Determination of systemically important companies with cross-shareholding network analysis: A case study from an emerging market

Das khan, Hossein; Gha neh, Nase Shams A icle De e mina ion o sys emically impo an companies wi h c oss-sha eholding ne wo k analysis: A case s udy om an eme ging ma ke In e na ional Jou nal o Financial S udies P o ided in Coope a ion wi h: MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel Sugges ed Ci a ion: Das khan, Hossein; Gha neh, Nase Shams (2016) : De e mina ion o sys emically impo an companies wi h c oss-sha eholding ne wo k analysis: A case s udy om an eme ging ma ke , In e na ional Jou nal o Financial S udies, ISSN 2227-7072, MDPI, Basel, Vol. 4, Iss. 3, pp. 1-17, h ps://doi.o g/10.3390/ij s4030013 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/167809 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. 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I he documen s ha e been made a ailable unde an Open Con en Licence (especially C ea i e Commons Licences), you may exe cise u he usage igh s as speci ied in he indica ed licence. h p://c ea i ecommons.o g/licenses/by/4.0/ In e na ional Jou nal o Financial S udies A icle De e mina ion o Sys emically Impo an Companies wi h C oss-Sha eholding Ne wo k Analysis: A Case S udy om an Eme ging Ma ke Hossein Das khan and Nase Shams Gha neh * Depa men o Indus ial Enginee ing and Managemen Sys ems, Ami Kabi Uni e si y o Technology, Ha ez S ., Teh an 15916-34311, I an; [email p o ec ed] *Co espondence: [email p o ec ed]; Tel.: +98-216-454-5363 Academic Edi o : Nicholas Ape gis Recei ed: 11 Feb ua y 2016; Accep ed: 17 June 2016; Published: 24 June 2016 Abs ac : Sys emic isk e en s cons i u e an impo an issue in cu en inancial sys ems. A leading cou se o ac ion used o mi iga e such e en s is iden i ica ion o sys emically impo an agen s in o de o implemen he p uden ial policies in a inancial sys em. In his pape , a bi-le el c oss-sha eholding ne wo k o he s ock ma ke is conside ed acco ding o di ec and in eg a ed owne ship s uc u e. Fu he mo e, di e en sys emic isk indices a e applied o iden i y sys emically impo an companies in an ea ly wa ning sys em. Resul s o applica ion o hese indices on c oss-sha eholding da a om Teh an S ock Exchange show ha in eg a ed ne wo k indices p oduce mo e eliable esul s. Mo eo e , esul s o s a is ical analysis o he ne wo ks indica ed he exis ence o scale- ee cha ac e is ics in he TSE c oss-sha eholding ne wo k. Keywo ds: inance; ne wo k heo y; sys emic isk; c oss-sha eholding; cen ali y measu es JEL: C6; E44; G10; G18; G32 1. In oduc ion Sys emic isk is ela ed o in e -connec ion and co ela ion o di e en pa s o a ma ke . In sys emic isk e en s, a inancial c isis occu s when an ini ial ailu e is ansmi ed o he whole ma ke [ 1 ]. Acco dingly, con ol and mi iga ion o sys emic isk is an impo an conside a ion in he inancial ma ke s. In e ms o sys emic isk, companies a e di e en in he ex en o hei con ibu ions o he sys emic e en s. In many coun ies, sys emically impo an companies a e iden i ied by inancial egula o y bodies and hese au ho i ies apply he p uden ial policies in he inancial sys ems. In he ecen sys emic isk s udies [ 1 – 3 ], size, in e connec edness and he co ela ion o companies a e in oduced as he main ac o s o sys emic isk. Some s udies ha e claimed ha la ge size companies a e mo e ulne able o sys emic isk and in oduced he “ oo big o ail” no ion. In he o he s udies, In e connec edness and co ela ion ha e a huge impac on he companies’ exposu e o each o he by causal and balance shee ela ions and co-mo emen o companies o e he same ime pe iod [3]. In ecen yea s, ne wo k heo y has been widely applied o analysis o inancial sys emic isk and inancial c isis [ 4 , 5 ]. Financial ne wo k s udies can be ca ego ized in o h ee main g oups. In he i s g oup, con agion heo y is applied o inancial sys ems in o de o simula e he beha io o a inancial sys em unde di e en ne wo k se ups [ 6 ]. The ocus o he second g oup is on he co ela ion-based ne wo ks and he analysis o he s uc u e o a inancial ma ke in di e en ime pe iods [ 7 , 8 ]. Mos esea ch in he ield alls in o he hi d g oup ha analyzes he s uc u e o in e -bank deb ne wo ks in di e en coun ies and p oposes a ious p uden ial policies o mi iga e inancial sys emic isk and i s associa ed cos s [ 9 – 12 ]. In his ype o s udies, in e -bank deb ne wo ks a e in oduced as scale- ee In . J. Financial S ud. 2016,4, 13; doi:10.3390/ij s4030013 www.mdpi.com/jou nal/ij s In . J. Financial S ud. 2016,4, 13 2 o 17 ne wo ks ha also ha e he “small wo ld” cha ac e is ics. These inancial ne wo ks a e also “ obus o agile”, which means ha hey ope a e as a isk sha ing mechanism o inc ease s abili y o a inancial sys em a he p e-c isis pe iod. Con e sely, in e connec edness would se e as a shock p opaga ion mechanism du ing a c isis pe iod and would hus inc ease he p obabili y o sys em ailu e [1]. E en hough c oss-sha eholding ne wo ks a e one o he main ypes o inancial ne wo ks, applica ion o c oss-sha eholding ne wo ks in he analysis o inancial sys emic isk has sca cely been epo ed. In o he wo ds, mos sys emic isk publica ions ha e analyzed in e connec edness in in e -bank deb ne wo ks. An excep ion is he s udy o Peco a and Spel a [ 1 ], which epo s use o he c oss-sha eholding ne wo k o Eu opean banks o analyze sys emic isk. Because o he limi ed a ailabili y o in o ma ion on bila e al exposu e be ween di e en companies, such as in e -bank deb s, c oss-sha eholding da a could be applied as an app op ia e p oxy o in o ma ion on in e connec edness. In addi ion o sys emic isk analysis in banking sys ems, applica ion o c oss-sha eholding in o ma ion makes sys emic isk analysis o be possible o o he companies in a inancial ma ke . Al hough a c oss-sha eholding ne wo k could ha e an impo an ole o main ain he s abili y in a inancial sys em, no pa icula analysis has applied he s ock ma ke c oss-sha eholding ne wo k o analyze sys emic isk. In his pape , he c oss-sha eholding ne wo k o he s ock ma ke is applied as a bi-le el ne wo k. A bi-le el c oss-sha eholding ne wo k means ha he ne wo k con ains companies lis ed in he s ock ma ke as well as ex e nal sha eholde s o companies and he sha eholde ’s sha eholde s and hei ela ionships. Fo his pu pose, he eme ging ma ke o Teh an S ock Exchange (TSE) was conside ed o analysis ( he ecen cancella ion o in e na ional sanc ions has s imula ed he in e es s om bo h in e nal and ex e nal in es o s). Based on he a ailable c oss-sha eholding in o ma ion, he di ec and in eg a ed c oss-sha eholding ne wo k o TSE is ep esen ed and di e en sys emic isk indices a e applied and s a is ically app aised. Mo eo e , he mos sys emically impo an companies and sec o s a e de e mined in acco d o di e en indices. Resul s show ha he c oss-sha eholding ne wo k is a good p oxy o analysis o in e connec edness and sys emic isk in inancial ma ke s. Conside a ion o in eg a ed owne ship in a c oss-sha eholding ne wo k leads o a mo e eliable ne wo k and be e unde s anding o he sys emic isk impo ance o di e en companies. Mo eo e , esul s o s a is ical analysis o hese ne wo ks show ha c oss-sha eholding ne wo k o TSE has a powe -law dis ibu ion. The powe -law dis ibu ion ampli ies he hypo hesis ha he ne wo k can be “ obus ye agile” and is e ile o c isis in he case o sys emic isk e en s. 2. Li e a u e Re iew E en hough mos inancial ne wo k s udies a e ca ego ized in o he h ee abo e-men ioned g oups, only a ew s udies ha e epo ed on applica ion o c oss-sha eholding ne wo ks o analyze owne ship s uc u e and con ol low [ 13 ]. The main ocus o hese s udies has been s a is ical analysis o c oss-sha eholding ne wo ks and con ol low in di e en coun ies. Fo example, Ba is on and Gla elde [ 14 ] applied he p inciples o ne wo k heo y in a c oss-sha eholding ne wo k o analyze owne ship con ol s uc u e in he wo ld inancial sys em. Gla elde and his co-wo ke s showed ha he c oss-sha eholding ne wo k has a bow- ie s uc u e. The bow- ie s uc u e is a co e-pe iphe y s uc u e wi h a cen al pa known as s ongly connec ed componen (SGC) and wo inpu and ou pu sec o s [ 14 , 15 ]. In ano he s udy, Ba is on [ 16 ] compa es he desc ip i e cha ac e is ics o c oss-sha eholding ne wo ks in Milan, New Yo k and London s ock exchanges. Ma, Zhuang and Li [ 17 ] analyzed mu ual in es men in he Shanghai S ock Exchange. A deg ee-equi alen measu e was in oduced o weigh ed di ec ed c oss-sha eholding ne wo k and analysis o s a is ical cha ac e is ics o he ne wo k in e ms o co po a e, sec o and p o ince was epo ed. In addi ion o he abo e, he e ha e been se e al o he s udies on he applica ion o c oss-sha eholding ( o example, see [ 18 – 21 ]). In conclusion, al hough li le s udies ha e been done on applica ion o c oss-sha eholding ne wo ks, In . J. Financial S ud. 2016,4, 13 3 o 17 no speci ic pape has ye been published on applica ion o he c oss-sha eholding ne wo k o sys emic isk analysis. 