emerging markets and the global economy || correlation and network structure of international...

CHAPTER3333Correlation and Network Structure ofInternational Financial Markets in Timesof CrisisLeonidas SandovalInsper Instituto de Ensino e Pesquisa, Brazil

1. INTRODUCTION

The last two largest global economic crises have their origins in the financial markets:the first one, which reached its peak in 2008, originated from the so-called suprimemortgages. This crisis was triggered by the default of a large number of mortgages inthe USA. Subprimes are loans to borrowers who have low credit scores. Most of themhad a small initial interest rate, adjustable for future payments, which led to many homeforeclosures after the rates climbed substantially. Meanwhile, the loans were transformedin pools that were then resold to interested investors. Since the returns of such investmentswere high, a financial bubble was created, inflating the subprime mortgage market untilthe defaults started to pop up.

Because of their underestimation of risk, financial institutions worldwide lost trillionsof dollars, and many of them declared bankruptcy, and credit lines tightened around theworld, taking the financial crisis to the so-called real economy and hitting many countriesin Europe with devastating effects.The so-called Credit Crisis is an ongoing one, and noone knows how much damage to the world economies it is yet to cause.

Other global financial crises that happened fairly recently were the Black Monday(1987), theAsian Crisis (1997), the Russian Crisis (1998), the burst of the dot.com bubble(2000 and 2001), and many minor crises. All of them should not have happened in all ofthe age of the Universe (about 13 billion years), according to the mainstream financialtheory and practice.

Economists have been studying the reasons why markets crash, and why there ispropagation of volatility from one market to another, since a long time. After the crashof 1987, many studies have been published on transmission of volatility (contagion)between markets using econometric models, on how the correlation between worldmarkets change with time, and on how the correlation tends to increase in times ofhigh volatility. This issue is of particular importance if one wishes to build portfolios ofinternational assets which can withstand times of crisis.

Emerging Markets and the Global Economy © 2014 Elsevier Inc.http://dx.doi.org/10.1016/B978-0-12-411549-1.00033-8 All rights reserved. 795

http://dx.doi.org/10.1016/B978-0-12-411549-1.00033-8

796 Leonidas Sandoval

One tool that was first developed in nuclear physics for studying complex systemswith unknown correlation structure is random matrix theory (see Mehta, 2004), whichconfronts the results obtained for the eigenvalues of the correlation matrix of a real systemwith those of the correlation matrix obtained from a pure random matrix.This approachhas been successfully applied to a large number of financial markets. In particular,Maslov(2001) applied Random Matrix Theory to the study of the relations between worldmarkets. This approach has also been used in the construction of hierarchical structuresbetween different assets of financial markets.

In this chapter, we shall focus on the time series of 79 benchmark indices across theworld during the 10 years from the beginning of 2003 to the end of 2012. The indicesand the countries they belong to are described in Section 2.The other crises listed abovehave been analyzed by Sandoval and Franca (2012), using basically the same tools as here,and their network structure was studied in Sandoval (2012a) (as Minimum SpanningTrees) and in Sandoval (2013) (as Asset Graphs). Section 3 explains how Random MatrixTheory is applied to the data;Section 4 deals with the connection between the correlationof indices and the volatility of the world market; Section 5 uses the correlation betweenindices in order to build networks based on asset graphs. Section 6 offers some finalconclusions.

2. THE DATA

The data consist on the time series of the closing values of the benchmark indices of79 stock exchanges across the globe, collected from the beginning of 2003 to the endof 2012. The indices and the countries to which they belong are: S&P 500 (UnitedStates of America), S&P/TSX Composite (Canada), IPC (Mexico), Bermuda SX Index(Bermuda), Jamaica SX Market Index (Jamaica), BCT Corp Costa Rica (Costa Rica),Bolsa deValores de Panama General, Merval (Argentina), Ibovespa (Brazil), IPSA (Chile),IGBC (Colombia), IGBVL (Peru), IBC (Venezuela),FTSE 100 (United Kingdom), ISEQ(Ireland),CAC 40 (France),DAX (Germany),ATX (Austria), SMI (Switzerland),BEL 20(Belgium),AEX (Netherlands), OMX Stockholm 30 (Sweden), OMX Copenhagen 20(Denmark), OMX Helsinki (Finland), OBX (Norway), OMX Iceland All-Share Index(Iceland), MIB-30 (Italy), IBEX 35 (Spain), PSI 20 (Portugal),Athens SX General Index(Greece),WIG (Poland), PX 50 (Czech Republic), SAX (Slovakia), CROBEX (Croa-tia), OMXT (Estonia), OMXR (Latvia), Budapest SX Index (Hungary), SOFIX (Bul-garia), PFTS (Ukraine), MICEX (Russia), ISE National 100 (Turkey), KASE (Kaza-khstan), Malta SX Index (Malta),Tel Aviv 25 (Israel), Al Quds (Palestine), ASE GeneralIndex (Jordan), BLOM (Lebanon), TASI (Saudi Arabia), MSM 30 (Oman), DSM 20(Qatar),ADX General Index (United Arab Emirates), KSE 100 (Pakistan), SENSEX 30(India),DSE General Index (Bangladesh), Sri Lanka Colombo Stock Exchange All-ShareIndex, Nikkei 225 (Japan), Hang Seng (Hong Kong), Shangai SE Composite (China),

