Asset Allocation and Risk Assessment with Gross Exposure Constraints for Vast Portfolios

Abstract

Markowitz (1952Markowitz ( , 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.

High dimensional covariance matrix estimation using a factor model

“…This quantity appears in risk assessment as in Fan, Zhang and Yu (2008). For any portfolio with allocation vector w, the true portfolio variance and the estimated one are given by w ′ Σw and w ′ Σ T w, respectively.…”

Pre-averaging estimators of the ex-post covariance matrix in noisy diffusion models with non-synchronous data

“…Still, there are some problems in financial economics, in which it is only the accuracy of the estimator that matters and positive semi-definiteness is less important, for example in asset allocation and risk management under gross exposure constraints (e.g. Fan et al, 2009). Moreover, our empirical results show that (HY [Y ] (k,l) n ) 1≤k,l≤d does not fail to be positive semi-definite on a single day for the d = 5-dimensional vector of asset prices considered there.…”

The Merit of High-Frequency Data in Portfolio Allocation

“…Finally, as in Carrasco and Noumon (2010), the portfolio allocation problem can be expanded to explicitly account for L 1 constraints on portfolio weights, as, e.g., studied by Fan et al (2008b), or to consider alternative regularization techniques. 43…”