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epl draft Anti-correlation and subsector structure in financial systems X.F. Jiang and B. Zheng(a) Department of Physics, Zhejiang University, Hangzhou 310027, P.R. China PACS 89.65.Gh–Economics; econophysics, financial markets, business and management 2 PACS 89.75.-k–Complex systems 1 0 Abstract – With therandom matrix theory,we studythespatial structureof theChinese stock 2 market,American stock marketandglobal marketindices. Aftertakingintoaccount thesignsof n thecomponentsintheeigenvectorsofthecross-correlation matrix,wedetectthesubsectorstruc- a tureof thefinancial systems. Thepositive and negative subsectors are anti-correlated each other J in the corresponding eigenmode. The subsector structure is strong in the Chinese stock market, 1 while somewhat weakerin theAmerican stock market andglobal market indices. Characteristics 3 of thesubsector structures in different markets are revealed. ] N G . n Introduction. – In recent years, much attention of vestigatedthe structureofinteractionsbetweenstocksfor i fphysicists has been attracted to the financial dynamics, the Chinese stock market based on the RMT method [6]. - which exhibits various collective behaviors [1–9]. Statisti- Asanimportantemergingmarket,theChinesemarketex- q [cal properties of price fluctuations and cross-correlations hibitsstrongercross-correlationsthanthematureones. At between individual stocks are of great interest, not only the same time, the effect of the standard business sectors 1 for quantitatively unveiling the complex structure of the is weak in the Chinese market. Instead, unusual sectors v 8financialsystems, but alsopracticallyfor the assetalloca- such as ST and Blue-chip sectors are detected. 1tionandportfolioriskestimation[10–12]. Theprobability In this paper, with the RMT method, we aim at fur- 4distribution of price returns usually exhibits a power-law ther understanding of the spatial structure. Our observa- 6 tail,andrepresentstherobustcharacteristicsinstockmar- tionisthatthecomponentsinaneigenvectorofthecross- . 1kets [13–15], while the higher-order time correlations and correlation matrix may show positive and negative signs. 0interactions between stocks are less universal [5–7,16]. In To the best of our knowledge, what roles the signs of the 2 somecases,pricereturnsmayalsoshowaPoisson-likedis- components play has not been explored. Our main find- 1 :tribution [17,18]. ing is that the signs of the components in an eigenvector v It is an important and challenging topic to explore the may classify a sector into two subsectors, which are anti- i X’spatial’ structure in financial systems. For example, the correlated each other within this eigenmode. This goes rhierarchical structure of stock markets has been investi- beyond what one may gain with standard methods such agated through the minimal spanning tree method and its as the minimal spanning tree and its variants in the anal- variants[19–23]. Withtherandommatrixtheory(RMT), ysis of the ’spatial’ structure in financial systems [19–23]. business sectors and topology communities may be iden- Methods and basics. – We have collected the daily tified [16,24–26]. The RMT method was firstly developed in the complex quantum systems where the interactions dataof259stockstradedintheShanghaiStockExchange between subunits are unknown [27,28]. The structure of (SSE) from Jan., 1997 to Nov., 2007, in total, 2633 days. business sectors have been examined for mature markets The daily data of 259 stocks in the NYSE are from Jan., such as the New York Stock Exchange (NYSE) and the 1990 to Dec., 2006, in total, 4286 days. Meanwhile, we KoreanStock Exchanges[24,25,29–32], andalso for some have collected the daily data of a set of 66 financial in- emerging markets such as the National Stock Exchange dices,including57indicesinstockmarketsand9treasury in India [16]. Very recently, the RMT method was ap- bond rates in US from Sep., 1997 to Oct., 2008, in total, plied to identify the dominant eigenmodes in the indices 2669 days. We name the 66 indices the global market