Correlation and Network Structure of International Financial Markets in Times of Crisis 797

TAIEX (Taiwan), KOSPI (South Korea), MSE TOP 20 (Mongolia), Straits Times (Sin-gapore), Jakarta Composite Index (Indonesia), KLCI (Malaysia), SET (Thailand), PSEI(Philippines),VN-Index (Vietnam), S&P/ASX 200 (Australia), NZX 50 (New Zealand),CFG 25 (Morocco), TUNINDEX (Tunisia), EGX 30 (Egypt), Nigeria SX All ShareIndex (Nigeria), Gaborone (Botswana), GSE All Share Index (Ghana), NSE 20 (Kenya),SEMDEX (Mauritius), and FTSE/JSE Africa All Share (South Africa). The data werecollected from a Bloomberg terminal.

Since we are dealing with countries that have different holidays and, in some cases,different weekends, we fixed the dates in which the New York Stock Exchange, by farthe most important in terms of volume of negotiations and in influence, operated. If anindex was not computed in one of the same dates, its value of the previous working daywas repeated; in case one index operated in a day the New York Stock Exchange didnot, then the index for that date was deleted. This contrasts with the common methodused, which is to delete all days to which one of the time series had no data assigned;that would lead to the deletion of too many days, and we then would have returns thatwere not true to the real variations of the indices. Indices that were most affected by ourmethod were the ones from Israel, Jordan, Saudi Arabia, Oman, Qatar, the United ArabEmirates, and Egypt, where weekends are not on Saturdays and Sundays. Nevertheless,we think that our choice preserves best the real returns in the major stock markets.

The data were used in order to calculate log-returns, defined as:

St = ln (Pt) − ln (Pt−1) ≈ Pt − Pt−1

Pt, (1)

where Pt is the value of an index on day t and Pt−1 is the value of the same index on theprevious day. Such measure is often used in order to avoid non-stationarity of data.

The log-returns are then used in order to calculate the Spearman rank correlationbetween each index, and the results are ordered in a correlation matrix. The reason forusing the Spearman rank correlation and not the usual Pearson correlation is because theformer is better at measuring nonlinear correlations, although both correlation measuresyield very similar results.

In order to study the effects of the crisis of 2008 and onwards,we divide the data intotwo blocks, the first pertaining the 5 years from 2003 to 2007, and the other containingthe years from 2008 to 2012. Then, Random Matrix Theory is used in order to findcorrelations that seem not to be random. This is explained in the next section.

3. RANDOMMATRIX THEORY

Random matrix theory had its origins in 1953, in the work of the Hungarian physicistEugeneWigner. He was studying the energy levels of complex atomic nuclei, such as ura-nium,and had no means of calculating the distance between those levels. He then assumed


that those distances were random, and arranged the random number in a matrix whichexpressed the connections between the many energy levels. Surprisingly, he could thenbe able to make sensible predictions about how the energy levels related to one another.

This method also found connections with the study of the Riemann zeta function,which is of primordial importance to the study of prime numbers, used for coding anddecoding information, for example. The theory was later developed, with many andsurprising results arising. Today, random matrix theory is applied to quantum physics,nanotechnology, quantum gravity, the study of the structure of crystals, and may haveapplications in ecology, linguistics, and many other fields where a large amount of appar-ently unrelated information may be understood as being somehow connected.The theoryhas also been applied to finance in a series of works dealing with the correlation matricesof stock prices, and to risk management in portfolios.

