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 problemwith 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 an \ell_1 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 na\"{i . 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 \ell_1 penalties and which can thus be tackled by algorithms similar to those discussed here .
We consider the problem of portfolio selection within the classical Markowitz mean-variance optimizing framework , which has served as the basis for modern portfolio theory for more than 50 years. To stabilize the problem, we propose to add to the Markowitz objective function a penalty which is proportional to the sum of the absolute values of the portfolio weights ( \ell_1 penalty ) . This penalty stabilizes the optimization problem, automatically encourages sparse portfolios, and facilitates an effective treatment of transaction costs . We implement our methodology using as our securities two sets of portfolios constructed by Fama and French: the 48 industry portfolios and 100 portfolios formed on size and book-to-market . In addition to their excellent performance, these portfolios have only a small number of active positions, a desirable feature for small investors, for whom the fixed overhead portion of the transaction cost is not negligible .
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We consider the problem of portfolio selection within the classical
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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 problemwith respect to the available data. In this paper we address these issues of estimation error by regularizing the
To stabilize the problem, we propose to add to the
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through the addition of an
a penalty which is proportional to the sum of the absolute values of the portfolio weights (
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automatically
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an effective
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in a transparent way
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this methodology using the Fama and French 48 industry portfolios
our methodology using
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. 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 na\"{i
two sets of portfolios constructed by Fama and French: the 48 industry portfolios and 100 portfolios formed on size and book-to-market
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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 \ell_1 penalties and which can thus be tackled by algorithms similar to those discussed here
desirable feature for small investors, for whom the fixed overhead portion of the transaction cost is not negligible