The Distributions of Funds' Active Shares

We consider the active share distributions of actively managed equity funds. Unlike the commonly used Manhattan active share metric, funds' Euclidean active share distributions are remarkably stable over time and across benchmarks, even when their concentration indices differ. Moreover, they are approximately Gaussian. We examine the properties of these distributions by modelling active portfolios as realisations of normal random vectors with varying standard deviations and a variable number of components representing the number of active holdings. The model establishes a one-to-one functional relationship between the two active share metrics.The distribution of funds' Euclidean active shares is represented by a scale mixture of chi distributions with varying degrees of freedom. The model explains the observed Gaussian-like distributions of US equity mutual funds' Euclidean active shares as the outcomes of managers' risk tolerances.We present a rule of thumb for linking a portfolio's active position sizes to its active position counts and its ex-ante tracking error. The model also provides a methodology for creating sample active portfolios when short-selling is permitted