Portfolio Optimization with Tracking-Error Constraints

This article 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. Such a constrained portfolio 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 total portfolio risk, which results in seriously inefficient portfolios unless some additional constraints are imposed. The development in this article shows that TEV-constrained portfolios are described by an ellipse on the traditional mean–variance plane. This finding yields a number of new insights. Because of the flat shape of this ellipse, adding a constraint on total portfolio volatility can substantially improve the performance of the active portfolio. In general, plan sponsors should concentrate on controlling total portfolio risk. In typical portfolio delegation, the investor assigns the management of assets to a portfolio manager who is given the task of beating a benchmark. To control for excessive risk, 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). Traditionally, TEV has been checked after the fact (i.e., from the volatility of historical excess returns), but recently, the advent of forward-looking measures of risk, such as value at risk, has allowed the industry to forecast TEV. With a distributional assumption for portfolio returns, excess-return VAR is equivalent to a forward-looking measure of TEV.Imposing TEV constraints is seriously inefficient, however, for the investor. When myopically focusing on excess returns, the active manager ignores the total risk of the portfolio. An optimization in excess-return space that includes the benchmark assets will always increase total portfolio risk relative to the benchmark. This observation is important because of the widespread emphasis on controlling tracking-error risk.The problem with a focus on TEV is exacerbated by the widespread use of information ratios (the ratio of expected excess return to TEV) as performance measures. Because information ratios consider only tracking-error risk, total portfolio risk is ignored when managers focus on information ratios. This issue has major consequences for performance measurement: Part of the value added by active managers acting in this fashion is illusory because it could be naively obtained by leveraging up the benchmark.In this article, I investigate whether this problem can be corrected with additional restrictions on the active portfolio without eliminating the usual TEV constraint. I derive the constant-TEV frontier in the original mean–variance space and show that it is described by an ellipse. The primary contribution of this paper is the derivation and interpretation of these analytical results, which yield several new insights. The implications of the results are illustrated with an example.The analytical solution allows exploration of the effect of imposing additional constraints on the active manager. The simplest constraint is to force the total portfolio volatility to be no greater than that of the benchmark. With the advent of forward-looking risk measures, such as VAR, such a constraint is easy to set up. The question is whether the addition of this constraint creates too large a penalty in terms of expected returns.I show that because of the flat shape of the ellipse, adding such a constraint can substantially improve the performance of the active portfolio. The constraint on total risk lowers risk for a small penalty in terms of expected returns. The cost will be small when the value of the admissible TEV is low or when the benchmark is relatively inefficient. Thus, if the active manager is confident that he or she can add value, the manager should have no objection to an additional constraint on total portfolio risk.In summary, my first prescription for investors is to discard TEV optimization and focus on controlling total risk. Some indications are that pension plans with advanced risk management systems are indeed now moving in this direction. Otherwise, if TEV constraints are kept in place, my recommendation is to impose an additional constraint on total volatility.