Market Correlation and Market Volatility in US BLue Chip Stocks

8/2/2019 Market Correlation and Market Volatility in US BLue Chip Stocks

1/23

Market Correlation and Market Volatility

in US Blue Chip Stocks

Craig Mounfield ([email protected])

and

Paul Ormerod ([email protected])

Volterra Consulting Ltd

The Old Power Station

121 Mortlake High Street

London SW14 8SN

Crowell Prize Submission

20th

March 2001
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


2/23

Abstract

We analyse the daily rates of return of US blue chip stocks over the 1993-2001 period.

Using the technique of random matrix theory, we show that the correlation matrix of

these rates of return is to a large extent dominated by noise rather than by true

information. These results confirm for this data set findings recently documented in the

econophysics literature.

However, the eigenvector associated with the principal eigenvalue of the correlationmatrix does contain true information and shows stability over time. This, the market

eigenvector, shows the extent to which the individual stocks tend to move together. We

quantify the fraction of total information contained within this eigenmode, which we

define asthe information index

We find a clear positive relationship between the absolute changes in the variability of

the information index and the absolute changes in the variability of the market index.

Further, the absolute change in the variability of the information index lagged one day

has statistically significant predictive power for the absolute change in the variability of

the market index.


3/23

1. Introduction

A precise quantification of the correlations between the returns of different assets

traded in financial markets is of fundamental importance to risk managementwhere one attempts to diversify as widely as possible the character of the portfolio

(reducing exposure to sector/industry specific shocks). As a consequence of this

the correlation matrix is one of the cornerstones of much of modern financial

engineering such as CAPM (Elton et al, 1995) and Value at Risk.

However it is well understood that empirical measurements of the correlations

between assets are subject to a number of significant sources of potential error.

The difficulties associated with determining the true correlations between

financial assets arise primarily due to :

Non-stationary correlations between assets (due, for example, to an

organisations profile changing over time)

A finite number of observations of asset price movements (the statistical

significance of spurious measurements becomes insignificant in the limit of an

infinite number of observations of asset pair price movements). In this case

the empirically measured correlations may be significantly noise dominated

masking the true correlations between asset returns.

The technique of Random Matrix Theory (RMT) has recently been applied to

financial market data to analyse the true degree of information content contained

within empirical correlation matrices formed from equity returns. RMT wasoriginally developed for the study of complex quantum mechanical systems.

In order to assess the degree to which an empirical correlation matrix is noise

dominated we can compare the eigenspectra properties of the empirical matrix

with the theoretical eigenspectra properties of a random matrix. Undertaking this


4/23

analysis will identify those eigenstates of the empirical correlation matrix which

contain genuine information content. The remaining eigenstates will be noise

dominated and hence unstable over time. This technique has recently been applied

by a number of researchers to financial market data (for example, Mantegna et al

1999, Laloux et al 1999, Plerou et al 1999, Gopikrishnan et al 2000, Plerou 2000,

Bouchaud et al 2000, Drozdz et al 2001) as well as to macroeconomic data

(Ormerod et al, 2000).

We apply this technique to daily returns on leading US blue chip stocks using

daily data over the 1993 - 2001 period. The results are consistent with those of

the recent literature, in that empirical financial correlation matrices are in general

dominated by noise, but there do exist some significant, stable deviations of

empirical financial correlation matrices from the universal predictions of RMT.

The main purpose of this paper is two-fold. First, to show that this technique may

also yield an information index which characterises the degree to which the

movements of assets in a portfolio are correlated. Second, that the temporal

evolution of this index is well correlated with the volatility of the overall market

index.

The structure of the paper is as follows. Section 2 outlines the relevant concepts of

RMT. Section 3 then applies these concepts to the analysis of a portfolio of US

blue chip equities. Finally the main results are summarised in section 4.

2. Random Matrix Theory

The problem of understanding the properties of matrices with stochastically

fluctuating entries is one which has been studied intensively since the 1950s in

the context of nuclear physics. In this context the problem was to understand the

empirically observed energy spectra of complex quantum mechanical systems

(specifically heavy nuclei composed of many interacting constituents).


