Research Article Partially Overlapping Ownership and Contagion in Financial Networks

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1 Hndaw Complexty Volume 17, Artcle ID , 16 pages Research Artcle Partally Overlappng Ownershp and Contagon n Fnancal Networks Mcah Pollak 1 and Yuanyng Guan 2 1 School of Busness and Economcs, Indana Unversty Northwest, 34 Broadway, Gary, IN 4648, USA 2 Department of Mathematcs and Actuaral Scence, Indana Unversty Northwest, 34 Broadway, Gary, IN 4648, USA Correspondence should be addressed to Mcah Pollak; mpollak@un.edu Receved 27 July 17; Accepted 8 October 17; Publshed 6 November 17 Academc Edtor: Ahmet Sensoy Copyrght 17 Mcah Pollak and Yuanyng Guan. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Usng hstorcal bankng data for the Unted States from the years to 15 we characterze the probablty and extent of a fnancal contagon usng a calbrated network model of heterogeneous nterbank exposures. Both the probablty and the average extent of a contagon begn to rse n 7 pror to the US fnancal crss. Includng a common asset n the model ncreases both the probablty and extent of contagon, especally durng the years of the fnancal crss. Based on rsng nsttutonal ownershp n the bankng ndustry, we ntroduce a partally overlappng ownershp asset that devalues endogenously. The addton of ths asset ncreases the extent of a fnancal contagon. Our results show that trends n captal buffers and the dstrbuton and type of assets have a sgnfcant effect on the predctons of fnancal network contagon models and that the rsng trend n ownershp of banks by banks amplfes shocks to the fnancal system. 1. Introducton Followng the 8-9 US and global fnancal crss, there has been a growng nterest n the role that the network structure of banks and the types and dstrbuton of ther assets have n determnng the probablty and extent of a potental fnancal contagon. Chnazz and Fagolo [1] provde a concse survey of recent lterature n ths area. Two lmtatons common n much of ths research relate to the complexty of the network and asset structure and the avalablty of data. Frst, the assets and network structures observed n real world fnancal systems tend to be more complcated than assumed n typcal fnancal network models. Second, detaled data on the structure of real world fnancal networks s often extremely lmted, especally for a major naton lke the Unted States. Emprcal papers tend to focus on natons other than the US and often employ data from a sngle or small number of years. In ths paper we mprove on these two lmtatons. Frst, n addton to the standard assumpton of drect exposures (nterbank loans) and ndependent external assets, we add an external common asset (smlar to [2]) and we ntroduce the concept of a partally overlappng ownershp asset, or an asset held by some banks that has value endogenously determned by falures wthn the bankng ndustry (.e., such as an nvestment n an ndexed portfolo of stock for the bankng ndustry). In addton to these assets, we consder a core-perphery network structure, whch s ncreasngly becomng the preferred representaton of the bankng ndustry. We contrast our results wth those from a scale-free network, whch s another common network structure. Fnally, we use hstorcal fnancal nformaton for depostory nsttutons n the Unted States, for the years pror to and followng the fnancal crss, to calbrate key fnancal network characterstcs such as the sze of the network, total assets, and captal buffers for ndvdual banks as well as to characterze and dstrbute the partally overlappng ownershp asset. These steps allow us to nvestgate how the predctons of a fnancal contagon model change based on observed trends n the bankng ndustry. Equty ownershp of banks by banks s becomng ncreasngly common n the Unted States. Between and 15 the number of banks wth ownershp n other banks doubled ntheuntedstates.overthesametme,thetotalvalueof