The first result of the theory that we shall mention is that, given an L × N matrixwith random numbers built on a Gaussian distribution with average zero and standarddeviation σ , then, in the limit L → ∞ and N → ∞ such that Q = L/N remains finiteand greater than 1, the eigenvalues λ of such a matrix will have the following probabilitydensity function, called a Marcenku-Pastur distribution (Marenko and Pastur, 1967):

ρ (λ) = Q2πσ 2

√(λ+ − λ)(λ − λ−)

λ, (2)

where

λ− = σ 2

(1 + 1

Q− 2

√1

Q

), λ+ = σ 2

(1 + 1

Q+ 2

√1

Q

), (3)

and λ is restricted to the interval [λ−, λ+].Figure 1 shows some Marcenku-Pastur distributions for some values of Q and σ .

Since the distribution (2) is only valid for the limit L → ∞ and N → ∞, finite distri-butions will present differences from this behavior. Another source of deviations is thefact that financial time series are better described by non-Gaussian distributions, suchas t Student or Tsallis distribution. Biroli et al. (2007) calculated a probability densityfunction analogous to the Marcenku-Pastur distribution, but based on a t Student dis-tribution for the randomized data. Their results show that there is a longer tale towardhigher eigenvalues which decays as a power law.

In Figure 2, we compare the theoretical distribution for Q = 10 and σ = 1 withdistributions of the eigenvalues of three correlation matrices generated from finite L ×Nmatrices such that Q = L/M = 10,and the elements of the matrices are random numberswith mean zero and standard deviation 1. Real data will deviate from the theoreticalprobability distribution. This may be taken into account by randomizing the order ofdata in each of the time series used so as to preserve its mean and standard deviation, butdestroy any correlation between the time series. The result of a large enough number of


λ

ρ(λ)

1 2

0, 5

1

Q = 3 and σ = 1

λ

ρ(λ)

1 2

0.5

1

Q = 5 and σ = 1

λ

ρ(λ)

1 2

0.5

1

Q = 12 and σ = 1

λ

ρ(λ)

0, 5 1

1

2

3

Q = 3 and σ = 0.5

λ

ρ(λ)

1 2

0.5

1

Q = 3 and σ = 0.8

λ

ρ(λ)

2 4 6 8 10

0.1

0.2

Q = 3 and σ = 2

Figure 1 Marcenku-Pastur distribution for fixed values of Q and σ .

λ

ρ(λ)

1 2

0, 5

1

N = 10 and L = 100

λ

ρ(λ)

1 2

0, 5

1

N = 30 and L = 300

λ

ρ(λ)

1 2

0, 5

1

N = 100 and L = 1000

Figure 2 Histograms of eigenvalues for generated correlationmatrix andMarcenku-Pastur theoreticaldistribution (solid line) for Q = 10 and σ = 1.

such simulations may then be compared with the results obtained with the original datainstead of comparing the latter with the theoretical Marcenku-Pastur distribution.

In order to apply Random Matrix Theory to the 79 stock exchanges, we dividetheir time series into two blocks: the first ranging from the beginning of 2003 to theend of 2007, so pre-crisis, and the second encompassing the years from 2008 to 2012(post-crisis). The first block of time series is formed by 1258 observations (days), so thatQ = L/M = 1258/79 ≈ 15, 92; the second block of time series is formed by 1259observations (days), so that Q = L/M = 1259/79 ≈ 15, 94. For these values, we haveλ− ≈ 0.56 and λ+ ≈ 1.56 for both blocks of time series.


λ

ρ(λ)

1 2 3 4 15 16

0.05

0.10

0.15

0.20

λ

ρ(λ)

1 2 3 4 5 6 21 22

0.05

0.10

0.15

0.20

Figure 3 Histograms of the eigenvalues of the correlationmatrices obtained from the log-return seriesfor 2003–2007 (left figure) and for 2008–2012 (right figure), in block graphs. The probability distribu-tions of the eigenvalues for 10,000 simulations with randomized data for each block of observationsare shown as solid lines.

The two blocks of log-returns are used in order to build correlation matrices (usingthe Spearman rank correlation) and then to calculate the eigenvalues and eigenvectorsof both matrices.The two graphs of Figure 3 show histograms for the eigenvalues of thecorrelation matrices for 2003–2007 (left graph) and for 2008–2012 (right graph).Togetherwith the graphs are plotted as solid lines the probability distributions of the eigenvaluesfor 10,000 simulations with randomized data for each block of observations. The resultsof the simulations are very similar to the theoretical Marcenku-Pastur distributions.

