Constructing optimal sparse portfolios using regularization methods

Mean-varianceportfolioshavebeencriticizedbecauseofunsatisfyingout- of-sample performance and the presence of extreme and unstable asset weights, espe- cially 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. l1-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, l1-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 l1-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.

The ideas of Markowitz indisputably constitute a milestone in portfolio theory, even though the resulting mean-variance portfolios typically exhibit an unsatisfying out-of-sample performance, especially when the number of securities is large and that of observations is not. The bad performance is caused by estimation errors in the covariance matrix and in the expected return vector that can deposit unhindered in the portfolio weights. Recent studies show that imposing a penalty in form of a l1-norm of the asset weights regularizes the problem, thereby improving the out-of-sample performance of the optimized portfolios. Simultaneously, l1-regularization selects a subset of assets to invest in from a pool of candidates that is often very large. However, l1-regularization might lead to the construction of biased solutions. We propose to tackle this issue by considering several alternative penalties proposed in non-financial contexts. Moreover we propose a simple new type of penalty that explicitly considers financial information. We show empirically that these alternative penalties can lead to the construction of portfolios with superior out-of-sample performance in comparison to the state-of-the-art l1-regularized portfolios and several standard benchmarks, especially in high dimensional problems. The empirical analysis is conducted with various U.S.-stock market datasets.