5/23

In order to characterise these properties it was assumed that the numerous many-

body interactions are in fact so complex that in the aggregate they may be

considered to be random. That is, the elements of the Hamiltonian matrix ijH

may be considered to be mutually independent random variables. Under thisassumption it was possible to derive the statistics of the eigenvalue distribution of

the Hamiltonian which were in remarkable agreement with experimental data (a

contemporary exposition of RMT may be found in Mehta, 1991).

It was also demonstrated that RMT predictions represent an average over all

possible interactions. Hence RMT predictions are universal predictions that will

apply to wide classes of systems. Deviations from the universal predictions of

RMT identify system-specific, non-random properties of the system under

consideration. These deviations provide clues about the underlying interactions

within the system (Mehta 1991).

In order to assess the degree to which an empirical correlation matrix is noise

dominated one may compare the eigenspectra properties of the empirical matrix

with the theoretical eigenspectra properties of a random matrix. Undertaking this

analysis will identify those eigenstates of the empirical matrix which contain

genuine information content. The remaining eigenstates are understood to be

noise dominated and hence potentially unstable over time. The eigenstates that

contain genuine information content are specific to the system under

consideration and are indicative of the presence of collective modes of motion.

2.1 Eigenspectra Properties of Random Matrices

Consider a matrix Mof Tobservations of price changes of Nassets (at a

frequency of e.g. inter-day observations). If the inter-period logarithmic returns

are defined as


6/23

)1(ln)(ln)( = tPtPtMiii

then the correlation matrix measuring the correlations between the Nassets is

given by

TMM

TC

1=

If the T observations are i.i.d random variables then in the limit N and

T the density of eigenvalues, , of the random correlation matrix C is

given by (Sengupta et al 1999)

))((

2)(

minmax

2

= Q

C

for ],[maxmin

where 1=N

TQ .

The upper and lower bounds on the theoretical eigenvalue distribution are given

by,

22

max)

11(

Q+=

22

min )

1

1( Q=

(2 is the variance of the elements of M, usually rescaled to unity). This

distribution is plotted below in figure 1 for Q = 3.22. As can be seen from this

figure there is a well-defined range of non-zero eigenvaluesmaxmin


7/23

This range of eigenvalues corresponds to a random, noisy subspace band where

the postulates of RMT hold. That is to say, the eigenvectors corresponding to

eigenvalues withinmaxmin


8/23

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

Eigenvalue

Density

Figure 1 : Theoretical Density of Eigenvalues for a Random Matrix


9/23

2.2 The Inverse Participation Ratio

To analyse the structure of the eigenvectors of the empirical correlation matrix the

inverse Participation Ratio (IPR) may be calculated. The IPR is commonly

utilised in localisation theory to quantify the contribution of the different

components of an eigenvector to the magnitude of that eigenvector (thus

determining if an eigenstate is localisedor extended) (Plerou et al 1999).

Component i of an eigenvector i

v corresponds to the contribution of time series

i to that eigenvector. That is to say, in this context, it corresponds to the

contribution of asset i to eigenvector . In order to quantify this we define the

IPR for eigenvector to be

=

=N

ii

vI1

4)(

Hence an eigenvector with identical components Nvi

1= will have

NI 1= and an eigenvector with one non-zero component will have 1=I .

Therefore the inverse participation ratio is the reciprocal of the number ofeigenvector components significantly different from zero (i.e. the number of

assets contributing to that eigenvector).

2.3 Temporal Stability of the Eigenvector Structure

For those eigenvectors that deviate from the theoretically predicted bounds of

RMT it is important to quantify the degree of stability of the information content

of the eigenmode (i.e. the stability of the correlations between the assets). This is

necessary since spurious correlations may be introduced by a particular choice of

data to calculate the correlation matrix from. We may assess this stability by

calculating the scalar product of eigenvectors in non-overlapping analysis periods.

That is for two analysis periodsA

T andB

T we form the overlap matrix


10/23

=

)()(...)()(

...

...

...

)()(...)()(

),(

111

1

BABN

A

BA

N

B

N

A

N

BA

TvTvTvTv

TvTvTvTv

TTO

Hence if the eigenvector structure remains perfectly stable in time (i.e. the

correlations between the assets contributing to that eigenvector remain stable from

period to period) then each element of the overlap matrix would be equal to

ijBAijTTO =),( . No inter-period stability would imply that 0),( =

BAijTTO .