3. Da a and Me hodology 3.1. The Da ase In his pape , he c oss-sha eholding ne wo k o he Teh an s ock exchange, as an eme ging ma ke , is applied. The sha eholding da a om 386 lis ed companies o he yea s 2013–2015 was ex ac ed om he Teh an S ock Exchange da ase [ 22 ]. Because many lis ed companies had a small amoun o ma ke alue, he da ase was limi ed o he sha eholding da a o 118 lis ed companies ha ha e mo e han 0.1 pe cen sha e o he o al ma ke alue. The missing da a o he c oss-sha eholding s uc u e a e di ec ly ex ac ed om he annual epo s o he companies. A gene al summa y o he companies, indus ial sec o s and hei ma ke sha e is ep esen ed in Table 1. As shown in Table 1, hese companies a e ca ego ized in o 27 di e en sec o s. The comple ed lis o he companies and sec o s has been ep esen ed in Appendix A. Fo he sake o simplici y, he da ase consis s o sha eholde s which ha e mo e han one pe cen o he lis ed companies. Mo eo e , as a signi ican numbe o lis ed companies had ex e nal sha eholde s, he da ase con ained sha eholde s’ da a and hei sha eholde ’s sha eholde in a bi-le el ne wo k. Table 1. Gene al summa y o he selec ed da ase . Row Sec o Numbe o Companies Ma ke Sha e (%) Row Sec o Numbe o Companies Ma ke Sha e (%) 1 Oil & Gas D illing 1 0.37 15 Re ailing 1 0.13 2 Me al Mining 5 4.99 16 Cemen 5 0.8 3 Oil P oduc s 11 10.63 17 Tou ism 1 0.2 4 Tie & Plas ic 2 0.34 18 In es men Co. 8 1.7 5 Me als 11 10.13 19 Banking 16 10.86 6 Me al P oduc s 1 0.19 20 Leasing 1 0.13 7Machines and Equipmen 1 0.12 21 T anspo a ion 3 1.63 8 Elec onic Machines 2 0.43 22 Telecommunica ion 2 6.58 9 Ca P oduc ion 6 2.26 23 Insu ance Co. 4 0.47 10 Indus ial Holdings 4 6.65 24 Housing 3 0.39 11 Elec ici y Supply 1 0.24 25 Compu e & IT 2 1.06 12 Food P oduc ion 2 0.53 26 Enginee ing Se ices 1 2 13 Medicine 7 1.34 27 Ex e nal Sha eholde s 13 - 14 Pe ochemical 17 27.34 3.2. Me hodology 3.2.1. Ne wo k No a ion A ne wo k is a se o ela ed nodes. In g aph heo y, ne wo ks a e ep esen ed as a se o e ices and edges. Gene ally, a ne wo k N = (V, E) is ep esen ed as a se o e ices V and edges E, which he se o edges can be shown as an adjacency ma ix. The ma ix elemen s `Aij˘ show he exis ence o a ela ion be ween he nodes iand jand each elemen can be 0 o 1. I he e is a ela ion be ween i and j, Aij is 1 and 0 o he wise. I he ne wo k is an indi ec ne wo k, he adjacency ma ix is symme ic. In weigh ed ne wo ks, Aij “wij. The numbe o edges o each node iis called he deg ee o node iand is ep esen ed as di . Fo di ec ed ne wo ks, he e is a di e ence be ween inpu and ou pu a cs. The numbe o inpu and ou pu a cs o each node iis known as din i and dou i , espec i ely. As owne ship ela ions in c oss-sha eholding ne wo ks a e asymme ic, hese c oss-sha eholding ne wo ks a e ep esen ed as a weigh ed di ec ed ne wo k. In . J. Financial S ud. 2016,4, 13 4 o 17 In addi ion o he abo e-men ioned no a ions, o each node in ne wo k N, he e a e alues such as i ha ep esen an in insic alue o a node and a e no ele an o he s uc u e o a ne wo k. In a c oss-sha eholding ne wo k, i could be he ma ke alue o he ope a ional e enue o each company. A simple ep esen a ion o a c oss-sha eholding ne wo k as a weigh ed di ec ed ne wo k is ep esen ed in Figu e 1. In . J. Financial S ud. 2016, 4, 13 4 o 18 In addi ion o he abo e-men ioned no a ions, o each node in ne wo k N, he e a e alues such as  ha ep esen an in insic alue o a node and a e no ele an o he s uc u e o a ne wo k. In a c oss-sha eholding ne wo k,  could be he ma ke alue o he ope a ional e enue o each company. A simple ep esen a ion o a c oss-sha eholding ne wo k as a weigh ed di ec ed ne wo k is ep esen ed in Figu e 1. j w ij Figu e 1. Rep esen a ion o a c oss-sha eholding ne wo k as a weigh ed di ec ed ne wo k. 3.2.2. Ne wo ks Based Sys emic Risk Indices One o he main challenges o ne wo k analysis o many esea che s is how o de e mine he impo ance o each node a he han o he s in a ne wo k. In he li e a u e ela ed o sys emic isk, he impo ance o nodes can be desc ibed as he con ibu ion o nodes in occu ence o a sys emic isk e en . Fo his pu pose, he e a e many measu es men ioned in ne wo k li e a u e [23–25] e med as “cen ali y measu es”. The wo main cen ali y measu es ha a e widely de eloped in he li e a u e o ne wo k heo y a e he deg ee measu e and he eigen ec o o eedback-based cen ali y measu e. Deg ee Cen ali y Measu e: The deg ee o each node in a c oss-sha eholding ne wo k is he numbe o companies wi h an owne ship ela ion wi h he node. d , as he ou -deg ee o node i, is he numbe o companies ha a e