Random matrix approach to cross correlations in financial data

Vasiliki Plerou, Parameswaran Gopikrishnan, Bernd Rosenow, Luís A. Nunes Amaral, Thomas Guhr, and H. Eugene Stanley
Phys. Rev. E 65, 066126 – Published 27 June 2002
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Abstract

We analyze cross correlations between price fluctuations of different stocks using methods of random matrix theory (RMT). Using two large databases, we calculate cross-correlation matrices C of returns constructed from (i) 30-min returns of 1000 US stocks for the 2-yr period 1994–1995, (ii) 30-min returns of 881 US stocks for the 2-yr period 1996–1997, and (iii) 1-day returns of 422 US stocks for the 35-yr period 1962–1996. We test the statistics of the eigenvalues λi of C against a “null hypothesis” — a random correlation matrix constructed from mutually uncorrelated time series. We find that a majority of the eigenvalues of C fall within the RMT bounds [λ,λ+] for the eigenvalues of random correlation matrices. We test the eigenvalues of C within the RMT bound for universal properties of random matrices and find good agreement with the results for the Gaussian orthogonal ensemble of random matrices—implying a large degree of randomness in the measured cross-correlation coefficients. Further, we find that the distribution of eigenvector components for the eigenvectors corresponding to the eigenvalues outside the RMT bound display systematic deviations from the RMT prediction. In addition, we find that these “deviating eigenvectors” are stable in time. We analyze the components of the deviating eigenvectors and find that the largest eigenvalue corresponds to an influence common to all stocks. Our analysis of the remaining deviating eigenvectors shows distinct groups, whose identities correspond to conventionally identified business sectors. Finally, we discuss applications to the construction of portfolios of stocks that have a stable ratio of risk to return.

  • Received 1 August 2001

DOI:https://doi.org/10.1103/PhysRevE.65.066126

©2002 American Physical Society

Authors & Affiliations

Vasiliki Plerou1,2,*, Parameswaran Gopikrishnan1, Bernd Rosenow1,3, Luís A. Nunes Amaral1, Thomas Guhr4,5, and H. Eugene Stanley1

  • 1Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215
  • 2Department of Physics, Boston College, Chestnut Hill, Massachusetts 02167
  • 3Department of Physics, Harvard University, Cambridge, Massachusetts 02138
  • 4Max-Planck-Institute for Nuclear Physics, D-69029 Heidelberg, Germany
  • 5Matematisk Fysik, LTH, Lunds Universitet, Lund, Sweden

  • *Corresponding author. Email address: plerou@cgl.bu.edu

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Issue

Vol. 65, Iss. 6 — June 2002

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