The first striking feature is that the largest eigenvalue for each block of data is muchlarger than the maximum limit expected from a distribution obtained from randomdata. There are also other eigenvalues that are located outside the region defined bythe Marcenku-Pastur theoretical distribution. This feature is enhanced if one plots theeigenvalues, like in Figure 4,where the eigenvalues are plotted as vertical lines and the grayareas are the region predicted for the Marcenku-Pastur distribution, and associated withnoise. On the left, we have the eigenvalues for data collected in 2003–2007, and on theright, the eigenvalues for data collected in 2008–2012. Note that the largest eigenvaluefor 2008–2012 is rather larger than the largest eigenvalue for 2003–2007. Also, for 2008–2012, there are three other eigenvalues that detach themselves from the bulk, while for2003–2007, there are two of them. Besides the eigenvalues that are above the valuespredicted for a random collection of data, there are many eigenvalues below the sameprediction. This happens in both periods being analyzed.

More information on the meaning of the largest eigenvalues may be obtained ifone plots the eigenvectors associated with them. Figure 5 shows the components of theeigenvectors, plotted as column bars, associated with the three largest eigenvalues for thedata collected in 2003–2007.The eigenvalue associated with the largest eigenvalue (e1) hasa very distinct structure, as nearly all indices appear with positive values. The exceptionsare indices from stock markets that are very small in terms of number of stocks and ofvolume of negotiations. This eigenvector is often associated with a market mode, and a


λ1 2 3 4 15 16

λ1 2 3 4 5 6 21 22

Figure 4 Eigenvalues of the correlationmatrices for the log-returns of 2003–2007 (left) and 2008–2012(right), represented as vertical lines. The gray areas are the regions predicted for the Marcenku-Pasturdistributon, and associated with noise.

e1

USAColombia

NetherlandsPoland

RussiaQatar

South KoreaMorocco

South Africa

0.1

0.2

e2

USAColombia

NetherlandsPoland

RussiaQatar

South KoreaMorocco

South Africa

0.1

0.2

e3

USAColombia

NetherlandsPoland

RussiaQatar

South KoreaMorocco

South Africa

0.1

0.2

0.3

0.4

e4

USAColombia

NetherlandsPoland

RussiaQatar

South KoreaMorocco

South Africa

0.1

0.2

0.3

0.4

Figure 5 Components of the eigenvectors of the four largest eigenvalues of the correlation matrix ofthe log-returns of indices collected in 2003–2007. White bars mean positive values and gray bars, neg-ative ones. The order of indices is the following: USA, Canada, Mexico, Bermuda, Jamaica, Costa Rica,Panama, Argentina, Brazil, Chile, Colombia, Peru, Venezuela, UK, Ireland, France, Germany, Austria,Switzerland, Belgium, Netherlands, Sweden, Denmark, Finland, Norway, Iceland, Italy, Spain, Portu-gal, Greece, Poland, Czech Republic, Slovakia, Croatia, Estonia, Latvia, Hungary, Bulgaria, Romania,Ukraine, Russia, Turkey, Kazakhstan, Malta, Israel, Palestine, Jordan, Lebanon, Saudi Arabia, Oman,Qatar, United Arab Emirates, Pakistan, India, Bangladesh, Sri Lanka, Japan, Hong Kong, China, Taiwan,SouthKorea, Mongolia, Singapore, Indonesia, Malaysia, Thailand, Philippines, Vietnam, Australia, NewZealand,Morocco, Tunisia, Egypt, Nigeria, Botswana, Ghana, Kenya, Mauritius, and South Africa.

portfolio built by taking the value of each index as its weight would follow very closelythe general movements of the international stock market. In fact, when compared withan index of the global stock market, like the MSCI World Index, the portfolio built interms of eigenvector e1 has Pearson correlation 0.77.

