Collective behavior of stock price movements in an emerging market
Collective behavior of stock price movements in an emerging market
Raj Kumar Pan∗ and Sitabhra Sinha†
The Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai - 600 113 India
(Dated: October 23, 2018)
To investigate the universality of the structure of interactions in different markets, we analyze
the cross-correlation matrix C of stock price fluctuations in the National Stock Exchange (NSE)
of India. We find that this emerging market exhibits strong correlations in the movement of stock
prices compared to developed markets, such as the New York Stock Exchange (NYSE). This is
shown to be due to the dominant influence of a common market mode on the stock prices. By
comparison, interactions between related stocks, e.g., those belonging to the same business sector,
are much weaker. This lack of distinct sector identity in emerging markets is explicitly shown by
reconstructing the network of mutually interacting stocks. Spectral analysis of C for NSE reveals
that, the few largest eigenvalues deviate from the bulk of the spectrum predicted by random matrix
theory, but they are far fewer in number compared to, e.g., NYSE. We show this to be due to
the relative weakness of intra-sector interactions between stocks, compared to the market mode, by
modeling stock price dynamics with a two-factor model. Our results suggest that the emergence of
an internal structure comprising multiple groups of strongly coupled components is a signature of
market development.
PACS numbers: 89.65.Gh,05.45.Tp,89.75.-k
I. INTRODUCTION
Financial markets can be considered as complex sys-
tems having many interacting elements and exhibiting
large fluctuations in their associated observable proper-
ties, such as stock price or market index [1, 2]. The
state of the market is governed by interactions among its
components, which can be either traders or stocks. In
addition, market activity is also influenced significantly
by the arrival of external information. Statistical proper-
ties of stock price fluctuations and correlations between
price movements of different stocks have been analyzed
by physicists in order to understand and model finan-
cial market dynamics [3, 4]. The fluctuation distribution
of stock prices is found to follow a power law with ex-
ponent α ∼ 3, the so-called “inverse cubic law” [5, 6].
This property is quite robust, and has been seen in de-
veloped as well as emerging markets [7]. On the other
hand, it is not yet known whether the cross-correlation
behavior between stock price fluctuations has a similar
universal nature. Although the existence of collective
modes have been inferred from the study of market dy-
namics, such studies have almost exclusively focussed on
developed markets, in particular, the New York Stock
Exchange (NYSE).
To uncover the structure of interactions among the
elements in a financial market, physicists primarily fo-
cus on the spectral properties of the correlation ma-
trix of stock price movements. Pioneering studies in-
vestigated whether the properties of the empirical cor-
relation matrix differed from those of a random matrix
∗Electronic address: rajkp@imsc.res.in
†Electronic address: sitabhra@imsc.res.in
that would have been obtained had the price movements
been uncorrelated [8, 9]. Such deviations from the pre-
dictions of random matrix theory (RMT) can provide
clues about the underlying interactions between various
stocks. It was observed that, while the bulk of the eigen-
value distribution for the correlation matrix of NYSE and
Tokyo Stock Exchange follow the spectrum predicted by
RMT [8, 9, 10, 11], the few largest eigenvalues deviate
significantly from this. The largest eigenvalue has been
identified as representing the influence of the entire mar-
ket, common for all stocks, whereas, the remaining large
eigenvalues are associated with the different business sec-
tors, as indicated by the composition of their correspond-
ing eigenvectors [10, 12]. The interaction structure of
stocks in NYSE have been reconstructed using filter-
ing techniques implementing matrix decomposition [13]
or maximum likelihood clustering [14]. Correlation ma-
trix analysis has applications in the area of financial risk
management, as mutually correlated price movements
may indicate the presence of strong interactions between
stocks [15]. Such analyses have been performed using as-
set trees and asset graphs to obtain the taxonomy of an
optimal portfolio of stocks [16, 17, 18].
While it is generally believed that stock prices in
emerging markets tend to be relatively more correlated
than the developed ones [19], there have been very few
studies of the former in terms of analysing the spectral
properties of correlation matrices [20, 21, 22, 23]. Most
studies of correlated price movements in emerging mar-
kets have looked at the synchronicity which measures the
incidence of similar (i.e., up or down) price movements
across stocks [19, 24]. Although related to correlation the
two measures are not same, as correlation also gives the
relative magnitude of similarity. In this paper, we an-
alyze the cross-correlations among stocks in the Indian
financial market, one of the largest emerging markets in
http://arxiv.org/abs/0704.0773v2
mailto:rajkp@imsc.res.in
mailto:sitabhra@imsc.res.in
the world. Our study spans over 1996-2006, a period
which coincides with the decade of rapid transformation
of the Indian economy after liberalization in the early
1990s.
