Principal Eigenportfolios for U.S. Equities

Abstract

We analyze portfolios constructed from the principal eigenvector of the equity re- turns’ correlation matrix and compare these portfolios with the capitalization weighted market portfolio. It is well known empirically that principal eigenportfolios are a good proxy for the market portfolio. We quantify this property through the large- dimensional asymptotic analysis of a spike model with diverging top eigenvalue, com- prised of a rank-1 matrix and a random matrix. We show that, in this limit, the top eigenvector of the correlation matrix is close to the vector of market betas divided component-wise by returns standard deviation. Historical returns data are generally consistent with this analysis of the correspondence between the top eigenportfolio and the market portfolio. We further examine this correspondence using eigenvectors ob- tained from hierarchically constructed tensors where stocks are separated into their respective industry sectors. This hierarchical approach results in a principal factor whose portfolio weights are all positive for a greater percentage of time compared to the weights of the vanilla eigenportfolio computed from the correlation matrix. Re- turns from hierarchical construction are also more robust with respect to the duration of the time window used for estimation. All principal eigenportfolios that we observe have returns that exceed those of the market portfolio between 1994 and 2020. We attribute these excess returns to the brief periods where short holdings are more than a small percentage of portfolio weight.

Keywords: Eigenportfolios, Principal Component Analysis, Tensor Decomposition

JEL Classification: C20, G10