Eigenvector e2, related with the second largest eigenvalue,has a structure that is typicalof stock markets that do not operate at the same time, as seen in Sandoval (2012b). Itgenerally shows positive values forWestern countries and negative values for Eastern ones,defining two basic blocks. Eigenvector e3, associated with the third largest eigenvalue,shows strong positive peaks in North America and South America, and negative peaks inEurope. Eigenvalue e4, associated with the fourth largest eigenvalue, has strong positivepeaks in Arab countries in the Middle East and North Africa, and smaller negative ones


e1

USAColombia

NetherlandsPoland

RussiaQatar

South KoreaMorocco

South Africa

0.1

0.2

e2

USAColombia

NetherlandsPoland

RussiaQatar

South KoreaMorocco

South Africa

0.1

0.2

e3

USAColombia

NetherlandsPoland

RussiaQatar

South KoreaMorocco

South Africa

0.1

0.2

0.3

0.4

e4

USAColombia

NetherlandsPoland

RussiaQatar

South KoreaMorocco

South Africa

0.1

0.2

0.3

0.4

Figure 6 Components of the eigenvectors of the four largest eigenvalues of the correlation matrixof the log-returns of indices collected in 2003–2007. White bars mean positive values and gray bars,negative ones. The order of indices is the same as in Figure 5.

for Pacific Asian countries. Other eigenvectors also show some structure, but it is quicklylost as their associated eigenvalues approach the region where noise dominates.

Figure 6 shows similar results for the eigenvectors associated with the four largesteigenvalues of the correlation matrix obtained from the log-returns of data in 2008–2012.

Eigenvector e1 still represents a market mode, with all significant indices appearingwith positive values, and eigenvector e2 shows two blocks, aWestern and an Eastern one.Eigenvector e3 shows a block formed by North and South American indices, a block ofEuropean indices, probably joined by the African indices that operate at the same hoursas the Central European markets, and a third block, of Pacific Asian indices, and it isprobably connected with a fine-tuning of the difference in operation hours of markets.Eigenvector e4 separates indices from the Americas, from Europe, Arab countries, andPacific Asian ones.

Comparing the eigenvectors in Figures 5 and 6, one may also notice that there wereno substantial changes to the structure shown by them from the period 2003–2007 to theperiod 2008–2012, leading to the belief that there is some stability on the world stockmarket structure, even during periods of crises.

4. CORRELATION AND VOLATILITY OF THEMARKET

It is a general consensus that, when markets are more volatile, they tend to behave moresimilarly, so that volatility and correlation of assets or indices should move relativelytogether. In Sandoval (2013), we made calculations of the average value of correlationmatrices and of the volatilities of the market mode for running windows during theperiods of some of the major financial crises of the past decades. Here, we come back tothat study, but now using our data of 79 indices for the years from 2003 to 2012. Theapproach is a little different than what has been done before, as we shall compare the


Date

< Corr >, < Vol. >

06/2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

1

2

3

4

5

Corr.

Vol.

0.1 0.2 0.3

0.01

0.02

0.03

Figure 7 Graph on the left: evolution in time of the average of the elements of the correlation matrixof the log-returns (black line) and of the volatility of the log-returns of theMSCIWorld Index (gray line),both normalized so as to have standard deviation one. The graph on the right is the dispersion graphof the average correlation and of the average volatility, not normalized.

average correlation of indices with the volatility of the MSCI World Index, which is astock market index of over 6000 stocks of companies in different stock markets acrossthe globe, and is often used as a benchmark for the global financial market.

The calculations are done in running windows of 100 days each, moving one day ata time. This gives us a total of 2415 windows running from 2003 to 2012. The averagecorrelation is calculated as the mean of the elements of the correlation matrix obtainedfrom each window, and it is assigned to the last day of the window (so, results start from100 days after the beginning of 2003). We also calculate the standard deviation of theelements of the correlation matrix for each window. The volatility of the MSCI WorldIndex is also calculated in moving windows of 100 days, with steps of one day, and it iscomputed as the standard deviation of the log-returns of the index for each window.Thisvolatility is also assigned to the last day of the window.

The results are plotted in Figure 7. The graph on the left is the evolution in timeof the average of the elements of the correlation matrix of the log-returns (black line)and of the average of the volatility of the MSCI World Index (gray line). In order tobest compare both measures, they are both normalized so as to have standard deviationone. The graph on the right is the dispersion graph of the average correlation and ofthe average volatility, not normalized. The Pearson correlation between both measuresis 0.84.

Note that,when volatility rises,mainly during and after the subprime crisis of 2008,thecorrelation between the indices also rises. This is evidence that correlation and volatilitygo hand in hand when it concerns world financial markets. This effect is bad news forinvestors who wish to avoid risk in times of crisis by diversifying their portfolios usingassets from different countries.

