## Abstract

We analyze the returns of stocks contained in the Standard & Poor’s 500 index from 1987 until 2011. We use covariance matrices of the firms’ returns determined in a time windows of several years. We find that the eigenvector belonging to the leading eigenvalue (the market) exhibits a phase transition. The market is in an ordered state from 1995 to 2005 and in a disordered state after 2005. We can relate this transition to an order parameter derived from the stocks’ beta and the trading volume. This order parameter can also be interpreted within an agent-based model.

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## Notes

Extending the window too much could in fact at some point lead to problems, even if one assumes that the betas are slowly varying, see for example Livan et al. (2012).

In this paper \(t\) is used to specific days, while \(\tau \) is used to index variables like \(C\) that are calculated for a time window.

The dominance of the IT sector and the change around 2005 can also be found for smaller window sizes of down to 2 years, revealing one sharp peak around 1995/96. For much smaller time windows the influence of single events like the 1987 stock market crash or the 9/11 attacks become rather large and distort long-run trends.

See any textbook on quantum mechanics, e.g., Messiah (1967).

## References

Ahlgren N, Antell J (2010) Stock market linkages and financial contagion: a cobreaking analysis. Q Rev Econ Finance 50:157–166

Alfarano S, Lux T, Wagner F (2005) Estimation of agent-based models: the case of an asymmetric herding model. Comp Econ 26:19–49

Adcock CJ, Cortez MC, Rocha Armada MJ, Silva F (2012) Time varying betas and the unconditional distribution of asset returns. Quant Finance 12(6):951–967

Beine M, Candelon B (2011) Liberalization and stock market co-movement between emerging economies. Quant Finance 12(2):299312

Black F, Jensen MC, Scholes M (1972) The capital asset pricing model: some empirical tests In: Jensen M (ed) Studies in the theory of capital markets. Praeger, New York

Bos T, Newbold P (1984) An empirical investigation of the possibility of stochastic systemic risk in the market model. J Bus 57(1.1):35–41

Bodurtha JN, Mark NC (1991) Testing the CAPM with time-varying risk and returns. J Finance

**46**(4):1485–1505Bollerslev T, Engle R, Wooldridge J (1988) A capital asset pricing model with time-varying covariances. J Polit Econ 96(1):116–131

Borland L, Hassid Y (2010) Market panic on different time-scales. arXiv:1010.4917v1

Bornholdt S (2001) Expectation bubbles in a spin model of markets. Int J Mod Phys C 12:667–674

Brock WA, Hommes CH (1998) Heterogeneous beliefs and routes to chaos in a simple asset pricing model. J Econ Dyn Control 22:12351274

Burda Z, Görlich A, Jarosz A, Jurkiewicz J (2004) Signal and noise in correlation matrix. Phys A 343:295

Chiarella C, Dieci R, He X (2010) A framework for CAPM with heterogeneous beliefs. In: Bischi G-I, Chiarella C, Gardini L (eds) Nonlinear dynamics in economics, finance and social sciences: essays in honour of John Barkley Rosser Jr. Springer, Berlin

Citeau P, Potters M, Bouchaud JP (2001) Correlation structure of extreme stock returns. Quant Finance 1:217–222

Cont R, Bouchaud JP (2000) Herd behaviour and aggregate fluctuations in financial markets. Macroecon Dyn 4:170

Cont R (2001) Empirical properties of asset returns: stylized facts and statistical issues. Quant Finance 1:223

Dichev ID, Huang K, Zhou D (2012) The dark side of trading. Emory law and economics research paper No. 11–95

Fama EF, French KR (1992) The cross-section of expected stock returns. J Finance 47(2):427–465

Galluccio S, Bouchaud JP, Potters M (1998) Rational decisions, random matrices and spin glasses. Phys A 259:449–456

Harvey CJ, Siddique A (2000) Conditional skewness in asset pricing tests. J Finance LV(3):1263–1295

Kenett DY, Shapira Y, Madi A, Bransburg-Zabary S, Gur-Gershgoren G, Ben-Jacob E (2011) Index cohesive force analysis reveals that the US market became prone to systemic collapses since 2002. PLoS ONE 6(4):e19378

Kenett DY, Preis T, Gur-Gershgoren G, Ben-Jacob E (2012a) Dependency network and node influence: application to the study of financial markets. Int J Bifurc Chaos 22. doi:10.1142/S0218127412501817

Kenett DY, Raddant M, Lux T, Ben-Jacob E (2012b) Evolvement of uniformity and volatility in the stressed global financial village. PLoS ONE 7(2):e31144

Kirman A (1993) Ants, rationality and recruitment. Q J Econ 108:137–156

Laloux L, Cizeau P, Bouchaud JP, Potters M (1999) Noise dressing of financial correlation matrices. Phys Rev Lett 83:1467

Lintner J (1965) The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Rev Econ Stat 47:13–37

Livan G, Alfarano S, Scalas E (2011) Fine structure of spectral properties for random correlation matrices. Phys Rev E 84:016113

Livan G, Inoue J, Scalas E (2012) On the non-stationarity of financial time series: impact on optimal portfolio selection. JSTAT. doi:10.1088/1742-5468/2012/07/P07025

Lux T, Ausloos M (2002) Market fluctuations I: scaling, multi-scaling and their possible origins. In: Bunde A, Kopp J, Schellhuber H (eds) Theory of desaster. Springer, Berlin, p 373

Marĉenko VA, Pastur LA (1967) Distribution of eigenvalues for some sets of random matrices. Sb Math 1:457

