演講摘要 : Markowitz (1952, 1959) laid down the ground-breaking work on mean-variance analysis without gross exposure constraints. Under this framework, the theoretical optimal allocation vector can be different from the estimated one due to intrinsic difficulty of estimating a large covariance matrix and return vector. This can result in adverse performance in portfolio selected based on empirical data due to noise accumulation on estimation errors (Jagannathan and Ma, 2003; Fan, Fan and Lv, 2008). We address this problem by introducing the gross-exposure constrained mean-variance portfolio selection. We show that with gross-exposure constraint the theoretical optimal portfolios have similar performance as empirically selected ones based on estimated covariance matrices and there is no noise accumulation effect from estimation of covariance matrices. This gives theoretical justification to the empirical results in Jagannathan and Ma (2003). As the constraint on short sales relaxes, the number of selected assets varies from 1 to all stocks, when tracking or selection portfolios. This achieves the optimal sparse portfolio selection, which has close performance to the theoretical optimal. Among 2000 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.
