Dynamics of market correlations: Taxonomy and portfolio analysis


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1 Dynamics of marke correlaions: Taxonomy and porfolio analysis J.P. Onnela, A. Chakrabori, and K. Kaski Laboraory of Compuaional Engineering, Helsinki Universiy of Technology, P.O. Box 9203, FIN HUT, Finland J. Kerész Deparmen of Theoreical Physics, Budapes Universiy of Technology & Economics, Budafoki ú 8, H1111 Budapes, Hungary and Laboraory of Compuaional Engineering, Helsinki Universiy of Technology, P.O. Box 9203, FIN HUT, Finland A. Kano Deparmen of Quaniaive Mehods in Economics and Managemen Science, Helsinki School of Economics, P.O. Box 1210, FIN Helsinki, Finland Received 26 February 2003; published 13 November 2003 The ime dependence of he recenly inroduced minimum spanning ree descripion of correlaions beween socks, called he asse ree has been sudied in order o reflec he financial marke axonomy. The nodes of he ree are idenified wih socks and he disance beween hem is a unique funcion of he corresponding elemen of he correlaion marix. By using he concep of a cenral verex, chosen as he mos srongly conneced node of he ree, an imporan characerisic is defined by he mean occupaion layer. During crashes, due o he srong global correlaion in he marke, he ree shrinks opologically, and his is shown by a low value of he mean occupaion layer. The ree seems o have a scalefree srucure where he scaling exponen of he degree disribuion is differen for business as usual and crash periods. The basic srucure of he ree opology is very robus wih respec o ime. We also poin ou ha he diversificaion aspec of porfolio opimizaion resuls in he fac ha he asses of he classic Markowiz porfolio are always locaed on he ouer leaves of he ree. Technical aspecs such as he window size dependence of he invesigaed quaniies are also discussed. DOI: /PhysRevE PACS number s : s, k, n I. INTRODUCTION In spie of he radiional wisdom Money does no grow on rees, here we wish o show ha he concep of rees graphs has poenial applicaions in financial marke analysis. This concep was recenly inroduced by Manegna as a mehod for finding a hierarchical arrangemen of socks hrough sudying he clusering of companies by using correlaions of asse reurns 1. Wih an appropriae meric, based on he correlaion marix, a fully conneced graph was defined in which he nodes are companies, or socks, and he disances beween hem are obained from he corresponding correlaion coefficiens. The minimum spanning ree MST was generaed from he graph by selecing he mos imporan correlaions and i is used o idenify clusers of companies. In his paper, we sudy he ime dependen properies of he minimum spanning ree and call i a dynamic asse ree. I should be menioned ha several aemps have been made o obain clusering from he huge correlaion marix, such as he Pos superparamagneic mehod 2, a mehod based on he maximum likelihood 3 or he comparison of he eigenvalues wih hose given by he random marix heory 4. We have chosen he MST because of is uniqueness and simpliciy. The differen mehods are compared in Ref. 3. Financial markes are ofen characerized as evolving complex sysems 5. The evoluion is a reflecion of he changing power srucure in he marke and i manifess he passing of differen producs and produc generaions, new echnologies, managemen eams, alliances and parnerships, among many oher facors. This is why exploring he asse ree dynamics can provide us new insighs o he marke. We believe ha dynamic asse rees can be used o simplify his complexiy in order o grasp he essence of he marke wihou drowning in he abundance of informaion. We aim o derive inuiively undersandable measures, which can be used o characerize he marke axonomy and is sae. A furher characerizaion of he asse ree is obained by sudying is degree disribuion 6. We will also sudy he robusness of ree opology and he consequences of he marke evens on is srucure. The minimum spanning ree, as a srongly pruned represenaive of asse correlaions, is found o be robus and descripive of sock marke evens. Furhermore, we aim o apply dynamic asse rees in he field of porfolio opimizaion. Many aemps have been made o solve his cenral problem from he classical approach of Markowiz 7 o more sophisicaed reamens, including spinglassype sudies 8. In all he aemps o solve his problem, correlaions beween asse prices play a crucial role and one migh, herefore, expec a connecion beween dynamic asse rees and he Markowiz porfolio opimizaion scheme. We demonsrae ha alhough he opological srucure of he ree changes wih ime, he companies of he minimum risk Markowiz porfolio are always locaed on he ouer leaves of he ree. Consequenly, asse rees in addiion o heir abiliy o form economically meaningful clusers, could poenially conribue o he porfolio opimizaion problem. Then wih a ligher key one could perhaps say ha some money may grow on rees, afer all X/2003/68 5 / /$ The American Physical Sociey
2 ONNELA e al. FIG. 1. Lef: Plo of he probabiliy densiy funcion of he correlaion coefficiens as a funcion of ime. Righ: The mean, sandard deviaion, skewness, and kurosis of he correlaion coefficiens as funcions of ime. II. RETURN CORRELATIONS AND DYNAMIC ASSET TREES The financial marke, for he larges par in his paper, refers o a se of daa commercially available from he Cener for Research in Securiy Prices CRSP of he Universiy of Chicago Graduae School of Business. Here we will sudy he spliadjused daily closure prices for a oal of N 477 socks raded a he New York Sock Exchange NYSE over he period of 20 years, from 02 Jan 1980 o 31 Dec This amouns o a oal of 5056 price quoes per sock, indexed by ime variable 1,2,...,5056. For analysis and smoohing purposes, he daa are divided imewise ino M windows 1,2,...,M of widh T corresponding o he number of daily reurns included in he window. Several consecuive windows overlap wih each oher, he exen of which is dicaed by he window sep lengh parameer T, describing he displacemen of he window, measured also in rading days. The