Mathematical Sciences Department Virtual Financial Math Seminar: Alex Shkolnik, University of California, Santa Barbara

Title: Portfolio Selection and Stein's paradox.

Abstract: 
We describe a version of Stein's paradox for eigenvectors of a sample covariance matrix to show, much like Charles Stein did for the sample mean in the 1950s, that in high dimensions, provably better estimators exist. At the core of the paradox, we find that relative to their population counterparts, sample eigenvectors carry a systematic bias that always accompanies high dimensional data. We prove that a recipe proposed by James and Stein more than six decades ago provides a bias correction to always produce a "better" set of eigenvectors. We apply these eigenvector estimators to the classic Markowitz problem in finance, which formulates portfolio construction as a trade-off between the mean and variance of return. It is proved that the risk of the resulting portfolios vanishes as the number of securities tends to infinity. We conclude with a discussion of the connections between Stein's paradox and portfolio selection. These two seemingly unrelated ideas, that date back to the 1950s, turn out to have much in common from the perspective of high dimensional statistics.

Bio:
Alex Shkolnik is an Assistant Professor at the Department of Statistics and Applied Probability at the University of California, Santa Barbara and a Research Fellow at the Consortium for Data Analytics in Risk at the University of California, Berkeley where he was a postdoctoral scholar. Alex received his PhD in computational mathematics and engineering from Stanford University in 2015. His research interests include Monte Carlo simulation, high-dimensional statistics and quantitative financial risk management.

Zoom link: https://wpi.zoom.us/j/93373231429