Another measure that is very much correlated with both the average of the elements ofthe correlation matrix and the volatility of the MSCIWorld Index is the largest eigenvalue


Date

Eigenvalues

06/20032004 2005 2006 2007 2008 2009 2010 2011 2012

10

20

30

Figure8 Three largest eigenvaluesof the correlationmatrix of the log-returnsof the indices, calculatedin running windows of 100 days with a step of one day.

of the correlation matrix. Figure 8 shows the first three largest eigenvalues of correlationmatrices calculated in running windows of 100 days with steps of one day from 2003 to2012. The first and the second largest eigenvalues are more sensitive to periods of highvolatility.

Next, we shall focus on the network structure of the stock market indices that can bederived from their correlations.

5. NETWORK STRUCTURE

The correlation matrix of the time series of financial data encodes a large amount ofinformation, and an even greater amount of noise. That information and noise must befiltered if one is to try to understand how the elements (in our case, indices) relate toeach other and how that relation evolves in time. One of the most common filteringprocedures is to represent those relations using a Minimum Spanning Tree (MST), whichis a graph containing all indices, connected by at least one edge, so that the sum of theedges is minimum, and which presents no loops. Another type of representation is that ofa Planar Maximally Filtered Graph (PMFG), which admits loops but must be representablein two-dimensional graphs without crossings.

Yet another type of representation is obtained by establishing a number which defineshow many connections (edges) are to be represented in a graph of the correlationsbetween nodes.There is no limitation with respect to the crossing of edges or to the for-mation of loops, and if the number is high enough, then one has a graph where all nodesare connected to one another. These are usually called Asset Trees, or Asset Graphs, sincethey are not trees in the network sense. Another way to build asset graphs is to establish avalue (threshold) such that distances above it are not considered.This eliminates connec-tions (edges) as well as indices (nodes), but also makes the diagrams more understandable


by filtering both information and noise. Some previous works using graphic represen-tations of correlations between international assets (indices or otherwise) can be foundin the works of Coelho et al. (2007), Coronnello et al. (2007), Ausloos and Lambiotte(2007), and Eryigit and Eryigit (2009).

We shall then represent the indices as asset graphs built from the correlation matricesobtained from their log-returns. Given a correlation coefficient cij in a correlation matrixof all indices,a distance measure,first used in the context of financial markets by Mantegna(1999), may be defined as:

dij =√

2(1 − cij

). (4)

As correlations between indices vary from −1 (anticorrelated) to 1 (completely cor-related), the distance between them vary from 0 (totally correlated) to 2 (completelyanticorrelated). Totally uncorrelated indices would have distance 1 between them.

Based on the distance measures, two-dimensional coordinates are assigned to eachindex using an algorithm called Classical Multidimensional Scaling, which is based onminimizing the stress function:

S =[∑n

i=1

∑nj>i

(δij −dij

)2∑ni=1

∑nj>i d2

ij

]1/2

, (5)

where δij is 1 for i = j and zero otherwise, n is the number of rows of the correlationmatrix, and dij is an m-dimensional Euclidean distance (which may be another type ofdistance for other types of multidimensional scaling) given by:

dij =[

m∑a=1

(xia − xja

)2]1/2

. (6)

The outputs of this optimization problem are the coordinates xij of each of the nodes,where i = 1, . . . , n is the number of nodes and j = 1, . . . , m is the number of eachdimension in an m-dimensional space. It is customary for the representation of the net-work to be well represented in smaller dimensions than m, and in the case of this articlewe shall consider m = 2 for a two-dimensional visualization of the network.The choiceis a compromise between fidelity to the original distances and the easiness of representingthe networks.

As some correlations may be the result of random noise,we ran some simulations basedon randomized data,consisting on randomly reordering the time series of each index so asto destroy any true correlations between them but maintain each frequency distributionintact. The result of 1000 simulations is a distance value above which correlations areprobably due to random noise, which, in our case, is d = 1.36.

Our asset graphs are built using distance measures as thresholds. As an example,for threshold T = 0.5, one builds an asset graph where all distances below 0.5 are


represented as edges (connections) between nodes (indices). All distances above thisthreshold are removed, and all indices that do not connect to any other indexbelow this threshold are also removed.

Before drawing them, there is a further innovation that must be discussed. The par-ticular nature of stock markets that operate at different time zones makes it difficult tomake a representation of them that shows the cyclical nature of the rotation of the planet,with markets from the East being more connected with their counterparts in the Westfrom the previous day. A solution to that, developed in Sandoval (2012b), is to build anetwork based on the stock market indices and on their lagged values of the previousday. So, one now has, instead of the 79 original indices, 79 more indices associated withtheir lagged values, for a total of 158 indices. Since it has already been shown Sandoval(2013) that the asset graphs of the stock market indices do not change much in time, weshall consider the asset graphs based on the whole period, from 2003 to 2012.