choice of window widh is a rade off beween oo noisy and oo smoohed daa for small and large window widhs, respecively. The resuls presened in his paper were calculaed from monhly sepped fouryear windows. Assuming 250 rading days a year, we used T 20.8 day and T 1000 day. We have explored a large scale of differen values for boh parameers, and he given values were found opimal 9. Wih hese choices, he overall number of windows is M 195. In order o invesigae correlaions beween socks we firs denoe he closure price of sock i a ime by P i ( ) Noe ha refers o a dae, no a ime window. We focus our aenion o he logarihmic reurn of sock i, given by r i ( ) ln P i ( ) ln P i ( 1) which, for a sequence of consecuive rading days, i.e., hose encompassing he given window, form he reurn vecor r i. In order o characerize he synchronous ime evoluion of asses, we use he equal ime correlaion coefficiens beween asses i and j defined as ij r i r j r i r j r 2 i r i 2 r 2 j r j 2, where indicaes a ime average over he consecuive rading days included in he reurn vecors. Due o Cauchy Schwarz inequaliy, hese correlaion coefficiens fulfill he condiion 1 ij 1 and form an N N correlaion marix C, which serves as he basis of dynamic asse rees o be discussed laer. Le us firs characerize he correlaion coefficien disribuion shown in Fig. 1, by is firs four momens and heir correlaions wih one anoher. The firs momen is he mean correlaion coefficien defined as 1 N N 1 /2 ij C 1 ij, 2 where we consider only he nondiagonal (i j) elemens ij of he upper or lower riangular marix. We also evaluae he higher order normalized momens for he correlaion coefficiens, so ha he variance is
3 DYNAMICS OF MARKET CORRELATIONS: TAXONOMY... he skewness is 1 2 N N 1 /2 (i, j) 1 3 N N 1 /2 (i, j) and he kurosis is ij 2, ij 3 / 3/2 2, 3 4 form a series hrough ime. Consequenly, his muliude of rees is inerpreed as a sequence of evoluionary seps of a single dynamic asse ree. As a simple measure of he emporal sae of he marke he asse ree we define he normalized ree lengh as L 1 N 1 d ij T d ij, N N 1 /2 (i, j) ij 4 / 2 2. The mean, sandard deviaion square roo of he variance, skewness, and kurosis of he correlaion coefficiens are ploed as funcions of ime in Fig. 1. In his figure he effec and repercussions of Black Monday Ocober 19, 1987 are clearly visible in he behavior of all hese quaniies. For example, he mean correlaion coefficien is clearly higher han average on he inerval beween 1986 and The lengh of his inerval corresponds o he window widh T, and Black Monday coincides wih he midpoin of he inerval 10. The increased value of he mean correlaion is in accordance wih he observaion by Drozdz e al. 11, who found ha he maximum eigenvalue of he correlaion marix, which carries mos of he correlaions, is very large during marke crashes. We also invesigaed wheher hese four differen measures are correlaed, as seems clear from he figure. For his we deermined he Pearson s linear and Spearman s rankorder correlaion coefficiens, which beween he mean and variance urned ou o be 0.97 and 0.90, and beween skewness and kurosis 0.93 and 0.96, respecively. Thus he firs wo and he las wo measures are very srongly correlaed. We now move on o consruc an asse ree. For his we use he nonlinear ransformaion d ij 2(1 ij ) o obain disances wih he propery 2 d ij 0, forming an N N disance marix D. A his poin an addiional hypohesis abou he opology of he meric space is required. The working hypohesis is ha a useful space for linking he socks is an ulrameric space, i.e., a space where all disances are ulrameric. This hypohesis is moivaed a poseriori by he finding ha he associaed axonomy is meaningful from an economic poin of view. The concep of ulramericiy is discussed in deail by Manegna 1, while he economic meaningfulness of he emerging axonomy is addressed laer in his paper. Ou of he several possible ulrameric spaces, he subdominan ulrameric is oped for due o is simpliciy and remarkable properies. In pracice, i is obained by using he disance marix D o deermine he MST of he disances, according o he mehodology of Ref. 1, denoed by T. This is a simply conneced graph ha connecs all N nodes of he graph wih N 1 edges such ha he sum of all edge weighs, dij T d ij, is minimum. Here ime window dependence of he ree is emphasized by he addiion of he superscrip o he noaion. Asse rees consruced for differen ime windows are no independen of each oher, bu 5 where again denoes he ime a which he ree is consruced, and N 1 is he number of edges presen in he MST. The probabiliy disribuion funcion of he N 1 disance elemens d ij in T as a funcion of ime is ploed in Fig. 2 cf. Ref. 12. Also he mean, sandard deviaion, skewness, and kurosis of normalized ree lenghs are depiced in Fig. 2. As expeced and as he plos show, he mean correlaion coefficien and he normalized ree lengh are very srongly anicorrelaed. Pearson s linear correlaion beween he mean correlaion coefficien () and normalized ree lengh L() is 0.98, and Spearman s rankorder correlaion coefficien is 0.92, hus boh indicaing very srong anicorrelaion. Anicorrelaion is o be expeced in view of how he disances d ij are consruced from correlaion coefficiens ij. However, he exen of his anicorrelaion is differen for differen inpu variables and is lower if, say, daily ransacion volumes are sudied insead of daily closure prices 13. I should be noed ha in consrucing he minimum spanning ree, we are effecively reducing he informaion space from N(N 1)/2 separae correlaion coefficiens o N 1 ree edges, in oher words, compressing he amoun of informaion dramaically. This follows because he correlaion marix C and disance marix D are boh N N dimensional, bu due o heir symmery, boh have N(N 1)/2 disinc upper or lower riangle elemens, while he spanning ree has only N 1 edges. So, in moving from correlaion or disance marix o he asse ree T, we have pruned he sysem from N(N 1)/2 o N 1 elemens of informaion. If we compare Figs. 1 and 2, we find ha disribuion of he disance elemens conained in he asse ree reain mos of he feaures of he correlaion coefficien disribuion. Their corresponding momens also bear sriking correlaion/ anicorrelaion, e.g., he Pearson s linear correlaion beween he skewness of he correlaion coefficiens and he skewness of he edge lenghs is 0.85, while he Spearman s rank order correlaion is Thus one may conemplae ha he minimum spanning ree as a srongly reduced represenaive of he whole correlaion marix, bears he essenial informaion abou asse correlaions. As furher evidence ha he MST reains he salien feaures of he sock marke, i is noed ha he 1987 marke crash can be quie accuraely seen from Figs. 1 and 2. The fac ha he marke, during crash, is moving ogeher is hus manifesed in wo ways. Firs, he ridge in he plo of he mean correlaion coefficien in Fig. 1 indicaes ha he whole