in- oftheindustrialproduction[33]. Inparticular,onehasin- dices (GMI). The data of the SSE are taken from ’Wind Financial Database’ (http://www.wind.com.cn) and the (a)correspondingauthor;email: [email protected] data of the NYSE and GMI are from ’Yahoo Finance’ p-1 X.F. Jiang and B. Zheng (http://finance.yahoo.com). thereexistsnon-randominteractions. Infact,inbothma- We assume that the price is the same as the preceding ture and emerging stock markets, the bulk of the eigen- day [34], if the price of a stock is absence in a particular valuespectrumP(λ)ofthecross-correlationmatrixissim- day. It has beenpointed outthat the missing data do not ilartoP (λ)oftheWishartmatrix,butsomelargeeigen- rm result in artifacts [16]. All markets concerned in this pa- values deviate significantly from the upper bound λran . max per have normal trading sessions in all days of the week, This scenario looks similar for the GMI. Let us arrange except for Saturdays, Sundays and holidays declared in thelargeeigenvaluesintheorderofλ >λ . Asshown α α+1 advance, excluding the Egyptian and Tel Aviv Stock Ex- in table 1, the largesteigenvalue λ of the SSE (China) is 0 change. For the latter two stock markets, trading takes 97.33,about56timesaslargeastheupperboundλran of max place from Sunday to Thursday. For the alignment of the P (λ),whileλ oftheNYSE(US)andGMIis45.61and rm 0 time series,wesimply movethe data onSundayto Friday 21.53,about29and16timesaslargeasλran respectively. max for these two markets. For comprehensive understand- According to the previous works [6,16,24,39], the large ing of the cross correlation of financial markets, the bond eigenvalues deviating from the bulk correspond to differ- rates in US are also added in our analysis, which include entmodesofmotioninstockmarkets. Thecomponentsin 9 indices ranging from the 3 month to 20 year rates. the eigenvector of the largest eigenvalue λ are uniformly 0 We define the logarithmic price return of the i-th stock distributed. Therefore, the largest eigenvalue represents over a time interval ∆t as themarketmode,whichisdrivenbyinteractionscommon for stocks in the entire market. The components in the R (t) lnP (t+∆t) lnP (t), (1) i i i eigenvectors of other large eigenvalues are localized. A ≡ − particular eigenvector is dominated by a sector of stocks, whereP (t)representsthe priceofthe stockprice attime i usually associated to a business sector. By u (λ ), we t. To ensure different stocks with an equal weight, we i α denote the component of the i-th stock in the eigenvec- introduce the normalized price return tor of λ . To identify the sector, one may introduce a α Ri(t) Ri(t) threshold uc, to select the dominating components in the r (t)= −h i, (2) i σi eigenvector by |ui(λα)|≥uc [6]. The threshold uc is de- terminedby two criteria. Firstly, if the matrix is random, wphheRrei2i−h··h·Riiii2s dtehneotaevsertahgee sotavnerdatridmedevt,iataionnd oσfi R=i. <sho|uul(dλ)b|e>la∼rg1e/r√thNanfo1r/√evNer.ySeeicgoenndmlyo,due.c sThhoeurledfonroet, buec Then, the elements of the cross-correlation matrix C are too large,otherwise there would be not so many stocks in defined by the equal-time correlations each sector. Inthispaper,weshowthatthecomponentsinaneigen- C r (t)r (t) . (3) ij ≡h i j i vectormaycarrypositiveandnegativesigns,andthecom- ponentswithoppositesignsareanti-correlatedwithinthis Bythedefinition,C isarealsymmetricmatrixwithC = ii eigenmode. Inspired by this observation, we investigate 1, and C is valued in the domain [ 1,1]. ij − the subsector structure of the financial markets, by tak- Themeanvalue C ofthe elements forthe SSE is0.37, ij ing into account the signs of the components. In other muchlargethan0.16and0.26for the NYSE andGMI re- words, we separate a sector into two subsectors by two spectively. It confirms that stock prices in emerging mar- thresholds u± = u : u (λ ) u+ and u (λ ) u−, ketsaremorecorrelatedthanmatureones[16,35,36]. The c ± c i α ≥ c i α ≤ c which correspond to the positive and negative subsectors correlationbetweenfinancialindicesintheGMIissmaller respectively. than that of