3. RMT Applied to Empirical Correlation Matrices

Having described the basic analysis tools of RMT we will now apply this

technology to financial correlation matrices.

3.1 Data Analysed

The data set is for 31 US equities (blue chips, mostly Dow Jones Industrial

Average constituents), daily closing data for the period 4th

January 1993 to 13th

March 2001. There are 2068 separate trading days (taking out holidays etc).

As a control this data set is also analysed after each of the time series of the

individual assets are shuffled at random 10000 times. This has the effect of

destroying any temporal correlations in the data, while at the same time

preserving the statistical properties of the distributions (e.g. mean and variance).

This randomly shuffled portfolio will act as a control to demonstrate there exists a

quantitative difference between the eigenspectra of random and empirical

correlation matrices.

3.2 Analysis of the Eigenspectra Properties

To demonstrate that RMT may yield genuine information as to the true

information content contained within an empirical correlation matrix we will


11/23

calculate the eigenspectra properties of the two portfolios described above. That is

to say we form the correlation matrix from the inter-day returns of the assets

(there are thus 2067 observations of daily price changes for the 31 assets). For a

matrix of this dimension the theoretical upper and lower bounds for the

eigenvalue distribution are 1.26 and 0.77 respectively.

For the correlation matrix (of dimension 31 x 31) formed from the shuffled data

we find that all 31 eigenvalues of the correlation matrix fall within the upper and

lower bounds. For the correlation matrix formed from the original data set we

observe that there are 17 eigenvalues below the lower bound, 4 eigenvalues above

the upper bound and therefore 10 eigenvalues which fall between the upper and

lower bounds.

We of course expect that for the shuffled data there should be no information

content contained within the time series since the process of shuffling the data

destroys any temporal correlations in the data. However the observation of a

significant number of eigenvalues outside the RMT bounds for the original, un-

shuffled, portfolio demonstrates that there does indeed exist genuine, non-random,

correlated movements between groups of assets within the portfolio.

We may also examine the stability of these correlations over time. Firstly we

choose two non-overlapping time periods of approximately 4 years in duration

and calculate the eigenspectra properties of the correlation matrices formed from

these two analysis periods. For matrices of these dimensions the theoretical upper

and lower eigenvalues are 1.38 and 0.68 respectively. For the two analysis periods

it is found that the numbers of eigenvalues below the theoretical minimum are 12

and 15 and above the theoretical maximum are identical (being 4). This indicates

that that large scale macrostructure of the portfolio remains unchanged over the

course of the 8 year total analysis period.


12/23

We can also calculate the overlap matrix between the two periods. This is shown

in figure 2. For these two analysis periods the overlap between the eigenvectors

corresponding to the largest eigenvalue is 0.99. In addition to this if we repeat the

analysis with 10 non-overlapping periods (each of 200 trading days in duration)

we also observe an average overlap for the eigenvectors corresponding to the

largest eigenvalue yields an average degree of overlap of 0.95. These numbers

represent a significant degree of temporal stability of the eigenvector structure.


13/23

Figure 2 : Colour coded plot of the degree of overlap of the eigenvectors

corresponding to 2 non-overlapping analysis periods for the US blue chip

portfolio. A white square corresponds to perfect overlap between the structure of

the 2 eigenvectors (perfect stability of the degree of information content in that

eigenmode) and black corresponds to no degree of overlap whatsoever. As can be

seen, the degree of stability of the market eigenmode (i.e. the dot product of

eigenvector 1 with itself in each of the two periods - bottom right hand corner) is

significantly different from that of any of the other overlaps.


14/23

3.3 Analysis of the Market Eigenmode

In terms of those eigenvalues which lie outside the noisy sub-space band the most

important is the largest eigenvalue. The application of RMT techniques to equities

traded in financial markets have demonstrated that this eigenmode corresponds to

the market (e.g. Gopikrishnan et al, 2000).

In particular, for this data set (2067 observations of daily returns for 31 assets),

the maximum eigenvalue of the correlation matrix is 7.05 (the remainder of the

eigenvalues are in the range 2.15 to 0.34). The theoretical maximum eigenvalue is

1.26 so it is clear that the largest empirically observed eigenvalue is significantly

above this threshold.

Analysis of the eigenvector corresponding to the largest eigenvalue demonstrates

that each of the 31 components of the eigenvector contribute approximately an

equal amount to the eigenvector. Indeed, the IPR for this eigenvector is 0.037.

