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Accepted Manuscript

One-factor model for cross-correlation matrix in the Vietnamese stock

market

Quang Nguyen

PII: S0378-4371(13)00178-7

DOI: http://dx.doi.org/10.1016/j.physa.2012.10.048

Reference: PHYSA 14302

To appear in: Physica A

Received date: 22 June 2012

Revised date: 17 October 2012

Please cite this article as: Q. Nguyen, One-factor model for cross-correlation matrix in the

Vietnamese stock market,Physica A(2013), http://dx.doi.org/10.1016/j.physa.2012.10.048

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One-factor model for cross-correlation matrix in the

Vietnamese stock market

Quang Nguyen

John von Neumann Institute, Vietnam National University - Hochiminh City,

Hochiminh city, Vietnam

Abstract

Random matrix theory (RMT) has been applied to the analysis of cross-correlation matrix of financial time series. The most important findings ofprevious studies using this method are: eigenvalue spectrum largely followthat of random matrices but the largest eigenvalue is at least one orderof magnitude higher than the maximum eigenvalue predicted by the RMT.In this work, we investigate the cross-correlation matrix in the Vietnamesestock market using RMT and find similar results with studies realized indeveloped market (US, European, Japan) [9, 10, 11, 12, 13, 14, 15, 16, 17, 18]

as well as in other emerging market [19, 20, 21, 22]. Importantly, we foundthat the largest eigenvalue could be approximated by the product of theaverage cross-correlation coefficient and the number of stocks studying. Wedemonstrate this dependence using a simple one-factor model. The modelcould be extended to describe other characteristics of the realistic data.

Keywords: cross-correlation, random matrix, eigenvalue

1. Introduction

Cross-correlation of financial time series is of great important in manag-

ing investment risk and in constructing an efficient portfolio [1]. When thenumber of studying time series is small and if non-stationary exists betweentime series, which is usually the case in finance, one may use the detrendedcross-correlation approach [2, 3, 4, 5, 6]. In contrast, when one studies thecross-correlation matrix of the whole stock market which composes of severalhundreds of individual stocks, it becomes a high-dimensional and complexsystem which is not easy to study. Recently, some method developed in sta-

Preprint submitted to Physica A October 17, 2012

*Manuscript

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tistical physics have been employed [7, 8], in particular the random matrixtheory (RMT) [9, 10]. These authors compare the eigenvalue spectrum of thefinancial correlation matrix with that of a random matrix and found theseimportant remarks:

Most of eigenvalue of the financial correlation matrix fall within atight range predicted by random matrix theory< < +, where theRMT boundsand+ are positive and close to 1 (explicit formula isgiven in the next section). In consequence, it was argued that most of

the structure of cross-correlation matrix are due to noise. This clusteredgroup of eigenvalue is called the bulk.

There is one largest eigenvaluemaxwhich is10 30 higher than themaximum expected value predicted by RMT + and its correspondingeigenvector is assigned to the market portfolio.

There are several other eigenvalue slightly greater than+which reflectthe sector behavior

There are a number of eigenvalues smaller than , the lower bound ofeigenvalue predicted by RMT, correspond to a specifically high corre-

lated pair of stocks.[10]

Other studies on developed stock market (US, European, Japan) showedaccordance with these findings [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 21]. Inaddition, a few of studies on emerging markets [19, 20, 22] showed some slightdifferent properties compare to the above developed markets, such as:

1. The average value of non-diagonal elements of C is higher and fluctuatesmore dynamically [19, 20, 22]

2. The largest eigenvaluemax is significantly higher (50) than + [20,22]

3. Excepting the largest eigenvalue, there are fewer number eigenvaluegreater than+[19, 20, 22]

4. In contrast, there are a large proportion of eigenvalues smaller than.In consequence, the total ratio of eigenvalues which fall within RMTbounds is far smaller[20, 22]

The point 1 is quite evident because in emerging markets, stocks movemore in tandem due to the low-diversification level of companies and the

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important impact of common macro economic factors. In consequence, theaverage correlation coefficient in emerging markets is usually higher thanthose in developed markets. Furthermore, as the market participants are lessprofessional, they create more volatility and higher fluctuation of correlationbetween stocks.