4 ONNELA e al. FIG. 2. Lef: The probabiliy disribuion funcion of he (N 1) disance elemens conained in he asse ree, as a funcion of ime. Righ: The mean, sandard deviaion, skewness, and kurosis of he normalized ree lenghs as funcions of ime. marke is excepionally srongly correlaed. Second, he corresponding well in he plo of he mean normalized ree lengh in Fig. 2 shows how his is refleced in considerably shorer han average lengh of he ree so ha he ree, on average, is very ighly packed. Upon leing he window widh T 0, he wo sides of he ridge converge o a single dae, which coincides wih Black Monday 10. III. TREE OCCUPATION AND CENTRAL VERTEX Nex we focus on characerizing he spread of nodes on he ree. In order o do so, we inroduce he quaniy of mean occupaion layer as l,v c 1 L v N i, i 1 where L(v i ) denoes he level of verex v i. The levels, no o be confused wih he disances d ij beween nodes, are measured in naural numbers in relaion o he cenral verex v c, whose level is aken o be zero. Here he mean occupaion layer indicaes he layer on which he mass of he ree, on average, is conceived o be locaed. Le us now examine he cenral verex in more deail, as he undersanding of he concep is a prerequisie for inerpreing mean occupaion layer resuls, o follow shorly. The cenral verex is considered he paren of all oher nodes in he ree, also known as he roo of he ree. I is used as he reference poin in he ree, agains which he locaions of all oher nodes are relaive. Thus all oher nodes in he ree are N 7 children of he cenral verex. Alhough here is arbirariness in he choice of he cenral verex, we propose ha i is cenral, or imporan, in he sense ha any change in is price srongly affecs he course of evens in he marke on he whole. We propose hree alernaive definiions for he cenral verex in our sudies, all yielding similar and, in mos cases, idenical oucomes. The firs and second definiions of he cenral verex are local in naure. The idea here is o find he node ha is mos srongly conneced o is neares neighbors. According o he firs definiion, his is he node wih he highes verex degree, i.e., he number of edges which are inciden wih neighbor of he verex. The obained resuls are shown in Fig. 3. The verex degree crierion leads o General Elecric GE dominaing 67.2% of he ime, followed by Merrill Lynch MER a 20.5%, and CBS a 8.2%. The combined share of hese hree verices is 95.9%. The second definiion, a modificaion of he firs, defines he cenral verex as he one wih he highes sum of hose correlaion coefficiens ha are associaed wih he inciden edges of he verex. Therefore, whereas he firs definiion weighs each deparing node equally, he second gives more weigh o shor edges, since a high value of ij corresponds o a low value of d ij. This is reasonable, as shor connecions link he verex more ighly o is neighborhood han long ones he same principle employed in consrucing he spanning ree. This weighed verex degree crierion resuls in GE dominaing 65.6% of he cases, followed by MER a 20.0%, and CBS a 8.7%, he share of he op hree being 94.3%. The hird definiion deals wih he global quaniy of cener of mass. In considering a ree T a ime, he verex v i
5 DYNAMICS OF MARKET CORRELATIONS: TAXONOMY... FIG. 4. Plo of mean occupaion layer l(,v c ) as a funcion of ime, wih saic solid and dynamic doed cenral verices. FIG. 3. Cenral verices according o 1 verex degree crierion op, 2 weighed verex degree crierion middle, and 3 cener of mass crierion boom. ha produces he lowes value for mean occupaion layer l(,v i ) is he cener of mass, given ha all nodes are assigned an equal weigh and consecuive layers levels are a equidisance from one anoher, in accordance wih he above definiion. Wih his cener of mass crierion we find ha he mos dominan company, again, is GE, as i is 52.8% of he ime he cener of mass of he graph, followed by MER a 15.4%, and Minnesoa Mining & MFG a 14.9%. These op hree candidaes consiue 83.1% of he oal. Should he weigh of he node be made proporional o he size e.g., revenue, profi, ec. of he company, i is obvious ha GE s dominance would increase. As Fig. 3 shows, he hree alernaive definiions for he cenral verex lead o very similar resuls. The verex degree and he weighed verex degree crieria coincide 91.8% of he ime. In addiion, he former coincides wih cener of mass 66.7% and he laer 64.6% of he ime, respecively. Overall, he hree crieria yield he same cenral verex in 63.6% of he cases, indicaing considerable muual agreemen. The exisence of a meaningful cener in he ree is no a rivial issue, and neiher is is coincidence wih he cener of mass. However, since he crieria applied, presen a mixure of boh local and global approaches, and he fac ha hey coincide almos 2/3 of he ime, does indicae he exisence of a welldefined cener in he ree. The reason for he coincidence of he crieria seems clear, inuiively speaking. A verex wih a high verex degree, he cenral verex, in paricular, carries a lo of weigh around i he neighboring nodes, which in urn may be highly conneced o ohers o heir children, and so on. Two differen inerpreaions may be given o hese resuls. One may have eiher i saic fixed a all imes or ii dynamic updaed a each ime sep cenral verex. If he firs approach