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Constructing optimal sparse portfolios using regularization methods

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Mean-variance portfolios have been criticized because of unsatisfying out-of-sample performance and the presence of extreme and unstable asset weights, especially when the number of securities is large. The bad performance is caused by estimation errors in inputs parameters, that is the covariance matrix and the expected return vector. Recent studies show that imposing a penalty on the 1-norm of the asset weights vector (i.e. \(\ell _{1}\)-regularization) not only regularizes the problem, thereby improving the out-of-sample performance, but also allows to automatically select a subset of assets to invest in. However, \(\ell _{1}\)-regularization might lead to the construction of biased solutions. We propose a new, simple type of penalty that explicitly considers financial information and then we consider several alternative penalties, that allow to improve on the \(\ell _{1}\)-regularization approach. By using U.S.-stock market data, we show empirically that the proposed penalties can lead to the construction of portfolios with an out-of-sample performance superior to several state-of-art benchmarks, especially in high dimensional problems.

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  1. Note that sample covariance matrix estimates are singular when \({T\!<\!K}\).

  2. The LASSO relies on imposing a constraint on the \(\ell _{1}\)-norm the regression coefficients. In this paper, \(\ell _{1}\)-regularization is used synonymously.

  3. They claimed that a good penalty function should result in an estimator with three properties: unbiasedness, sparsity, and continuity. They also coined the term oracle property, which, in a nutshell, means that a strategy performs as well as when the true underlying model is known a priori.

  4. This is a very useful approach for practitioners, among whom the so-called 130/30, 120/20, and 110/10 portfolios are very popular. These portfolios consist of, e.g., 130 percent long and 30 percent short positions.

  5. Obviously, \(w8\) abbreviates weighted and \(Las\) LASSO.

  6. It is relatively easy to choose a value for \(\eta \) because it can be directly interpreted as the threshold that determines how large an absolute portfolio weight must (at least) be to be constantly penalized.

  7. The parameter \(\phi \) is simply a very small increment that prevents division by zero.

  8. In the latter case firms were sorted according to the market (ME) and the book-to-market (BE/ME) value to obtain the firm portfolios. The data sets may be downloaded from the homepage of Kenneth French (, where information on the portfolio construction methods is also provided.

  9. The largest data set consists of 1,036 assets, because in the course of data preparation, penny stocks and series with missing data were excluded.

  10. This is different for the 98 firm portfolios (see the caption of Table 2).

  11. The abscissa denotes the number of active positions instead of the values of \(\lambda \), since the scaling of the latter differs among the penalties.

  12. Note that the latter contain detailed results for five data sets and three covariance matrix estimators. These results do not only support the above findings but also provide various additional insights.


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Correspondence to S. Paterlini.

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Fastrich, B., Paterlini, S. & Winker, P. Constructing optimal sparse portfolios using regularization methods. Comput Manag Sci 12, 417–434 (2015).

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