2 2 Complexty ths ownershp across the bankng system has been growng, especally followng the fnancal crss. Between 11 and 15 the total value of ownershp of banks by banks ncreased by211%(overthssametmeperodthevalueofthes&p5 rose by 55%). Ths growth n bank ownershp by banks has the potental to add a new and sgnfcant feedback channel that amplfes shocks. For example, a negatve shock to the bankng ndustry reduces the value of ownershp n banks, whch, n turn, further negatvely affects balance sheets of banks, further reducng the value of bank ownershp and addng addtonal stress to the system. To nvestgate ths trend n nsttutonal ownershp, we ntroduce a partally overlappng ownershpasset,tocapturethersenequtyownershpof banks by banks, and demonstrate how the addton of ths asset amplfes the effects of fnancal contagons. Our approach s related to Sensoy s paper [3], n whch the author nvestgates the effects of frm sze and ownershp structure on commonalty n lqudty usng unque ownershp data for Turkey. The author analyzes both nsttutonal and ndvdual ownershp and fnds that, n addton to the amount of ownershp and frm sze, the characterstcs of nvestors, and the mcrostructure of the market are mportant n determnng systematc lqudty. Whle related, n our case we focus on the role of nsttutonal ownershp n bankng networks n order to study fnancal contagon. Our results show that for the US bankng system between and15theaveragecaptalbuffersheldbybanks ncreased overall and the probablty of contagon declned, wth some sgnfcant exceptons n 7 and 8. Based on a network of drect exposures only, n 7 and 8, pror to the US fnancal crss, the probablty of contagon resultng from the falure of a core bank rose from 19.7% to 24.6% (a 24% ncrease) and the average extent of a potental contagon more than trpled. The addton of a common asset greatly ncreases the probablty of a contagon n all years, brngng t close to 1% from a random core bank falure between 8 and 1 and close to 1% n 9 from a random perphery bank falure. A common asset also more than doubles the average extent of contagon from a core bank falure. Whle addng a partally overlappng ownershp asset ncreases the probablty of contagon from a core bank falure by only about 5%, t doubles the extent of a contagon on average across all years. These results are smlar for both core-perphery and scale-free network structures. Before focusng on the data and model we begn wth a bref revew of lterature related to ths topc. Caccol et al. [4] study the Australan nterbank network and show that the nterplay of multple channels of exposures s a major contrbutor to systemc rsk and contagon. They conduct stress tests to analyze contagon through drect exposures, overlappng portfolos, and the combnaton of thesetwochannels.theyconcludethatcontagondueto counterpartyrskcanbestronglyamplfedbytheaddton of a common portfolo. In another paper, Caccol et al. [2] extend the analyss of contagon caused by overlappng portfolos to a scenaro wth multple assets. They characterze how the average level of dversfcaton n bank portfolos, theratoofthenumberofbankstothenumberofassets, and the leverage attaned by banks all affect system stablty wthrespecttoanntalshockonasngleassetorbank.by conductng analytcal smulatons on a stylzed network (a random network wth Posson degree dstrbutons for both banks and assets), they estmate the regon of parameter space where global cascades occur. Poledna et al. [5] analyze a four-layered nterbank network and show that a tradtonal measure of systemc rsk based only on a sngle layer of deposts and loans, whch s common n most studes, dramatcally underestmates (by as much as 9%) the rsk nherent n a fnancal system. Whle earler lterature provdes emprcal evdence of real world fnancal networks followng a scale-free network structure [6 8], more recent papers argue that evdence from nterbank markets suggest that a core-perphery network structure better represents nterbank exposures for varous countes, such as for the nterbank systems of Netherlands [9], Italy [1], Germany [11], UK [12], Brazl [13], and Mexco [14]. A core-perphery network classfes nodes nto two dfferent types: core nodes and perphery nodes. Core nodes are well connected to each other and to perphery nodes, whle perphery nodes have connectons only wth core nodes. Several papers have nvestgated the core-perphery structure n the context