the SSE, but bigger than that of the NYSE. Wenowcomputetheeigenvaluesofthecross-correlation Subsectors. – According to reference [6], standard matrix C, in comparison with those of the so-called business sectors can hardly be detected in the SSE Wishart matrix,whichisderivedfromnon-correlatedtime (China). Instead,onefindsthatthereexiststhreeunusual series. Assuming N time series with length T, and in sectors, i.e., the ST, Blue-chip and SHRE sectors, corre- the large-N and large-T limit with Q T/N 1, the ≡ ≥ sponding to the second, third and forth largest eigenval- probabilitydistributionP (λ)oftheeigenvalueλforthe rm ues respectively. What are the dominating stocks for the Wishart matrix is give by [37,38] eigenvectors of other large eigenvalues remains puzzling. Q (λran λ)(λ λran) In the SSE, a company will be specially treated if its Prm(λ)= p max− − min , (4) financial situation is abnormal. Then a prefix of the 2π λ acronym “ST” will be added to the stock ticker. The with the upper and lower bounds acronym “ST” will be removed when the financial situ- ations becomes normal. In reference [6], the so-called ST 2 λran = 1 1/ Q . (5) sector consists of the “ST” stocks. On the other hand, min(max) h ±(cid:16) p (cid:17)i the Blue-chip sector is referred to those companies with For a dynamic system, large eigenvalues of the cross- a national reputation, and with good performance, i.e., a correlationmatrix,whichdeviatefromP (λ),implythat reasonablepositive profit in a period of time. Meanwhile, rm p-2 Anti-correlation and subsector structure in financial systems theSHRErepresentsthecompaniesregisteredinShanghai are rarelyobserved. From the view of the behavioralpsy- with the real estate business. chology,theinvestorsinChinaareextraordinarilylooking Now we introduce two thresholds u± = u to sepa- at the performance of the companies and the dominating c ± c ratethedominatingcomponentsinaneigenvectorintotwo business and areas, etc. Therefore, unusual sectors such parts, i.e., u (λ ) u+ and u (λ ) u−, which are re- as the ST, Blue-chip, and SHRE emerge. i α ≥ c i α ≤ c ferredto the positive and negative subsectors respectively. Forcomparison,wealsoapplythismethodtostudythe With this method, we are able to identify the subsectors subsector structure in the NYSE (US). The results are ofthe SSEup tothe seventhlargesteigenvalueλ , andto listed in table 3. The subsector structure in the NYSE is 6 achievedeeper understanding onthe unusualsectors such somewhatdifferentfromthatintheSSE.Fromgeneralbe- asthe ST andBlue-chipsectors. The resultsareshownin lieving,the standardbusiness subsectorsshoulddominate table 2. The market mode described by the largesteigen- the eigenvectors of the large eigenvalues. Additionally, it valueλ isnotincludedinthetable,whereallcomponents would be expected that there exists only one dominating 0 in the eigenvector posses a same sign. subsectorinaneigenvector,probablyundercertaincondi- The negative components in the eigenvector of the sec- tions, e.g., when the total number of stocks is sufficiently ondlargesteigenvaluesλ aredominatedbytheSTstocks. large. To clarify these issues, our results are presented up 1 Withthethresholdu− = 0.10,forexample,23dominat- to the thresholds u± = 0.12. As shown in table 3, most c − c ± ing stocks are selected, and 20 of them are the ST stocks. subsectors are indeed the standard business subsectors. Therefore this subsector is called the ST subsector. For For λ , λ , λ andλ , only one dominating subsector re- 1 2 6 11 thepositivecomponentsintheeigenvectorofλ ,wecould mains for sufficiently large thresholds u±. For λ , λ , λ 1 c 3 7 8 not identify a common feature for the dominating stocks. andλ ,however,therearetwodominatingsubsectors. For 9 In fact, as the threshold u+ increases, the number of the our dataset of the NYSE, our method does also provide a c dominating stocks shrinks. For example, with the thresh- deeper understanding on the spatial structure. old u− = 0.10, there are only 7 dominating stocks, and Finally, as shown in table 4, the subsectors in the GMI c − halfarealso the ST stocks. In reference[6], therefore,the can be identified with the threshold u = 0.15, exclu- c ± whole sector of λ is called the ST sector. The negative sively in terms of the areas to which the indices belong. 