Narrow Results

We study the Markowitz mean-variance portfolio selection model under three types of regularizations: single-norm regularizations on individual stocks, mixed-norm regularizations on stock groups, and composite regularizations that combine the single-norm and mixed-norm regularizations. With mixed-norm regularizations incorporated, our model can accomplish group and stock selections simultaneously. Our empirical results using both US and global equity market data show that compared to the classical mean-variance portfolio, almost all regularized portfolios have better out-of-sample risk-adjusted performance measured by Sharpe ratio. In addition, stock selection and group screening accomplished by adding ${\ell }_1$ and ${\ell }_{2,1}$ regularizations respectively can lead to decreased volatility, turnover rate, and leverage ratio. Yet there are instances in which diversifying across different groups is more favorable, depending on the grouping methods. Moreover, we find a positive correlation between portfolio turnover and leverage. Heavily leveraged portfolios also have high turnover rates and thus high transaction costs.