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# Random matrix theory and financial correlations

International Journal of Theoretical and Applied Finance, (2011)

Abstract

We show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of multivariate financial time series. We find a remarkable agreement between the theoretical prediction (based on the assumption that the correlatio...More

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Introduction

- The authors show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of multivariate financial time series.
- As the authors shall show below, the smallest eigenvalues of this matrix are the most sensitive to this ‘noise’, the corresponding eigenvectors being precisely the ones that determine the least risky portfolios.
- It is important to devise methods which allows one to distinguish ‘signal’ from ‘noise’, i.e. eigenvectors and eigenvalues of the correlation matrix containing real information, from those which are devoid of any useful information, and, as such, unstable in time.

Highlights

- We show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of multivariate financial time series
- The study of correlation matrices has a long history in finance and is one of the cornerstone of Markowitz’s theory of optimal portfolios [1,2]: given a set of financial assets characterized by their average return and risk, what is the optimal weight of each asset, such that the overall portfolio provides the best return for a fixed level of risk, or the smallest risk for a given overall return?
- From this point of view, it is interesting to compare the properties of an empirical correlation matrix C to a ‘null hypothesis’ purely random matrix as one could obtain from a finite time series of strictly independent assets
- We want to compare the empirical distribution of the eigenvalues of the correlation matrix of stocks corresponding to different markets with the theoretical prediction given by Eq (0.3), based on the assumption that the correlation matrix
- The results are shown in Fig. 2 : one sees very clearly that using the empirical correlation matrix leads to a dramatic underestimation of the real risk, by overinvesting in artificially low-risk eigenvectors
- We have shown that results from the theory of random matrices is of great interest to understand the statistical structure of the empirical correlation matrices

Results

- From this point of view, it is interesting to compare the properties of an empirical correlation matrix C to a ‘null hypothesis’ purely random matrix as one could obtain from a finite time series of strictly independent assets.
- BNote that even if the ‘true’ correlation matrix Ctrue is the identity matrix, its empirical determination from a finite time series will generate non trivial eigenvectors and eigenvalues, see 8,9 .
- The authors want to compare the empirical distribution of the eigenvalues of the correlation matrix of stocks corresponding to different markets with the theoretical prediction given by Eq (0.3), based on the assumption that the correlation matrix
- The authors have studied numerically the density of eigenvalues of the correlation matrix of N = 406 assets of the S&P 500, based on daily variations during the years 1991-96, for a total of T = 1309 days.
- Several eigenvalues are still above λmax and might contain some information, thereby reducing the variance of the effectively random part of the correlation matrix.
- Since the eigenstates corresponding to the ‘noise band’ are not expected to contain real information, one should not distinguish the different eigenvalues and eigenvectors in this sector.
- The authors determine the correlation matrix using the first subperiod, ‘clean’ it, and construct the family of optimal portfolios and the corresponding efficient frontiers.

Conclusion

- The results are shown in Fig. 2 : one sees very clearly that using the empirical correlation matrix leads to a dramatic underestimation of the real risk, by overinvesting in artificially low-risk eigenvectors.
- The risk of the optimized portfolio obtained using a cleaned correlation matrix is more reliable, the real risk is always larger than the predicted one.
- The central result of the present study is the remarkable agreement between the theoretical prediction and empirical data concerning both the density of eigenvalues and the structure of eigenvectors of the empirical correlation matrices corresponding to several major stock markets.

Funding

- Less than 6% of the eigenvectors which are responsible of 26% of the total volatility, appear to carry some information

Reference

- E.J. Elton and M.J. Gruber, Modern Portfolio Theory and Investment Analysis (J.Wiley and Sons, New York, 1995); H. Markowitz, Portfolio Selection: Efficient Diversification of Investments (J.Wiley and Sons, New York, 1959).
- J.P. Bouchaud and M. Potters, Theory of Financial Risk, (Alea-Saclay, Eyrolles, Paris, 1997) (in French, English translation to appear in Cambridge University Press, Spring 2000).
- For a review, see: O. Bohigas, M. J. Giannoni, Mathematical and computational methods in nuclear physics, Lecture Notes in Physics, Vol. 209, Springer-Verlag (1983)
- M. Mehta, Random Matrices (Academic Press, New York, 1995).
- S. Galluccio, J.P. Bouchaud and M. Potters, Physica A 259, 449 (1998).
- L. Laloux, P. Cizeau, J.P. Bouchaud and M. Potters, Phys. Rev. Lett. 83, 1467 (1999).
- V. Plerou, P. Gopikrishnan, B. Rosenow, L. A. Amaral, H. E. Stanley, Phys. Rev. Lett. 83, 1471 (1999).
- A. Edelman, SIAM J. Matrix Anal. Appl. 9, 543 (1988) and references therein.
- A.M. Sengupta and P.P. Mitra Distribution of Singular Values for Some Random Matrices, cond-mat/9709283 preprint, where the authors generalise the results of random Wishart matrices in presence of multivariate correlations both in space and time.
- T.H. Baker, P.J. Forrester and P.A. Pearce, J. Phys. A 31, 6087 (1998).
- M.J. Bowick and E. Brezin, Phys. Lett. B 268, 21 (1991), see also M.K. Sener and J.J.M. Verbaarschot, Phys. Rev. Lett. 81, 248 (1998) and, J. Feinberg and A. Zee, J. Stat. Phys. 87 473 (1997).
- L. Laloux, P. Cizeau, J.P. Bouchaud and M. Potters, in preparation. 13. for a review, see: Hull, J. (1997) Options, futures and other derivative securities, Prentice-Hall. For a physicist approach, see 2.

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