Estimation risk and the implicit value of index-tracking

Abstract

We study [Roll, R., A mean/variance analysis of tracking error. J. Portfolio Manage., 1992, 18, 13–22.] conjecture that there exists an implicit value in index-tracking (IVIT) relative to forming mean-variance (MV) optimal portfolios under estimation error. We derive an analytical definition for the opportunity cost facing the MV investor who does not index-track. Our findings indicate that the opportunity cost is positive and statistically significant. The existence of an IVIT (positive opportunity cost) is strongly associated with a reduction in the portfolio's induced estimation risk under index-tracking relative to an MV-efficient portfolio of equal target mean return. Under high estimation error cases, increased IVIT translates to higher risk-adjusted returns, lower volatility, higher Sharpe-ratio, lower turnover, and larger certainty equivalent returns. Empirically, a one standard deviation increase in IVIT translates to an annual increase of 4%–5% in the out-of-sample Sharpe-ratio and a 6%–15% decrease in the monthly turnover.

Keywords:

• Portfolio theory
• Mean-variance analysis
• Shrinkage
• Tracking error

JEL Codes:

• C13
• C44
• C46
• G11

Acknowledgments

We thank an anonymous reviewer for calling our attention to the utility framework studied in this paper along with other valuable suggestions in terms of shrinkage. We also thank Thomas Shohfi, Aparna Gupta, Bill Francis, Iftekhar Hasan, Alexandre Baptista (discussant), Yusif Simaan, Alberto Martin-Utrera, German Creamer, and Steve Yang for comments and feedback. Also, special thanks go to participants at the Finance & Financial Engineering Brownbag Series at Stevens Institute of Technology, the 2020 Leir Research Seminar Series at NJIT, the FMA 2018 Annual Meeting, and the 2016 R in Finance meeting at UIC. All errors remain our responsibility.

Disclosure statement

No potential conflict of interest was reported by the authors.

Supplemental data

Supplemental data for this article can be accessed at http://dx.doi.org/10.1080/14697688.2021.1959631.

Notes

1 A number of approaches have been proposed in the literature to mitigate the ex-ante MV sub-optimality of the MTEP (see e.g. Jorion 2003, Alexander and Baptista 2010, Alexander et al. 2017, Rossbach and Karlow 2019). However, these approaches do not pursue the estimation risk investigation undertaken in this paper.

2 Haugen and Baker 1991 compare the GMVP with capitalization-weighted and other risk-parity strategies to claim that GMVP has superior out-of-sample performance.

3 By definition, it follows that ${\eta }_1={\sigma }_1^2>0$ since $\mathbf{B}$ is positive definite.

4 The MVEP can be viewed as the one that maximizes the Sharpe-ratio, whereas the MTEP can be viewed as the fund that maximizes the information ratio instead of the Sharpe ratio. Hence, the latter denotes a portfolio that maximizes a relative rather than absolute performance.

5 It can also be shown that the MTEP's efficient frontier increases at a faster rate than that of the MVEP's as portfolio variance increases, i.e. $\frac{\mathrm{d}{\mu }_A}{\mathrm{d}{\sigma }_A}{|}_{{\sigma }_A=\overline{\sigma }}\le \frac{\mathrm{d}{\mu }_B}{\mathrm{d}{\sigma }_B}{|}_{{\sigma }_B=\overline{\sigma }}$, for all $\overline{\sigma }$. Moreover, as risk tolerance decreases, the two efficient frontiers converge with ${\mu }_B\to {\mu }_A$, as ${\kappa }_A,{\kappa }_B\to 0$. See Edirisinghe 2013 for an alternative exposition of these ex-ante properties.

6 As section 5 demonstrates, the more general choice $ϵ\in [0,1]$ is equivalent to partial index-tracking.

7 In section 5, we relate the portfolio problem studied in this paper in relation to partial-tracking and shrinkage.

8 Different from DeMiguel et al. 2013, we control for the portfolio growth in the calculation of the portfolio turnover as an additional robustness. Nonetheless, in unreported results we find that there no significant difference between the two approaches and that our main findings are robust using either turnover definition.

9 See e.g. Jorion 1986, Ledoit and Wolf 2003, DeMiguel et al. 2009a, Xing et al. 2014, Li 2015, Kremer et al. 2020, Ledoit and Wolf 2017.