This is to be compared with the value of 0.032 (1/N) that we would expect if all of

the assets contributed equally to the eigenvector. This indicates that this

eigenmode is extended. Hence the behaviour of this eigenmode is indicative of

large-scale correlated movements of all of the assets within the portfolio.

In order to quantify this overall collective motion of the portfolios asset price

dynamics we may exploit the fact that the trace of the correlation matrix is

preserved. That is, for the US blue chip portfolio of 31 assets, the trace of the

correlation matrix is equal to 31 (since there are 31 independent time series). The

closer the 'market' eigenmode (i.e. the maximum eigenvalue) is to this value the

more information is contained within this mode and the more correlated the

movements of the price changes of the assets within the portfolio are. We may

therefore quantify the fraction of total information contained within this

eigenmode the information index - expressed as a percentage by the following

formula


15/23

NtQ max100)(

=

If the assets in the portfolio move together very closely, then we would expect

%100)( tQ . Conversely, if the asset price movements are completely

uncorrelated then we would expect %0)( tQ (corresponding to no collective

dynamics).

3.4 Temporal Evolution of the Market Eigenmode

We have seen that the eigenmode of the empirical correlation matrix

corresponding to the maximum eigenvalue represents a collective motion of all of

the assets within the portfolio. What is of interest is to determine how this

eigenmode evolves temporally.

The analysis is undertaken with a fixed window of data. Within this window, the

spectral properties of the correlation matrix formed from the constituent elements

of the US blue chip portfolio are calculated. In particular, the maximum

eigenvalue is calculated. This window is then advanced by one period

(corresponding, in this data set, to one trading day) and the maximum eigenvalue

noted for each period. The same procedure is followed for the Dow Jones

Industrial Index (DJIA) itself. A window of 250 periods, which corresponds to

approximately one year in terms of elapsed time, was chosen for the analysis. As

previously the correlation matrix is formed from the returns on the assets.

Plotted in figures 3a and 3b respectively are the absolute values of the logarithmicdifferences of DJIA and the information index, Q. The absolute value of the

logarithmic differences represents a proxy for the volatility of the time series

(Ponzi, 2000) (with a window of 250 trading days). Figure 3a (for the DJIA)

demonstrates that there exists periods of bursts of volatility interspersed by

periods of low volatility (so-called volatility clustering characteristic of the


16/23


17/23


18/23

Volatility of DJIA

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

29/12/93 29/12/94 29/12/95 29/12/96 29/12/97 29/12/98 29/12/99 29/12/00

Figure 3a : Plot of volatility of the DJIA for the period 4th January 1998 13th

March2001

Volatility of Information Index

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

29/12/93 29/12/94 29/12/95 29/12/96 29/12/97 29/12/98 29/12/99 29/12/00

Figure 3b : Plot of volatility of the information index for the period 4th January 1998

13th March 2001


19/23


20/23

A scatter plot of the two variables, set out in Figure 4, does suggest a positive

relationship between them. The simple correlation coefficient, , is in fact 0.462,

highly statistically significantly different from zero.

Volatility of DJIA

VolatilityofInformationIndex

0.0 0.02 0.04 0.06

0.0

0.0

2

0.0

4

0.0

6

0.0

8

0.1

0

0.1

2

Figure 4 : Scatter plot demonstrating the relationship between the volatility of the

returns on the DJIA with the volatility of the returns on the information index

Using the full data set from the windowing, we have N = 1818 trading days. The

significant positive correlation persists even when the large potential outliers are

trimmed from the data set. For example, using only those observations where theabsolute value of the maximum eigenvalue is < 0.03 gives N = 1789 and =

0.371. Figure 4 shows that the overwhelming bulk of the data is concentrated at

low values of the variables, but even choosing only those observations where the

absolute value of the maximum eigenvalue is < 0.01 gives = 0.283, with N =

1610.


21/23

These results suggest that the volatility of the Dow Jones Industrial Average is

positively correlated with the volatility of the degree of information in the

eigenvector associated with the 'market' eigenvalue.

We then examined the possibility that the volatility of the degree of information in

the market eigenvector - the degree to which the constituent stocks move together

- might have some predictive power as far as the volatility of the overall index is

concerned.

