Title: Maximum likelihood estimation of covariance matrices with constraints on the efficient frontier

Authors: Hilal Yilmaz; Neil D. Pearson

Addresses: Department of Finance and Economics, St. Edward's University, 3001 Congress Ave., Austin, Texas, 78704, USA ' Department of Finance, University of Illinois at Urbana-Champaign, 419 Wohlers Hall, 1206 South Sixth Street, Champaign, IL, 61820, USA

Abstract: This paper develops an improved covariance matrix estimator in the mean-variance optimisation setting. Well-known problems with the sample covariance matrix are that it is singular when the number of observations is less than the number of assets, and can be nearly singular when the number of observations exceeds the number of assets. Therefore, using the sample covariance matrix as an input in mean-variance optimisation can result in unreasonable optimal portfolios and badly biased estimates of Sharpe ratios. We address this problem by imposing structure on the estimated covariance matrix by putting constraints on the Sharpe ratio, asset return variances, and the variance of the global minimum variance portfolio. We show that the constrained maximum likelihood estimator (CMLE) performs better than the sample covariance matrix. Moreover, when the shrinkage approach is applied to the CMLE and single index covariance matrix, it performs better than the shrinkage estimator of Ledoit and Wolf (2004).

Keywords: covariance matrix; mean-variance optimisation; constrained MLE; maximum likelihood estimation; Sharpe ratio; shrinkage estimator; global minimum variance portfolio; Monte Carlo simulation; singular; quadratic matrix equation; sample covariance matrix; efficient frontier.

DOI: 10.1504/IJCEE.2016.073352

International Journal of Computational Economics and Econometrics, 2016 Vol.6 No.1, pp.71 - 92

Available online: 15 Nov 2015 *

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