1 components in the eigenvector of the third largest eigen- Different from the SSE and NYSE, the eigenvector of the value λ well define the high technology subsector, while largest eigenvalue λ of the GMI does not describe the 2 0 the positive ones are dominated by the traditional indus- so-called ’market mode’, which represents the global mo- try stocks. Stocks in both subsectors are the Blue-chip tionofthefinancialsystem. Thismayreflectthefactthat stocks. Therefore, these two subsectors together are as- all the financial markets in the world have not been in cribed to the Blues-chip sector in reference [6]. For the such a unified status. The first, second and third largest fourth largesteigenvalue λ , the SHRE sector detected in eigenvalues correspond to the US, Asia-Pacific and Bond 3 reference[6]splitsintotwosubsectors,i.e.,the SHREand sectors, with only a single dominating subsector. The US ST subsectors. Consistent with the result in reference [6], sector mainly consists of the indices in US, except for the half of the ST stocks are also the SHRE stocks. But the GSPTSE from Canada and GDAXI from Germany. This ST stocks here are different from those for λ . result reflects that US is the dominating economy in the 1 Inreference[6],the sectorstructureis exploredonly up world. From λ to λ , there emerge two dominating sub- 3 7 toλ . Withtheexplorationofthesubsectorstructure,we sectors. One important feature of the subsector structure 3 areable to step further. For λ , the positive and negative isthattheindicesinthemainlandofChinaorinHongkong 4 subsectors are identified to be the weakly and strongly alwaysformanindependentsubsector. Ontheotherhand, cyclical industry respectively. The former includes the the US bond rates do not mix with the indices in stock stockswhichfluctuate little withthe economiccycle,such markets. Forλ ,theshort-termbondratesandlong-term 6 asthe daily consumergoods andservices,while the latter bond rates are separated into the positive and negative isbloomingordepressingwiththeeconomiccycle,includ- subsectors respectively. ingthebasicmaterialsandenergyresources. Thepositive Anti-correlation between subsectors. – What is components in the eigenvectors of λ and λ are domi- 5 6 the physical meaning of the positive and negative subsec- nated by the finance and non-daily consumer subsectors, tors? The cross-correlation between two stocks can be although the negative ones remain unknown. written as Taking into account the signs of the components in the eigenvector,onemayexplorethesubsectorstructureinthe N C = λ Cα, Cα =uαuα (6) SSE up to λ6. A number of standard business subsectors ij X α ij ij i j suchasthe hightechnologyandfinancearealsoobserved. α=1 But the SSE is indeed dominated by unusual sectors and where λ is the α-th eigenvalue, uα is the i-th component α i subsectors such as the ST, Blue-chip, traditional indus- in the eigenvector of λ , and Cα represents the cross- α ij try,SHRE,weaklyandstrongcyclicalindustry. InChina, correlation in the α-th eigenmode. In other words, the the companies are not operated strictly within the regis- cross-correlation between two stocks can be decomposed tered business. Therefore, standard business subsectors intothosefromdifferenteigenmodes. Sincetheeigenvalue p-3 X.F. Jiang and B. Zheng λ isalwayspositive,itgivestheweightoftheα-theigen- aretheindicesintheAmericanstockmarkets. onesubsec- α mode,and the signofCα is essentialin the sum. Accord- tor is composed of the IIX, IXIC, NDX, NWX, PSE and ij ing to Eq. (6), Cα is positive if the components uα and SOXX.Mostoftheseindicesarerelatedtotheinformation ij i uα have the same sign in a particular eigenmode. Other- technology,semiconductorindustry,internetindustry,etc, j wise, it is negative. When Cα is negative, two stocks are with a potentially high payoff and asset risk. The other ij referredto be anti-correlated in this