In Figure 9, we plot all indices on their coordinates. The indices are represented bythe four first letters of each country, with the exception of Austria, which is written Autrso as to differentiate it from Australia, and by three or fours letters in case of countrieswith multiple words, like CzRe for the Czech Republic or USA for the United Statesof America. Lagged indices are represented by black boxes and the original ones arerepresented in white boxes. Note that there is a clear distinction between original andlagged indices, forming two separate clusters with some interaction occurring for indiceswhich are not very correlated to any other indices in the network, like some Arab andAfrican indices. There is also a clear clustering of indices according to geography (or bytime zones), particularly forWestern European indices,American, and Pacific Asian ones.

Figure 10 shows the asset graphs for the world stock market indices,displaying only theindices that are connected below certain distance thresholds (T = 0.5, T = 0.7, T =0.9, and T = 1.1) and their connections. For T = 0.5, the only countries that areconnected are France, Germany, the Netherlands, and Italy, original and lagged ones,forming the core of the Central European countries. For T = 0.7, the UK, Switzerland,Belgium, Sweden, Finland, and Spain join the European cluster, which is represented inits two versions (original and lagged).

At T = 0.9,two more clusters are formed:one ofAmerican indices,made by the USA,Canada,Mexico, and Brazil, and one of Pacific Asian indices,made by Japan,Hong Kong,South Korea,Singapore, andTaiwan. For T = 1.1, the clusters join and add to themselvesmany more indices. Something to be noticed is that Israel and South Africa are part ofthe European cluster, and that Australia and New Zealand join the Pacific Asian cluster.Most important, there are now connections between lagged and original indices, withthe lagged indices of the USA, Canada, Mexico, and Brazil forming connections withthe next day indices of Japan, Hong Kong, the Philippines,Australia, and New Zealand.If one makes asset graphs including indices that are lagged by two and three days, a


All

USA

CanaMexi

BermJamaCoRiPana

ArgeBrazChil

ColoPeru

Vene

UK IrelFranGerm

AutrSwitBelgNeth

SwedDenmFinlNorw

Icel

ItalSpaiPort GreePola

CzRe

Slov

CroaEsto

Latv

Hung

Bulg

RomaUkraRuss

Turk

Kaza

Malt

Isra

PaleJordLeba

SaArOmanQata

UAEPaki

Indi

Bang

SrLa

JapaHoKo

Chin

TaiwSoKo

Mong

Sing

IndoMala

Thai

Phil

Viet

Aust

NeZe

Moro

Tuni

Egyp

NigeBots

GhanKeny

Maur

SoAf

USA

CanaMexi

BermJama

CoRiPana

Arge

Braz

Chil

Colo

Peru

Vene

UK

Irel

FranGerm

Autr

SwitBelgNeth

Swed

Denm

FinlNorw

Icel

ItalSpai

Port

Gree

Pola

CzRe

Slov

Croa

Esto

Latv

Hung

Bulg

Roma

Ukra

Russ

Turk

KazaMalt

Isra

PaleJord

LebaSaAr

OmanQataUAE

Paki

Indi

BangSrLa

Japa

HoKo

Chin

TaiwSoKo

Mong

Sing

IndoMala

Thai

PhilViet

Aust

NeZeMoroTuniEgypNige

BotsGhanKenyMaur

SoAf

Figure 9 Two-dimensional representation of the node structure (without the connections) of theworld stock market indices. White boxes represent the original indices and black boxes represent thelagged indices.

T = 0.5

FranGermNeth

Ital

FranGermNethItal

T = 0.7

UKFranGermSwitBelgNeth

SwedFinl

ItalSpai

UKFranGerm

BelgNeth

SwedFinl

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T = 0.9

USA

CanaMexiBraz

UK IrelFranGerm

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SwedDenmFinlNorw

ItalSpaiPort

CzRe

JapaHoKo

TaiwSoKoSing

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Irel

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SwitBelgNeth

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T = 1.1

USA

CanaMexiArgeBrazChilPeru

UK IrelFranGerm

AutrSwitBelgNeth

SwedDenmFinlNorw

ItalSpaiPort GreePola

CzRe

Hung

RussTurkIsra

Indi

JapaHoKo

TaiwSoKoSing

IndoMala

Thai

Phil

Aust

NeZeSoAf

USA

CanaMexi

Arge

Braz

Chil

Peru

UK

Irel

FranGerm

Autr

SwitBelgNeth

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Gree

Pola

CzReHungRuss

Turk

IsraIndi

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TaiwSoKo

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IndoMala

Thai

Aust

NeZe

SoAf

Figure 10 Asset graphs of the world market indices for distance threshold values T = 0.5, T =0.7, T = 0.9, and T = 1.1. White boxes represent the original indices and black boxes represent thelagged indices.


similar structure appears, with American indices being correlated with Pacific Asian andOceanian ones.