We find that, in terms of the properties of its collec-
tive modes, the Indian market shows significant devia-
tions from developed markets. As the fluctuation distri-
bution of stocks in the Indian market [7, 21, 25] follows
the “inverse cubic law” seen in NYSE [6, 26], the devi-
ations observed in the correlation properties should be
almost entirely due to differences in the nature of inter-
action structure in the two markets. Our observation of
a higher degree of correlation in the Indian market com-
pared to developed markets is found to be the result of
a dominant market mode affecting all the stocks, which
is further accentuated by the relative absence of clusters
of mutually interacting stocks. This is explicitly shown
by reconstructing the network of interactions within the
market, using a filtered correlation matrix from which
the common market influence and random noise has been
removed. This procedure give a more accurate represen-
tation of the intra-market structure than the commonly
used method of constructing minimal spanning tree from
the unfiltered correlation matrix [13, 16, 17]. To sup-
port the interpretation of our empirical observations, we
present a two-factor model of market dynamics in Sec-
tion IV. Multi-factor models of market behavior have
been used by other groups for explaining various spectral
properties of empirical correlation matrices [27, 28, 29].
In this model, we assume the market to consist of several
correlated groups of stocks which are also influenced by
a common external signal, i.e., market mode. By vary-
ing the relative strength of the factor associated with
the market mode to that associated with the groups, we
show that decreasing the intra-group interactions result
in spectral distribution properties similar to that seen
for the Indian market. Our results imply that one of the
key features signifying the transition of a market from
emerging to developed status is the appearance and con-
solidation of distinct group identities.
II. DATA ANALYZED
The National Stock Exchange (NSE) is the largest
stock market in India. Having commenced operations
from Nov 1994, it is already the world’s third largest
stock exchange (after NASDAQ and NYSE) in terms of
transactions [30]. It is thus an excellent source of data
for studying the correlation structure of price movements
in an emerging market.
We have considered the daily closing price data of 201
stocks (see Table I) traded in NSE from Jan 1996 to May
2006, which corresponds to 2607 days. This data is ob-
tained from the NSE web-site [31] and has been manually
corrected by us for stock splitting. The selected stocks
were traded over the entire period 1996-2006 and had
the minimum number of missing data points (i.e., days
for which no price data is available). If the price value of
a stock is missing on a particular day, a problem common
to data from emerging markets [20], it is assumed that
no trading took place on that day, i.e, the price remained
the same as the preceding day. For comparison we also
consider the daily closing price of 434 stocks of NYSE
belonging to the S&P 500 index over the same period
as the Indian data. However, the total number of work-
ing days is slightly different, viz., 2622 days. This data
was obtained from the Yahoo! Finance website [32]. In
all our analysis, while comparing with the NSE data, we
have used multiple random samples of 201 stocks each,
from the set of 434 NYSE stocks. We verified that the re-
sults obtained were independent of the particular sample
of 201 stocks chosen.
To ensure that the missing closing prices in the Indian
market data do not result in artifacts leading to spurious
divergence from the US market, we have also performed
our analysis on synthetic US market data containing the
same number of missing data points. Multiple sets of
such data were generated from the actual closing price
time series by randomly choosing the required number
of data points and replacing them with the same value
as the preceding day. The resulting analysis showed no
significant difference from the results obtained with the
original US data.
III. THE RETURN CROSS-CORRELATION
MATRIX
To observe correlation between the price movements
of different stocks, we first measure the price fluctua-
tions such that the result is independent of the scale of
measurement. If Pi(t) is price of the stock i = 1, . . . , N
at time t, then the (logarithmic) price return of the ith
stock over a time interval ∆t is defined as
Ri(t,∆t) ≡ lnPi(t+∆t)− lnPi(t). (1)
As different stocks have varying levels of volatility (mea-
sured by the standard deviation of its returns) we define
the normalized return,
ri(t,∆t) ≡
Ri − 〈Ri〉
, (2)
where σi ≡
〈R2i 〉 − 〈Ri〉2, is the standard deviation
of Ri and 〈. . .〉 represents time average over the period
of observation. We then compute the equal time cross-
correlation matrix C, whose element
Cij ≡ 〈rirj〉, (3)
represents the correlation between returns for stocks i
and j. By construction, C is symmetric with Cii = 1
and Cij has a value in the domain [−1, 1]. Fig. 1 shows
that, the correlation among stocks in NSE is larger on
the average compared to that among the stocks in NYSE.