is oped for, he above evidence well subsaniaes he use of GE as he cenral verex. In he second approach, he resuls will vary somewha depending on which of he hree crieria are used in deermining he cenral verex. The mean occupaion layer l() is depiced in Fig. 4, where also he effec of differen cenral verices is demonsraed. The solid curve resuls from he saic cenral verex, i.e., GE, and he doed one o dynamic cenral verex evaluaed using he verex degree crierion. The wo curves coincide where only he solid curve is drawn. This is rue mos of he ime, as he above cenral verex consideraions lead us o expec. The wo dips a 1986 and 1990, locaed symmerically a half a window widh from Black Monday, correspond o he opological shrinking of he ree associaed wih he famous marke crash of Roughly beween 1993 and 1997, l() reaches very high values, which is in concordance wih our earlier resuls obained for a differen se of daa 14. High values of l() are considered o reflec a finer marke srucure, whereas in he oher exreme low dips are conneced o marke crashes, where he behavior of he sysem is very homogeneous. The finer srucure may resul from general seady growh in asse prices during ha period as can be seen, for example, from he S&P 500 index. IV. TREE CLUSTERS AND THEIR ECONOMIC MEANINGFULNESS As menioned earlier, Manegna s idea of linking socks in an ulrameric space was moivaed a poseriori by he propery of such a space o provide a meaningful economic ax
6 ONNELA e al. FIG. 5. Color online Snapsho of a dynamic asse ree connecing he examined 116 socks of he S&P 500 index. The ree was produced using fouryear window widh and i is cenered on January 1, Business secors are indicaed according o Ref. 15. In his ree, General Elecric GE was used as he cenral verex and eigh layers can be idenified. onomy. We will now explore his issue furher, as he meaningfulness of he emerging economic axonomy is he key jusificaion for he use of he curren mehodology. In Ref. 1, Manegna examined he meaningfulness of he axonomy by comparing he grouping of socks in he ree wih a hird pary reference grouping of socks by heir indusry, ec., classificaions. In his case, he reference was provided by Forbes 15, which uses is own classificaion sysem, assigning each sock wih a secor higher level and indusry lower level caegory. In order o visualize he grouping of socks, we consruced a sample asse ree for a smaller daase 14, shown in Fig. 5. This was obained by sudying our previous daase 14, which consiss of 116 S&P 500 socks, exending from he beginning of 1982 o he end of 2000, resuling in a oal of 4787 price quoes per sock 16. Before evaluaing he economic meaningfulness of grouping socks, we wish o esablish some erminology. We use he erm secor exclusively o refer o he given hird pary classificaion sysem of socks. The erm branch refers o a subse of he ree, o all he nodes ha share he specified common paren. In addiion o he paren, we need o have a reference poin o indicae he generaional direcion i.e., who is who s paren in order for a branch o be well defined. Wihou his reference here is no way o deermine where one branch ends and he oher begins. In our case, he reference is he cenral node. There are some branches in he ree, in which mos of he socks belong o jus one secor, indicaing ha he branch is fairly homogeneous wih respec o business secors. This finding is in accordance wih hose of Manegna 1, alhough here are branches ha are fairly heerogeneous, such as he one exending direcly downwards from he cenral verex, see Fig. 5. Since he grouping of socks is no perfec a he branch level, we define a smaller subse whose members are more homogeneous as measured by he uniformiy of heir secor classificaions. The erm cluser is defined, broadly speaking, as a subse of a branch. Le us now examine some of he clusers ha have been formed in he sample ree. We use he erms complee and incomplee o describe, in raher sric erms, he success of clusering. A complee cluser conains all he companies of he sudied se belonging o he corresponding business secor, so ha none are lef ouside he cluser. In pracice, however, clusers are mosly incomplee, conaining mos, bu no all, of he companies of he given business secor, and he res are o be found somewhere else in he ree. Only he Energy cluser was found complee, bu many ohers come very close, ypically missing jus one or wo members of he cluser. Building upon he normalized ree lengh concep, we can characerize he srengh of clusers in a similar manner, as hey are simply subses of he ree. These clusers, wheher complee or incomplee, are characerized by he normalized cluser lengh, defined for a cluser c as follows: L c 1 N c d ij d ij c, 8 where N c is he number of socks in he cluser. This can be compared wih he normalized ree lengh, which for he sample ree in Fig. 5 a ime * is L(*) A full accoun of he resuls is o be found in Ref. 16, bu as a shor summary of resuls we sae he following. The Energy companies form he mos ighly packed cluser resuling in L Energy (*) 0.92, followed by he Healhcare cluser wih L Healh care (*) For he Uiliies cluser we have L Uiliies (*) 1.01 and for he diverse Basic Maerials cluser L Basic maerials (*) Even hough he Technology cluser has he fewes number of members, is mean disance is he highes of he examined groups of clusers being L Technology (*) Thus, mos of he examined clusers seem o be more ighly packed han he ree on average
