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 .
We consider the problem of portfolio selection within the classical Markowitz mean-variance framework, reformulated as a constrained least-squares regression problem. We propose to add to the objective function a penalty proportional to the sum of the absolute values of the portfolio weights . This penalty regularizes (stabilizes) the optimization problem, encourages sparse portfolios (i.e. portfolios with only few active positions), and allows to account for transaction costs. Our approach recovers as special cases the no-short-positions portfolios, but does allow for short positions in limited number. We implement this methodology on two benchmark data sets constructed by Fama and French . 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 naive evenly-weighted portfolio which constitutes, as shown in recent literature, a very tough benchmark .
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optimizing framework, which has served as the basis for modern portfolio theory for more than 50 years. To stabilize the problem, we
framework, reformulated as a constrained least-squares regression problem. We
D
Markowitz
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which is
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(\ell_1 penalty)
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stabilizes
regularizes (stabilizes)
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automatically
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, and facilitates an effective treatment of
(i.e. portfolios with only few active positions), and allows to account for
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We implement our methodology using as our securities two sets of portfolios
Our approach recovers as special cases the no-short-positions portfolios, but does allow for short positions in limited number. We implement this methodology on two benchmark data sets
R
: 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
. 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 naive evenly-weighted portfolio which constitutes, as shown in recent literature, a very tough benchmark