We carried out classical least squares regression of the volatility of the Dow Jones

index on lagged values of the volatility of the maximum eigenvalue. Empirically

only the first lagged value was statistically significant. The estimated coefficient

was 0.0965 with a standard error of 0.0213, so the coefficient is significantly

different from zero at p< 0.0001.

We examined this relationship using the general non-linear least squares

technique of local regression (available in the program S-Plus, for example). This

technique fits a curve to the data points locally, so that any point on the curve at

that point depends only on the observations at that point and some specified

neighbouring points. For any given data point, x(t) say, we choose the k nearest

neighbours of x(t), which constitute a neighbourhood N(x(t)). The number of

neighbours k is specified as a percentage of the total available number of data

points. This percentage is called the span.

By choosing a sufficiently large value for the span, all the points in the data set

are in the neighbourhood of every single point. In other words, in the limit the

local regression technique is identical to that of classical least squares. This

enables us to carry out standard analysis of variance on the results for different

choices of the span. In this case, a value of 0.8 represents the best choice of the

span. The reduction in the residual sum of squares compared to that obtained with


22/23

classical least squares is significantly different from zero at p = 0.00012.

However, the degree of non-linearity is not strong. The equivalent number of

parameters in the local regression model with span = 0.8 is only 2.5, indicating

that the local regression model is somewhere between linear and a quadratic one

in complexity [ref S-Plus Modern Statistics and Advances Graphics, Guide to

Statistics vol. 1, Mathsoft, Seattle, 2000]

4. Conclusions

The correlation matrix of returns is of fundamental importance to much modern

portfolio analysis. However, recent literature in the physics journals using the

technique of random matrix theory has shown that such empirical correlation

matrices contain substantial amounts of noise rather than true information. The

results presented here confirm these findings with a data set of daily returns on

US blue chip stocks over the 1993-2001 period.

However, the correlation matrix does contain a certain amount of true

information. In particular, the eigenvector associated with the principal

eigenvalue of the correlation matrix enables us to identify the extent to which the

individual stocks are genuinely moving together over time. We use the term

'market eigenmode' to characterise this eigenvalue and vector. We demonstrate

that the market eigenmode is stable over time. We define the information index to

be the fraction of total information contained within this eigenmode

We analyse the temporal movements of variability of the information index and of

the variability of the index formed from the component stocks and find a clear

positive correlation between their absolute values. Further, the variability of the

information index lagged one day has statistically significant power in accounting

for movements in the current variability of the index.


23/23

5. References

J.-P. Bouchaud and M. Potters, Theory of Financial Risks From StatisticalPhysics to Risk Management, Cambridge University Press (2000)

S. Drozdz, J. Kwapien, F. Grummer, F. Ruf, J. Speth, Quantifying the Dynamicsof Financial Correlations, cond-mat/0102402 (2001)

E. J. Elton and M.J.Gruber, Modern Portfolio Theory and Investment Analysis,

J.Wiley and Sons, New York (1995)

P. Gopikrishnan, B. Rosenow, V. Plerou, and H.E. Stanley Identifying Business

Sectors from Stock Price Fluctuations, cond-mat/0011145 (2000)

L. Laloux, P. Cizeau, J.-P Bouchaud and M. Potters Noise Dressing of Financial

Correlation Matrices, Phys Rev Lett 83, 1467 (1999)

R. N. Mantegna and H. E. Stanley, An Introduction to Econophysics, Cambridge

University Press (2000)

M. Mehta,Random Matrices, Academic Press (1991)

P. Ormerod and C. Mounfield, Random Matrix Theory and the Failure of

Macroeconomic Forecasts, Physica A 280, 497 (2000)

V. Plerou, P. Gopikrishnan, B. Rosenow, L.A.N. Amaral and H.E. StanleyUniversal and Non-universal Properties of Cross-correlations in Financial Time

Series, Phys Rev Lett 83, 1471 (1999)

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

Random Matrix Theory Approach to Financial Cross-Correlations, Physica A

287, 374 (2000)

A. Ponzi, The Volatility in a Multi-share Financial Market Model, cond-mat/0012309 (2000) To appear in European Physical Journal

A.M Sengupta and P. P. Mitra, Phys Rev E 60 3389 (1999)

Documents

Market Correlation and Market Volatility in US BLue Chip Stocks