sha eholde s o i. Con e sely, d  as he in-deg ee o node i, is he numbe o companies belonging o he in es men po olio o company i. The e o e, i could be no ed ha d  is a p oxy o he le el o in eg a ion o company i in he o he companies and d  is he deg ee o po olio di e si ica ion o company i. Mic o-in es o s a e excluded in he es ima ion o d  and d  o each company. E en hough his migh cause a bias in d  and d  es ima ion, bu because his is he case o es ima ion o all companies, he bias e ec could be igno ed a he compa a i e analysis o companies. In addi ion, i  ep esen s he ma ke alue o company i, P is in oduced as he po olio alue o company i and conside ed as a sys emic isk index in [19]. P= w     (1) whe e w is he pe cen o company j ha is owned by company i and  is he ma ke alue o company j. Since he c oss-sha eholding ne wo k is a weigh ed ne wo k, i is equi ed o conside he weigh s in o de o calcula e cen ali y as well as sys emic isk impo ance o companies. In-deg ee and ou -deg ee indices can also be p esen ed in weigh ed ne wo ks. Fo he i s ime, Newman [26] in oduced he deg ee o each node in a weigh ed ne wo k as he s eng h o a node. Then, a mixed index o cen ali y based on s eng h and deg ee o each node was in oduced by Opsahl and hei co-wo ke s [27]. Al hough he li e a u e epo s ha such indices a e used widely, hey canno be applied in his case because o he na u e o weigh s in c oss-sha eholding ne wo ks. The e o e in his pape , he wo ep esen ed measu es o Ba is on ha e been applied o measu e weigh ed Figu e 1. Rep esen a ion o a c oss-sha eholding ne wo k as a weigh ed di ec ed ne wo k. 3.2.2. Ne wo ks Based Sys emic Risk Indices One o he main challenges o ne wo k analysis o many esea che s is how o de e mine he impo ance o each node a he han o he s in a ne wo k. In he li e a u e ela ed o sys emic isk, he impo ance o nodes can be desc ibed as he con ibu ion o nodes in occu ence o a sys emic isk e en . Fo his pu pose, he e a e many measu es men ioned in ne wo k li e a u e [ 23 – 25 ] e med as “cen ali y measu es”. The wo main cen ali y measu es ha a e widely de eloped in he li e a u e o ne wo k heo y a e he deg ee measu e and he eigen ec o o eedback-based cen ali y measu e. Deg ee Cen ali y Measu e: The deg ee o each node in a c oss-sha eholding ne wo k is he numbe o companies wi h an owne ship ela ion wi h he node. dou i , as he ou -deg ee o node i, is he numbe o companies ha a e sha eholde s o i. Con e sely, din i as he in-deg ee o node i, is he numbe o companies belonging o he in es men po olio o company i. The e o e, i could be no ed ha dou i is a p oxy o he le el o in eg a ion o company iin he o he companies and din i is he deg ee o po olio di e si ica ion o company i. Mic o-in es o s a e excluded in he es ima ion o dou i and din i o each company. E en hough his migh cause a bias in dou i and din i es ima ion, bu because his is he case o es ima ion o all companies, he bias e ec could be igno ed a he compa a i e analysis o companies. In addi ion, i i ep esen s he ma ke alue o company i, Pi is in oduced as he po olio alue o company iand conside ed as a sys emic isk index in [19]. Pi“ÿjNeighbo s o iwij j(1) whe e wij is he pe cen o company j ha is owned by company iand j is he ma ke alue o company j. Since he c oss-sha eholding ne wo k is a weigh ed ne wo k, i is equi ed o conside he weigh s in o de o calcula e cen ali y as well as sys emic isk impo ance o companies. In-deg ee and ou -deg ee indices can also be p esen ed in weigh ed ne wo ks. Fo he i s ime, Newman [ 26 ] in oduced he deg ee o each node in a weigh ed ne wo k as he s eng h o a node. Then, a mixed index o cen ali y based on s eng h and deg ee o each node was in oduced by Opsahl and hei co-wo ke s [ 27 ]. Al hough he li e a u e epo s ha such indices a e used widely, hey canno be applied in his case because o he na u e o weigh s in c oss-sha eholding ne wo ks. The e o e in his In . J. Financial S ud. 2016,4, 13 5 o 17 pape , he wo ep esen ed measu es o Ba is on ha e been applied o measu e weigh ed in-deg ee and ou -deg ee measu es [ 16 ]. sj , which is an in-deg ee equi alen cen ali y measu e, ep esen ed as ollows: sj“ˆřdin j i“1wij˙2 řdin j i“1wij2 (2) whe e sj shows he numbe o inwa d a cs o he node j. In o he wo ds, sj calcula es he e ec i e numbe o companies owned by company j. On he o he hand, Hi is in oduced as an equi alen ou -deg ee measu e and desc ibed as ollows: Hi“ kou i ÿ j“1 hij (3) hij which shows he amoun o con ol ha company jhas on company i, is