−0.2 0 0.2 0.4 0.6 0.8
  ) NYSE
FIG. 1: The probability density function of the elements of
the correlation matrix C for 201 stocks in the NSE of India
and NYSE for the period Jan 1996-May 2006. The mean
value of elements of C for NSE and NYSE, 〈Cij〉, are 0.22
and 0.20 respectively.
This supports the general belief that developing markets
tend to be more correlated than developed ones. To un-
derstand the reason behind this excess correlation, we
perform an eigenvalue analysis of the correlation matrix.
A. Eigenvalue spectrum of correlation matrix
If the N return time series of length T are mutually un-
correlated, then the resulting random correlation matrix
is called a Wishart matrix, whose statistical properties
are well known [33]. In the limit N → ∞, T → ∞, such
that Q ≡ T/N ≥ 1, the eigenvalue distribution of this
random correlation matrix is given by
Prm(λ) =
(λmax − λ)(λ − λmin)
, (4)
for λmin ≤ λ ≤ λmax and, 0 otherwise. The bounds of
the distribution are given by λmax,min = [1± (1/
Q)]2.
We now compare this with the statistical properties of
the empirical correlation matrix for the NSE. In the
NSE data, there are N = 201 stocks each containing
T = 2606 returns; as a result Q = 12.97. Therefore,
it follows that, in the absence of any correlation among
the stocks, the distribution should be bounded between
λmin = 0.52 and λmax = 1.63. As observed in developed
markets [8, 9, 10, 11], the bulk of the eigenvalue spectrum
P (λ) for the empirical correlation matrix is in agreement
with the properties of a random correlation matrix spec-
trum Prm(λ), but a few of the largest eigenvalues deviate
significantly from the RMT bound (Fig. 2). However, the
number of these deviating eigenvalues are relatively few
for NSE compared to NYSE. To verify that these outliers
are not an artifact of the finite length of the observation
period, we have randomly shuffled the return time se-
ries for each stock, and then re-calculated the resulting
correlation matrix. The eigenvalue distribution for this
surrogate matrix matches exactly with the random ma-
trix spectrum Prm(λ), indicating that the outliers are not
0 2 4 6 8
Eigenvalue λ
0 20 40
 max 
0 2 4 6 8
Eigenvalue λ
0 20 40
FIG. 2: The probability density function of the eigenvalues
of the correlation matrix C for NSE (top) and NYSE (bot-
tom). For comparison, the theoretical distribution predicted
by Eq. (4) is shown using broken curves, which overlaps with
the distribution obtained from the surrogate correlation ma-
trix generated by randomly shuffling each time series. In both
figures, the inset shows the largest eigenvalue.
due to “measurement noise” but are genuine indicators
of correlated movement among the stocks. Therefore, by
analyzing the deviating eigenvalues, we may be able to
obtain an understanding of the structure of interactions
between the stocks in the market.
B. Properties of the “deviating” eigenvalues
The largest eigenvalue λ0 for the NSE cross-correlation
matrix is more than 28 times greater than the maximum
predicted by RMT. This is comparable to NYSE, where
λ0 is about 26 times greater than the random matrix up-
per bound. Upon testing with synthetic US data contain-
ing same number of missing data points as in the Indian
market, we observed that λ0 remains almost unchanged
compared to the value obtained from the original US
data. The corresponding eigenvector shows a relatively
uniform composition, with all stocks contributing to it
and all elements having the same sign (Fig. 3, top). As
this is indicative of a common factor that affects all the
stocks with the same bias, the largest eigenvalue is associ-
ated with the market mode, i.e., the collective response of
the entire market to external information [8, 10]. Of more
interest for understanding the market structure are the
intermediate eigenvalues, i.e., those occurring between
the largest eigenvalue and the bulk of the distribution
predicted by RMT. For the NYSE, it was shown that
corresponding eigenvectors of these eigenvalues are local-
ized, i.e., only a small number of stocks, belonging to
similar or related businesses, contribute significantly to
each of these modes [10, 12]. However, for NSE, although
the Technology and the IT & Telecom stocks are domi-
nant contributors to the eigenvector corresponding to the
third largest eigenvalue, a direct inspection of eigenvector
composition does not yield a straightforward interpreta-
tion in terms of a related group of stocks corresponding
to any particular eigenvalue (Fig. 3).