of fnancal networks, both from a theoretcal and dynamc perspectve. Lux [15] develops a smple dynamc model of an nterbank market where banks ntally choose tradng partners randomly due to dosyncratc lqudty shocks. He shows that wth heterogenety n balance sheets and a smple renforcement learnng scheme governng potental tradng counterparts, the system quckly converges to a core-perphery network structure. In the paper of van der Lej et al. [16], they propose a smple model of the overnght nterbank lendng market n whch banks compete for ntermedaton benefts. They fnd that a complete coreperphery network s not stable whle an ncomplete coreperphery network may be stable wth heterogenety between banks and nequalty n payoffs correspondng to nequalty n szes. In ther paper, the banks are ex antehomogeneous and they show that heterogenety plays a key role n formng a stable core-perphery network. Slva et al. [13] brdges the emprcal and theoretcal lterature by developng a method of measurng how close a fnancal network s to a perfect core-perphery structure and then applyng ths measure to the Brazlan nterbank market. Galeott et al. [17] study how fnancal lnkages n networks affect ndvdual payoffs and rsk to the system by constructng an ownershp matrx and explorng the effects ofchangestothsnetwork.theyfndthattheeffectsofntegraton (strengthenng of current lnks) and dversfcaton (spreadng of lnks to more neghbors) depend crucally on the topology of the network. Specfcally, they show that, n a core-perphery network, core banks take more rsk than perphery banks, whch s consstent wth our data, results, and other lteratures [9]. They also show that when the network s homogenous, ndvduals take on too lttle rsk relatve to the socally optmal portfolos, whle when the network s homogenous, they take on too much rsk. In addton to the emprcal lterature, whch focuses prmarly on non-us fnancal networks, and the theoretcal lterature, there have been several studes on the structure of

3 Complexty % All others 15.8% C 4.8% WFC 11.3% BAC 12.6% JPM % All others 17.2% C 11.7% WFC.7% BAC 18.9% JPM % All others 13.7% C 14.2% WFC 17.% BAC 18.7% JPM Bllons of year 15 dollars ($) Fgure 1: Combned total assets and dstrbuton for years, 9, and 15 among the largest four banks: JPMorgan & Chase Co. (JPM), Bank of Amerca Corporaton (BAC), Wells Fargo & Company (WFC), and Ctgroup (C). the US fnancal market followng the fnancal crss. McCord and Prescott [18] dentfy sgnfcant changes to the structure of the US bankng system after the fnancal crss. They fnd a sharp declne n the number of banks, mostly among smaller banks. They provde evdence that ths decrease s not due to more banks extng the market, but rather to a declne n the number of new entres. They also fnd trends n asset concentraton whch are consstent wth our data. Whle we focusonbankswth$1bllonormorentotalassets,kowalk et al. [19] studed mergers of US banks wth $1 bllon or less n total assets followng the fnancal crss. They also fnd adeclnenthenumberofbanksandarguethatsncethe end of the recesson, voluntary mergers have been the man reason for ths declne. They fnd that larger banks tend to acqure smaller banks n order to more quckly expand loan operatons and gan access to cash and deposts to support future loan growth. The rest of ths paper s organzed n the followng way. Secton 2 presents data from to 15 on the US bankng system and dscusses key trends n the number of banks, total assets, captal buffers, and equty ownershp. Secton 3 explans the underlyng model, network algorthm, and calbraton to the observed data. Secton 4 presents the results based on dfferent combnatons of the three assets types: drect exposures, a common asset and a partally overlappng ownershp asset. Secton 5 provdes a bref summary of our results and some concludng remarks. 