eigenmode: when the subsector consists of the XMI, DJA, DJI, DJU and DJX. price return of the i-th stock is positive, the price return Mostofthem arefor the weightedandtraditionalcompa- of the j-th stock tends to be negative in the statistical nies, which share the general feature of a stable currency sense. Therefore, all stocks in a same subsector are pos- flowand mature business mode, but a lowerprofit. These itively correlated in this eigenmode, while the stocks in two subsectors are anti-correlatedin this eigenmode. The different subsectors are anti-correlated. This is the phys- second example is the subsectors of λ , which are obvi- 6 ical meaning of the subsectors. For the NYSE, however, ouslyanti-correlatedfortheyarejustshort-termandlong- only a number of sectors split into two subsectors. This term bond rates in US. For λ , λ and λ , the subsector 3 4 7 suggeststhatthespatialstructureandinteractionsamong structure indicates that the stock markets in China are the stocks in the SSE are more complicated. somewhat special. Let us examine some examples in the SSE. The sector To quantitatively measure the anti-correlationbetween of λ is composed of the traditional industry and high the positive and negative subsectors, we construct the 2 ± technologysubsectors. The former representsthose tradi- combinations of stocks in the two subsectors, I (t) = α ± tionalindustry companieswith a long-termandstable in- u (α)r (t), and compute the cross-correlation Pi i i terest, but a lower asset risk and expected revenue, while the latter includes the high technology companies with C+−(α)=hIα+(t)Iα−(t)i. (7) novel business and conceptions, but a higher asset risk ± Here u (α) is the i-th positive or negative component and potential profit. In a particular period, for exam- i in the α-th eigenmode selected by the threshold, e.g., ple, the stock market is uncertain, and investors prefer the traditional industries with a lower risk, then their u±c = ±0.08. In figure 1, the cross-correlation C+−(α) is shownfor the SSE and NYSE, in comparisonwith that stock prices rise up higher than those of the high tech- between two random combinations of stocks. This result nology companies. In another period, however, the stock isnotqualitativelysensitivetowhetheroneintroducesthe market is booming, and the situation is reverse. Thus, thresholds u± to select the dominating components. these two subsectors are anti-correlatedin the eigenmode c ofλ . Thesectorofλ consistsoftheweaklyandstrongly In figure 1, we observe that C+−(α) monotonically in- 2 4 creases, and gradually approaches that for two random cyclicalindustrysubsectors. Bothsubsectorsareunusual, but their anti-correlation seems obvious. The weakly combinations of stocks. C+−(α) computed with Iα±(t) is smallerthanthatwithtworandomcombinationsofstocks and strongly cyclical industries are weakly and strongly because of the anti-correlation between the positive and correlated with the macro-economy environment respec- negative subsectors. tively. Thus, investors prefer the strongly cyclical indus- What matrix structure results in the subsector struc- try when the macro-economy is booming. Instead, in- ture? Let us consider a 4 4 cross-correlationmatrix, vestors rather choose the weakly cyclical industry when × themacro-economydeclines. Inreference[6],thesectorof 1 0.55 0.15 0.11 λ isidentified asthe SHRE sector. Nowthis sectorsplits   3 0.55 1 0.39 0.34 into the ST and SHRE subsectors. In fact, half of the C4×4 = 0.15 0.39 1 0.95 , (8) STstocksalsobelongstothe SHRE stocks. Thissuggests    0.11 0.34 0.95 1  that the investors care much the normal and abnormal financial situation, even for the SHRE companies. which is taken from the λ sector of the GMI. The 1-th 6 In the NYSE, the subsector structure of λ , λ and λ and2-thindicesrepresentthe3-monthand6-monthbond 3 7 9 isunderstandable. Thedailyconsumergoodsandservices ratesrespectively,identifiedasthepositivesubsector. The areconsideredasthetraditionalindustries,whilethehigh 3-th and 4-th indices are the 10-year and 20-year bond technologyandfinancebelongto