On the Estimation of the Global Minimum Variance Portfolio

Abstract

The implementation of the Markowitz optimization requires the knowledge of the parameters of the return distribution. These parameters cannot be observed, but have to be estimated. Merton (1980) and Jorion (1985) point out that especially the expected returns are hard to estimate from time series data. The estimation risk is huge. The global minimum variance portfolio is the only efficient stock portfolio whose weights do not depend on the expected returns. Therefore, one can avoid extreme estimation risk by investing into this portfolio. Nevertheless, there remains a considerable estimation risk with respect to the covariance matrix. This article deals with the estimation of the weights of the global minimum variance portfolio. The literature suggests a two-step approach to determine the optimal portfolio weights. In the first step one estimates the return distribution parameters, and in the second step one optimizes the portfolio weights using the estimated parameters. The main contribution of our paper is to suggest new one-step approaches to estimate optimal portfolio weights. Our paper has four main results: 1) Our one-step regression approach is the best unbiased weight estimator. 2) The estimation risk for this best unbiased estimator is large. 3) (Biased) shrinkage estimators lead to portfolios with smaller out-of-sample return variances. 4) Our one-step shrinkage estimator beats the two step shrinkage approach proposed by Ledoit and Wolf (2003) significantly. The results 1 and 2 are shown analytically. The results 3 and 4 are derived from an extensive simulation study.

Keywords: Global Minimum Variance Portfolio, Estimation Risk

JEL Classification: C22, G11