measure the downside loss for current portfolio positions, based on the best risk forecast. With a distributional assumption for portfolio returns, excess returns VAR is equivalent to a forward-looking measure of TEV. Nowadays, VAR limits are increasingly used to ensure that the active portfolio does not stray too much from the benchmark. Furthermore, the VAR methodology is now implemented as “risk budgets,” which can be deÞned as the conversion of optimal mean-variance allocations to VAR assignments for active managers. The spreading use of VAR systems makes it possible to consider other ex-ante restrictions on the active portfolio. To do this, this paper explores the risk and return relationship of active portfolios subject to a TEV constraint. I show that the TEV constraint is described by an ellipse in the usual mean-variance space. This yields several new insights. For instance, we can explore the effect of adding a constraint that the total portfolio risk be equal to that of the benchmark. Such constraint can be imposed easily using a VAR system, along with the usual TEV constraint. The question is whether the addition of this constraint creates too large a penalty in terms of expected returns. I show that the penalty depends on the “slope” of the constant TEV ellipse and explore the driving factors behind this slope. See Jorion (2000) for a detailed analysis of VAR. Another issue is who should be given authority to control tracking risk. Possible candidates are the investment manager, the plan sponsor, the custodian, or outside consultants. One could argue that risk should be controlled by the portfolio manager. After all, he or she should already have in place a risk measurement system, which gives the tracking error of the active portfolio. The manager should also have the best understanding of the instruments in the portfolio. Thomas (2000), however, argues that there is a consict of interest for the manager to control VAR and that this function is best performed by a disinterested party. For an introduction to risk budgeting, see Chow and Kritzman (2001). Lucas and Klaasen (1998) also discuss the link between portfolio optimization and VAR. P.Jorion-Portfolio Optimization with TEV Constraints 3 The primary contribution of this paper is the derivation and interpretation of these analytical results. Apart from Roll’s seminal paper, there are only few publications on this important, and practical, topic. The implications of these analytical results are illustrated with an example. This paper is structured as follows. Section 1 reviews optimization results for the efficient frontiers in absolute and relative spaces. The main result is that TEV optimization creates an increase in the fund’s risk. Section 2 then shows that diversifying across active managers is not likely to reduce much this effect. Next, Section 3 describes the relationship between absolute risk and return of portfolios subject to this TEV constraint. Based on these results, Section 4 then discusses how imposing additional restrictions on the active manager could bring the portfolio closer to the efficient frontier. Finally, Section 5 provides some concluding comments. The closest paper is that of Leibowitz, Kogelman, and Bader (1992), who discuss the application of the shortfall approach to portfolio choice for a pension fund. In this case, the tracking error volatility is replaced by the “surplus return,” deÞned relative to the liabilities. Their paper, however, involves another constraint, which is a linear relationship between expected returns and volatility and has a simple setup with two risky assets only. In addition, no closed-form solutions are presented. Chow (1995) argues that the objective function should account for total risk but also tracking error risk. Rudolf, Wolter, and Zimmermann (1999) compare various linear models to minimize tracking error. Ammann and Zimmermann (2001) examine the relationship between limits on TEV and deviations from benchmark weights. P.Jorion-Portfolio Optimization with TEV Constraints 4 1 Efficient Frontiers in Absolute and Relative Space