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Phase transition in the S&P stock market

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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

  1. 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).

  2. 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.

  3. 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.

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

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Correspondence to Matthias Raddant.

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 \(r_M^2\)

$$\begin{aligned}{}[r_M^2]_{\tau ,t_W}=\frac{\lambda _0(\tau )}{N} \end{aligned}$$
(24)

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

$$\begin{aligned}{}[r_{av}^2]_{\tau ,t_W}=\frac{1}{N}\sum _{\mu =0}^{N-1}a_\mu ^2(\tau )\lambda _\mu (\tau ) \end{aligned}$$
(25)

with

$$\begin{aligned} a_\mu (\tau )=\frac{1}{\sqrt{N}}\sum _ie_i^\mu (\tau ) \end{aligned}$$
(26)

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

$$\begin{aligned}{}[r_M\cdot r_{av}]_{\tau ,t_W}=\frac{\lambda _0(\tau )}{N}\cdot \bar{\beta } \end{aligned}$$
(27)

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

$$\begin{aligned} \varDelta ^2=(1-\bar{\beta })^2 + \sum _{\mu >0}\frac{\lambda _\mu (\tau )}{\lambda _0(\tau )}a_\mu ^2(\tau ) \end{aligned}$$
(28)

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

$$\begin{aligned} a_\mu (\tau )=\frac{1}{\sqrt{N}}\sum _i e_i^\mu (1-\sqrt{N}e_i^0) \end{aligned}$$
(29)

Applying the Schwartz inequality to (29) we get

$$\begin{aligned} a_\mu ^2(\tau )\le 2(1-\bar{\beta }) \end{aligned}$$
(30)

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

$$\begin{aligned} (f^0,f^\mu )=0 \end{aligned}$$
(31)

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 \(C^1_{ij}=\delta _{ij}\gamma _i^2\)

$$\begin{aligned} (f^\nu ,C^1\; f^\mu )=0 \quad \text{ for } \; \mu \ne \nu \quad \text{ and }\; \mu , \nu \ne \; 0 \end{aligned}$$
(32)

We apply standard second order Rayleigh Schrödinger perturbation theoryFootnote 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

$$\begin{aligned} \lambda _0=E_0+(f^0,C^1f^0)+\frac{1}{E_0}[(f^0,(C^1)^2f^0) -(f^0,C^1f^0)^2] \end{aligned}$$
(33)

and its eigenvector

$$\begin{aligned} e^0_i=f^0_i+\frac{1}{E_0}[(C^1f^0)_i-(f^0,C^1f^0)f^0_i] \end{aligned}$$
(34)

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

$$\begin{aligned} \lambda _\nu =(f^\nu ,C^1\;f^\nu )-\frac{1}{E_0}(f^0,C^1f^\nu )^2 \end{aligned}$$
(35)

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

$$\begin{aligned} (f^0,C^1f^0)= & {} \frac{1}{N}\sum _i(\beta ^0_i)^2\gamma _i^2=\langle \gamma ^2\rangle _\beta \end{aligned}$$
(36)
$$\begin{aligned} (f^0,(C^1)^2f^0)= & {} \frac{1}{N}\sum _i(\beta ^0_i)^4\gamma _i^2=\langle \gamma ^4\rangle _\beta \end{aligned}$$
(37)
$$\begin{aligned} \lambda _0= & {} \gamma _M^2N+\langle \gamma ^2 \rangle _\beta +\frac{1}{\gamma _M^2N} \left[ \langle \gamma ^4 \rangle _\beta -(\langle \gamma ^2 \rangle _\beta )^2 \right] \end{aligned}$$
(38)
$$\begin{aligned} \beta _i= & {} \beta ^0_i \left( 1+\frac{1}{\gamma _M^2N} \left( \gamma ^2_i-\langle \gamma ^2 \rangle \right) \right) \end{aligned}$$
(39)

\(\langle \;\rangle _\beta \) 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.

Fig. 6
figure 6

The risk parameter \(m(\tau ,s)\) with the same \(\beta _i\) and \(v_i\) as in Eq. (20) but for shuffled sector affiliations \(s\)

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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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