calcula ed as ollows: hij “w2 ij řkin j l“1w2 lj (4) In o he wo ds, hij shows he impo ance o company iwi hin he companies ha belong o company j. Hi measu es he o al impo ance o company i o companies ha ha e in es ed in company iand acco dingly shows he e ec i e numbe o in es o s o company i. Eigen Vec o Cen ali y Measu e: Eigen ec o cen ali y measu es a e based on he concep ha impo ance o each node is ela ed o he impo ance o i s neighbo s. This concep leads o a se ies o equa ions ha should be sol ed simul aneously. In a weigh ed di ec ed ne wo k, he Hubbell measu e [ 28 ] is he mos amilia Eigen ec o measu e. In his measu e, each node could ha e an in insic impo ance, such as c0 and impo ance is ela ed o exis ing ela ions o o he nodes. The ollowing ela ion shows he Hubbell index: cH“AcH`c0(5) whe e cH is he Hubbell cen ali y measu e and A is he adjacency ma ix o he ne wo k. The solu ion o Equa ion (5) is ep esen ed as: cH“ pI´Aq´1c0(6) The abo e solu ion is ob ained, i he pI´Aq ma ix is in e ible o equi alen ly none o he eigen alues o A is equal o 1. The e o e, choosing he pa ame e s c0 and cH in a c oss-sha eholding ne wo k, he Hubbell measu e could be used o de e mine companies ha a e sys emically impo an . In his pape , ma ke alues and he in eg a ed ma ke alues a e chosen as c0and cH. 3.2.3. In eg a ed Ma ke Value and In eg a ed C oss-Sha eholding Ma ix In his sec ion, cen ali y and impo ance o each company in a c oss-sha eholding ne wo k is e alua ed based on in eg a ed owne ship o companies. Acco ding o B ioschi [ 29 ], in eg a ed owne ship is de e mined by agg ega ed amoun s o di ec and indi ec owne ship o a sha eholde on an asse . Figu e 2illus a es a simple compa ison o di ec and indi ec owne ship in a c oss-sha eholding ne wo k. In . J. Financial S ud. 2016,4, 13 6 o 17 In . J. Financial S ud. 2016, 4, 13 6 o 18 i j k l w ij w jk w jl wik ~ wil ~ Figu e 2. Rep esen a ion o di ec and indi ec owne ship in a c oss-sha eholding ne wo k. As inancial agen s in a s ock ma ke can be ca ego ized in o lis ed companies and ex e nal sha eholde s, he po olio alue o a lis ed company is calcula ed as Equa ion (7). P=w  ()  (7) whe e w shows he pe cen o company j ha is owned by company i, Γ() is he se o neighbo s o company i and  is he ma ke alue o company j. The ma ix ep esen a ion o Equa ion (7) could be ep esen ed as ollows: P=WV (8) whe eP shows ec o o po olio alues, W is an n × n adjacency ma ix,V is he ec o o ma ke alues and n is he numbe o lis ed companies. Con e sely as he main ex e nal sha eholde s o he lis ed companies a e conside ed in his s udy, he po olio alue o an ex e nal sha eholde k (P,) is ep esen ed as ollows: P, =d و=1,2,…, (9) whe e m is he numbe o ex e nal sha eholde s, P, ep esen s ma ke alues o he lis ed companies and d is he owne ship ac ion o ex e nal sha eholde k on he lis ed companies. The ma ix ep esen a ion o he alue o ex e nal sha eholde s po olio is de e mined om Equa ion (10). P =d (10) When he c oss-sha eholding ne wo k is shown as a combina ion o ex e nal sha eholde s and lis ed companies, he ollowing ma ix A is ep esen ed: A=W0 󰇍  d0 󰇍   (11) whe e 0 󰇍  is a ze o ma ix and A is a (m + n) × (m + n) ma ix. Acco dingly, he in eg a ed ma ke alue o each company could be calcula ed as a combina ion o in insic alues and amoun s o in e connec edness as ollows: V =AV +V (12) whe e V and V show in insic ma ke alues and he in eg a ed ma ke alues o companies, espec i ely. The solu ion o Equa ion (13) is as ollows: V =(I−A)V (13) The V, as he in eg a ed alues o companies could be applied as a sys emic isk index ha shows impo ance in e ms o sys emic isk. I in he Equa ion (6), he c=V and V=c, i is ob ious ha he V is a kind o Hubbell cen ali y measu e. Figu e 2. Rep esen a ion o di ec and indi ec owne ship in a c oss-sha eholding ne wo k. As inancial agen s in a s ock ma ke can be ca ego ized in o lis ed companies and ex e nal sha eholde s, he po olio alue o a lis ed company is calcula ed as Equa ion (7). Pi“ÿ jeΓpiq wij j(7) whe e wij shows he pe cen o company j ha is owned by company i, Γpiq is he se o neighbo s o company iand j is he ma ke alue o company j. The ma ix ep esen a ion o Equa ion (7) could be ep esen ed as ollows: P“WV (8) whe e P shows ec o o po olio alues, W is an n ˆ nadjacency ma ix, V is he ec o o ma ke alues and nis he numbe o lis ed companies. Con e sely as he main ex e nal sha eholde s o he lis ed companies a e conside ed in his s udy, he po olio alue o an ex e nal sha eholde k `Pk,ex ˘ is