Markowitz (1952, 1959) laid down the ground-breaking work on themean-variance analysis. Under his framework, the theoretical optimal allocationvector can be very different from the estimated one for large portfolios due tothe intrinsic difficulty of estimating a vast covariance matrix and returnvector. This can result in adverse performance in portfolio selected based onempirical data due to the accumulation of estimation errors. We address thisproblem by introducing the gross-exposure constrained mean-variance portfolioselection. We show that with gross-exposure constraint the theoretical optimalportfolios have similar performance to the empirically selected ones based onestimated covariance matrices and there is no error accumulation effect fromestimation of vast covariance matrices. This gives theoretical justification tothe empirical results in Jagannathan and Ma (2003). We also show that theno-short-sale portfolio is not diversified enough and can be improved byallowing some short positions. As the constraint on short sales relaxes, thenumber of selected assets varies from a small number to the total number ofstocks, when tracking portfolios or selecting assets. This achieves the optimalsparse portfolio selection, which has close performance to the theoreticaloptimal one. Among 1000 stocks, for example, we are able to identify alloptimal subsets of portfolios of different sizes, their associated allocationvectors, and their estimated risks. The utility of our new approach isillustrated by simulation and empirical studies on the 100 Fama-Frenchindustrial portfolios and the 400 stocks randomly selected from Russell 3000.
展开▼
