Sparse Minimax Portfolio and Sharpe Ratio Models

In this paper, we investigate sparse portfolio selection models with a regularized lp-norm term (0 < p ≤ 1) and negatively bounded shorting constraints. We obtain some basic properties of several linear lp-sparse minimax portfolio models in terms of the regularization parameter. In particular, we introduce an l1-sparse minimax Sharpe ratio model by guaranteeing a positive denominator with a pre-selected parameter and design a parametric algorithm for finding its global solution. We carry out numerical experiments of linear lp-sparse minimax portfolio models with 1200 stocks from Hang Seng Index, Shanghai Securities Composite Index, and NASDAQ Index and compare their performance with lp-sparse mean-variance models. We test the effect of the regularization parameter and the negatively bounded shorting parameter on the level of sparsity, risk, and rate of return respectively and find that portfolios including fewer stocks of the linear lp-sparse minimax models tend to have lower risks and lower rates of return. However, for the lp-sparse mean-variance models, the corresponding changes are not so significant.