To obtain a quantitative measure of the number of
stocks contributing to a given eigenmode, we calculate
the inverse participation ratio (IPR), defined for the kth
eigenvector as Ik ≡
[uki]
4, where uki are the com-
ponents of eigenvector k. An eigenvector having compo-
nents with equal value, i.e., uki = 1/
N for all i, has
Ik = 1/N . We find this to be approximately true for
the eigenvector corresponding to the largest eigenvalue,
which represents the market mode. To see how different
stocks contribute to the remaining eigenvectors, we note
that if a single stock had a dominant contribution in any
eigenvector, e.g., uk1 = 1 and uki = 0 for i 6= 1, then
Ik = 1 for that eigenvector. Thus, IPR gives the recipro-
cal of the number of eigenvector components (and there-
fore, stocks) with significant contribution. On the other
hand, the average value of Ik, for eigenvectors of a ran-
dom correlation matrix obtained by randomly shuffling
the time series of each stock, is 〈I〉 = 3/N ≈ 1.49×10−2.
Fig. 4 shows that the eigenvalues belonging to the bulk
of the spectrum indeed have this value of IPR. But at
the lower and higher end of eigenvalues, both the US
and Indian markets show deviations, suggesting the ex-
istence of localized modes. However, these deviations
are much less significant and fewer in number in the lat-
ter compared to the former. This implies that distinct
groups, whose members are mutually correlated in their
price movement, do exist in NYSE, while their existence
is far less clear in NSE.
C. Filtering the correlation matrix
The above analysis suggests the existence of a market-
induced correlation across all stocks, which makes it dif-
ficult to observe the correlations that might be due to
interactions between stocks belonging to the same sec-
tor. Therefore, we now use a filtering method to remove
market mode, as well as the random noise [13]. The cor-
relation matrix is first decomposed as
λiuiu
i , (5)
where λi are the eigenvalues of C sorted in descending
order and ui are corresponding eigenvectors. As only the
eigenvectors corresponding to the few largest eigenvalues
are believed to contain information on significantly cor-
related stock groups, the contribution of the intra-group
correlations to the C matrix can be written as a partial
sum of λαuαu
α , where α is the index of the correspond-
ing eigenvalue. Thus, the correlation matrix can be de-
composed into three parts, corresponding to the market,
group and random components:
C = Cmarket +Cgroup +Crandom
= λ0u0u
λiuiu
i=Ng+1
λiuiu
i , (6)
where, Ng is the number of eigenvalues (other than the
largest one) which deviates from the bulk of the eigen-
value spectrum. For NSE we have chosen Ng = 5. How-
ever, the exact value of this choice is not crucial as small
changes in Ng do not alter the results, the error involved
being limited to the eigenvalues closest to the bulk that
have the smallest contribution to Cgroup. Fig. 5 shows
the result of decomposing the correlation matrix into the
three components, for both the Indian and US markets.
Compared to the latter, the distribution of matrix ele-
ments of Cgroup in the former shows a significantly trun-
cated tail. This indicates that intra-group correlations
are not prominent in NSE, whereas they are compara-
ble with the overall market correlations in NYSE. It fol-
lows that the collective behavior in the Indian market is
dominated by external information that affects all stocks.
Correspondingly, correlations generated by interactions
between stocks, as would be the case for stocks in a given
business sector, are much weaker, and hence, such corre-
lated sectors would be difficult to observe.
We indeed find this to be true when we use the infor-
mation in the group correlation matrix to construct the
network of interacting stocks [13]. The adjacency matrix
A of this network is generated from the group correlation
matrix Cgroup by using a threshold cth such that Aij = 1
group
> cth, and Aij = 0 otherwise. Thus, a pair
of stocks are connected if the group correlation coeffi-
cient C
group
is larger than a preassigned threshold value,
cth. To determine an appropriate choice of cth = c
observe the number of isolated clusters (a cluster being
defined as a group of connected nodes) in the network for
a given cth (Fig. 6). We found this number to be much
less in NSE compared to that observed in NYSE for any
value of cth [13]. Fig. 7 shows the resultant network for
c∗ = 0.09, for which the largest number of isolated clus-
ters of stocks are obtained. The network has 52 nodes
and 298 links partitioned into 3 isolated clusters. From
these clusters, only two business sectors can be properly
identified, namely the Technology and the Pharmaceuti-
cal sectors. The fact that the majority of the NSE stocks
cannot be arranged into well-segregated groups reflect-
ing business sectors illustrates our conclusion that intra-
group interaction is much weaker than the market-wide
correlation in the Indian market.
B C D E F G H I J K L
0 20 40 60 80 100 120 140 160 180 200
Stocks ( i )
FIG. 3: The absolute values of the eigenvector components ui(λ) of stock i corresponding to the first four largest eigenvalues of
C for NSE. The stocks i are arranged by business sectors separated by broken lines. A: Automobile & transport, B: Financial,
C: Technology D: Energy, E: Basic materials, F: Consumer goods, G: Consumer discretionary, H: Industrial, I: IT & Telecom,
J: Services, K: Healthcare & Pharmaceutical, L: Miscellaneous.