2. Data Ourdataarebasedonbalancesheetandownershpnformaton from the FactSet Fundamentals database for the years to 15 or approxmately seven years before and after the 8/9 US fnancal crss. We restrct our attenton to frms (henceforth banks ) classfed as depostory credt ntermedaton nsttutons (NAICS code 5221) and headquarteredntheuntedstatesandwthtotalassetsexceedng $1 bllon n year 15 dollars (adjusted for nflaton usng the Bureau of Labor Statstcs Consumer Prce Index for all urban consumers, US cty average seres for all tems). Between and 15 there are a total of 696 banks that satsfy ths crtera wth an average of 361 actve banks n a gven year. These data represent an unbalanced panel n whch nsttutons may enter and ext the sample based on mergers, acqustons, and bankruptces. A large proporton of nsttutons, 145 out of the average 361 or 4% n a gven year, are actve and fled fnancal statements across all 16 years.whleacompleteanalyssofthestructureoftheus bankng system s beyond the scope of ths paper (for a more detaled characterzaton of some of changes to the US bankng system over ths tme see [18, 19]), we focus on major trendsnthenumberofbanks,szeanddstrbutonoftotal assets, changes n captal buffers, and trends n ownershp of other banks. Table 1 presents the summary statstcs for number of banks, total assets adjusted for nflaton and captal buffers. The total number of banks n the data peaks n 3 at 39 declnes to a low of 333 n 12 consstent wth [18, 19] and thenreturnsby15toroughlythesamelevelasn. Between and 15 mean total assets rose consstently, growng 63.3% (3.3% on average annually). Over the same tme perod, medan total assets generally declned untl 9 before rsng agan for a total growth between and 15 of 17.1% (1.1% on average annually). The large dfference between mean and medan s drven prmarly by three outlers. Between and 15 the assets of Wells Fargo & Company grew by $1.43 trllon (+385%), JPMorgan Chase & Co. by $1.39 trllon (+142%), and Bank of Amerca Corporaton by $1.27 trllon (+144%). The growth of these banks s prmarly the result of consoldaton n the years leadng up to and durng the 7/8 fnancal crss (among the most sgnfcant were the 4 mergers of Bank One Corporaton wth JPMorgan & Chase Co. and FleetBoston Fnancal Corporaton wth Bank of Amerca Corporaton and then n 8 the purchase of Washngton MutualoutofsezurebyJPMorgan&ChaseCo.andthe acquston of Wachova Corp. by Wells Fargo). Ths can also be seen n Fgure 1, whch shows the combned total assets n the data, hghlghtng the largest four banks, for the years of, 9, and 15. These data show extreme consoldaton andgrowthamongthefourlargestbanksuptoanddurng the US fnancal crss. In the four largest banks owned 44.4% (or $3.46 bllon) of all assets n the system, n 9 ths grew to 68.5% (or $8.12 bllon), and n year 15 they owned 63.6% (or $8.6 bllon) Trends n Captal Buffers. Table 1 also shows summary statstcs by year for captal buffers. Consstent wth Ga

4 4 Complexty Table 1: Summary statstcs (observatons, total assets, and captal buffers) Obs Total assets (bllons of year 15 dollars) Mean Medan Top 5% Top 25% Bottom 25% Bottom 5% Captal buffer (%, net-worth/total assets) Mean Medan Top 5% Top 25% Bottom 25% Bottom 5%

5 Complexty Table 2: Number of banks wth ownershp and average ownershp (as a percent of total assets) by bank sze n 15. Total Assets Range Banks (#) Owner-banks (#) Mean share of assets $1 bllon % $1 1 bllon % $5 1 bllon % Under $5 bllon % Mean Medan Upper/lower 5% Fgure 2: Captal buffer levels by year (as a percent of total assets). Total ownershp value (bllons of year 15 dollars) ($) Number of banks wth bank ownershp and Kapada s specfcaton n [] we defne an nsttuton s captal buffer as ther net-worth (total assets mnus total labltes) dvded by total assets. Captal buffers show a trend smlar to mean total assets and are generally ncreasng, from a mean of 8.9% n to 11.1% n 15. The overall rse n captal buffers s lkely n part due to the tghtenng of the Basel Accords durng ths tme (see Appendx A for dscusson on the role of the Basel Accords over ths tme perod). One sgnfcant excepton to the trend occurs n 7 and 8 just before the US fnancal crss and can be seen n Fgure 2. Mean captal buffers fell from 1.3% n 7 to 9.7% n 8 and for the hghest 5% percentle, captal buffers declned from 27.2% to 22.8%. For banks n the lowest 5% percentle of captal buffers these buffers declned from 6.