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Sparse Portfolio Selection via Quasi-Norm Regularization

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  • Caihua Chen
  • Xindan Li
  • Caleb Tolman
  • Suyang Wang
  • Yinyu Ye
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    Abstract

    In this paper, we propose $\ell_p$-norm regularized models to seek near-optimal sparse portfolios. These sparse solutions reduce the complexity of portfolio implementation and management. Theoretical results are established to guarantee the sparsity of the second-order KKT points of the $\ell_p$-norm regularized models. More interestingly, we present a theory that relates sparsity of the KKT points with Projected correlation and Projected Sharpe ratio. We also design an interior point algorithm to obtain an approximate second-order KKT solution of the $\ell_p$-norm models in polynomial time with a fixed error tolerance, and then test our $\ell_p$-norm modes on S&P 500 (2008-2012) data and international market data.\ The computational results illustrate that the $\ell_p$-norm regularized models can generate portfolios of any desired sparsity with portfolio variance and portfolio return comparable to those of the unregularized Markowitz model with cardinality constraint. Our analysis of a combined model lead us to conclude that sparsity is not directly related to overfitting at all. Instead, we find that sparsity moderates overfitting only indirectly. A combined $\ell_1$-$\ell_p$ model shows that the proper choose of leverage, which is the amount of additional buying-power generated by selling short can mitigate overfitting; A combined $\ell_2$-$\ell_p$ model is able to produce extremely high performing portfolios that exceeded the 1/N strategy and all $\ell_1$ and $\ell_2$ regularized portfolios.

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    File URL: http://arxiv.org/pdf/1312.6350
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1312.6350.

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    Date of creation: Dec 2013
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    Handle: RePEc:arx:papers:1312.6350

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    Web page: http://arxiv.org/

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    1. Robert C. Merton, 1980. "On Estimating the Expected Return on the Market: An Exploratory Investigation," NBER Working Papers 0444, National Bureau of Economic Research, Inc.
    2. Victor DeMiguel & Lorenzo Garlappi & Raman Uppal, 2009. "Optimal Versus Naive Diversification: How Inefficient is the 1-N Portfolio Strategy?," Review of Financial Studies, Society for Financial Studies, vol. 22(5), pages 1915-1953, May.
    3. Brodie, Joshua & Daubechies, Ingrid & De Mol, Christine & Giannone, Domenico & Loris, Ignace, 2008. "Sparse and stable Markowitz portfolios," Working Paper Series 0936, European Central Bank.
    4. Shefrin, Hersh & Statman, Meir, 2000. "Behavioral Portfolio Theory," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 35(02), pages 127-151, June.
    5. 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.
    6. Massimo Guidolin & Francesca Rinaldi, 2010. "Ambiguity in asset pricing and portfolio choice: a review of the literature," Working Papers 2010-028, Federal Reserve Bank of St. Louis.
    7. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, 03.
    8. 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.
    9. Dimitris Bertsimas & Romy Shioda, 2009. "Algorithm for cardinality-constrained quadratic optimization," Computational Optimization and Applications, Springer, vol. 43(1), pages 1-22, May.
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