In investigating the largest eigenvalue time evolution, Kulkarniet al. [19]found positive correlation between the largest eigenvalue and the marketvolatility. Because it is commonly known that correlation tends to increaseduring volatile period, the correlation between the largest eigenvalue and the

correlation level is implicitly mentioned. In subsequence section, we will shownumerically that the largest eigenvalue is proportioned to the average value ofcorrelation matrix elements. Together with the point 1, the largest eigenvaluein emerging markets turns out to be higher than that in developed markets,or the point 2. In [21], the author mentioned a temporal opposite movementbetween the largest eigenvalue and the bulk of small eigenvalues due to thepropulsion effect: as the sum of all eigenvalues is always constant, a change invalue of the largest eigenvalue must be compensated by an opposite changeof the others, or the shift of the bulk. This remark explained the point 4because a high value of the largest eigenvalue in emerging markets will resultin a shift of the eigenvalue bulk out of the lower RMT bound .

A part from these deviations from the conventional RMT, there are moresubtle but systematic differences also in the bulk of eigenspectrum as dis-cussed in [23, 24]. These differences, masked by noise, may contain usefulinformation on the cross-correlation structure. In addition, RMT is also usedin order to study time-lag cross correlations in multiple time series of finance[25] as well as biology and atmospheric geophysics [26]. These authors foundlong-range power-law cross correlations in the absolute values of returns thatquantify risk and find that they decay much more slowly than cross correla-tions between the returns.

In this paper, we investigate the statistical properties of cross-correlation

matrix of N = 90 stocks traded in the Ho Chi Minh city stock exchange(HCMCSE) from 1 January 2007 to 2 May 2012 using RMT method. Ourstudies are accorded to the previous findings in both developed and emergingmarkets. In addition, we try to quantify the magnitude of the largest eigen-value by analyzing its dependencies on the stock number and the averagecross-correlation coefficient. We employ the one-factor model similar to [21]in order to demonstrate these behaviors. We present our method and theone factor model in section 2, then we introduce briefly the data in the Viet-

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namese stock market in section 3. In section 4 we look at the result obtainedfrom empirical data as well as the simple model of correlation matrix. Weconclude in Section 5 with some potential extensions of the model.

2. Method

We consider a set ofNstocks over a period ofT trading days. Let Si(t)the price of stock i (i = 1,...,N) at day t (t = 1,...,T). The daily returnRi(t) of stock i is defined by

Ri(t) = ln(Si(t+ 1)) ln(Si(t)) (1)Since different stocks may have different volatility, we normalize the re-

turn by

ri(t) =Ri(t) Ri(t)

i(2)

wherei=

R2i (t) Ri(t)2 is the standard deviation or volatility ofRi(t)and... denotes the average return (or trend) over the period studied. Theequal-time cross-correlation matrix C between N stocks is a N N matrixwith elements

Cij = ri(t)rj(t) (3)Cij denotes the correlation coefficient between stock i and j, which has valuebetween -1 and 1. Cij = 0 means that there is no correlation between stocki and j while 0 < Cij 1(1Cij



with independent random elements drawn from a known probability distri-bution p. For the case of financial time series, a random cross-correlationmatrix CRMT is obtained from N time series of length L of random re-turn with zero mean and unit variance, using the same calculation of (2,3).Statistical properties of such random cross-correlation matrices are solved byDyson and Mitra in [34, 35, 36]. In particular whenN andL suchthatQ L/N >1 is fixed, one obtain an analytical density of eigenvalue ofCRMTgiven by the Marchenko-Pastur formula [36]

RMT() = Q2

(+ )( )

(5)

where lies between and+, the minimum and maximum eigenvalues ofCRMT, respectively given by

= 1 + 1

Q 2

1

Q (6)

and equal to zero elsewhere. As mentioned above, eigenvalues ofCthat fallin between these bounds are interpreted as noise.

In our one-factor model, we propose a simple form of trueNNcross-correlation matrix E, where all non-diagonal elements are constant, similarto [21], of value 0 between (1, 1) (its diagonal elements are 1). We gener-ate independently Ttimes a jointly standard normal distributed N-vector.Combining these Tvectors we obtain Nsimulated time series of length T,which replace the N stocks empirical time series of return. The simulatedcorrelation matrix Cis calculated using function 2. With Tis large enough,non-diagonal elements ofCwill be normally distributed around0. It is easyto deduce the eigenvalues ofE: one large eigenvalue equals to 1 + (N 1)0( N 0 whenNis large enough) and (N 1) folddegenerate eigenvaluesequal to 10< 1. Therefore, we expect that the largest eigenvalue ofCwillbe of order

N 0 and there will be a bulk of small eigenvalues distributed

around 1 0



3. Data

Vietnamese stock market is a frontier market which dated only since 2000when HCMCSE was established. In 2006, the second exchange was openedin Hanoi which trades relatively small stocks. The number of quoted stocks,hence the market size, has increased gradually, and became considerable since2007. As the market size of HCMCSE is 3 higher than that of the other,we analyze the daily stock return in HCMCSE from 1 January 2007 to 2 May2012. During this period there are N = 90 stocks which were continuouslytrading for a total of L = 1324 trading days. We use the closing price tocalculate the daily return.