7 DYNAMICS OF MARKET CORRELATIONS: TAXONOMY... One could find and examine several oher clusers in he ree, bu he ones ha were idenified are quie convincing. The minimum spanning ree, indeed, seems o provide a axonomy ha is well compaible wih he secor classificaion provided by an ouside insiuion, Forbes in his case. This is a srong voe for he use of he curren mehodology in sock marke analysis. Some furher analysis of he idenified clusers is presened in Ref. 16. There are, however, some observed deviaions o he classificaion, which call for an explanaion. For hem he following poins are raised. i The seemingly random asse price flucuaions sem no only from sandard economic facors, bu also from psychological facors, inroducing noise in he correlaion marix. Therefore, i is no reasonable o expec a oneoone mapping beween business secors and MST clusers. ii Business secor definiions are no unique, bu vary by he organizaion issuing hem. In his work, we used he classificaion sysem by Forbes 15, where he sudied companies are divided ino 12 business secors and 51 indusries. Forbes has is own classificaion principle, based on company dynamics raher han size alone. Alernaively, one could have used, say, he Global Indusry Classificaion Sandard GICS, released on January 2, 2001, by Sandard & Poor s 17. Wihin his framework, companies are divided ino 10 secors, 23 indusry groups, 59 indusries, and 122 subindusries. Therefore, he classificaion sysem clearly makes a difference, and here are discrepancies even a he opmos level of business secors amongs differen sysems. iii Hisorical price ime series is, by definiion, old. Therefore, one should use conemporary definiions for business secors, ec., as hose mos accuraely characerize he company. Since hese were no available o he auhors, he curren classificaion scheme by Forbes was used. The error caused by his approach varies for differen companies. iv In many classificaion sysems, companies engaged in subsanially differen business aciviies are classified according o where he majoriy of revenues and profis comes from. For highly diversified companies, hese classificaions are more ambiguous and, herefore, less informaive. As a consequence, classificaion of hese ypes of companies should be viewed wih some skepicism. This problem has is roos in he desire o caegorize companies by a single label, and he approach fails where his division is unnaural. v Some cluser ouliers can be explained hrough he MST clusering mechanism, which is based on correlaions beween asse reurns. Therefore, one would expec, for example, invesmen banks o be grouped wih heir invesmens raher han wih oher similar insiuions. Through porfolio diversificaion, hese banks disance hemselves from he price flucuaions risks of a singlebusiness secor. Consequenly, i would be more surprising o find a oally homogeneous financial cluser han a fairly heerogeneous one currenly observed. vi The risks imposed on he companies by he exernal environmen vary in heir degree of uniformiy from one business secor o anoher. For example, companies in he Energy secor price of heir socks are prone o flucuaions in he world marke price of oil, whereas i is difficul o hink of one facor having equal influence on, say, companies in he Consumer/Noncyclical business secor. This uniformiy of exernal risks influences he sock price of hese companies, in coarse erms, leading o heir more complee clusering han ha of companies facing less uniform exernal risks. In conclusion, regarding all he above lised facors, he success of he applied mehod in idenifying marke axonomy is remarkable. V. SCALEFREE STRUCTURE OF THE ASSET TREE So far we have characerized he asse ree as an imporan subgraph of he fully conneced graph derived from all he elemens of he correlaion marix. Since he asse ree is expeced o reflec some aspecs of he marke and is sae, i is herefore of ineres o learn more abou is srucure. During he las few years, much aenion has been devoed o he degree disribuion of graphs. I has become clear ha he so called scalefree graphs, where his disribuion obeys a power law, are very frequen in many fields, ranging from human relaionships hrough cell meabolism o he Inerne 18,19. Scalefree rees have also been exensively sudied see, e.g., Ref. 20. Recenly, examples for scalefree neworks in economy and finance have been found 6,21,22. Vandewalle e al. 6 found scalefree behavior for he asse ree in a limied one year, 1999 ime window for 6358 socks raded a he NYSE, NASDAQ, and AMEX. They proposed he disribuion of he verex degrees f (n) o follow a power law behavior: f n n, wih he exponen 2.2, where n is he verex degree or number of neighbors of a node. This exponen implies ha he second momen of he disribuion would diverge in he infinie marke limi, or in oher words, he second momen of he disribuion is always dominaed by he rare bu exremely highly conneced verices. Our aim here is o sudy he propery of scale freeness in he ligh of asse ree dynamics. Firs, we conclude ha he asse ree has, mos of he ime, scalefree properies wih a raher robus exponen for normal opology i.e., ouside crash periods of business as usual, a resul close o ha given in Ref. 6. For mos of he ime he disribuion behaves in a universal manner, meaning ha he exponen is a consan wihin he error limis. However, when he behavior of he marke is no business as usual i.e., wihin crash periods, he exponen also changes, alhough he scalefree characer of he ree is sill mainained. For he Black Monday period, we have This resul is in full agreemen wih he observaion of he shrinking of he ree during marke crashes, which is accompanied by an increase in he degree, hus explaining he lower value of he exponen. The observaion concerning he change in he value of he exponen for normal and crash period is exemplified in Fig. 6. When fiing he daa, in many cases we found one or wo ouliers, i.e., verices whose degrees did no fi o he overall power law behavior since hey were much oo high. In all cases hese socks corresponded eiher o he highes con
