Markowitz (1952, 1959) laid down the ground-breaking work on the mean-variance analysis. Under his framework, the theoretical optimal allocation vector can be very different from the estimated one for large portfolios due to the intrinsic difficulty of estimating a vast covariance matrix and return vector. This can result in adverse performance in portfolio selected based on empirical data due to the accumulation of estimation errors. We address this problem by introducing the gross-exposure constrained mean-variance portfolio selection. We show that with gross-exposure constraint the empirically selected optimal portfolios based on estimated covariance matrices have similar performance to the theoretical optimal portfolios and there is no error accumulation effect from estimation of vast covariance matrices. This gives theoretical justification to the empirical results in Jagannathan and Ma (2003). We also show that the no-short-sale portfolio is not diversified enough and can be improved by allowing some short positions. As the constraint on short sales relaxes, the number of selected assets gradually increases and finally reaches the total number of stocks when tracking portfolios or selecting assets. This achieves the optimal sparse portfolio selection, which has close performance to the theoretical optimal one. Among 1000 stocks, for example, we are able to identify all optimal subsets of portfolios of different sizes, their associated allocation vectors, and their estimated risks. The utility of our new approach is illustrated by simulation and empirical studies on the 100 Fama-French industrial portfolios and the 600 stocks randomly selected from Russell 3000. We also test our theory using different risk measure such as Least Absolute Deviation (LAD) and CVaR. We found in general imposing gross exposure constraint useful in obtaining the optimal portfolio.;Fama-French (1993) 3-factor model has been very successful in explaining the cross-sectional risks of the U.S. equity market. However, the selection of the 3 factors is somewhat ad-hoc. Not much research in finance has been done to find a procedure to correctly identify the most important factors from a pool of underlying factors. In this paper, we consider two extensions to Fama-French 3-factor model. Firstly, we propose a max-SCAD estimator to identify the important underlying factors for a portfolio of assets and at the same time estimate the factor loading correctly with properly chosen regularization parameters. We show the max-SCAD one step estimator enjoys the Oracle Property. We test the max-SCAD one step estimator in identifying the most important factors before the financial crisis (2006) and during the financial crisis (2008). We found in different periods, the most important underlying factors were indeed different. We also propose a local-SCAD estimator to dynamically select the most important underlying factors as time changes. It can be applied to index tracking and replication.
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