% n 7 to 5.1% n 8 to 4.1% n 9. The sharp declne durng 7 and 8 for these lowest 5% of banks wll play an mportant n our results, as the defnton of a fnancal contagon s when at least 5% of banks n the system fal. Based on ths defnton, n 9 an exogenous loss of just over 4% of total assets at these banks (gnorng any subsequent losses due to drect exposures and other common and overlappng assets) would alone be suffcent to trgger a contagon. Followng 9, mean and medan captal buffers begntoclmbagan.fgure2showsthstrendncaptal buffers over tme. There s a weak correlaton between total assets and captal buffers that s negatve from to 6 and postve from 7 to 15 ths correlaton s statstcally sgnfcant at the 5% level and negatve n years and 1 and postve n years 11, 12, and 15, based on an OLS regresson of captal buffer = α+βlog(total Assets).In year a 1% ncrease n total assets results n a.25- percentage pont decrease n captal buffer whle n year 15 a 1% ncrease n total assets results n a.27-percentage pontncreasencaptalbuffer.thssuggeststhatprorto the US fnancal crss larger banks tended to carry relatvely smaller captal buffers and after the crses relatvely larger buffers. Total ownershp value Banks wth ownershp Fgure 3: Trends n bank ownershp n other banks. Total value of ownershp n bllons of year 15 dollars (left axs) and number of owner-banks (rght axs) by year Trends n Ownershp and Owner-Banks. In addton to balance sheet nformaton, we collect equty ownershp data foreachbankfromthefactsetfundamentalsdatabase.we then match ths ownershp data to the banks n our data to construct a matrx of bank equty ownershp n other banks. To smplfy dscusson, banks wth ownershp n other banks wll be referred to as owner-banks. Between and 15 there was a marked ncrease n both the number of ownerbanksandthetotalvalueofthatownershp.intheyear, there were 25 banks wth ownershp n other banks that totaled $31.3 mllon n value. By year 15 there were 44 banks wth ownershp that totaled $38.9 mllon. The ncrease n the total value of ownershp s especally pronounced snce the end of the fnancal crss. Between 11 and 15 the total system value of bank ownershp by ownerbanks ncreased from $12.5 bllon to $38.9 bllon (a 211% ncrease). Fgure 3 shows the total value of ownershp wthn thesystemandthenumberofowner-banks.whlesomeof thsncreasecanbeattrbutedtothersenthestockmarket (the S&P 5 rose 55.1% over the same perod), ths only explans a small porton of the ncrease n ownershp value and does not explan the sustaned ncrease n the number of owner-banks (between 8 and 15 the total value of bank ownershp ncreased by 299% whle the S&P 5 rose only 44.7%). Whle ownershp n other banks s more lkely among larger banks, t s not exclusve to them. Table 2 shows for year 15 the number of owner-banks and the proporton of owner-bank total assets n ownershp, broken down by varous szes of total assets. Of the 14 largest banks (those

6 6 Complexty wth over $1 bllon n total assets) fve were owner-banks. Forthesefve,themeanvalueofthsownershpwasequalto.47% of ther total assets. For smaller banks, those wth total assets between $5 and $1 bllon and those wth assets below $5 bllon, the value of ther ownershp, as a proporton of total assets, was much hgher (1.16% and 2.24%, resp.). Thus, nvestng n ownershp of other banks s not lmted to large banks, and smaller banks leverage a larger porton of ther assets n bank ownershp. For most owner-banks ths ownershp s relatvely dversfed. For example, n 15 owner-banks had, on average, ownershp n over 27 other banks (medan 12.5), wth larger banks generally havng ownershp n a larger number of other banks. Owner-banks also tend to lmt ther exposure to ownershp n a sngle bank, wth the average owner-bank n 15 nvestng at most 4% of ther ownershp assets n a sngle bank. In 15, the average ownershp amount per bank was 12% of ther total ownershp value. There are smlartes n the banks that owner-banks choose to nvest n. In 15, more than half of total system ownershp value was n just two banks, (33% of the total system ownershp value was n Bank of Amerca and 17.7% n Wells Fargo). In addton, outofthe44owner-banks,42haveashareofownershpn Wells Fargo, 4 n JPMorgan, and 33 