ep esen ed as ollows: Pk,ex “dk In . J. Financial S ud. 2016, 4, 13 6 o 18 i j k l w ij w jk w jl wik ~ wil ~ Figu e 2. Rep esen a ion o di ec and indi ec owne ship in a c oss-sha eholding ne wo k. As inancial agen s in a s ock ma ke can be ca ego ized in o lis ed companies and ex e nal sha eholde s, he po olio alue o a lis ed company is calcula ed as Equa ion (7). P=w  ()  (7) whe e w shows he pe cen o company j ha is owned by company i, Γ() is he se o neighbo s o company i and  is he ma ke alue o company j. The ma ix ep esen a ion o Equa ion (7) could be ep esen ed as ollows: P=WV (8) whe eP shows ec o o po olio alues, W is an n × n adjacency ma ix,V is he ec o o ma ke alues and n is he numbe o lis ed companies. Con e sely as he main ex e nal sha eholde s o he lis ed companies a e conside ed in his s udy, he po olio alue o an ex e nal sha eholde k (P,) is ep esen ed as ollows: P, =d و=1,2,…, (9) whe e m is he numbe o ex e nal sha eholde s, P, ep esen s ma ke alues o he lis ed companies and d is he owne ship ac ion o ex e nal sha eholde k on he lis ed companies. The ma ix ep esen a ion o he alue o ex e nal sha eholde s po olio is de e mined om Equa ion (10). P =d (10) When he c oss-sha eholding ne wo k is shown as a combina ion o ex e nal sha eholde s and lis ed companies, he ollowing ma ix A is ep esen ed: A=W0 󰇍  d0 󰇍   (11) whe e 0 󰇍  is a ze o ma ix and A is a (m + n) × (m + n) ma ix. Acco dingly, he in eg a ed ma ke alue o each company could be calcula ed as a combina ion o in insic alues and amoun s o in e connec edness as ollows: V =AV +V (12) whe e V and V show in insic ma ke alues and he in eg a ed ma ke alues o companies, espec i ely. The solu ion o Equa ion (13) is as ollows: V =(I−A)V (13) The V, as he in eg a ed alues o companies could be applied as a sys emic isk index ha shows impo ance in e ms o sys emic isk. I in he Equa ion (6), he c=V and V=c, i is ob ious ha he V is a kind o Hubbell cen ali y measu e. k“1, 2, . . . , m(9) whe e mis he numbe o ex e nal sha eholde s, Pk,ex ep esen s ma ke alues o he lis ed companies and dk is he owne ship ac ion o ex e nal sha eholde kon he lis ed companies. The ma ix ep esen a ion o he alue o ex e nal sha eholde s po olio is de e mined om Equa ion (10). Pex “d (10) When he c oss-sha eholding ne wo k is shown as a combina ion o ex e nal sha eholde s and lis ed companies, he ollowing ma ix A is ep esen ed: A“¨ ˝ WÑ 0 dÑ 0˛ ‚(11) whe e Ñ 0 is a ze o ma ix and A is a (m+n) ˆ (m+n) ma ix. Acco dingly, he in eg a ed ma ke alue o each company could be calcula ed as a combina ion o in insic alues and amoun s o in e connec edness as ollows: Vin “AVin `V (12) In . J. Financial S ud. 2016,4, 13 7 o 17 whe e V and Vin show in insic ma ke alues and he in eg a ed ma ke alues o companies, espec i ely. The solu ion o Equa ion (13) is as ollows: Vin “ pI´Aq´1V (13) The Vin , as he in eg a ed alues o companies could be applied as a sys emic isk index ha shows impo ance in e ms o sys emic isk. I in he Equa ion (6), he cH“Vin and V“c0 , i is ob ious ha he Vin is a kind o Hubbell cen ali y measu e. Howe e , he in eg a ed owne ship ma ix can also be ep esen ed as ollows: Ain “Ain A`A (14) whe e Ain conside s all di ec and indi ec pa hs o calcula e amoun s o in eg a ed owne ship. Fo example, he in eg a ed pe cen o company j ha is owned by company iis ep esen ed as ollows: ain ij “aij `ÿ k aikain kj (15) The solu ion o he Equa ion (14) is as ollows: Ain “ pI´Aq´1A (16) Acco ding o he abo e-men ioned equa ions, i is possible o each a dual ela ion o each company as he Equa ion (17), whe e each side o he equa ion shows he in eg a ed po olio alues o companies. Ain V“AVin “Pin (17) The Pin , which is a complemen o he po olio alue indices ( Pi ), can been used as ano he sys emic isk index. 4. Resul s and Analysis 4.1. Di ec C oss-Sha eholding Ne wo k Using ne wo k heo y and Pajek so wa e, a ep esen a ion o c oss-sha eholding ne wo k o companies lis ed on he Teh an S ock Exchange is shown in Figu e 3. As i is ob ious, nodes a e ep esen a i e o lis ed companies and ex e nal sha eholde s. Acco ding o he sec o classi ica ion o TSE, he conside ed companies a e ca ego ized acco ding o 27 di e en sec o s dis inguished by colo s. As can be seen in Figu e 3, he e a e some companies and ex e nal sha eholde s ha ha e mo e owne ship ela ions wi h he o he agen s. The size o node (Figu e 3) ep esen s he amoun o a node’s deg ee, whe e he “go e nmen ”, “Tamin o ganiza ion” and he “na ional pension und” anked highes wi