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

Abstract

The portfolio problem for high dimensional data when the dimension and size are both large is considered. The traditional Markowitz mean-variance (MV) portfolio by large dimension matrix theory is analyzed, and it is found that 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. Therefore, a new spectrally corrected method is introduced to correct the spectral elements of the sample covariance to a sample spectrally-corrected covariance, by which the spectrally-corrected portfolio and the corresponding return and risk are provided naturally. Moreover, the limiting behavior of the expected return and risk on the spectrally-corrected MV portfolio is deduced and the superior properties of the spectrally-corrected MV portfolio are illustrated. In simulations, the spectrally-corrected estimates get the best performance in both portfolio return and portfolio risk. The comparisons of their performance by using the S&P 500 data show the superiority of the proposed spectrally-corrected estimates get over the traditional and bootstrap-corrected estimates. Then, the empirical analysis shows that consistent with the theory developed, the proposed spectrally-corrected estimates outperform both the traditional and bootstrap-corrected estimates. Further, the findings show that all risk-averters will get improvement in portfolio returns or risk-adjusted portfolio returns, by adopting our proposed methods.