4. Results and discussion

4.1. Distribution and dynamics of correlation coefficient

In this section, we analyze the statistical properties of the elements ofcross-correlation matrix C. Figure 1(a) shows the distribution P(Cij) of90 stocks over the whole analyzing period from 1 January 2007 to 2 May2012. Other descriptive statistics are presented in table 1. We found thatthe average value

Cij

of 0.3663 is relatively large in comparison to others

studies, suggesting that stocks in Vietnamese market are strongly correlated.Furthermore, the whole period correlation matrix elements are all positive,suggesting that the diversification level within the Vietnamese stock marketis quite low. In consequence, its systematic risk is rather high. We also foundthat over the whole period the distribution is relatively closed to normal, witha small positive skewness of 0.13 (there is a few number of high elementshigher than 0.6) and kurtosis of 2.7 (3.0 is considered as normal). Thesefindings support our above assumptions in the one-factor model.

In figure 1(b) we plot the time-varying distribution of cross-correlationmatrices using a sliding window of 250 days with five days lag time. We foundthat this distribution changes considerably over time. Figure 1(c) shows thetemporal dynamic of the average value Cij and others descriptive statisticsof elements of C using the same slide windows. The average correlationcoefficients peaked at the crisis period of end 2008, then gradually decreases.Until August 2010, its distribution standard deviation is relatively low, itsskewness is negative and kurtosis is closed to the corresponding normal value.After August 2010, the distribution became positively skew and its shapedeviated from that of normal distribution.

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Figure 1: (a) Probability density ofCij for the whole period 2007-2012. (b) Temporalevolution of cross-correlation distribution using a sliding window of 250 days with fivedays lag time. One notices strong fluctuation of the average valueCij as well as thedistribution shape. (c) Summary statistics of the distributions in (b) plotted in functionof times

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http:///reader/full/one-factor-model-for-cross-correlation-matrix-in-the-vietname

Mean Standard deviation Skewness Kurtosis

0.3664 0.0910 0.1347 2.7147

Table 1: Statistics for cross-correlation of Vietnamese stock between Jan 2007 - May 2012

4.2. Eigenvalue spectrum

In this section, we decompose the cross-correlation matrix and calcu-late its eigenvalues and eigenvectors. The eigenvalue spectrum is showed in2(a) together with the spectrum predicted by RMT theory. We find similar

characteristics as most other studies: the largest eigenvalue max = 34.4,while N Cij = 90 0.3664 = 32.9. The similarity between max andN Cij again support arguments in our simple one-factor model. In ourstudy, the theoretical and + are 0.546 and 1.589, respectively. In theinsert graph, we found that apart from the largest eigenvalue, there are twoother eigenvalues which are beyond the maximum RMT value+. This devi-ating eigenvalue number is small in comparison to other studies in developedmarkets [9, 10], but in line with those in emerging markets [20]. These eigen-values are assigned to the sector group in the correlation matrix [10]. Thatmeans stocks in emerging markets are less sector-specific than in developedmarkets.

Nb. Groups Market L N Q Cij max +1 V. Plerou US 8685 422 20.58 0.12 46.3 1.49*2 A. Utsugi US 2598 297 8.75 - 52.2 1.793 A. Utsugi Tokyo 1848 493 3.75 - 121.6 2.34 S. Cukur Turkey 1516 206 7.36 0.35 83.3 1.875 G. Oh Korean 2845 473 6.01 0.20 96 1.86 D. Wilcoz S. Africa 1304 244 5.34 - 21.2 2.237 V. Kulkarni India 80 70 1.14 - 9.17 3.638 Our group Vietnam 1324 90 14.7 0.366 34.4 1.59

Table 2: Data summary and largest eigenvalue statistics of some previous studies; ():estimated from publication; (*): calculated using formula 6. Remark that in studies 1 and5, Nare closed but asCij is 2 higher in 5, max is also 2 higher in 5. In 4 and 8,Cijare similar, but N is 2.3 higher in 4 resulting in max of 2.4 higher

