Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2025-01-05T14:01:00.716Z Has data issue: false hasContentIssue false
  • English
  • Français

Improving Minimum-Variance Portfolios by Alleviating Overdispersion of Eigenvalues

Published online by Cambridge University Press:  24 October 2019

Fangquan Shi ,
Lianjie Shu Open the ORCID record for Lianjie Shu [Opens in a new window] ,
Aijun Yang  and
Fangyi He
Fangquan Shi
Affiliation:
Shi, [email protected], University of Macau Faculty of Business Administration
Lianjie Shu*
Affiliation:
Shu, [email protected], University of Macau Faculty of Business Administration
Aijun Yang
Affiliation:
Yang, [email protected], Nanjing Forest University College of Economics and Management
Fangyi He
Affiliation:
He, [email protected], Southwestern University of Finance and Economics School of Finance
*
Shu (corresponding author), [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In portfolio risk minimization, the inverse covariance matrix of returns is often unknown and has to be estimated in practice. Yet the eigenvalues of the sample covariance matrix are often overdispersed, leading to severe estimation errors in the inverse covariance matrix. To deal with this problem, we propose a general framework by shrinking the sample eigenvalues based on the Schatten norm. The proposed framework has the advantage of being computationally efficient as well as structure-free. The comparative studies show that our approach behaves reasonably well in terms of reducing out-of-sample portfolio risk and turnover.

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 55 , Issue 8 , December 2020 , pp. 2700 - 2731
Copyright
Copyright © Michael G. Foster School of Business, University of Washington 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

We thank the editor and an anonymous referee for their valuable comments. Shu’s work was supported in part by the Macau Science and Technology Development Fund (FDCT/0064/2018/A2) and the University Research Grant (MYRG2018-00087-FBA). He’s work was supported by the National Natural Science Foundation of China (No. 71772153).