Messiah A (1967) Quantum mechanics volume I. North Holland, Amsterdam

Münnix MC, Shimada T, Schäfer R, Leyvraz F, Seligman T, Guhr T, Stanley HE (2012) Identifying states of a financial market. Sci Rep 2(644). http://www.nature.com/srep/2012/120910/srep00644/full/srep00644.html

Pearson K (1885) Notes on regression and inheritance in the case of two parents. Proc R Soc Lond 58(240):242

Plerou V et al (2002) Random matrix approach to cross correlations in financial data. Phys Rev E 65:066126

Preis T, Schneider JJ, Stanley HE (2011) Switching processes in financial markets. Proc Natl Acad Sci. doi:10.1073/pnas.1019484108

Sharpe W (1964) Capital asset prices: a theory of market equilibrium under conditions of risk. J Finance 19:425–442

Stauffer D, Sornette D (1999) Self-organized percolation model for stock market fluctuation. Phys A 271:496–506

Tumminello M, Aste T, Di Matteo T, Mantegna RN (2005) A tool for filtering information in complex systems. Proc Natl Acad Sci USA 102(30):10421–10426

Wagner F (2006) Application of Zhangs square root law and herding to financial markets. Phys A 364:369–384

Wishart J (1928) The generalized product moment distribution in samples from a normal multivariate population. Biometrica 20A(1928):32–52

Wu FY (1982) The Potts model. Rev Mod Phys 54(1):235268

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## Appendices

### Appendix 1: Correlation matrix

Denoting the time average as in Eq. (3) by \([\;]_{\tau ,t_W}\) we get from Eqs. (4) and (5) the average of \(r_M^2\)

Similarity we get for \([r_{av}^2]_{\tau ,t_W}\) and \([r_M\cdot r_{av}]_{\tau ,t_W}\)

with

\(a_0\) corresponds to the mean value \(\bar{\beta }\) of \(\beta _i\) over \(i\).

Inserting Eqs. (24), (25) and (27) into \(\varDelta ^2\) from (4) we get

Since \(e_i^\mu \) and \(e_i^0\) are orthogonal for \(\mu >0\) we can write \(a_\mu \) as

Applying the Schwartz inequality to (29) we get

Together with \(\sum _{\mu =0}\lambda _\mu =trace(C)\) this leads to the inequality (13) for \(\varDelta ^2\). Since the average \(C_{av}\) is a function \([r_{av}^2]_t\) insertion of \(\varDelta ^2\) into Eq. (25) leads to Eq. (15).

### Appendix 2: Perturbation expansion

The matrix \(C\) in Eq. (17) is a sum of two matrices. The first \(C^0_{ij}=\beta ^0_i\beta ^0_j\gamma _M^2\) has one large eigenvalue \(E_0=\gamma _M^2N\) with an eigenvector \(f^0_i=\beta ^0_i/{\sqrt{N}}\) and \(N-1\) degenerate zero eigenvalues with vectors \(f^\mu _i\) with \(\mu >0\).These must satisfy only the orthogonality relation

with (\(a,b)\) denoting the scalar product. To obtain a complete basis we impose on \(f^\mu \) in the \(N-1\) dimensional subspace the following conditions with the second matrix \(C^1_{ij}=\delta _{ij}\gamma _i^2\)

We apply standard second order Rayleigh Schrödinger perturbation theory^{Footnote 4} with \(C^1\) as perturbation. For general matrices \(C^0\) and \(C^1\) using only the spectrum of \(C^0\) and condition (31) we get up to order \(1/E_0^2\) for the leading eigenvalue

and its eigenvector

The other eigenvalues require the in general complicated solution of Eq. (17) for \(f^\mu _j\). They are given by

Inserting the specific form of \(C^1\) we obtain for \(\lambda _0\) and \(\beta _i\) with

\(\langle \;\rangle _\beta \) denotes the average over \(i\) weighted with \((\beta ^0_i)^2\). Since neglected terms are of order \(1/N^2\) these formulae describe \(\lambda _0\) and \(\beta _i\) fairly accurate already for moderate \(N\). Due to the degeneracy the general formalism of Marĉenko and Pastur (1967) for the modification due to noise does not apply. Using the Wishart (1928) formula we find the relative error in \(\lambda _0\) due to a finite observation window \(T\) is of order \(1/\sqrt{T}\) instead of \(\sqrt{N/T}\) expected from Marĉenko and Pastur (1967). The non-leading eigenvalue will be changed considerably if the spread of \(\gamma _i\) is small, see Burda et al. (2004).

### Appendix 3: Significance of the order parameter

We analyzed the significance of the order parameter by assigning a random value for the sector to each stock. In Fig. 6 we show one representative result of such a random assignment. Compared with Fig. 5, the large values of \(m\) disappear. Before and after 2006 \(m\) behaves similar as in a disordered phase. The structure observed near 2006 cannot be removed by re-shuffling, because a substantial part of the stocks change their values of \(\beta _i v_i\). If there would be only one stock in each IT or financial sector responsible for the transition seen in Fig. 5, one would get the same behavior. The fluctuation size of \(m\) in the order of \(\pm 0.15\) indicates the error on the estimation of \(m\). This shows that the observed values for the IT and financial sector are far out of the range of random fluctuations.

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Raddant, M., Wagner, F. Phase transition in the S&P stock market.
*J Econ Interact Coord* **11**, 229–246 (2016). https://doi.org/10.1007/s11403-015-0160-x

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DOI: https://doi.org/10.1007/s11403-015-0160-x