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

  • Olivier Ledoit
  • Michael Wolf

Markowitz (1952) portfolio selection requires estimates of (i) the vector of expected returns and (ii) the covariance matrix of returns. Many successful proposals to address the first estimation problem exist by now. This paper addresses the second estimation problem. We promote a nonlinear shrinkage estimator of the covariance matrix that is more flexible than previous linear shrinkage estimators and has 'just the right number' of free parameters to estimate (that is, the Goldilocks principle). It turns out that this number is the same as the number of assets in the investment universe. Under certain high-level assumptions, we show that our nonlinear shrinkage estimator is asymptotically optimal for portfolio selection in the setting where the number of assets is of the same magnitude as the sample size. For example, this is the relevant setting for mutual fund managers who invest in a large universe of stocks. In addition to theoretical analysis, we study the real-life performance of our new estimator using backtest exercises on historical stock return data. We find that it performs better than previous proposals for portfolio selection from the literature and, in particular, that it dominates linear shrinkage.

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Paper provided by Department of Economics - University of Zurich in its series ECON - Working Papers with number 137.

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Date of creation: Jan 2014
Date of revision: Dec 2014
Handle: RePEc:zur:econwp:137
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  1. Ravi Jagannathan & Tongshu Ma, 2002. "Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps," NBER Working Papers 8922, National Bureau of Economic Research, Inc.
  2. Kan, Raymond & Zhou, Guofu, 2007. "Optimal Portfolio Choice with Parameter Uncertainty," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 42(03), pages 621-656, September.
  3. Gabriel Frahm & Christoph Memmel, 2010. "Dominating Estimators for Minimum-Variance Portfolios," Post-Print peer-00741629, HAL.
  4. 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.
  5. 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.
  6. 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.
  7. Olivier Ledoit & Michael Wolf, 2013. "Spectrum estimation: a unified framework for covariance matrix estimation and PCA in large dimensions," ECON - Working Papers 105, Department of Economics - University of Zurich, revised Jul 2013.
  8. 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.
  9. Michael W. Brandt & Pedro Santa-Clara & Rossen Valkanov, 2009. "Parametric Portfolio Policies: Exploiting Characteristics in the Cross-Section of Equity Returns," Review of Financial Studies, Society for Financial Studies, vol. 22(9), pages 3411-3447, September.
  10. 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.
  11. Ross, Stephen A., 1976. "The arbitrage theory of capital asset pricing," Journal of Economic Theory, Elsevier, vol. 13(3), pages 341-360, December.
  12. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, 03.
  13. 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.
  14. 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 Nov 2014.
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