Sparse and Stable Markowitz Portfolios

20 Pages Posted: 30 May 2008

See all articles by Joshua Brodie

Joshua Brodie

Princeton University

Ingrid Daubechies

Princeton University - Department of Mathematics

Christine De Mol

Free University of Brussels (VUB/ULB) - European Center for Advanced Research in Economics and Statistics (ECARES)

Domenico Giannone

Federal Reserve Banks - Federal Reserve Bank of New York; Centre for Economic Policy Research (CEPR)

Multiple version iconThere are 2 versions of this paper

Date Written: September 2007

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.

Keywords: Penalized Regression, Portfolio Choice, Sparse Portfolio

JEL Classification: C00, G11

Suggested Citation

Brodie, Joshua and Daubechies, Ingrid and De Mol, Christine and Giannone, Domenico, Sparse and Stable Markowitz Portfolios (September 2007). CEPR Discussion Paper No. DP6474. Available at SSRN: https://ssrn.com/abstract=1138587

Joshua Brodie

Princeton University ( email )

22 Chambers Street
Princeton, NJ 08544-0708
United States

Ingrid Daubechies

Princeton University - Department of Mathematics ( email )

Princeton, NJ 08544-1021
United States

Christine De Mol

Free University of Brussels (VUB/ULB) - European Center for Advanced Research in Economics and Statistics (ECARES) ( email )

Ave. Franklin D Roosevelt, 50 - C.P. 114
Brussels, B-1050
Belgium

Domenico Giannone (Contact Author)

Federal Reserve Banks - Federal Reserve Bank of New York ( email )

33 Liberty Street
New York, NY 10045
United States

Centre for Economic Policy Research (CEPR)

London
United Kingdom

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