Sparse and Stable Markowitz Portfolios
AbstractThe 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.
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Bibliographic InfoPaper provided by C.E.P.R. Discussion Papers in its series CEPR Discussion Papers with number 6474.
Date of creation: Sep 2007
Date of revision:
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Other versions of this item:
- Brodie, Joshua & Daubechies, Ingrid & De Mol, Christine & Giannone, Domenico & Loris, Ignace, 2008. "Sparse and stable Markowitz portfolios," Working Paper Series 0936, European Central Bank.
- Joshua Brodie & Ingrid Daubechies & Christine De Mol & Domenico Giannone & Ignace Loris, 2007. "Sparse and stable Markowitz portfolios," Papers 0708.0046, arXiv.org, revised May 2008.
- C00 - Mathematical and Quantitative Methods - - General - - - General
- G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
This paper has been announced in the following NEP Reports:
- NEP-ALL-2007-09-24 (All new papers)
- NEP-CMP-2007-09-24 (Computational Economics)
- NEP-FMK-2007-09-24 (Financial Markets)
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