8 ONNELA e al. FIG. 6. Typical plos of verex degree for normal lef and crash opology righ, for which he exponens and goodness of fi are 2.15, R and 1.75, R , respecively. The plo on he lef is cenered a and he righ one a , and for boh T 1000 days, i.e., 4 years. neced node i.e., he cenral verex or were nodes wih very high degrees. This resul suggess ha i could be useful o handle hese nodes wih special care, hus providing furher suppor o he concep of he cenral node. However, for he purpose of fiing he observed verex degree daa, such nodes were considered ouliers. To give an overall measure of goodness of he fis, we calculaed he R 2 coefficien of deerminaion, which can be inerpreed as he fracion of he oal variaion ha is explained by he leassquares regression line. We obained, on average, values of R for he enire daase wih ouliers included and R wih ouliers excluded. Furher, he fis for he normal marke period were beer han hose obained for he crash period as characerized by he average values of R and R , respecively, wih ouliers excluded. In addiion o he marke period based dependence, he exponen was also found o depend on he window widh. We examined a range of values for he window widh T beween 2 and 8 yr and found, wihou excluding he ouliers, he fied exponen o depend linearly on T. In conclusion, we have found he scaling exponen o depend on he marke period, i.e., crash vs normal marke circumsances and on he window widh. These resuls also raise he quesion of wheher i is reasonable o assume ha differen markes share he scaling exponen. In case hey do no, one should be careful when pooling socks ogeher from differen markes for he purpose of verex degree analysis. can reflec real changes in he asse axonomy, ohers may simply be due o noise. On leing T 0, we find ha () 1, indicaing ha he rees are sable in his limi 9. A sample plo of singlesep survival raio for T 1000 days and T 20.8 days is shown in Fig. 7. The following observaions are made. i A large majoriy of connecions survives from one ime window o he nex. ii The wo prominen dips indicae a srong ree reconfiguraion aking place, and hey are window widh T apar, posiioned symmerically around Black Monday, and hus imply opological reorganizaion of he ree during he marke crash 10. iii Singlesep survival raio () increases as he window widh T increases while T is kep consan. Thus an increase in window widh renders he rees more sable wih respec o singlesep survival of connecions. We also find ha he rae of change of he survival raio decreases as he window widh increases and, in he limi, as he window widh is increased owards infiniy T, () 1 for all. The survival raio seems o decrease very rapidly once he window widh is reduced below roughly 1 yr. As he window widh is decreased furher owards zero, in he limi as T 0, () 0 for all. VI. ASSET TREE EVOLUTION In order o invesigae he robusness of asse ree opology, we define he singlesep survival raio of ree edges as he fracion of edges found common in wo consecuive rees a imes and 1 as 1 N 1 E E In his E() refers o he se of edges of he ree a ime, is he inersecion operaor, and gives he number of elemens in he se. Under normal circumsances, he ree for wo consecuive ime seps should look very similar, a leas for small values of window sep lengh parameer T. Wih his measure i is expeced ha while some of he differences FIG. 7. Singlesep survival raio () as a funcion of ime. The average value is indicaed by he horizonal line
9 DYNAMICS OF MARKET CORRELATIONS: TAXONOMY... FIG. 8. Mulisep survival raio (,k) as a funcion of ime for differen parameric values of T in days. iv Variance of flucuaions around he mean is consan over ime, excep for he exreme evens and he inerim period, and i ges less as he window widh increases. In order o sudy he long erm evoluion of he rees, we inroduce he mulisep survival raio a ime as,k 1 N 1 E E 1 E k 1 E k, 11 where only hose connecions ha have persised for he whole ime period wihou any inerrupions are aken ino accoun. According o his formula, when a bond beween wo companies breaks even once in k seps and hen reappears, i is no couned as a survived connecion. I is found ha many connecions in he asse rees evaporae quie rapidly in he early ime horizon. However, his rae decreases significanly wih ime, and even afer several years here are some connecions ha are lef inac. This indicaes ha some companies remain closely bonded for imes longer han a decade. The behavior of he mulisep survival raio for hree differen values of window widh 2, 4, and 6 yr is shown in Fig. 8, ogeher wih he associaed fis. In his figure he horizonal axis can be divided ino wo regions. Wihin he firs region, decaying of connecions is faser han exponenial, and akes place a differen raes for differen values of he window widh. Laer, wihin he second region, when mos connecions have decayed and only some 20% 30% remain for he shown values of T), here is a crossover o power law behavior. The exponens obained for he window widhs of T 500, T 1000, and T 1500, in days, are 1.15, 1.19, and 1.17, respecively, and so remains he same wihin error margins. Thus, ineresingly, he power law decay in he second region seems independen of he window widh. FIG. 9. Plo of halflife 1/2 as a funcion of window widh T. We can also define a characerisic ime, he so called halflife of he survival raio 1/2,orree halflife for shor, as he ime inerval in which half he number of iniial connecions have decayed, i.e., (, 1/2 / T) 0.5. The behavior of 1/2 as a funcion of he window widh is depiced in Fig. 9 and i is seen o follow a clean linear dependence for values of T being beween 1 and 5 yr, afer which i begins o grow faser han a linear funcion. For he linear region, he ree halflife exhibis 1/2 0.12T dependence. This can also be seen in Fig. 8, where he dashed horizonal line indicaes he level a which half of he connecions have decayed. For he sudied values of he window widh, ree halflife occurs wihin he firs region of he mulisep survival plo, where decaying was found o depend on he window widh. Consequenly, he dependence of halflife on window widh T does no conradic he window widh independen power law decaying of connecions, as he wo occur in differen regions. In general, he number of socks N, as well as he heir ype, is likely o affec he halflives. Earlier, for a se of N 116 S&P 500 socks, halflife was found o depend on he window widh as 1/2 0.20T 9. A smaller ree, consising primarily of imporan indusry gians, would be expeced o decay more slowly han he larger se of NYSEraded socks sudied in his paper. VII. PORTFOLIO ANALYSIS Nex, we apply he above discussed conceps and measures o he porfolio opimizaion problem, a basic problem of financial analysis. This is done in he hope ha he asse ree could serve as anoher ype of quaniaive approach o and/or visualizaion aid of he highly inerconneced marke, hus acing as a ool supporing he decision making process. We consider a general Markowiz porfolio P() wih he asse weighs w 1,w 2,...,w N. In he classic Markowiz porfolio opimizaion scheme, financial asses are characerized by heir average risk and reurn, where he risk associaed wih an asse is measured by he sandard deviaion of reurns. The Markowiz opimizaion is usually carried ou
