title: Risks of Large Portfolios.
abstract: The risk of a large portfolio is often estimated by substituting a good estimator of the volatility matrix. However, the accuracy of such a risk estimator is largely unknown. We study factor-based risk estimators under a large amount of assets, and introduce a high-confidence level upper bound (H-CLUB) to assess the estimation. The H-CLUB is constructed using the confidence interval of risk estimators with either known or unknown factors. We derive the limiting distribution of the estimated risks in high dimensionality. We find that when the dimension is large, the factor-based risk estimators have the same asymptotic variance no matter whether the factors are known or not, which is slightly smaller than that of the sample covariance-based estimator. Numerically, H-CLUB outperforms the traditional crude bounds, and provides an insightful risk assessment. In addition, our simulated results quantify the relative error in the risk estimation, which is usually negligible using 3-month daily data.
authors.affiliation: Department of Operations Research and Financial Engineering, Princeton University ; Bendheim Center for Finance, Princeton University.
authors.forename: Jianqing
authors.initials: J
authors.lastname: Fan
authors.affiliation: Department of Mathematics, University of Maryland.
authors.forename: Yuan
authors.initials: Y
authors.lastname: Liao
authors.affiliation: Department of Operations Research and Financial Engineering, Princeton University.
authors.forename: Xiaofeng
authors.initials: X
authors.lastname: Shi
keywords: High dimensionality; factor models; principal components; sparse matrix; volatility