References

Behr, P.; Guettler, A.; and Miebs, F.. “On Portfolio Optimization: Imposing the Right Constraints.” Journal of Banking and Finance, 37 (2013), 1232–1242.CrossRefGoogle Scholar
Best, M. J., and Grauer, R. R.. “On the Sensitivity of Mean-Variance-Efficient Portfolios to Changes in Asset Means: Some Analytical and Computational Results.” Review of Financial Studies, 4 (1991), 315–342.CrossRefGoogle Scholar
Cai, T. T.; Liu, W.; and Xia, Y.. “Two-Sample Test of High Dimensional Means under Dependence.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 76 (2014), 349–372.Google Scholar
Candelon, B.; Hurlin, C.; and Tokpavi, S.. “Sampling Error and Double Shrinkage Estimation of Minimum Variance Portfolios.” Journal of Empirical Finance, 19 (2012), 511–527.CrossRefGoogle Scholar
Daniels, M. J., and Kass, R. E.. “Shrinkage Estimators for Covariance Matrices.” Biometrics, 57 (2001), 1173–1184.CrossRefGoogle ScholarPubMed
DeMiguel, V.; Garlappi, L.; Nogales, F. J.; and Uppal, R.. “A Generalized Approach to Portfolio Optimization: Improving Performance by Constraining Portfolio Norms.” Management Science, 55 (2009), 798–812.CrossRefGoogle Scholar
DeMiguel, V.; Martin-Utrera, A.; and Nogales, F. J.. “Size Matters: Optimal Calibration of Shrinkage Estimators for Portfolio Selection.” Journal of Banking and Finance, 37 (2013), 3018–3034.CrossRefGoogle Scholar
Farrell, R. H.“Multivariate Inequalities.” In Multivariate Calculation. New York, NY: Springer (1985).CrossRefGoogle Scholar
Gillen, B. J.“An Empirical Bayesian Approach to Stein-Optimal Covariance Matrix Estimation.” Journal of Empirical Finance, 29 (2014), 402–420.CrossRefGoogle Scholar
Goto, S., and Xu, Y.. “Improving Mean Variance Optimization through Sparse Hedging Restrictions.” Journal of Financial and Quantitative Analysis, 50 (2015), 1415–1441.CrossRefGoogle Scholar
Jagannathan, R., and Ma, T.. “Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps.” Journal of Finance, 58 (2003), 1651–1683.CrossRefGoogle Scholar
Jobson, J. D., and Korkie, B.. “Estimation for Markowitz Efficient Portfolios.” Journal of the American Statistical Association, 75 (1980), 544–554.CrossRefGoogle Scholar
Jobson, J. D., and Korkie, R. M.. “Putting Markowitz Theory to Work.” Journal of Portfolio Management, 7 (1981), 70–74.CrossRefGoogle Scholar
Johnstone, I. M., and Lu, A. Y.. “On Consistency and Sparsity for Principal Components Analysis in High Dimensions.” Journal of the American Statistical Association, 104 (2009), 682–693.CrossRefGoogle ScholarPubMed
Jorion, P.“Bayes-Stein Estimation for Portfolio Analysis.” Journal of Financial and Quantitative Analysis, 21 (1986), 279–292.CrossRefGoogle Scholar
Kan, R., and Zhou, G.. “Optimal Portfolio Choice with Parameter Uncertainty.” Journal of Financial and Quantitative Analysis, 42 (2007), 621–656.CrossRefGoogle Scholar
Ledoit, O., and Wolf, M.. “Improved Estimation of the Covariance Matrix of Stock Returns with an Application to Portfolio Selection.” Journal of Empirical Finance, 10 (2003), 603–621.CrossRefGoogle Scholar
Ledoit, O., and Wolf, M.. “A Well-Conditioned Estimator for Large-Dimensional Covariance Matrices.” J. Multivariate Analysis, 88 (2004), 365–411.CrossRefGoogle Scholar
Ledoit, O., and Wolf, M.. “Robust Performance Hypothesis Testing with the Sharpe Ratio.” Journal of Empirical Finance, 15 (2008), 850–859.CrossRefGoogle Scholar
Ledoit, O., and Wolf, M.. “Robust Performances Hypothesis Testing with the Variance.” Working Paper No. 516, University of Zurich Institute for Empirical Research in Economics (2011).CrossRefGoogle Scholar
Levy, H., and Levy, M.. “The Benefits of Differential Variance-Based Constraints in Portfolio Optimization.” European Journal of Operational Research, 234 (2014), 372–381.CrossRefGoogle Scholar
Marčenko, V. A., and Pastur, L. A.. “Distribution of Eigenvalues for Some Sets of Random Matrices.” Mathematics of the USSR-Sbornik, 1 (1967), 457–483.CrossRefGoogle Scholar
Markowitz, H.“Portfolio Selection.” Journal of Finance, 7 (1952), 77–91.Google Scholar
Scherer, B., and Martin, R. D.. Modern Portfolio Optimization with NuOPT™, S-PLUS ® , and S+ Bayes™. New York, NY: Springer Science & Business Media (2005).CrossRefGoogle Scholar
Stein, C.“Estimation of a Covariance Matrix.” Rietz Lecture, 39th Annual IMS Meeting, Atlanta, GA, (1975).Google Scholar
Tu, J., and Zhou, G.. “Markowitz Meets Talmud: A Combination of Sophisticated and Naive Diversification Strategies.” Journal of Financial Economics, 99 (2011), 204–215.CrossRefGoogle Scholar
Warton, D. I.“Penalized Normal Likelihood and Ridge Regularization of Correlation and Covariance Matrices.” Journal of the American Statistical Association, 103 (2008), 340–349.CrossRefGoogle Scholar
Witten, D. M., and Tibshirani, R.. “Covariance-Regularized Regression and Classification for High Dimensional Problems.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 71 (2009), 615–636.CrossRefGoogle ScholarPubMed
Won, J.-H.; Lim, J.; Kim, S.-J.; and Rajaratnam, B.. “Condition-Number-Regularized Covariance Estimation.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75 (2013), 427–450.CrossRefGoogle ScholarPubMed
Xing, X.; Hu, J.; and Yang, Y.. “Robust Minimum Variance Portfolio with L-Infinity Constraints.” Journal of Banking and Finance, 46 (2014), 107–117.CrossRefGoogle Scholar