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A random matrix theory approach to financial cross-correlations

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  • Plerou, V
  • Gopikrishnan, P
  • Rosenow, B
  • Amaral, L.A.N
  • Stanley, H.E

Abstract

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.

Suggested Citation

  • Plerou, V & Gopikrishnan, P & Rosenow, B & Amaral, L.A.N & Stanley, H.E, 2000. "A random matrix theory approach to financial cross-correlations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 374-382.
  • Handle: RePEc:eee:phsmap:v:287:y:2000:i:3:p:374-382
    DOI: 10.1016/S0378-4371(00)00376-9
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    1. repec:kap:compec:v:50:y:2017:i:3:d:10.1007_s10614-016-9589-9 is not listed on IDEAS
    2. Sharifi, S. & Crane, M. & Shamaie, A. & Ruskin, H., 2004. "Random matrix theory for portfolio optimization: a stability approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 335(3), pages 629-643.
    3. Barunik, Jozef & Vacha, Lukas, 2010. "Monte Carlo-based tail exponent estimator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(21), pages 4863-4874.
    4. Eterovic, Nicolas A. & Eterovic, Dalibor S., 2013. "Separating the wheat from the chaff: Understanding portfolio returns in an emerging market," Emerging Markets Review, Elsevier, vol. 16(C), pages 145-169.
    5. Fabio Caccioli & Imre Kondor & G'abor Papp, 2015. "Portfolio Optimization under Expected Shortfall: Contour Maps of Estimation Error," Papers 1510.04943, arXiv.org.
    6. Frank W. K. Firk, 2014. "Analyses of Statistical Structures in Economic Indices," Papers 1501.02216, arXiv.org.
    7. Thomas Lux, 2009. "Applications of Statistical Physics in Finance and Economics," Chapters,in: Handbook of Research on Complexity, chapter 9 Edward Elgar Publishing.
    8. Lux, Thomas, 2008. "Applications of statistical physics in finance and economics," Kiel Working Papers 1425, Kiel Institute for the World Economy (IfW).
    9. Ormerod, Paul & Mounfield, Craig, 2002. "The convergence of European business cycles 1978–2000," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 307(3), pages 494-504.
    10. Sandoval, Leonidas & Franca, Italo De Paula, 2012. "Correlation of financial markets in times of crisis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(1), pages 187-208.
    11. Ormerod, Paul, 2008. "Random Matrix Theory and Macro-Economic Time-Series: An Illustration Using the Evolution of Business Cycle Synchronisation, 1886-2006," Economics - The Open-Access, Open-Assessment E-Journal, Kiel Institute for the World Economy (IfW), vol. 2, pages 1-10.
    12. Daly, J. & Crane, M. & Ruskin, H.J., 2008. "Random matrix theory filters in portfolio optimisation: A stability and risk assessment," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(16), pages 4248-4260.
    13. Kenett, Dror Y. & Raddant, Matthias & Lux, Thomas & Ben-Jacob, Eshel, 2011. "Evolvement of uniformity and volatility in the stressed global financial village," Kiel Working Papers 1739, Kiel Institute for the World Economy (IfW).
    14. Jushan Bai & Shuzhong Shi, 2011. "Estimating High Dimensional Covariance Matrices and its Applications," Annals of Economics and Finance, Society for AEF, vol. 12(2), pages 199-215, November.
    15. Istvan Varga-Haszonits & Fabio Caccioli & Imre Kondor, 2016. "Replica approach to mean-variance portfolio optimization," Papers 1606.08679, arXiv.org.
    16. Varga-Haszonits, Istvan & Caccioli, Fabio & Kondor, Imre, 2016. "Replica approach to mean-variance portfolio optimization," LSE Research Online Documents on Economics 68955, London School of Economics and Political Science, LSE Library.
    17. G'abor Papp & Fabio Caccioli & Imre Kondor, 2016. "Fluctuation-bias trade-off in portfolio optimization under Expected Shortfall with $\ell_2$ regularization," Papers 1602.08297, arXiv.org.

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