h 40, 31 and 27 a cs, espec i ely. Al hough i is no possible o ecognize he gene al s uc u e o he TSE c oss-sha eholding ne wo k, he exis ence o a bunch o nodes wi h many in-deg ee and ou -deg ee as well as some nodes wi h only inwa d a cs and ou wa d a cs shows ha he exis ence o a bow- ie s uc u e in he TSE is likely. Table 2 ep esen s a summa y o he s a is ical analysis o he di ec c oss-sha eholding ne wo k and he ele an sys emic isk indices in he company and sec o le el. As shown in Table 2, he e was no signi ican di e ence be ween he op i e companies o deg ee, in-deg ee, sj and Pi in he di ec c oss-sha eholding ne wo k; howe e , he di e ence o hese indices is mo e e iden in he lowe o de companies. Excep o di e ence in he o de o companies, he only a ia ion was acco ding o exis ence o “Adala Sha es” in he op i e companies based on sj and Pi , which indica es ha e en hough he “Adala Sha es” has only a ew ela ions wi h he o he companies, i has a signi ican sha e in i s own companies. Sec o analysis shows some di e ences be ween he ou indices. In . J. Financial S ud. 2016,4, 13 8 o 17 In . J. Financial S ud. 2016, 4, 13 8 o 18 Figu e 3. Rep esen a ion o c oss-sha eholding ne wo k o Teh an S ock Exchange. The size o nodes is ela ed o he deg ee o each node and he deg ee is he sum o in-deg ee and ou -deg ee o nodes. Table 2. Compa ison o di e en sys emic isk indices o di ec ne wo k in he Teh an S ock Exchange. Deg ee In-Deg ee Ou -Deg ee Min. Deg ee 0 Min. In-Deg ee 0 Min. Ou -Deg ee 0 Max. Deg ee 40 Max. In-Deg ee 40 Max. Ou -Deg ee 8 A g. Deg ee 5.88 A g. In-Deg ee 2.94 A g. Ou -Deg ee 2.94 Top Fi e Nodes Deg ee Top Fi e Nodes In-Deg ee Top Fi e Nodes Ou -Deg ee Go . 40 Go . 40 MADN 8 Tamin O g 31 Tamin O g 31 MS022 8 SA3A1 27 SA3A1 24 PK061 8 Oilcopen 23 Oilcopen 23 GD021 7 NIKX1 22 NIKX1 17 FO041 7 Top Fi e Sec o A g Deg ee Top Fi e Sec o A g In-Deg ee Top Fi e Sec o A g Ou -Deg ee Indus ial Holdings 17.25 Indus ial Holdings 13.5 Me al Mining 5.6 Ex Sha eholde s 13 Ex Sha eholde s 13 Insu ance Co. 5.25 In es men Co. 10.125 In es men Co. 6.75 D illing 5 Me al Mining 8 Me al Mining 2.4 Cemen 4.4 Insu ance Co. 6 Bank 2 Basic Me als 4   (Million Rials) Min. 0 Min. 0 Min. 0 Max. 20.1 Max. 2.36 Max. 424,000,000 A g. 1.63 A g. 0.51 A g. 17,600,000 Top Fi e Nodes  Top Fi e Nodes Top Fi e Nodes  (Million Rials) Go . 20.1 TORZ1 2.36 Adala Sha e 424,000,000 Adala Sha e 13.33 ARFZ1 2.03 Go . 354,000,000 Tamin O g 11.66 BDYZ1 2.02 Tamin O g 218,000,000 Oilcopen 9.75 SMAZ1 2.01 Oilcopen 142,000,000 SA3A1 8.78 KS121 2 SA3A1 115,000,000 Top Fi e Sec o A g.  Top Fi e Sec o A g  Top Fi e Sec o A g  (Million Rials) Ex . Sha eholde s 6.48 Insu ance Co. 1.29 Ex Sha eholde s 109,000,000 Figu e 3. Rep esen a ion o c oss-sha eholding ne wo k o Teh an S ock Exchange. The size o nodes is ela ed o he deg ee o each node and he deg ee is he sum o in-deg ee and ou -deg ee o nodes. Table 2. Compa ison o di e en sys emic isk indices o di ec ne wo k in he Teh an S ock Exchange. Deg ee In-Deg ee Ou -Deg ee Min. Deg ee 0 Min. In-Deg ee 0 Min. Ou -Deg ee 0 Max. Deg ee 40 Max. In-Deg ee 40 Max. Ou -Deg ee 8 A g. Deg ee 5.88 A g. In-Deg ee 2.94 A g. Ou -Deg ee 2.94 Top Fi e Nodes Deg ee Top Fi e Nodes In-Deg ee Top Fi e Nodes Ou -Deg ee Go . 40 Go . 40 MADN 8 Tamin O g 31 Tamin O g 31 MS022 8 SA3A1 27 SA3A1 24 PK061 8 Oilcopen 23 Oilcopen 23 GD021 7 NIKX1 22 NIKX1 17 FO041 7 Top Fi e Sec o A g Deg ee Top Fi e Sec o A g In-Deg ee Top Fi e Sec o A g Ou -Deg ee Indus ial Holdings 17.25 Indus ial Holdings 13.5 Me al Mining 5.6 Ex Sha eholde s 13 Ex Sha eholde s 13 Insu ance Co. 5.25 In es men Co. 10.125 In es men Co. 6.75 D illing 5 Me al Mining 8 Me al Mining 2.4 Cemen 4.4 Insu ance Co. 6 Bank 2 Basic Me als 4 sjHiPi(Million Rials) Min. 0 Min. 0 Min. 0 Max. 20.1 Max. 2.36 Max. 424,000,000 A g. 1.63 A g. 0.51 A g. 17,600,000 Top Fi e Nodes sjTop Fi e Nodes HiTop Fi e Nodes Pi(Million Rials) Go . 20.1 TORZ1 2.36 Adala Sha e 424,000,000 Adala Sha e 13.33 ARFZ1 2.03 Go . 