The Gerber Statistic: A Robust Co-Movement Measure for Portfolio Optimization

Abstract

The purpose of this paper is to introduce the Gerber statistic, a robust co-movement measure for covariance matrix estimation for the purpose of portfolio construction. The Gerber statistic extends Kendall's Tau by counting the proportion of simultaneous co-movements in series when their amplitudes exceed data-dependent thresholds. Since the statistic is neither affected by extremely large or extremely small movements, it is especially well-suited for financial time series, which often exhibit extreme movements as well as a great amount of noise. Operating within the mean-variance portfolio optimization framework of Markowitz (1952,1959) we consider the performance of the Gerber statistic against two other commonly used methods for estimating the covariance matrix of stock returns: the sample covariance matrix (also called the historical covariance matrix) and shrinkage of the sample covariance matrix as formulated in Ledoit and Wolf (2004). Using a well-diversified portfolio of nine assets over a thirty year time period (January 1990-December 2020), we empirically find that, for almost all scenarios considered, the Gerber statistic's returns dominate those achieved by both historical covariance and by the shrinkage method of Ledoit and Wolf (2004).