Active Risk and Information Ratio

Abstract

One of the underlying assumptions of the Fundamental Law of Active Management is that the active risk of an active investment strategy equates estimated tracking error by a risk model. We show there is an additional source of active risk that is unique to each strategy. This strategy risk is caused by variability of the strategy's information coefficient over time. This implies that true active risk is often different from, and in many cases, significantly higher than the estimated tracking error given by a risk model. We show that a more consistent estimation of information ratio is the ratio of average information coefficient to the standard deviation of information coefficient. We further demonstrate how the interaction between information coefficient and investment opportunity, in terms of cross sectional dispersion of actual returns, influences the IR. We then provide supporting empirical evidence and offer possible explanations to illustrate the practicality of our findings when applied to active portfolio management.

The Fundamental Law of Active Management: Time Series Dynamics and Cross-Sectional Properties

“…They will be the same only under some strong conditions. I will also show that different forms of fundamental laws are a result of either unrealistic assumptions (Grinold (1989)) or mis-specified residual return covariance matrices for the expected residual return used (Grinold (1989), Qian and Hua (2004), and Ye(2008)). When the more relevant conditional residual return covariance matrices are used, we will arrive at the more general form of the fundamental law presented in this paper.…”

“…If this is not the case, then the forecast provides no additional information and the lagged information set, 1 I − t , is useless. This is the major difference between the risk model used in this paper and the risk models used in Grinold (1989Grinold ( , 1994, Grinold and Kahn (2000), Clarke et al (2002), Qian and Hua (2004), and Ye (2008). Of course, the assumption of stock return normality may not be valid in practice, and the return and risk models one uses are very likely misspecified, which may cause theoretically derived results not to reflect what one gets in reality.…”

“…As will be shown later in this paper, the cross-sectional IC can be quite different from the time series IC even if all the securities have a same time series IC. Qian and Hua (2004) show that a more appropriate IR to use is average IC divided by the standard deviation of IC…”

“…where IC σ is the standard deviation of IC that Qian and Hua (2004) call "the strategy risk." In statistics, the quantity 2 IC / 1 σ is a measure of how close (precise) the realized information coefficient at time t, t IC , is to the mean IC.…”

“…In a more recent paper, Ye (2008) goes one step further to bridge the gap between the original Grinold (1989) formula and the Qian and Hua (2004) formula. Based on her assumptions, she establishes that…”