Financial applications of random matrix theory: a short review

Abstract

This article reviews some applications of random matrix theory (RMT) in the context of financial markets and econometric models, with emphasis on various theoretical results (for example, the Marčenko-Pastur spectrum and its various generalizations, random singular value decomposition, free matrices, largest eigenvalue statistics) as well as some concrete applications to portfolio optimization and out-of-sample risk estimation. The discussion begins with an overview of principal component analysis (PCA) of the correlation matrix, followed by an analysis of return statistics and portfolio theory. In particular, the article considers single asset returns, multivariate distribution of returns, risk and portfolio theory, and nonequal time correlations and more general rectangular correlation matrices. It also presents several RMT results on the bulk density of states that can be obtained using the concept of matrix freeness before concluding with a description of empirical correlation matrices of stock returns.