The structure of the network of stock market indices may be used to study thepropagation of crises. There is a vast literature on contagion between banks or othermarkets via networks. In the majority of these works, the interbank loans and liabilitiesare used in order to build networks of financial institutions. A general result is thatcontagion may occur for a certain critical level of integration between institutions, andthat high integration may be both a curse and a blessing for the propagation of crises.This will not be discussed in this chapter, but we stress that certain measures of centralitymay be useful in the identification of the main agents in the propagation of a crisis.

Some of the main centrality measures used in the theory of Complex Networks arenode centrality, which measures the number of connections a node has, eigenvector centrality,which measures how connected a node is, and how connected its neighbors are, andbetweenness centrality, which measures if a node is in the shortest path connecting all theother nodes in the network.The application of these measures of centrality to asset graphsgives results that depend on the threshold level that is chosen, as Sandoval (2013). As anexample, for low enough values of the threshold, there are no connections between nodes,and for high enough threshold values, all nodes are connected to every other node in thenetwork.

If these measures are applied to the asset graph for T = 1.1,we obtain the results listedin Table 1, which contains the nodes with the ten best scores in each of the centralitymeasures. Central European indices occupy the first positions both for node centrality andeigenvector centrality, since they are very connected and in a region of highly connectednodes. So, Europe would be a center for the propagation of crises in a variety of models.

Table 1 Highest ranking stockmarket indices according to node centrality, eigenvector centrality, andbetweenness centrality.

Node degree Eigenvector centrality Betweenness centrality

France UK, France SingaporeUK Netherlands JapanNetherlands Belgium, Italy, Germany USAFinland, Italy, Belgium, Germany,Norway

Finland France

Sweden, Spain Sweden, Spain AustraliaAustralia Norway NetherlandsSwitzerland Austria FinlandDenmark, Czech Republic, USA Switzerland Hong KongPoland Czech Republic AustriaSouth Africa, Hungary, Ireland Denmark, Poland Denmark


Now for betweenness centrality, we have Singapore and Japan occupying the first twopositions, mainly due to their role as connections between Pacific Asian and Europeanindices to the next day indices of the Americas. The USA also occupies a high rank dueto the same reason.

6. CONCLUSION

Based on the time series of 79 stock indices taken from a variety of countries from 2003to 2012, we built correlation matrices based on the log-returns of those indices, firstcomparing the results of 2003–2007 (a period of low volatility) with the ones of 2008–2012 (a time of high volatility) under the light of Random MatrixTheory.We identifieda large eigenvalue which represents a market mode, which conveys the general behaviorof the world stock market, and a second largest one, which highlights the differencesbetween stock markets operating in different time zones.We also use correlation matricescalculated in running windows of 100 days with time step of one day in order to comparethe average correlation of stock markets with the volatility of the world stock marketsrepresented by the MSCI World Index, showing there is a strong correlation betweenthem so that, in times of high volatility, stock markets tend to behave in the same way.Next, we built networks of the stock market indices, as represented by asset graphs, inwhich a structure highly dependent on geographical position and/or on operation hoursis clear.We could see that some Central European markets are highly correlated, and thattwo other cluster, one of American markets, and another of Pacific Asian ones, formsunder lower correlation values. We could also see that stock market indices in PacificAsia are correlated with the previous day’s indices of America, and that Central Europeanindices occupy a position of high centrality in a network of stock market indices, and thatSingapore and Japan present a high number of betweenness centrality, being the mainchannels between Eastern and Western indices.

ACKNOWLEDGMENTSThe author acknowledges the support of this work by a grant from Insper, Instituto de Ensino e Pesquisa.This chapter was written using LATEX, all figures were made using PSTricks, and the calculations were madeusing Matlab, Excel, and Ucinet. All data and algorithms used are freely available upon request to the authorat [email protected].

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