Sparse and stable Markowitz portfolios
Abstract
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.Keywords
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