Abstract: Portfolio managers need to estimate risk for many assets simultaneously with a limited number of useful observations. The standard approach is to do this using factor models, which reduce the number of variables that need to be estimated in the resulting structured covariance matrix. Even in a one-factor setting, there remains the open problem of finding a good estimate for the leading eigenvector – usually called beta -- representing the loadings on the single factor.

We describe how to apply a statistical approach known as shrinkage to the novel setting of eigenvectors of unknown matrices. We can do so in a way that is customized to the particular constraints of a portfolio optimization problem, resulting in an estimated portfolio that is quantifiably better than one obtained by standard principal component analysis. This is joint work with Lisa Goldberg and Hubeyb Gurdogan, and closely connected to work of Alex Shkolnik and Youhong Lee.