n US Bank. As a result, the falure of any of these hghly owned banks would have a sgnfcant effect on the value of ownershp assets across the system. Overall, ownershp of banks by banks stll represents a relatvely small porton of total assets. In 15, for the ten banks wth the largest ownershp postons (as a percent of total assets) represented on average only 3.6% of ther total assetsandacrossallowner-banksn15theaveragevalueof ownershp was only 1.26% of total assets. Whle nvestment n bank ownershp s a relatvely small asset class, t s growng. Furthermore, there may be sgnfcant ownershp through prvate equty and debt that we cannot easly observe. The growth of bank ownershp by banks s potentally concernng gven that t has the potental to feedback nto and amplfy any shocks to the bankng ndustry. 3. Model and Methods The model we use n our analyss s based on Ga and Kapada s model from [], wth the addton of heterogeneous lnk-weghts as n [21] and a common asset as n [4]. We also ntroduce a new type of asset, a partally overlappng ownershp asset, whch s held by only some banks and has value affected endogenously by bank falures. We begn by gong over the detals of the model and assets before dscussng the generaton of the fnancal network, calbraton to US bankng data, and the method behnd the fnancal contagon smulatons Banks and the Fnancal Envronment. Assume that there are N fnancal nsttutons (banks) n a network and each bank s represented by a node n the network. Let A TOT and L TOT bethetotalassetsandlabltesforbank. Each bank holds nterbank assets, A I, as well as some combnaton of acommonasset,a C, a partally overlappng ownershp asset, A O, and other external assets, AM.Eachbankhastwotypesof labltes, customer deposts, D, and nterbank labltes, L I. If a bank becomes nsolvent at any pont (.e., A TOT <L TOT )t mmedately fals and defaults on ts nterbank labltes. The value of any nterbank assets at a faled bank becomes zero. The fnancal network s defned by a network of nterbank lendng and borrowng. Let A R 2 + be N by N matrx that represents the network of nterbank exposures. Each element a j represents the amount of assets bank loans to bank j. Byconventon,a =. Therefore A I = N j=1 a j s the total nterbank lendng by bank and L I = N j=1 a j s the total nterbank borrowng of bank. MatrxA represents a weghted drected network wth heterogeneous lnk-weghts. Incomng lnks of a node reflect the nterbank assets of the node. Outgong lnks of a node represent the nterbank labltes of that node. Let w j =a j /A I be the proporton of nterbank assets belongng to bank held by bank j. In addton to nterbank assets, each bank nvests ts remanng assets across a combnaton of a common asset, a partally overlappng ownershp asset and external assets. The rsks and exposures assocated wth each of these assets are dfferent. The common asset reflects an asset held by all banks wth a common value determned exogenously (e.g., mortgage-backed securtes). The common asset decreases n value only due to an exogenous shock, however, any decrease n value wll affect the balance sheets of all banks. The partally overlappng ownershp asset reflects equty ownershp n a portfolo of banks wthn the system and may declne n value followng the falure of any bank ncluded n the portfolo. Unlke the common asset, nether do all banks hold the partally overlappng ownershp asset, nor all banks hold necessarly ncluded n the portfolo. The partally overlappng ownershp asset can be nterpreted as a sngle nvestment fund ndexed to select bankng stocks. The network of ownershp generated by ths asset can be thought of as a second, completely dstnct, structure from the network of nterbank exposures. In a gven year, we defne the share of ths portfolo nvested n bank j as P j.ifbankj fals, then the value of ts stock falls to zero and the partally overlappng ownershp asset wll devalue to reflect ths or suffer a 1 P j loss. Ths loss, n turn, s reflected n the value of thepartallyoverlappngownershpassetheldbysomebanks. All banks that choose to nvest n the partally overlappng ownershp asset purchase the same portfolo. Fnally, any remanng assets not