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Sparse and Stable Markowitz Portfolios

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  • Brodie, Joshua
  • Daubechies, Ingrid
  • De Mol, Christine
  • Giannone, Domenico

Abstract

The Markowitz mean-variance optimizing framework has served as the basis for modern portfolio theory for more than 50 years. However, efforts to translate this theoretical foundation into a viable portfolio construction algorithm have been plagued by technical difficulties stemming from the instability of the original optimization problem with respect to the available data. In this paper we address these issues of estimation error by regularizing the Markowitz objective function through the addition of a penalty proportional to the sum of the absolute values of the portfolio weights (l1 penalty). This penalty stabilizes the optimization problem, encourages sparse portfolios, and facilitates treatment of transaction costs in a transparent way. We implement this methodology using the Fama and French 48 industry portfolios as our securities. Using only a modest amount of training data, we construct portfolios whose out-of-sample performance, as measured by Sharpe ratio, is consistently and significantly better than that of the naïve portfolio comprising equal investments in each available asset. In addition to their excellent performance, these portfolios have only a small number of active positions, a highly desirable attribute for real life applications. We conclude by discussing a collection of portfolio construction problems which can be naturally translated into optimizations involving l1 penalties and which can thus be tackled by algorithms similar to those discussed here.

Suggested Citation

  • Brodie, Joshua & Daubechies, Ingrid & De Mol, Christine & Giannone, Domenico, 2007. "Sparse and Stable Markowitz Portfolios," CEPR Discussion Papers 6474, C.E.P.R. Discussion Papers.
  • Handle: RePEc:cpr:ceprdp:6474
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    1. Brodie, Joshua & Daubechies, Ingrid & De Mol, Christine & Giannone, Domenico, 2007. "Sparse and Stable Markowitz Portfolios," CEPR Discussion Papers 6474, C.E.P.R. Discussion Papers.
    2. 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 1651-1684, August.
    3. De Mol, Christine & Giannone, Domenico & Reichlin, Lucrezia, 2006. "Forecasting Using a Large Number of Predictors: Is Bayesian Regression a Valid Alternative to Principal Components?," CEPR Discussion Papers 5829, C.E.P.R. Discussion Papers.
    4. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, March.
    5. De Mol, Christine & Giannone, Domenico & Reichlin, Lucrezia, 2008. "Forecasting using a large number of predictors: Is Bayesian shrinkage a valid alternative to principal components?," Journal of Econometrics, Elsevier, vol. 146(2), pages 318-328, October.
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    Citations

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    Cited by:

    1. Nikolaus Hautsch & Stefan Voigt, 2017. "Large-Scale Portfolio Allocation Under Transaction Costs and Model Uncertainty," Papers 1709.06296, arXiv.org.
    2. Brodie, Joshua & Daubechies, Ingrid & De Mol, Christine & Giannone, Domenico, 2007. "Sparse and Stable Markowitz Portfolios," CEPR Discussion Papers 6474, C.E.P.R. Discussion Papers.
    3. Gianbiagio Curato & Jim Gatheral & Fabrizio Lillo, 2014. "Optimal execution with nonlinear transient market impact," Papers 1412.4839, arXiv.org.
    4. Briec, Walter & Kerstens, Kristiaan & Van de Woestyne, Ignace, 2013. "Portfolio selection with skewness: A comparison of methods and a generalized one fund result," European Journal of Operational Research, Elsevier, vol. 230(2), pages 412-421.
    5. Giovanni Bonaccolto & Massimiliano Caporin & Sandra Paterlini, 2018. "Asset allocation strategies based on penalized quantile regression," Computational Management Science, Springer, vol. 15(1), pages 1-32, January.
    6. Mohammed Bouaddi & Abderrahim Taamouti, 2012. "Portfolio risk management in a data-rich environment," Financial Markets and Portfolio Management, Springer;Swiss Society for Financial Market Research, vol. 26(4), pages 469-494, December.
    7. Jianqing Fan & Jingjin Zhang & Ke Yu, 2008. "Asset Allocation and Risk Assessment with Gross Exposure Constraints for Vast Portfolios," Papers 0812.2604, arXiv.org.
    8. Fan, Jianqing & Liao, Yuan & Shi, Xiaofeng, 2015. "Risks of large portfolios," Journal of Econometrics, Elsevier, vol. 186(2), pages 367-387.
    9. Fabio Caccioli & Imre Kondor & Matteo Marsili & Susanne Still, 2014. "$L_p$ regularized portfolio optimization," Papers 1404.4040, arXiv.org.
    10. Imre Kondor, 2014. "Estimation Error of Expected Shortfall," Papers 1402.5534, arXiv.org.
    11. Akiko Takeda & Mahesan Niranjan & Jun-ya Gotoh & Yoshinobu Kawahara, 2013. "Simultaneous pursuit of out-of-sample performance and sparsity in index tracking portfolios," Computational Management Science, Springer, vol. 10(1), pages 21-49, February.
    12. Björn Fastrich & Sandra Paterlini & Peter Winker, 2014. "Cardinality versus q -norm constraints for index tracking," Quantitative Finance, Taylor & Francis Journals, vol. 14(11), pages 2019-2032, November.
    13. Carlos Castro, 2010. "Portfolio choice under local industry and country factors," Financial Markets and Portfolio Management, Springer;Swiss Society for Financial Market Research, vol. 24(4), pages 353-393, December.
    14. Ignace Loris & Caroline Verhoeven, 2013. "An iterative algorithm for sparse and constrained recovery with applications to divergence-free current reconstructions in magneto-encephalography," Computational Optimization and Applications, Springer, vol. 54(2), pages 399-416, March.
    15. Jun-ya Gotoh & Akiko Takeda, 2011. "On the role of norm constraints in portfolio selection," Computational Management Science, Springer, vol. 8(4), pages 323-353, November.
    16. C. Gourieroux & A. Monfort, 2013. "Granularity Adjustment for Efficient Portfolios," Econometric Reviews, Taylor & Francis Journals, vol. 32(4), pages 449-468, December.
    17. Mayr, Johannes, 2010. "Forecasting Macroeconomic Aggregates," Munich Dissertations in Economics 11140, University of Munich, Department of Economics.
    18. Oliver Hülsewig & Johannes Mayr & Timo Wollmershäuser, 2008. "Forecasting Euro Area Real GDP: Optimal Pooling of Information," CESifo Working Paper Series 2371, CESifo Group Munich.
    19. Enzo Busseti & Fabrizio Lillo, 2012. "Calibration of optimal execution of financial transactions in the presence of transient market impact," Papers 1206.0682, arXiv.org.
    20. Caihua Chen & Xindan Li & Caleb Tolman & Suyang Wang & Yinyu Ye, 2013. "Sparse Portfolio Selection via Quasi-Norm Regularization," Papers 1312.6350, arXiv.org.
    21. Serge Darolles & Christian Gouriéroux & Emmanuelle Jay, 2012. "Robust Portfolio Allocation with Systematic Risk Contribution Restrictions," Working Papers 2012-35, Center for Research in Economics and Statistics.
    22. Björn Fastrich & Peter Winker, 2014. "Combining Forecasts with Missing Data: Making Use of Portfolio Theory," Computational Economics, Springer;Society for Computational Economics, vol. 44(2), pages 127-152, August.
    23. Yu-Min Yen, 2010. "A Note on Sparse Minimum Variance Portfolios and Coordinate-Wise Descent Algorithms," Papers 1005.5082, arXiv.org, revised Sep 2013.
    24. Briec, Walter & Kerstens, Kristiaan & Van de Woestyne, Ignace, 2011. "Portfolio Selection with Skewness: A Comparison and a Generalized Two Fund Separation Result," Working Papers 2011/09, Hogeschool-Universiteit Brussel, Faculteit Economie en Management.
    25. Jun-ya Gotoh & Akiko Takeda & Rei Yamamoto, 2014. "Interaction between financial risk measures and machine learning methods," Computational Management Science, Springer, vol. 11(4), pages 365-402, October.

    More about this item

    Keywords

    Penalized Regression; Portfolio Choice; Sparse Portfolio;

    JEL classification:

    • C00 - Mathematical and Quantitative Methods - - General - - - General
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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