An Approximate Formula linking Active Share And Tracking Error

Abstract

The concept of Active Share in portfolio management has gained widespread acceptance since it was proposed in 2009 by Cremers and Petajisto (2009). An explicit relationship between Active Share and the other popular measure of relative risk, Tracking Error, was proposed by Stowe in 2014 but necessitates the full knowledge of all the weights of the portfolio and of the complete covariance matrix of the assets. I derive an approximation in a simple setup where the exact weights are unknown, the pairwise correlation between the returns of the assets is fixed, and the volatility of the assets is described by its mean and variance. This approximation is really a “toy model” and works for the purpose of understanding changes in Tracking error stemming from changes in a portfolio, and to show what the drivers of the link between Active Share and Tracking error are. Using the approximation and the Fundamental Law of Active Management, I compute the total expected return and variance of a fund with a given active share in our setup. A full derivation of the main results is provided in the appendix.