10 ONNELA e al. FIG. 10. Plo of he weighed minimum risk porfolio layer l P (, 0) wih no shor selling doed and mean occupaion layer l(,v c ) solid agains ime. Top saic cenral verex, boom dynamic cenral verex according o he verex degree crierion. by using hisorical daa. The aim is o opimize he asse weighs so ha he overall porfolio risk is minimized for a given porfolio reurn r P 23. In he dynamic asse ree framework, however, he ask is o deermine how he asses are locaed wih respec o he cenral verex. Le r m and r M denoe he reurns of he minimum and maximum reurn porfolios, respecively. The expeced porfolio reurn varies beween hese wo exremes, and can be expressed as r P, (1 )r m r M, where is a fracion beween 0 and 1. Hence, when 0, we have he minimum risk porfolio, and when 1, we have he maximum reurn maximum risk porfolio. The higher he value of, he higher he expeced porfolio reurn r P, and, consequenly, he higher he risk he invesor is willing o absorb. We define a single measure, he weighed porfolio layer as l P, w i L v i, i P(, ) 12 where N i 1 w i 1 and furher, as a saring poin, he consrain w i 0 for all i, which is equivalen o assuming ha here is no shor selling. The purpose of his consrain is o preven negaive values for l P (), which would no have a meaningful inerpreaion in our framework of rees wih cenral verex. This resricion will shorly be discussed furher. Figure 10 shows he behavior of he mean occupaion layer l() and he weighed minimum risk porfolio layer l P (, 0). We find ha he porfolio layer is higher han he mean layer a all imes. The difference beween he layers depends on he window widh, here se a T 1000, and he ype of cenral verex used. The upper plo in Fig. 10 is produced using he saic cenral verex GE, and he difference in layers is found o be The lower one is produced FIG. 11. Plo of he weighed minimum risk porfolio layer l P (, 0) wih shor selling allowed doed and mean occupaion layer l(,v c ) solid agains ime. Top saic cenral verex, boom dynamic cenral verex according o he verex degree crierion. by using a dynamic cenral verex, seleced wih he verex degree crierion, in which case he difference of 1.39 is found. Above we assumed he no shorselling condiion. However, i urns ou ha, in pracice, he weighed porfolio layer never assumes negaive values and he shorselling condiion, in fac, is no necessary. Fig. 11 repeas he earlier plo, his ime allowing for shor selling. The weighed porfolio layer is now 99.5% of he ime higher han he mean occupaion layer and, wih he same cenral verex configuraion as before, he difference beween he wo is 1.18 and 1.14 in he upper and lower plos, respecively. Thus we conclude ha only minor differences are observed in he previous plos beween banning and allowing shor selling, alhough he difference beween weighed porfolio layer and mean occupaion layer is somewha larger in he firs case. Furher, he difference in layers is also slighly larger for saic han dynamic cenral verex, alhough no by much. As he socks of he minimum risk porfolio are found on he ouskirs of he ree, we expec larger rees higher L) o have greaer diversificaion poenial, i.e., he scope of he sock marke o eliminae specific risk of he minimum risk porfolio. In order o look a his, we calculaed he meanvariance froniers for he ensemble of 477 socks using T 1000 as he window widh. In Fig. 12, we plo he level of porfolio risk as a funcion of ime, and find a similariy beween he risk curve and he curves of he mean correlaion coefficien and normalized ree lengh L. Earlier, in Ref. 14, when he smaller daase of 116 socks consising of primarily imporan indusry gians was used, we found Pearson s linear correlaion beween he risk and he mean correlaion coefficien () o be 0.82, while ha beween he risk and he normalized ree lengh L() was Therefore, for ha daase, he normalized ree lengh was able o explain he diversificaion poenial of he marke
11 DYNAMICS OF MARKET CORRELATIONS: TAXONOMY... FIG. 12. Plos of a he mean correlaion coefficien (), b he normalized ree lengh L(), and c he risk of he minimum risk porfolio, as funcions of ime. beer han he mean correlaion coefficien. For he curren se of 477 socks, which includes also less influenial companies, he Pearson s linear and Spearman s rankorder correlaion coefficiens beween he risk and he mean correlaion coefficien are 0.86 and 0.77, and hose beween he risk and he normalized ree lengh are 0.78 and 0.65, respecively. So far, we have only examined he locaion of socks in he minimum risk porfolio, for which 0. As we increase owards uniy, porfolio risk as a funcion of ime soon sars behaving very differenly from he mean correlaion coefficien and normalized ree lengh. Consequenly, i is no longer useful in describing diversificaion poenial of he marke. However, anoher ineresing resul emerges: The average weighed porfolio layer l P (, ) decreases for increasing values of, as shown in Fig. 13. This means ha ou of all he possible Markowiz porfolios, he minimum risk porfolio socks are locaed furhes away from he cenral verex, and as we move owards porfolios wih higher expeced reurn, he socks included in hese porfolios are locaed closer o he cenral verex. When saic cenral node is used, he average values of he weighed porfolio layer l P (, ) for 0, 1/4, 1/2, and 3/4 are 6.03, 5.70, 5.11, and 4.72, respecively. Similarly, for a dynamic cenral node, we obain