Tracking error volatility and relative risk budgets

ABSTRACT

This paper uses a pooled cross-sectional sample of actively managed US equity mutual funds from 1991–2022 to show that tracking error volatility (TEV) is characterised by reversion. Mutual funds with relatively high (low) TEV tend to reduce (increase) their TEV in subsequent periods, and the degree of reversion is determined by the degree to which TEV is relatively high or low. This suggests that TEV is managed over time to satisfy relative risk budgets. This paper also shows that the previous literature’s finding that mutual funds increase their TEV as their performance declines holds even when taking into consideration that both performance and change in TEV may be jointly determined by TEV level. The results are robust to a variety of measurement periods and methodologies.

KEYWORDS:

• tracking error volatility
• risk budgets
• risk-shifting
• alpha

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 The risk/return tradeoff is captured in the information ratio, which is the ratio of excess returns to TEV (see Goodwin, 1998).

2 E.g., see Jensen (1968), Fama (1970), Grinblatt and Titman (1993), and Fama and French (2010).

3 E.g., see Gupta et al. (1999), Wermers (2003), Kacperczyk et al. (2005), Israelsen and Cogswell (2006), Cremers and Petajisto (2009), Amihud and Goyenko (2013), and Doshi et al. (2015).

4 E.g., Brown et al. (1996), Koski and Pontiff (1999), Busse (2001), Goriaev et al. (2005), and Huang et al. (2011).

5 Some papers argue that other factors can explain risk-shifting behavior. Schwarz (2011) identifies a sorting bias in which sorting based on performance will inevitably result in sorting based on risk as well, resulting in mean reversion of risk levels unrelated to risk-shifting. Lee et al. (2019) explain risk-shifting as driven by the incentives within option-like compensation contracts, and present evidence that those funds with performance close to the benchmark engage in TEV risk-shifting the most.

6 The factors are the value weighted market return in excess of the risk-free rate on all NYSE/AMEX/NASDAQ firms; the difference in returns across small and big stock portfolios that controls for the same weighted average book-to-market equity in the two portfolios; the difference in returns between high and low book-to-market equity portfolios; the difference in returns across robust and weak operating profitability portfolios; the difference in returns between conservative and aggressive investment portfolios; and the momentum factor, which is the average return on two high prior return portfolios minus the average return on two low prior portfolios. All factors are extracted from Kenneth French’s website (https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html).

7 The nine non-overlapping in-sample periods are 1991–1993, 1994–1996, 1997–1999, 2000–2002, 2003–2005, 2006–2008, 2009–2011, 2012–2014, and 2015–2017.

8 To further explore the nature of TEV over time, the Hurst exponent for the time series of each fund’s TEV is estimated. The Hurst exponent is a classification measure that ranges between 0 and 1. A value of 0.5 indicates that the time series is a random series, a value between 0 and 0.5 indicates that the time series is mean reverting, and a value between 0.5 and 1 means that the time series is trend-reinforcing (See Hurst, 1951; Peters, 1991). Anis and Lloyd (1976) provide a size-corrected estimate of the rescaled range. To estimate the exponent, the monthly TEV is calculated for each fund for each month using the fund’s daily returns, creating a separate time-series of monthly TEVs for each fund. The Anis-Lloyd size-corrected rescaled range Hurst exponent is then estimated, using the NumXL time series analysis software tool, separately for the log difference of each fund’s time series. The average value and standard deviation of the exponent across all funds are 0.2762 and 0.0725, respectively, with approximately 97% falling below 0.5. Similar estimation using time-series of annual TEV, limited to those funds for which at least ten consecutive annual TEV observation exists, finds average value and standard deviation of the exponent of 0.3281 and 0.2855, respectively, with approximately 84% falling below 0.5. The author thanks an anonymous referee for recommending that this analysis be performed.