High-dimensional Markowitz portfolio optimization problem: empirical comparison of covariance matrix estimators

ABSTRACT

We compare the performance of recently developed regularized covariance matrix estimators for Markowitz's portfolio optimization and of the minimum variance portfolio (MVP) problem in particular. We focus on seven estimators that are applied to the MVP problem in the literature; three regularize the eigenvalues of the sample covariance matrix, and the other four assume the sparsity of the true covariance matrix or its inverse. Comparisons are made with two sets of long-term S&P 500 stock return data that represent two extreme scenarios of active and passive management. The results show that the MVPs with sparse covariance estimators have high Sharpe ratios but that the naive diversification (also known as the ‘uniform (on market share) portfolio’) still performs well in terms of wealth growth.

KEYWORDS:

• Markowitz's portfolio optimization
• minimum variance portfolio
• high-dimensional covariance matrix
• S&P500 data

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 62H99
• 62P20

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 This subtly tailored problem requires the covariance matrix term to be positive definite in principle, which highlights the usefulness of the estimator that is presented in Xue et al. [8].

Additional information

Funding

This work was supported by National Research Foundation of Korea [ NRF-2017R1A2B2012264].