Random matrix theory approach to financial cross-correlations

It is common knowledge that any two firms in the economy are correlated. Even firms belonging to different sectors of an industry may be correlated because of `indirect' correlations. How can we analyze and understand these correlations? This article reviews recent results regarding cross-correlations between stocks. Specifically, we use methods of random matrix theory (RMT), which originated from the need to understand the interactions between the constituent elements of complex interacting systems, to analyze the cross-correlation matrix C of returns. We analyze 30-min returns of the largest 1000 US stocks for the 2-year period 1994-1995. We find that the statistics of approximately 20 of the largest eigenvalues (2%) show deviations from the predictions of RMT. To test that the rest of the eigenvalues are genuinely random, we test for universal properties such as eigenvalue spacings and eigenvalue correlations, and demonstrate that C shares universal properties with the Gaussian orthogonal ensemble of random matrices. The statistics of the eigenvectors of C confirm the deviations of the largest few eigenvalues from the RMT prediction. We also find that these deviating eigenvectors are stable in time. In addition, we quantify the number of firms that participate significantly to an eigenvector using the concept of inverse participation ratio, borrowed from localization theory.