nvested n nterbank assets, the common asset or the partally overlappng ownershp asset, are nvested n a rsk-free external asset. Ths external asset represents other assets outsde the scope of the model. Our man nterest s n determnng the probablty and extent of a fnancal contagon stemmng from a sngle shock to the fnancal system. All scenaros we study begn by mposng the falure of a random bank on the fnancal network. In scenaros n whch the common asset s present, we smultaneously mpose an exogenous shock of sze φ to ths common asset n addton to the ntal falure of a random bank. Followng ths ntal falure (and possbly

7 Complexty 7 thedeclnenvalueofthecommonasset),theshockmay then be transmtted across the fnancal network through two channels. Frst, the falure and default of the ntal bank wll elmnate any nterbank assets of other banks held at ths bank. Second, f the faled bank s contaned n the portfolo of the partally overlappng ownershp asset then the value of ths asset wll declne for banks holdng ths asset. These two effects combne to elmnate or reduce the value of assets for some banks across the network, whch n turn may cause addtonal banks to fal. To formalze the condton under whch bank fals followng the ntal falure of another bank (and possble smultaneous devalung of the common asset) let K =A TOT L TOT = (A I +A M +A C +A O ) (L +D ) represent the captal buffer of bank. Followng an ntal falure of bank j,thesolvencycondtonforbanks L +D <A M and bank wll fal f +(1 w j )A I +(1 φ)ac +(1 P j )A O w j > K φac P j A O A I. (2) Should bank fal, they then default on ther nterbank labltes and the partally overlappng ownershp asset may devalue. Ths process contnues (wth the solvency condton updated to reflect pror falures) untl no further banks fal. Consstent wth [], f over 5% of all banks n the system fal then we defne ths as a fnancal contagon. The extent of such a fnancal contagon s defned as the proporton of banks that fal or the expected proporton of banks that fal condtoned on at least 5% of banks falng Calbraton and Smulaton Methods. For each year we begn by calbratng the number of banks, as well as the total assets and total labltes for ndvdual banks to the observed data. The prvate nature of nterbank loans n the Unted States means that data on the structure of the nterbank asset networkslmted.asaresult,foreachyearwesmulate1 core-perphery network structures to represent the network of nterbank assets, usng an algorthm calbrated to match the typcal characterstcs of the nterbank market of other natons (whle 1 smulatons may seem low, because assets and labltes of ndvdual banks are drawn drectly from the data and because the network generaton algorthm s based n part on these data, t s suffcent to allow our results to converge. Increasng smulatons to 1 per year does not have a sgnfcant effect on our results). See Secton 3.3 for addtonal detals on the algorthm and parameters used to generate the network. Whle the network of nterbank exposures s based on underlyng balance sheet data, the partally overlappng asset s generated from separate data on the equty ownershp of banks by banks that s not drectly related to nterbank exposures (for a bref dscusson of when exposures from ownershp may be equvalent to exposures from nterbank (1) lendng and when t s approprate to treat these as two dstnct assets see Secton 5). To generate the partally overlappng ownershp asset, n each year we construct a sngle asset that reflects a representatve portfolo of all observed ownershp n banks. The share of ths portfolo nvested n bank j s equal to the total value of equty ownershp by all banks n bank j as a fracton of the total value of equty ownershp by all banks n all other banks, whch can be expressed as P j = N =1 ( P j / N P k=1 k ),where P j stheobserveddollarvalueofownershpheldbybank n bank j. After constructng the ownershp asset portfolo P={P 1,P 2,...,P N },wethendstrbutetheamountofassets; we observe each bank nvestng n equty ownershp to ths portfolo. For owner-banks or those banks whch we observe havng ownershpassetsnthedata,wemultplytheamountofthe observed ownershp, or A O = N P =1 