Constructing optimal sparse portfolios using regularization methods

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.