In addition, we found that only about half of small eigenvalue bulk liesbetween RMT bounds, considerably lower than results in developed mar-kets [10]. We account this fact to the repulsion effect between the largest

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Figure 2: (a) Probability density ofi in comparison with RMT density (the red solidline). Remark that the largest eigenvalue max is 21 times higher than +. Insert graphshow a zoom into the bulk. We found 2 other eigenvalues larger than +. The bulk issignificantly deviated from RMT, with only half fall within the RMT bounds. (b) Inverseparticipant ratio of eigenvalues (c) The time-varying comparision ofCij, the largesteigenvalue, percentage of deviating eigenvalue and the average of 80 small eigenvalues

using the sliding windows of 250 days with 5 days lag time.

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eigenvalue and the small eigenvalues [21] as the sum of all eigenvalues remainsconstant: as+is higher in emerging markets as discuss previously, the smalleigenvalues bulk is repulse further to the left resulting in a high ratio of RMToff-limit. We demonstrate this effect in figure 2(c) where the movements ofthe largest eigenvalue and the average of 80 smallest eigenvalues are oppositewith correlation factor of -0.97. Combined with the previous findings, wededuce that the ratio of deviating eigenvalue from RMT bounds, mainly dueto the shift of small eigenvalues bulk beyond , is caused principally by therepulsion effect when there exists a positive average correlation in C. The

correlation between the ratio of deviating eigenvalue (which are considerednon-noise eigenvalue in some studies) and the average correlation coefficientis 0.97 as showed in 2(c).

In the previous studies, the number of stocks Nand the average correla-tion coefficients varied between groups. According to statistics presented intable 2, we find that when N is similar between groups, the market whichhas a higher average value of correlation coefficient has higher max. In ad-dition, when the average correlation is similar, group who analyzes a highernumber of stocks obtains higher value ofmax. As groups usually chose avalueQ = 5, . . . , 10 1, according to function 6, + varies slightly between1.5-3. Those authors estimated the deviation effect of the correlation struc-

ture from noise by the ratio max to + and there is no magnification effectof the stock numberN in+(though it still depends onNthroughQ, but toa certain limit). In reality when this assumption is in general no longer true,i.e.Cij is different from zero, the value ofmax will linearly depend on thestock number N. We examined this relation by plotting the time-varyingevolution of max and the average of the cross-correlation matrix using asliding window of 250 days with five days lag time in figure 2(c). We foundalmost identical dynamic between these two quantities as found in [37, 19],with correlation of 0.99.

In figure 3, we show the component distribution of some eigenvectors cor-

responding to: the largest eigenvalue 1, the second largest deviating eigen-value2 (larger than+), one eigenvalue of RMT bulk 45 and the smallesteigenvalue 90. We find similar result to other studies: the components ofthe eigenvector correspond to the largest eigenvalue are almost equal. Thisvector is argued to represent the market portfolio. In contrast, the eigen-vector corresponding to the smallest eigenvector 90 has only 3 significantcomponents. We identified that these 3 stocks [39] are the most active stocksduring the studying period and has the highest cross-correlation coefficients

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in the market. Portfolios constructed from these smallest eigenvectors willhave smallest volatility and therefore, could be used to determine statisticalarbitrage strategy. The component spectrum of2 and45, however, do notreveal any remarkable characteristics. We further investigate the eigenvectorby calculating the Inversed participation ratio (IPR) defined as

Ik =Nl=1

[ukl]4

(7)

whereuk

l, l= 1,....,Nare the components of thekth

eigenvector. The inverseof IPR represents the number of components that contribute significantly inthe portfolio corresponding to that eigenvector. The IPR spectrum is showedin figure 2(b). As expected, the value of IPR of the largest eigenvalue is1/82.5, showing that almost stocks participate. The IPR of the second andthird largest eigenvectors that deviate from RMT are also low, suggestingthat these two eigenvalues may representing two large groups (it could besector specific or other reason) of stocks that have a higher degree of corre-lation. On the other hand, the two smallest eigenvectors have the highestIPR which show that only a few stocks participate in them, as we have seenin the previous paragraph. These small eigenvalues are the results of some

particular high correlated pair or triple stocks[10]. It is interesting to notethat theIk is higher than that of the average bulk at the high and low endsof the small eigenvalue bulk (see also [20]), suggesting that there are someparticular correlation structure of small group of stocks yet to be identified.It is remarkable that the IPR at the high end of the bulk is higher than theaverage bulk while their corresponding eigenvalues are around 1 and are stillinside the range predicted by RMT. One may suspect that the eigenvaluesfall within the RMT bound do not need to be pure noise [40].

