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Nonlinear shrinkage of the covariance matrix for portfolio selection: Markowitz meets Goldilocks

Author

Listed:
  • Olivier Ledoit
  • Michael Wolf

Abstract

Markowitz (1952) portfolio selection requires an estimator of the covariance matrix of returns. To address this problem, we promote a nonlinear shrinkage estimator that is more flexible than previous linear shrinkage estimators and has just the right number of free parameters (that is, the Goldilocks principle). This number is the same as the number of assets. Our nonlinear shrinkage estimator is asymptotically optimal for portfolio selection when the number of assets is of the same magnitude as the sample size. In backtests with historical stock return data, it performs better than previous proposals and, in particular, it dominates linear shrinkage.

Suggested Citation

  • Olivier Ledoit & Michael Wolf, 2014. "Nonlinear shrinkage of the covariance matrix for portfolio selection: Markowitz meets Goldilocks," ECON - Working Papers 137, Department of Economics - University of Zurich, revised Feb 2017.
    • Handle: RePEc:zur:econwp:137
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    File URL: https://www.zora.uzh.ch/id/eprint/90273/26/econwp137.pdf
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    References listed on IDEAS

    as
    1. DeMiguel, Victor & Martin-Utrera, Alberto & Nogales, Francisco J., 2013. "Size matters: Optimal calibration of shrinkage estimators for portfolio selection," Journal of Banking & Finance, Elsevier, vol. 37(8), pages 3018-3034.
    2. Ledoit, Olivier & Wolf, Michael, 2004. "A well-conditioned estimator for large-dimensional covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 365-411, February.
    3. Ravi Jagannathan & Tongshu Ma, 2003. "Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps," Journal of Finance, American Finance Association, vol. 58(4), pages 1651-1683, August.
    4. Kan, Raymond & Zhou, Guofu, 2007. "Optimal Portfolio Choice with Parameter Uncertainty," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 42(3), pages 621-656, September.
    5. Victor DeMiguel & Lorenzo Garlappi & Francisco J. Nogales & Raman Uppal, 2009. "A Generalized Approach to Portfolio Optimization: Improving Performance by Constraining Portfolio Norms," Management Science, INFORMS, vol. 55(5), pages 798-812, May.
    6. Michael W. Brandt & Pedro Santa-Clara & Rossen Valkanov, 2009. "Parametric Portfolio Policies: Exploiting Characteristics in the Cross-Section of Equity Returns," The Review of Financial Studies, Society for Financial Studies, vol. 22(9), pages 3411-3447, September.
    7. Stephen A. Ross, 2013. "The Arbitrage Theory of Capital Asset Pricing," World Scientific Book Chapters, in: Leonard C MacLean & William T Ziemba (ed.), HANDBOOK OF THE FUNDAMENTALS OF FINANCIAL DECISION MAKING Part I, chapter 1, pages 11-30, World Scientific Publishing Co. Pte. Ltd..
    8. Frahm, Gabriel & Memmel, Christoph, 2010. "Dominating estimators for minimum-variance portfolios," Journal of Econometrics, Elsevier, vol. 159(2), pages 289-302, December.
    9. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, March.
    10. Olivier Ledoit & Michael Wolf, 2013. "Optimal estimation of a large-dimensional covariance matrix under Stein’s loss," ECON - Working Papers 122, Department of Economics - University of Zurich, revised Mar 2017.
    11. Avanidhar Subrahmanyam, 2010. "The Cross†Section of Expected Stock Returns: What Have We Learnt from the Past Twenty†Five Years of Research?," European Financial Management, European Financial Management Association, vol. 16(1), pages 27-42, January.
    12. Ledoit, Oliver & Wolf, Michael, 2008. "Robust performance hypothesis testing with the Sharpe ratio," Journal of Empirical Finance, Elsevier, vol. 15(5), pages 850-859, December.
    13. Ledoit, Olivier & Wolf, Michael, 2003. "Improved estimation of the covariance matrix of stock returns with an application to portfolio selection," Journal of Empirical Finance, Elsevier, vol. 10(5), pages 603-621, December.
    14. repec:bla:jfinan:v:58:y:2003:i:4:p:1651-1684 is not listed on IDEAS
    15. Jegadeesh, Narasimhan & Titman, Sheridan, 1993. "Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency," Journal of Finance, American Finance Association, vol. 48(1), pages 65-91, March.
    16. repec:hal:journl:peer-00741629 is not listed on IDEAS
    17. Tu, Jun & Zhou, Guofu, 2011. "Markowitz meets Talmud: A combination of sophisticated and naive diversification strategies," Journal of Financial Economics, Elsevier, vol. 99(1), pages 204-215, January.
    18. Ledoit, Olivier & Wolf, Michael, 2015. "Spectrum estimation: A unified framework for covariance matrix estimation and PCA in large dimensions," Journal of Multivariate Analysis, Elsevier, vol. 139(C), pages 360-384.
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    Cited by:

    1. Hsu, Po-Hsuan & Han, Qiheng & Wu, Wensheng & Cao, Zhiguang, 2018. "Asset allocation strategies, data snooping, and the 1 / N rule," Journal of Banking & Finance, Elsevier, vol. 97(C), pages 257-269.
    2. Francesco Lautizi, 2015. "Large Scale Covariance Estimates for Portfolio Selection," CEIS Research Paper 353, Tor Vergata University, CEIS, revised 07 Aug 2015.
    3. Ledoit, Olivier & Wolf, Michael, 2017. "Numerical implementation of the QuEST function," Computational Statistics & Data Analysis, Elsevier, vol. 115(C), pages 199-223.

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    More about this item

    Keywords

    Large-dimensional asymptotics; Markowitz portfolio selection; nonlinear shrinkage;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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