Spectrally-Corrected Estimation for High-Dimensional Markowitz Mean-Variance Optimization

This paper considers the portfolio problem for high dimensional data when the dimension and size are both large.We analyze the traditional Markowitz mean-variance (MV) portfolio by large dimension matrix theory, and find the spectral distribution of the sample covariance is the main factor to make the expected return of the traditional MV portfolio overestimate the theoretical MV portfolio. A correction is suggested to the spectral construction of the sample covariances to be the sample spectrally corrected covariance, and to improve the traditional MV portfolio to be spectrally corrected. In the expressions of the expected return and risk on the MV portfolio, the population covariance matrix is always a quadratic form, which will direct MV portfolio estimation. We provide the limiting behavior of the quadratic form with the sample spectrally-corrected covariance matrix, and explain the superior performance to the sample covariance as the dimension increases to infinity proportionally with the sample size. Moreover, this paper deduces the limiting behavior of the expected return and risk on the spectrally-corrected MV portfolio, and illustrates the superior properties of the spectrally-corrected MV portfolio. In simulations, we compare the spectrally-corrected estimates with the traditional and bootstrap-corrected estimates, and show the performance of the spectrally-corrected estimates are the best in portfolio returns and portfolio risk. We also compare the performance of the new proposed estimation with different optimal portfolio estimates for real data from S&P 500. The empirical findings are consistent with the theory developed in the paper.