4.3. One-factor model

In this section, we present simulation result of the one factor model forthe cross-correlation matrix described earlier. Our model takes N, L and0 as input, and generates the cross-correlation matrix Cand its eigenvaluespectrum as output. Firstly, we simulate the empirical matrix C by takingthe same number ofN, L and let 0 equal toCij. We present the eigen-value spectrum ofC in 4 (a) and the inverse participation ratio of randomeigenvalue in 4 (b). AsN, L are the same as in Section 4.2, the RMT the-oretical spectrum is the same and is plotted altogether. Figure 5 displays

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Figure 3: Components of eigenvectors u1, u2, u45 and u90. In u1 the components ofall stocks are positive and relatively uniform, this eigenvector characterizes the marketportfolio. In the smallest u90 there are only 3 significant components. These three stocksare found to have strong correlation with each other. Other two eigenvectors do not showremarkable characteristic.

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Figure 4: The largest eigenvalue and the small eigenvalue bulk could be well approximatedto empirical data by eigenvalue spectrum (a) and by the inverse participant ratio (b)

components of some eigenvectors. The eigenvalue statistics of simulated andempirical data are showed in table 3.

Sample Cij max min > + < % noiseReal data 0.3664 34.45 0.1874 3 44 0.48

Simulation 0.3712 34.05 0.3461 1 34 0.63

Table 3: Eigenvalue statistics of real and simulated data

We found that the largest eigenvalue of simulated data is closed to thatof empirical data, suggesting that the high magnitude of max, the mostimportant deviation from RMT, could be explained by an unique factor: thepositive average correlation between stocks. The high value of stock numberNwill enhanced this effect but is not the original cause. On the other hand,we found that the small eigenvalue bulk also shifts to the left of the RMTbounds, as a consequence of the repulsion effect by the largest eigenvalue.

The model predicts about 70% of eigenvalues out the RMT low limit. Themodel also explains the IPR value of the largest and the bulk eigenvector.However, it does not explain the appearance of few eigenvalues higher thanthe RMT high +; the component spectrum of eigenvector corresponding tothe smallest eigenvalue (where there is a few high components as discussedin 4.2; the high IPR number of some eigenvalues at both ends of the smalleigenvalue bulk as shown in figure 2(b). Addition features are needed toinclude into the model in order to demonstrate these behaviors.

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Figure 5: Components of eigenvectors u1,u2,u45 of random generated data. Componentsof largestu1are identical and equal to 1/

N, while components of the others are randomly

distributed.

Finally, we simulate the dependence of max on two factors,Cij andthe stock number N using our model. On each simulation, we kept 2 fixedparameters as that of empirical data and varied the rest. Figure 6 shows thedependence ofmax onCij (a) and on N (b). We found that whenCijvaries from 0 to 1, the max goes from the RMT high bound + to N. Thedependence is positive but sublinear and an explicit analytical function is onour future research plan. In contrast, the dependence on Nis almost linear.This result supports our discussion in the previous section.

5. Conclusion

In conclusion, we have analyzed the statistical characteristics of Viet-namese stock cross-correlation matrix using RMT theory. Our result resem-bles most of previous studies on the eigenvalue spectrum and eigenvectorcomponents. It also shares some specific characteristics of emerging mar-kets as: high correlation level, weak sector grouping effect. We do explainthe largest eigenvalue magnitude by the appearance of positive average cor-relation coefficient among stocks and the high number of stocks analyzed.

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Figure 6: Simulation result of largest eigenvalue maxgenerated by the model in functionof average correlation coefficient 0 (a) and stock number N (b). max is found to be aconvex function of0 and linearly depend on N

We demonstrate this argument by a simple model. Despite its simplicity,our model can explain the value of the largest eigenvalue, the shift of thesmall value bulk, the approximate ratio of deviating eigenvalue. Other non-explained characteristics such as other deviating eigenvalue beyond the max-imum RMT, components of the smallest eigenvalue, the high IPR of some

eigenvalues both in the bulk and beyond the bulk are expected to be identifiedin more sophisticated models.

Acknowledgement

We thank the referees of Physica A for their pertinent comments andreferences, such as [2, 23, 25, 36, 37, 38]. The work has been financiallysupported by the Grant B2011-42-01TD from Vietnam National University- Ho Chi Minh City.

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