354,000,000 Tamin O g 11.66 BDYZ1 2.02 Tamin O g 218,000,000 Oilcopen 9.75 SMAZ1 2.01 Oilcopen 142,000,000 SA3A1 8.78 KS121 2 SA3A1 115,000,000 Top Fi e Sec o A g. sjTop Fi e Sec o A g HiTop Fi e Sec o A g Pi (Million Rials) Ex . Sha eholde s 6.48 Insu ance Co. 1.29 Ex Sha eholde s 109,000,000 Ind. Holdings 6.22 Me al Mining 1.16 Ind. Holdings 72,100,000 In es men Co. 3.42 T anspo a ion 1.09 Telecommunica ion 54,400,000 Me al Mining 1.76 Elec ici y 1 Pe ochemical 13,100,000 Ca P oduc ion 1.65 Basic Me als 0.78 Me al Mining 11,100,000 In . J. Financial S ud. 2016,4, 13 15 o 17 Company Symbol Sec o Company Symbol Sec o 44 Omid In . Mng OIMC1 Ind. Holdings 110 Pa sian IPAR1 Insu ance Co. 45 Ba gh Mapna Co. BMAZ1 Elec ici y Supp. 111 Mella Insu . BMEX1 Insu ance co. 46 Behshah TS3A1 Food P od. 112 Pasa gad Insu . BIPZ1 Insu ance co. 47 Behshah Ind. SBEX1 Food P od. 113 Shahed In . SAHX1 Housing 48 Tamin Da oo DTIP1 Medicine 114 Housing In . MSKX1 Housing 49 Albo z In . ALBZ1 Medicine 115 In . Cons . BSTE1 Housing 50 Sobhan Pha m. DSOZ1 Medicine 116 In . Se ices INFX1 Compu e & IT 51 Os ah Pha m. DOSE1 Medicine 117 Pa sian E-Comme ce PRSX1 Compu e & IT 52 Razak Lab. DRZX1 Medicine 118 MAPNA MAPX1 Compu e & IT 53 Zah a i Pha . DZAX1 Medicine 119 Go e men Go Ex . Sha eholde 54 Da oupakhsh DARO1 Medicine 120 Tamin O ganiza ion Tamin O g Ex . Sha eholde 55 Tamin Pe o PT3A1 Pe ochemical 121 A my Pen. Fund Sa a Ex . Sha eholde 56 I .In .Pe . IPTZ1 Pe ochemical 122 Adala Sha es Adasha e Ex . Sha eholde 57 KhalijFa s PL081 Pe ochemical 123 Mos aza in Founda ion Mos ound Ex . Sha eholde 58 Pa sianOil PA021 Pe ochemical 124 Oil Comp Pen. und Oilcopen Ex . Sha eholde 59 Shazand Pe . PARK1 Pe ochemical 125 Tadbi In . Tadbi in Ex . Sha eholde 60 Pa dis Pe . PRDZ1 Pe ochemical 126 Fa hangian In . Fa hin Ex . Sha eholde 61 Paksho PASH1 Pe ochemical 127 Village Pen. Fund Villpen Ex . Sha eholde 62 Kho asan Pe o PSKZ1 Pe ochemical 128 Foolad Pen. Fund Fooladpen Ex . Sha eholde 63 Kha k Pe . PK061 Pe ochemical 129 Meh In . Co. Meh in Ex . Sha eholde 64 I an Chem. Ind. SSIN1 Pe ochemical 130 Banks Pen Find Bankpen Ex . Sha eholde 65 Jam Pe . PJMZ1 Pe ochemical 131 Meh eayande In . Meh ayande Ex . Sha eholde 66 Ke man PK3A1 Pe ochemical Appendix B. Appendix B: His og am and Cumula i e Dis ibu ion o Sys emic Risk Measu es In . J. Financial S ud. 2016, 4, 13 16 o 18 Appendix B: His og am and Cumula i e Dis ibu ion o Sys emic Risk Measu es         Appendix C: Complemen o Cumula i e Dis ibu ion o Sys emic Risk in Log-Log Scale (  ) (  ) (    ) (   ) 0.00% 20.00% 40.00% 60.00% 80.00% 100.00% 120.00% 0 10 20 30 40 50 60 70 F equency Cumula i e % 1-CDF(%) 0.00% 20.00% 40.00% 60.00% 80.00% 100.00% 120.00% 0 5 10 15 20 25 30 35 F equency Cumula i e % 1-CDF (%) 0.00% 20.00% 40.00% 60.00% 80.00% 100.00% 120.00% 0 10 20 30 40 50 60 70 F equency Cumula i e % 1-CDF % 0.00% 20.00% 40.00% 60.00% 80.00% 100.00% 120.00% 0 5 10 15 20 25 30 35 40 45 F equency Cumula i e % 1-CDF % -2.5 -2 -1.5 -1 -0.5 0 24.82 27.69 28.35 28.90 29.15 29.43 29.99 30.24 30.31 30.58 30.79 31.11 31.38 31.49 31.89 32.16 32.32 32.65 33.49 33.86 -6 -5 -4 -3 -2 -1 0 28.97 30.18 30.71 31.06 31.31 31.52 31.74 31.96 32.11 32.45 32.59 32.95 33.28 33.98 -6 -5 -4 -3 -2 -1 0 0.0 0.7 1.1 1.4 1.6 1.8 1.9 2.1 2.2 2.4 2.5 2.6 2.6 2.8 2.9 3.0 3.2 3.6 -6 -5 -4 -3 -2 -1 0 -1.39 -0.69 -0.29 0.00 0.22 0.41 0.56 0.69 0.81 0.92 =0.033 (0.000) R-Squa e=0.887 p- alue=0.000 =0.083 (0.000) R-Squa e=0.92 p - alue=0.000 =0.194 (0.000) R-Squa e=0.963 p- alue=0.000 =0.48 (0.000) R-Squa e=0.944 p- alue=0.000 In . J. Financial S ud. 2016,4, 13 16 o 17 Appendix C. Appendix C: Complemen o Cumula i e Dis ibu ion o Sys emic Risk in Log-Log Scale In . J. Financial S ud. 2016, 4, 13 16 o 18 Appendix B: His og am and Cumula i e Dis ibu ion o Sys emic Risk Measu es         Appendix C: Complemen o Cumula i e Dis ibu ion o Sys emic Risk in Log-Log Scale (  ) (  ) (    ) (   ) 0.00% 20.00% 40.00% 60.00% 80.00% 100.00% 120.00% 0 10 20 30 40 50 60 70 F equency Cumula i e % 1-CDF(%) 0.00% 20.00% 40.00% 60.00% 80.00% 100.00% 120.00% 0 5 10 15 20 25 30 35 F equency Cumula i e % 1-CDF (%) 0.00% 20.00% 40.00% 60.00% 80.00% 100.00% 120.00% 0 10 20 30 40 50 60 70 F equency Cumula i e % 1-CDF % 0.00% 20.00% 40.00% 60.00% 80.00% 100.00% 120.00% 0 5 10 15 20 25 30 35 40 45 F equency Cumula i e % 1-CDF % -2.5 -2 -1.5 -1 -0.5 0 24.82 27.69 28.35 28.90 29.15 29.43 29.99 30.24 30.31 30.58 30.79 31.11 31.38 31.49 31.89 32.16 32.32 32.65 33.49 33.86 -6 -5 -4 -3 -2 -1 0 28.97 30.18 30.71 31.06 31.31 31.52 31.74 31.96 32.11 32.45 32.59 32.95 33.28 33.98 -6 -5 -4 -3 -2 -1 0 0.0 0.7 1.1 1.4 1.6 1.8 1.9 2.1 2.2 2.4 2.5 2.6 2.6 2.8 2.9 3.0 3.2 3.6 -6 -5 -4 -3 -2 -1 0 -1.39 -0.69 -0.29 0.00 0.22 0.41 0.56 0.69 0.81 0.92 =0.033 (0.000) R-Squa e=0.887 p- alue=0.000 =0.083 (0.000) R-Squa e=0.92 p - alue=0.000 =0.194 (0.000) R-Squa e=0.963 p- alue=0.000 =0.48 (0.000) R-Squa e=0.944 p- alue=0.000 Re e ences 1. 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