We analyze portfolios constructed from the principal eigenvector of the equity returns' correlation matrix and compare how well these portfolios track 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, which is comprised 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 variance. Historical returns data supports this analytical explanation for the correspondence between the top eigenportfolio and the market portfolio. We further examine this correspondence using eigenvectors obtained from hierarchically constructed tensors where stocks are separated into their respective industry sectors. This hierarchical approach provides robustness in eigenportfolio construction for a large number of equity returns when a shortened time window is used. For portfolios constructed using a rolling window of only one month of daily returns, our study shows improved tracking between the returns of the market portfolio and those from hierarchically constructed portfolios.
Bio: Andrew Papanicolaou is an assistant professor in the Department of Mathematics at North Carolina State University. He has previously held position at NYU's Tandon School of Engineering, The School of Mathematics at The University of Sydney, and a postdoc position in the Department of ORFE at Princeton. His PhD is in applied mathematics from Brown University. He holds a MS in Financial Mathematics from the University of Southern California and a BS in Mathematical Sciences from the University of California at Santa Barbara.