Can Covariance Matrix Refinements Alleviate the Contradiction of Mean-Variance Efficiency and Diversification of Portfolio Selection?

In this paper, through the prism of refining covariance matrices, we study the mean-variance efficiency and diversification contradiction that the high concentration in a few securities in portfolio selection implicates the incompatibility of efficiency and diversification which are two corner stones of modern finance. Typical treatments include refining covariance matrices and imposing portfolio constraints. Unfortunately, the existing studies usually suffer from simplified models, small-scale computation, and crude representation of efficient frontiers. To make thing worse, inequality constraints of portfolio optimization such as restrictions of short sales knock out the possibility of closed-form type of optima, so computational methodology based on empirical data can be the only way to study the contradiction. We sample 13, 26, 52, 125, 326, and 687 Chinese stocks from 2003-2005 and 2006-2008 periods, deploy factor models and equal correlation coefficient models to refine covariance matrices, and utilize parametric quadratic programming to obtain precise and complete efficient frontiers of portfolio optimization. We find that contrary to existing beliefs, the refinements can not alleviate the contradiction. Our systematic sampling, large-scale computation, and precise representations of efficient frontiers are original in the area in China, which alone make the paper unique. The research can help both individual and institutional investors balance efficiency and diversification and consequently serve portfolio theory and finance industry of our motherland, especially in the turmoil world-wide financial crises. Moreover, our methodology is based on the latest computational extensions of portfolio optimization and can be deployed to stock markets worldwide to draw more comprehensive conclusions.