Eigenvalue λ
Eigenvalue λ
FIG. 4: Inverse participation ratio as a function of eigen-
value for the correlation matrix C of NSE (top) and NYSE
(bottom). The broken line indicates the average value of
〈I〉 = 1.49×10−2 for the eigenvectors of a matrix constructed
by randomly shuffling each of the N time series.
D. Relating correlation with market evolution
We now compare between two different time intervals
in the NSE data. For convenience we divide the data set
into two non-overlapping parts corresponding to the pe-
riods between Jan 1996-Dec 2000 (Period I) and between
Jan 2001-May 2006 (Period II). The corresponding corre-
lation matrices C are generated following the same set of
steps as for the entire data set. The average value for the
−0.2 0 0.2 0.4 0.6
Matrix elements
C group
C market 
C random 
−0.2 0 0.2 0.4 0.6
Matrix elements
C random 
C market 
C group 
FIG. 5: The distribution of elements of correlation matrix
corresponding to the market, Cmarket, the group, Cgroup,
and the random interaction, Crandom. For NSE (top) Ng = 5
whereas for NYSE (bottom) Ng = 10. The short tail for the
distribution of the Cgroup elements in NSE indicates that the
correlation generated by mutual interaction among stocks is
relatively weak.
elements of the correlation matrix is slightly lower for the
later period, suggesting a greater homogeneity between
the stocks at the earlier period (Fig. 8).
Next, we look at the eigenvalue distribution of C for
the two periods (Fig. 9. The Q value for Period I is 6.21,
while for Period II it is 6.77. Thus the bounds for the
random distribution is almost same in the two cases. In
contrast, the largest deviating eigenvalues, λ0, are dif-
0 0.05 0.1 0.15 0.2 0.25 0.3
FIG. 6: The number of isolated clusters in the interaction net-
work for NSE stocks as a function of the threshold value cth.
For low cth the network consist of a single cluster which con-
tains all the nodes, whereas for high cth the network consists
only of isolated nodes.
ferent: 48.56 for Period I and 45.88 for Period II. This
implies the relative dominance of the market mode in
the earlier period, again suggesting that with time the
market has become less homogeneous. The number of
deviating eigenvalues remain the same for the two peri-
When the interaction networks between stocks are gen-
erated for the two periods, they show less distinction into
clearly defined sectors than was obtained with the data
for the entire period. This is possibly because the shorter
data sets create larger fluctuations in the correlation val-
ues, thereby making it difficult to segregate the existing
market sectors. However, we do observe that, using the
same threshold value for generating networks in the two
periods yield, for the later period, isolated clusters that
are distinguishable into distinct sub-clusters connected
to each other via a few links only, whereas in the earlier
period the clusters are much more homogeneous. This
implies that as the Indian market is evolving, the inter-
actions between stocks are tending to get arranged into
clearly identifiable groups. We propose that such struc-
tural re-arrangement in the interactions is a hallmark of
emerging markets as they evolve into developed ones.
IV. MODEL OF MARKET DYNAMICS
To understand the relation between the interaction
structure among stocks and the eigenvalues of the cor-
relation matrix, we perform a multivariate time series
analysis using a simple two-factor model of market dy-
namics. We assume that the normalized return at time
t of the ith stock from the kth business sector can be
decomposed into (i) a market factor rm(t), that contains
information or signal common to all stocks, (ii) a sector
factor rkg (t), representing effects exclusive to stocks in the
kth sector, and (iii) an idiosyncratic term, ηi(t), which
corresponds to random variations unique for that stock.
Thus,
rki (t) = βirm(t) + γ
g (t) + σiηi(t), (7)
where βi, γ
i and σi represent relative strengths of the
three terms mentioned above, respectively. For simplic-
ity, these strengths are assumed to be time independent.
We choose rm(t), r
g (t) and ηi(t) from a zero mean and
unit variance Gaussian distribution. We further assume
that the normalized returns ri, also follow Gaussian dis-
tribution with zero mean and unit variance. Although
the empirically observed return distributions have power
law tails, as these distributions are not Levy stable,
they will converge to Gaussian if the returns are calcu-
lated over sufficiently long intervals. The assumption of
unit variance for the returns ensures that the relative
strengths of the three terms will follow the relation:
2 + (γki )
2 + σi
2 = 1. (8)
As a result, for each stock we can assign σi and γi in-
dependently, and obtain βi from Eq. (8). We choose σi
and γi from a uniform distribution having width δ and
centered about the mean values σ and γ, respectively.
