Large Scale Covariance Estimates for Portfolio Selection

Abstract

The Mean-Variance (MV) portfolio by Markowitz (1952) and the Global Minimum Variance (GMV) portfolio represent standard rules for investment decisions and both rely on estimates of the inverse of the covariance matrix of stock returns. When working in a large-scale framework, where the cross-section of assets (N) is comparably big to the length of the time series (T), the sample covariance matrix can no longer be used because the data do not contain enough information to estimate the great number of parameters (O(N2)) of the covariance matrix. We present a slight improvement of the estimator first developed by Mu nnix et al. (2014) that proves to improve its performance in a large scale framework, by combining it with the shrinkage estimator by Ledoit and Wolf (2003). We run a comparison between alternative estimators for the covariance matrix when used to estimate the optimal weights of the GMV portfolio. Our investment universe is the set of constituents of the S&P 500, and we assess the out-of-sample performance of the different approaches both on the basis of statistical and economic measures. Moreover, among the statistical measures, we employ the Model Confidence Set by Hansen et al. (2011) in order to check which of the compared methods ends up in the confidence set. Our results suggest that, in line with DeMiguel et al. (2009) and Sharpe (1963), it is very hard to beat the naive strategy, but the proposed estimators provides slight improvements upon both Mu nnix et al. (2014) and Ledoit and Wolf (2003). Finally, we extensively analyse the sensitivity of both the statistical and economic measures to the length of the estimation window, and analyse the behaviour of the estimators to this choice.