Estimation Risk and Implicit Value of Index-Tracking

Abstract

We study Roll’s (1992) conjecture that there may exist an implicit value in index-tracking (IVIT) relative to forming mean-variance (MV) optimal portfolios under estimation error. While index-tracking portfolios are deemed MV inefficient ex-ante, it is unclear whether this is the case when taking into account estimation error in the parameters of the portfolio selection problem. To investigate this, we derive an analytical definition for the opportunity cost facing the MV investor who does not index-track, building on the expected out-of-sample utility framework of Kan & Zhou (2007). Our findings, both analytical and empirical, indicate that such 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 cases of high estimation error, we show that increased IVIT translates to improved metrics of portfolio performance, such as higher risk-adjusted returns, lower volatility, higher Sharpe-ratio, lower turnover, and larger certainty equivalent returns. Moreover, our paper reconciles the theoretical propositions with the out-of-sample performance empirically. The cross-sectional regression results indicate that a one standard deviation increase in the proposed IVIT translates to an annual increase of 4%-5% in the out-of-sample Sharpe-ratio and a 6%-15% decrease in the monthly portfolio turnover.

Keywords: Portfolio Optimization, Index Tracking, Parameter Estimation Risk

JEL Classification: C13, C61, G11, G12