We now simulate an artificial market with N stocks be-
longing to K sectors by generating time series of length
T for returns rki from the above model. These K sec-
tors are composed of n1, n2, . . . , nK stocks such that
n1 + n2 + · · ·+ nK = N . The collective behavior is then
analysed by constructing the resultant correlation matrix
C and obtaining its eigenvalues. Our aim is to relate the
spectral properties of C with the underlying structure of
the market given by the relative strength of the factors.
We first consider the simple case, where the contribution
due to market factor is neglected, i.e., βi = 0 for all i, and
the strength of sector factor is equal for all stocks within
a sector, i.e., γki = γ
k, is independent of i. In this case,
the spectrum of the correlation matrix is composed of K
large eigenvalues, 1+(nj−1)(γj)2, where j = 1 . . .K, and
N−K small eigenvalues, 1−(γj)2, each with degeneracy
nj − 1, where j = 1 . . .K [28]. Now, we consider nonzero
market factor which is equal for all stocks i.e., βi = β for
all i, and the strength of sector factor is also same for
all stocks, i.e., γki = γ (independent of i and k). In this
case too, there are K large eigenvalues and N −K small
eigenvalues. Our numerical simulations suggest that the
largest and the second largest eigenvalues are
λ0 ∼ Nβ2,
λ1 ∼ nl(1− β2), (9)
respectively, where nl is the size of the largest sector,
while the N − K small degenerate eigenvalues are 1 −
β2 − γ2. We now choose the strength γki and σi from a
uniform distribution with mean γ and σ respectively and
with width δ = 0.05. Fig. 10 shows the variation of the
largest and second largest eigenvalues with σ and γ. The
strength of the market factor is determined from Eq.8.
Note that, decreasing the strength of the sector factor
relative to the market factor results in decreasing the
second largest eigenvalue λ1. As Q = T/N is fixed, the
RMT bounds for the bulk of the eigenvalue distribution,
[λmin, λmax], remain unchanged. Therefore, a decrease in
Pajek
FIG. 7: The structure of interaction network in the Indian financial market at threshold c∗ = 0.09. The left cluster comprises
of mostly Technology stocks, while the middle cluster is composed almost entirely of Healthcare & Pharmaceutical stocks. By
contrast, the cluster on the right is not dominated by any particular sector. The node labels indicate the business sector to
which a stock belongs and are as specified in the caption to Fig 3.
−0.2 0 0.2 0.4 0.6 0.8
NSE ’01−’06
NSE ’96−’00
FIG. 8: The probability density functions of the elements in
the correlation matrix C for NSE during (a) the period Jan
1996-Dec 2000 and (b) Jan 2001-May 2006. The mean value
of the elements of C for the two periods are 0.23 and 0.21,
respectively.
λ1 implies that the large intermediate eigenvalues occur
closer to the bulk of the spectrum predicted by RMT,
as is seen in the case of NSE. The analysis of the model
supports our hypothesis that the spectral properties of
the correlation matrix for the NSE are consistent with a
market in which the effect of information common for all
stocks (i.e., the market mode) is dominant, resulting in
all stocks exhibiting a significant degree of correlation.
V. CONCLUSIONS
In conclusion, we demonstrate that the stocks in
emerging market are much more correlated than in de-
veloped markets. Although, the bulk of the eigenvalue
spectrum of the correlation matrix of stocks C in emerg-
ing market is similar to that observed for developed mar-
kets, the number of eigenvalues deviating from the RMT
upper bound are smaller in number. Further, most of
the observed correlation among stocks is found to be due
to effects common to the entire market, whereas correla-
tion due to interaction between stocks belonging to the
same business sector are weak. This dominance of the
market mode relative to modes arising through interac-
0 1 2 3 4 5
Eigenvalue λ
0 20 40
0 1 2 3 4 5
Eigenvalue λ
0 20 40
FIG. 9: The probability density function of the eigenvalues
of the NSE correlation matrix C for the periods (top) Jan
1996-Dec 2000 and (bottom) Jan 2001-May 2006. For com-
parison, the theoretical distribution predicted by Eq. (4) is
shown using broken curves. In both figures, the inset shows
the largest eigenvalue.
tions between stocks makes an emerging market appear
more correlated than developed markets. Using a simple
two-factor model we show that a dominant market factor,
relative to the sector factor, results in spectral properties
similar to that observed empirically for the Indian mar-
ket. Our study helps in understanding the evolution of
markets as complex systems, suggesting that strong in-
teractions may emerge within groups of stocks as a mar-
ket evolves over time. How such self-organization occurs
and its relation to other changes that a market undergoes
during its development, e.g., large increase in transaction
volume, is a question worth pursuing in the future with
the tools available to econophysicists.
