Portfolio performance under tracking error and benchmark volatility constraints.
| Autor | Hausner, Jan Frederick |
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Introduction
Within the asset management sector, tracking error (TE) is a fundamental performance evaluation measure that has been used by the industry to ensure portfolio managers adhere to their given investment policy statement (IPS). TE is an active risk measure that stipulates the standard deviation of the difference between the portfolio return and the benchmark return (or excess return). In practice, TE gauges how consistently a portfolio outperforms or underperforms its designated benchmark, and it is used in conjunction with several other metrics to evaluate a portfolio's performance, such as the value at risk and information ratio. Generally, TE results from security selection, taxes, factor tilts, transaction costs, cash management and general market volatility (Thomas et al., 2013).
Portfolio managers pursue positive returns that are above the benchmark, while simultaneously adhering to mandated constraints as they have an incentive to outperform the benchmark to earn additional revenue from performance fees. Investors thus enforce a TE constraint, but this can result in portfolio managers focusing only on optimising excess returns and ignoring an investor's overall portfolio risk (Roll, 1992). TE is directly agnostic: it only determines the deviation of excess returns but articulates essentially nothing with regards to the direction of returns (e.g. positive or negative). Many investors incorrectly believe or assume that a higher TE is associated with a higher potential return, which may not always be true (Thomas et al., 2013).
Other performance metrics such as the information ratio [a performance measure which does not account for total risk, only relative risk - see equation (1) - and is generally used expost performance] also ignores investors' overall portfolio risk and exacerbate agency issues:
[Please download the PDF to view the mathematical expression] (1)
Higher information ratios could indicate higher excess returns but could also indicate an increase in absolute risk (Jorion, 1992).
Jorion (2003) developed a constant TE frontier which incorporated the risk of the benchmark. This frontier is an ellipse in mean/variance space, and which identifies a portfolio's permissible return/risk profile given a TE constraint. Portfolio managers will ideally try to maximise the portfolio's return, even if this incurs extra risk. This suggests that portfolio managers would essentially create a portfolio that lies on the uppermost point of this ellipse. However, Jorion's (2003) suggestion of limiting the absolute risk of the portfolio to that of the benchmark would produce a portfolio with a lower coefficient of variation, thus optimising the portfolio in a mean-variance sense. Jorion (2003) noted that the shape of the ellipse was relatively flat; therefore, the reduction in expected return should be marginal. This article explores the effectiveness of Jorion's (2003) proposal is in a practical situation by using a standard combination of equities and bonds in both bull and bear market conditions. The idea is to test, practically, whether investors would benefit from Jorion's (2003) proposal, especially in challenging market conditions. Two periods are tested: a low volatility period (LV), to act as a basis for comparison and a high volatility period (HV) to represent a stressed market condition.
This article proceeds as follows: Section 2 presents the relevant literature and Section 3 introduces the variables and assumptions (e.g. transaction costs, rebalancing periods, asset weights, periods and TE constraint) which are required. The empirical results and discussion follow in Section 4 and Section 5 concludes the paper.
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Literature review
Most asset managers are evaluated by their total return performance relative to a given benchmark, for example, the S&P 500. This leads to monthly performance reviews, in which portfolio managers and investors assess total returns after fees/expenses and compare these results to the allocated benchmark. Positive returns above benchmark returns (after fees/expenses) generally lead to the retention of the portfolio manager. Many investors thus focus on fund outperformance over the benchmark, which consistently results in positive expected
TE. Unfortunately, asset returns are generally noisy and require several months of return data before one can reliably measure the average portfolio performance, after which investors can make an educated decision on the statistical significance of the portfolio's return and determine if the portfolio manager is adding value. When portfolios are initiated, overall performance is not necessarily used, plus performance fees incentivise portfolio managers to take on additional risk to earn a higher return, which further detracts investors from focusing on the initial overall performance figure. This, in turn, causes investors to focus heavily on TE because a lower TE indicates fewer months of excessive negative returns, but also fewer months of excessive positive returns (Roll, 1992).
Roll (1992) argued that a portfolio manager's objective of outperforming the benchmark, while simultaneously attempting to reduce the portfolio's TE, was comparable to the typical Markowitz mean-variance objectives with modifications. Roll's (1992) work established the TE-constrained frontier - the locus of points in mean/risk space which represents the maximum return (MR) (and associated risk) for each level of TE (grey line in Figure 1). Jorion (2003) extended Roll's (1992) work and derived a constant TE frontier - the locus of points in return/risk space which represent all combinations of return and risk for a given TE. The shape of the ellipse led Jorion (2003) to propose the extra constraint that portfolio managers observe the TE and match the portfolio risk (aP) to that of the benchmark (erB) i.e. that [Please download the PDF to view the mathematical expression]. TE frontiers are not necessarily efficient.
Jorion (2003) accepted the TE constraint as given, even though it is not optimal, as the TE constraint is widely used within the asset management industry and found that limiting a portfolio's absolute risk to that of the benchmark, substantially improved portfolio performance.
El-Hassan and Kofman (2003) explored the impact of inefficient benchmarks - which Jorion (2003) excludes - and introduced a short-selling constraint. Australian stocks were used during a period characterised by a strong bull market, followed by a severe, long bear market. El-Hassan and Kofman (2003) concluded that a short-selling constraint on a portfolio that exhibited Jorion's (2003) absolute risk constraint caused the unrestricted weights to form entirely impractical values throughout the bear market conditions with considerable short-selling repercussions. The opportunity set was thus severely reduced, leading to constant rebalancing to maintain control over total risk in actively managed portfolios (Plaxco and Arnott, 2002). This, in turn, meant that a short-selling constraint changes an actively managed portfolio into more of a passively managed portfolio (El-Hassan and Kofman, 2003).
Bertrand (2009) extended the work of Bertrand et al. (2001) by creating portfolios in which the TE varied, but risk aversion remained fixed. These portfolios would then lie on what Bertrand et al. (2001) called an iso-aversion frontier. Bertrand (2009) demonstrated that optimal portfolios, which lay on the iso-aversion frontier had many desirable properties for portfolio managers, most importantly, all portfolios had the same information ratio. Portfolios that lay on this frontier had (3
Bertrand (2010) noted that performance attribution alone was inadequate and misleading. Volatility is also insufficient because of asset correlations. Bertrand's (2010) study examines the information ratio breakdown suggested by Menchero (2007) with regards to the analysis of risk-adjusted performance attribution developed by Bertrand (2005). Bertrand (2009) showed that optimising portfolios only with a TE constraint was like the risk-adjusted performance attribution process proposed by Menchero (2007). If extra constraints...
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