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This is a summary of "Approaching Mean–Variance Efficiency for Large Portfolios," by Mengmeng Ao, Yingying Li, and Xinghua Zheng, published in the Review of Financial Studies.


The authors propose a new methodology, MAXSER, for estimating the optimal mean–variance portfolio for large portfolios with relatively small samples. This approach provides a new unconstrained regression representation of the mean–variance portfolio problem, yielding high Sharpe ratios with better control of risk. Using MAXSER, investing in individual equities can substantially improve portfolio performance.

What Is the Investment Issue?

To construct mean–variance efficient portfolios, investors need estimates for expected returns and covariances. In practice, sample estimators are used, including linear and nonlinear shrinkage estimators. These approaches produce portfolios that typically underestimate risk and produce suboptimal Sharpe ratios, however, especially in situations where the number of assets is large relative to the sample size.

The authors propose a new approach, called the maximum-Sharpe-ratio estimated and sparse regression (MAXSER) method, to estimate mean–variance efficient portfolios when the number of assets is large compared with the sample size. The advantage of MAXSER is that it is an unconstrained regression, although equivalent to the traditional mean–variance portfolio problem, which aids large portfolio selection because it can take advantage of sparse regression techniques (such as LASSO) that may set some portfolio weights to zero. Using simulation, the authors show that the MAXSER portfolio significantly outperforms other methods, including the “plug-in” portfolio (that uses historical sample means and covariances), the equally weighted portfolio, and the linear and nonlinear shrinkage portfolios, as well as several other variations of mean–variance and global minimum variance portfolios with constraints on portfolio weights.

How Did the Authors Conduct This Research?

The authors assess the performance of the MAXSER strategy based on two stock universes drawn from the Dow Jones Industrial Average (DJIA) 30 index constituents and the S&P 500 Index constituents, and for the purpose of factor investing, the Fama–French three (FF3) factors. The authors obtain the lists of DJIA 30 index constituents from CRSP and Compustat.

The authors conduct their evaluation using a rolling-window scheme in which at the beginning of each month, an asset pool is formed by including the contemporaneous DJIA constituents and the FF3 factors. For the S&P 500 constituents, the authors conduct the analysis using a rolling-window scheme similar to that for the DJIA constituents. The portfolios under comparison are rebalanced monthly, and the out-of-sample returns over the 1977 to 2016 period are recorded. At the beginning of each year, the authors randomly construct pools of 100 stocks from the S&P 500 that have complete price data for the prior 120 months and construct portfolios (weights updated every half year) using the excess returns of the 100 stocks and FF3 factors.

The authors make four specific contributions: 1) establishment of an equivalent unconstrained regression representation of the mean–variance portfolio problem; 2) estimation of the optimal portfolio when the asset pool includes only individual assets; 3) estimation of the optimal portfolio when factors are included in the asset pool; and 4) conducting simulation and empirical analysis to demonstrate that the MAXSER portfolio nearly achieves mean–variance efficiency and significantly outperforms other approaches.

What Are the Findings and Implications for Investors and Investment Professionals?

The MAXSER portfolio outperforms other portfolio construction methods, irrespective of whether transaction costs are considered: Its risk is close to the risk constraint, and it exhibits the highest Sharpe ratio. The presence of small p-values indicates that the MAXSER portfolio has a statistically significant advantage compared with the other portfolios.

About the Author(s)

Mark K. Bhasin CFA

Mark K. Bhasin, CFA, is senior vice president of Basis Investment Group, LLC, New York City.