Our paper also makes a significant point regarding the
0.4 00.1
0.4 00.1
FIG. 10: The variation of the largest (top) and second largest
(bottom) eigenvalues of the correlation matrix of simulated
return in the two-factor model (Eq. 7) with the model pa-
rameters γ and σ (corresponding to strength of the sector
and idiosyncratic effects, respectively). The matrix is con-
structed for N = 200 stocks each with return time series of
length T = 2000 days. We assume there to be 10 sectors, each
having 20 stocks.
physical understanding of markets as complex dynami-
cal systems. In recent times, the role of the interaction
structure within a market in governing its overall dy-
namical properties has come under increasing scrutiny.
However, such intra-market interactions affect only very
weakly certain market properties, which is underlined
by the observation of identical fluctuation behaviour in
markets having very different interaction structures, viz.,
NYSE and NSE [7, 25]. The system can be considered as
a single homogeneous entity responding only to external
signals in explaining these statistical features, e.g., the
price fluctuation distribution. This suggests that the ba-
sic assumption behind the earlier approach of studying
financial markets as essentially executing random walks
in response to independent external shocks [34], which ig-
nored the internal structure, may still be considered to be
accurate for explaining market fluctuation phenomena.
In other words, complex interacting systems like finan-
cial markets can have simple mean field-like description
for some of their properties.
Acknowledgments
We thank N. Vishwanathan for assistance in preparing
the data for analysis. We also thank M. Marsili and
M.S. Santhanam for helpful discussions.
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TABLE I: The list of 201 stocks of NSE analyzed in this paper.
i Company Sector i Company Sector i Company Sector
1 UCALFUEL Automobiles Transport 68 IBP Energy 135 HIMATSEIDE Industrial
2 MICO Automobiles Transport 69 ESSAROIL Energy 136 BOMDYEING Industrial
3 SHANTIGEAR Automobiles Transport 70 VESUVIUS Energy 137 NAHAREXP Industrial
4 LUMAXIND Automobiles Transport 71 NOCIL Basic Materials 138 MAHAVIRSPG Industrial
5 BAJAJAUTO Automobiles Transport 72 GOODLASNER Basic Materials 139 MARALOVER Industrial
6 HEROHONDA Automobiles Transport 73 SPIC Basic Materials 140 GARDENSILK Industrial
7 MAHSCOOTER Automobiles Transport 74 TIRUMALCHM Basic Materials 141 NAHARSPG Industrial
8 ESCORTS Automobiles Transport 75 TATACHEM Basic Materials 142 SRF Industrial
9 ASHOKLEY Automobiles Transport 76 GHCL Basic Materials 143 CENTENKA Industrial
10 M&M Automobiles Transport 77 GUJALKALI Basic Materials 144 GUJAMBCEM Industrial
11 EICHERMOT Automobiles Transport 78 PIDILITIND Basic Materials 145 GRASIM Industrial
12 HINDMOTOR Automobiles Transport 79 FOSECOIND Basic Materials 146 ACC Industrial
13 PUNJABTRAC Automobiles Transport 80 BASF Basic Materials 147 INDIACEM Industrial
14 SWARAJMAZD Automobiles Transport 81 NIPPONDENR Basic Materials 148 MADRASCEM Industrial
15 SWARAJENG Automobiles Transport 82 LLOYDSTEEL Basic Materials 149 UNITECH Industrial
16 LML Automobiles Transport 83 HINDALC0 Basic Materials 150 HINDSANIT Industrial
17 VARUNSHIP Automobiles Transport 84 SAIL Basic Materials 151 MYSORECEM Industrial
18 APOLLOTYRE Automobiles Transport 85 TATAMETALI Basic Materials 152 HINDCONS Industrial
19 CEAT Automobiles Transport 86 MAHSEAMLES Basic Materials 153 CARBORUNIV Industrial
