Constructing optimal sparse portfolios using regularization methods

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. $$\ell _{1}$$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, $$\ell _{1}$$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 $$\ell _{1}$$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.