he values of 5.68, 5.34, 4.78, and We have no included he weighed porfolio layer for 1, as i is no very informaive. This is due o he fac ha he maximum reurn porfolio comprises only one asse he maximum reurn asse in he curren ime window and, herefore, l P (, 1) flucuaes wildly as he maximum reurn asse changes over ime. We believe hese resuls o have poenial for pracical applicaion. Due o he clusering properies of he MST, as well as he overlap of ree clusers wih business secors as defined by a hird pary insiuion, i seems plausible ha companies of he same cluser face similar risks, imposed by he exernal economic environmen. These dynamic risks influence he sock prices of he companies, in coarse erms, leading o heir clusering in he MST. In addiion, he radial locaion of socks depends on he chosen porfolio risk level, characerized by he value of. Socks included in lowrisk porfolios are consisenly locaed furher away from he cenral node han hose included in highrisk porfolios. Consequenly, he radial disance of a node, i.e., is occupaion layer, is meaningful. Thus, i can be conjecured ha he locaion of a company wihin he cluser reflecs is posiion wih regard o inernal, or cluser specific, risk. Characerizaion of socks by heir branch, as well as heir locaion wihin he branch, enables us o idenify he degree of inerchangeabiliy of differen socks in he porfolio. For example, in mos cases we could pick wo socks from differen asse ree clusers, bu from nearby layers, and inerchange hem in he porfolio wihou considerably alering he characerisics of he porfolio. Therefore, dynamic asse rees provide an inuiionfriendly approach o and faciliae incorporaion of subjecive judgmen in he porfolio opimizaion problem. VIII. SUMMARY AND CONCLUSION FIG. 13. Color online Plos of he weighed minimum risk porfolio layer l P (, ) for differen values of. In summary, we have sudied he disribuion of correlaion coefficiens and is momens. We have also sudied he dynamics of asse rees: he ree evolves over ime and he normalized ree lengh decreases and remains low during a crash, hus implying he shrinking of he asse ree paricularly srongly during a sock marke crisis. We have also found ha he mean occupaion layer flucuaes as a funcion of ime, and experiences a downfall a he ime of marke crisis due o opological changes in he asse ree. Furher, our sudies of he scalefree srucure of he MST show ha his graph is no only hierarchical in he sense of a ree bu here are special, highly conneced nodes and he hierarchical srucure is buil up from hese. As for he porfolio analysis, i was found ha he socks included in he minimum risk porfolio end o lie on he ouskirs of he asse ree: on average he weighed porfolio layer can be almos one and a half levels higher, or furher away from he cenral verex,
12 ONNELA e al. han he mean occupaion layer for window widh of four years. For many of he quaniies we have sudied, he behavior is significanly differen for hose daa windows conaining he daes around Ocober 19, 1987 Black Monday from windows wihou hem. We have sudied he effecs of his crash, more specifically in Ref. 10. We should clarify ha he period which has shown a crashlike behavior is an arifac of he fouryear window widh used o analyze he daa and excep for he daes around Ocober 19, 1987 his period was normal. Correlaion beween he risk and he normalized ree lengh was found o be srong, hough no as srong as he correlaion beween he risk and he mean correlaion coefficien. Thus we conclude ha he diversificaion poenial of he marke is very closely relaed also o he behavior of he normalized ree lengh. Finally, he asse ree can be viewed as a highly graphical ool, and even hough i is srongly pruned, i sill reains all he essenial informaion of he marke and can be used o add subjecive judgmen o he porfolio opimizaion problem. ACKNOWLEDGMENTS J.P.O. is graeful o European Science Foundaion for REACTOR gran o visi Hungary, he Budapes Universiy of Technology and Economics for he warm hospialiy, and he Graduae School in Compuaional Mehods of Informaion Technology ComMIT, Finland. The role of Harri Toivonen a he Deparmen of Accouning, Helsinki School of Economics, is acknowledged for carrying ou CRSP daabase exracions. We are also graeful o R. N. Manegna for very useful discussions and suggesions. This research was parially suppored by he Academy of Finland, Research Cener for Compuaional Science and Engineering, Projec No Finnish Cener of Excellence Program and OTKA Gran No. T R.N. Manegna, Eur. Phys. J. B 11, L. Kullmann, J. Kerész, and R.N. Manegna, Physica A 287, L. Giada and M. Marsili, Physica A 315, L. Laloux e al., Phys. Rev. Le. 83, ; V. Plerou e al., ibid. 83, The Economy as an Evolving Complex Sysem II, edied by W.B. Arhur, S.N. Durlauf, and D.A. Lane AddisonWesley, Reading, MA, N. Vandewalle, F. Brisbois, and X. Tordoir, Quan. Finance 1, H.M. Markowiz, J. Finance 7, S. Gallucio, J.P. Bouchaud, and M. Poers, Physica A 259, ; A. Gabor and I. Kondor, ibid. 274, ; L. Bongini e al., Eur. Phys. J. B 27, J.P. Onnela, M. Sc. hesis, Helsinki Universiy of Technology, Finland, J.P. Onnela, A. Chakrabori, K. Kaski, and J. Kerész, Physica A 324, S. Drozdz e al., Physica A 287, J.P. Onnela, A. Chakrabori, K. Kaski, and J. Kerész, Phys. Scr. T 106, J.P. Onnela, A. Chakrabori, K. Kaski, and J. Kerész unpublished. 14 J.P. Onnela, A. Chakrabori, K. Kaski, and J. Kerész, Eur. Phys. J. B 30, Forbes a hp:// referenced in MarchApril, Supplemenary maerial is available a hp:// jonnela/ 17 Sandard & Poor s 500 index a hp:// referenced in June, R. Alber and A.L. Barabasi, Rev. Mod. Phys. 74, S.N. Dorogovsev and J.F.F. Mendes, Adv. Phys. 51, G. Szabó, M. Alava, and J. Kerész, Phys. Rev. E 66, M. Marsili, Quan. Finance 2, I. Yang, H. Jeong, B. Kahng, and A.L. Barabasi, eprin condma/ ; H.J. Kim, Y. Lee, B. Kahng, and I. Kim, J. Phys. Soc. Jpn. 71, Several sofware packages based on sandard procedures are available. We used MATLAB wih Financial Toolbox
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