20 GOETZEIND Automobiles Transport 87 SURYAROSNI Basic Materials 154 SUPREMEIND Industrial
21 MRF Automobiles Transport 88 BILT Basic Materials 155 RUCHISOYA Industrial
22 IDBI Financial 89 TNPL Basic Materials 156 BHARATFORG Industrial
23 HDFCBANK Financial 90 ITC Consumer Goods 157 GESHIPPING Industrial
24 SBIN Financial 91 VSTIND Consumer Goods 158 SUNDRMFAST Industrial
25 ORIENTBANK Financial 92 GODFRYPHLP Consumer Goods 159 SHYAMTELE Telecom
26 KARURVYSYA Financial 93 TATATEA Consumer Goods 160 ITI Telecom
27 LAKSHVILAS Financial 94 HARRMALAYA Consumer Goods 161 HIMACHLFUT Telecom
28 IFCI Financial 95 BALRAMCHIN Consumer Goods 162 MTNL Telecom
29 BANKRAJAS Financial 96 RAJSREESUG Consumer Goods 163 BIRLAERIC Telecom
30 RELCAPITAL Financial 97 KAKATCEM Consumer Goods 164 INDHOTEL Services
31 CHOLAINV Financial 98 SAKHTISUG Consumer Goods 165 EIHOTEL Services
32 FIRSTLEASE Financial 99 DHAMPURSUG Consumer Goods 166 ASIANHOTEL Services
33 BAJAUTOFIN Financial 100 BRITANNIA Consumer Goods 167 HOTELEELA Services
34 SUNDARMFIN Financial 101 SATNAMOVER Consumer Goods 168 FLEX Services
35 HDFC Financial 102 INDSHAVING Consumer Goods 169 ESSELPACK Services
36 LICHSGFIN Financial 103 MIRCELECTR Consumer Discretonary 170 MAX Services
37 CANFINHOME Financial 104 SURAJDIAMN Consumer Discretonary 171 COSMOFILMS Services
38 GICHSGFIN Financial 105 SAMTEL Consumer Discretonary 172 DABUR Health Care
39 TFCILTD Financial 106 VDOCONAPPL Consumer Discretonary 173 COLGATE Health Care
40 TATAELXSI Technology 107 VDOCONINTL Consumer Discretonary 174 GLAXO Health Care
41 MOSERBAER Technology 108 INGERRAND Consumer Discretonary 175 DRREDDY Health Care
42 SATYAMCOMP Technology 109 ELGIEQUIP Consumer Discretonary 176 CIPLA Health Care
43 ROLTA Technology 110 KSBPUMPS Consumer Discretonary 177 RANBAXY Health Care
44 INFOSYSTCH Technology 111 NIRMA Consumer Discretonary 178 SUNPHARMA Health Care
45 MASTEK Technology 112 VOLTAS Consumer Discretonary 179 IPCALAB Health Care
46 WIPRO Technology 113 KECINTL Consumer Discretonary 180 PFIZER Health Care
47 BEML Technology 114 TUBEINVEST Consumer Discretonary 181 EMERCK Health Care
48 ALFALAVAL Technology 115 TITAN Consumer Discretonary 182 NICOLASPIR Health Care
49 RIIL Technology 116 ABB Industrial 183 SHASUNCHEM Health Care
50 GIPCL Energy 117 BHEL Industrial 184 AUROPHARMA Health Care
51 CESC Energy 118 THERMAX Industrial 185 NATCOPHARM Health Care
52 TATAPOWER Energy 119 SIEMENS Industrial 186 HINDLEVER Miscellaneous
53 GUJRATGAS Energy 120 CROMPGREAV Industrial 187 CENTURYTEX Miscellaneous
54 GUJFLUORO Energy 121 HEG Industrial 188 EIDPARRY Miscellaneous
55 HINDOILEXP Energy 122 ESABINDIA Industrial 189 KESORAMIND Miscellaneous
56 ONGC Energy 123 BATAINDIA Industrial 190 ADANIEXPO Miscellaneous
57 COCHINREFN Energy 124 ASIANPAINT Industrial 191 ZEETELE Miscellaneous
58 IPCL Energy 125 ICI Industrial 192 FINCABLES Miscellaneous
59 FINPIPE Energy 126 BERGEPAINT Industrial 193 RAMANEWSPR Miscellaneous
60 TNPETRO Energy 127 GNFC Industrial 194 APOLLOHOSP Miscellaneous
61 SUPPETRO Energy 128 NAGARFERT Industrial 195 THOMASCOOK Miscellaneous
62 DCW Energy 129 DEEPAKFERT Industrial 196 POLYPLEX Miscellaneous
63 CHEMPLAST Energy 130 GSFC Industrial 197 BLUEDART Miscellaneous
64 RELIANCE Energy 131 ZUARIAGRO Industrial 198 GTCIND Miscellaneous
65 HINDPETRO Energy 132 GODAVRFERT Industrial 199 TATAVASHIS Miscellaneous
66 BONGAIREFN Energy 133 ARVINDMILL Industrial 200 CRISIL Miscellaneous
67 BPCL Energy 134 RAYMOND Industrial 201 INDRAYON Miscellaneous