anothercategory. These rates, identified as the negative subsector. Obviously, the two sorts of stocks may show an anti-correlation, consis- matrix elements C within the same subsectors, i.e., in ij tentwith the subsectorstructure ofλ inthe SSE.For λ the diagonal blocks, are larger than the ones between the 2 2 with the threshold u± =0.08, a weak subsector structure positive and negative subsectors, i.e., in the off-diagonal c is observed in the NYSE. From their intrinsic properties, blocks. thedailyconsumergoodsandbasicmaterialsareclassified Toverifytheanti-correlationmoreintuitively,therefore, astheweaklyandstronglycyclicalindustriesrespectively. we may calculate the average C within the positive or ij This is similar to the case of λ in the SSE. For λ , the negative subsector, and between the positive and nega- 4 8 subsectors may be also explained along the lines above. tive subsectors. The results for the NYSE are shown in In the GMI, two examples are typical. The first one is figure 2, and those for the SSE are similar. 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Bouchaud and M. Potters, Theory of Financial Risk and Derivative Pricing: From Statisitcal Physics to p-5 X.F. Jiang and B. Zheng Table 2: The subsectors in theSSE. The fraction is the numberof well identified stocks over thetotal numberof stocks in the subsector. Null: noobviouscategory;ST:speciallytreated;Trad: traditionalindustry;Tech: hightechnology;SHRE:Shanghai real estate; Weak: weakly cyclical industry; Stro: strongly cyclical industry; Fin: finance; IG: industrial goods; Util: utility; Basic: basic materials; Heal: health care; CG: daily consumer goods; Serv: services. λ1 λ2 λ3 λ4 λ5 λ6 Sign + − + − + − + − + − + − Sector Null ST Trad Tech ST SHRE Weak Stro Fin Null IG Null u±c =±0.08 26 31/35 22/23 23/25 24/27 27/27 23/26 24/26 14/18 25 15/17 25 u±c =±0.10 7 20/23 16/17 12/13 11/12 20/20 13/15 15/16 10/14 17 8/9 18 Table 3: The subsectors in theNYSE.The abbreviations can beseen in thecaption of table 2. λi λ1 λ2 λ3 λ4 λ5 λ6 Sign + − + − + − + − + − + − Sector Util Tech CG Basi Tech CG Null Null Basi Null Null Fin u±c =±0.08 26/26 3/4 9/16 23/26 15/26 14/32 26 25 6/6 9 14 16/20 u±c =±0.10 25/25 0/0 0/0 19/21 6/13 13/19 18 12 0/0 8 8 16/18 u±c =±0.12 21/21 0/0 0/0 19/21 5/7 5/6 7 6 0/0 4 2 8/11 λi λ7 λ8 λ9 λ10 λ11 Sign + − + − + − + − + − Sector Tech Serv Serv Heal Fin Serv Null CG Null Serv u±c =±0.08 11/24 12/29 9/19 9/16 11/25 13/24 12 2/4 7 9/12 u±c =±0.10 7/13 10/18 8/11 8/13 6/11 11/19 6 2/3 6 9/10 u±c =±0.12 4/7 8/11 5/5 7/7 4/6 10/10 2 2/3 1 7/8 Table 1: λran denote the lower (upper) bound of the 1 min(max) eigenvalues of the Wishart matrix, while λreal, λ , λ and λ SSE (1997-2007) min 0 1 2 represents the lower bound of the eigenvalues and the three largest eigenvalues of the real systems respectively. 0.9 λrmainn λrmaanx λrmeianl λ0 λ1 λ2 Cij SSE 0.47 1.73 0.18 97.3 4.17 3.35 0.37 0.8 C (α) NYSE 0.54 1.55 0.20 45.6 8.71 6.24 0.16 +- GMI 0.72 1.33 0.00 21.5 6.65 5.40 0.26 0.8 RRaenaldom 0.4 NYSE (1990-2006) Table 4: The subsector structurein theGMI . The thresholds areu± =±0.15. TheboldItalicitemsarethosenotbelonging 0.7 c 0 totheareas. NorA refers totheNorthAmerica, and b3m and 50 100 b1y are the3 month and 1 yearbonds. 20 40 α 60 80 100 Sign Area λ0 VLIC XMI DJA DJI DJT DJX IIX US Fig.1: C+−(α)fortheSSEandNYSEarecomparedwiththat IXICMIDNDXNWX between two random combinations of stocks. OEX PSE RUA RUI RUT SML SPC GDAXIGSPTSE λ1 AORDHSIHSNC HSNFHSNPHSNU Asia JKSEKS11N225NZ50PSISTI ATX λλ23 + Ab6EmXbB1yFXb2yFCbH3yIbF5TySbE7yGbD2A0yXbI2M0yIB- BEUond 0.6 WWBeiitttwhhiienne npn oepgsoiatstiiivtvieev sesu uabbnssdee cnctteoogrrative subsectors TELSSMI - SHASZASHBSZB China λ4 + HSIHSNCHSNP HK 0.4 NYSE (1990-2006) - AEX BFX FCHI FTSE GDAXI MIB- EU C ij TELSSMI SHA SZA SHB SZB λ5 + IIXIXICNDXNWXPSESOXX US 0.2 - XMIDJADJIDJUDJXb3m US λ6 + b3mb6mb1y Bond - b7yb10yb20y HSNU Bond λ7 + ATOWRIID JKSE KS11 N225 NZ50 PSI Asia 0 20 α 40 - HSIHSNCHSNFHSNPHSNUb3m HK λ8 + BVSP IPSA MERV MXX XAX NorA Fig. 2: The average cross-correlation Cij for the NYSE. GSPTSE p-6