j /A TOT, by a factor of 15,cappedat4%ofthertotalassets.Thefactorof15was chosen to ncrease the average ownershp by owner-banks to be equal to approxmately % of total assets or half of theproportonweassgntothecommonasset(theobserved ownershp by owner-banks as a percent of ther total assets n 15 was equal to 1.26% on average. See Secton 2.2 for more detals). Banks that we observe havng no assets n equty ownershp nvest zero n the ownershp asset. We use ths form of a representatve portfolo of ownershp, rather than the observed ownershp network structure, and amplfy the value of ths ownershp, to reflect the possblty of other unobserved sources of ownershp, such as prvate equty and debt ownershp, as well as to more clearly capture the effect of ths asset class on fnancal contagons. In the results presented n Secton 4 we consder four scenaros. In all four scenaros each bank holds % of ther total assets n nterbank assets (A I =.2A TOT )andthe dfferences are n how the remanng 8% of total assets are nvested. The four scenaros are as follows: (1) Drect exposures only: n Secton 4.1 we frst consder the smplest case wth only drect exposures from nterbank assets (A C =A O =). (2) Common asset: next, n Secton 4.2, n addton to drect exposures, we add a common asset equal to 4% of total assets to all banks whch deprecates by 1% smultaneously wth the ntal shock (A C =.4A TOT,A O =). (3) Partally overlappng ownershp asset: nsecton4.3 nstead of the common asset, we add to drect exposures the partally overlappng asset, whch s held by some banks and may reflect up to 4% of ther total assets (A C =,.4A TOT A O ). (4) Common and partally overlappng ownershp assets: fnally, n Secton 4.4 we consder the combned effects of these assets by combnng the partally overlappng asset wth the common asset (A C =.4A TOT,.4A TOT A O ). In each scenaro, any assets not nvested n drect exposures (nterbank assets), the common asset or the partally

8 8 Complexty Table 3: Comparng the CP model for the Dutch nterbank market to Germany (Crag and Von Peter, 14 [11]), Italy (Frcke and Lux, 14 [1]), UK (Langfeld et al., 14 [12]), and Mexco (Sols-Montes, 13 [14]). Country Netherlands Germany Italy UK Mexco The number of banks Average number of core banks ±15 ±45 ±3 16 ±16 Average core sze ±15% ±2.5% ±25% 9.1% ±35% Error frequency, as % of lnks 29% 12% 42% 47% 25% overlappng ownershp asset, are nvested n other external assets (A M ) Network Structure. To generate the network of nterbank exposures we follow the defnton of a core-perphery network n n t Veld and van Lelyveld [9] and van der Lej et al. [16]. Defnton 1. A network s a perfect core-perphery network f there exsts a set of core nodes K Nand perphery nodes P=N\Csuch that g j =1, g j =1, g j =, g j =, g j =1, g h =1,, j K,, j P, K, j P, K, h P, where g j represents a drected lnk between nodes and j. Ths defnes a perfect drected core-perphery network. If we form a network of nterbank assets as a perfect coreperphery structure, usng the matrx A to represent the lnks between dfferent banks, we wll get (3) CC CP A=[ PC PP ]=[1 CP ]. (4) PC A number of recent papers [9 14] fnd that real fnancal networks exhbt a smlar structure to a perfect coreperphery networks, only wth the addton of lnk errors. These errors are mssng lnks between core nodes as well as extra lnks between perphery nodes. Table 3 represents data from n t Veld and van Lelyveld [9] characterzng the number of core banks and error frequency of bankng systems n dfferent countres. To smulate a network of nterbank exposures to reflect the US bankng system we frst we assume that core banks are those banks wth the hghest total assets n the system. Wefxthenumberofcorebanksat25,whchstheaverage number of core banks across the countres n Table 3. After constructng the perfect core-perphery bank network, we calbrate error lnks. The error frequency of our network s set at 31%, whch s the average of error frequences of dfferent countres n Table 3. In addton to smulatng the network of nterbank exposures as a core-perphery network structure, we also consder a scale-free structure as n [21] and fnd results very smlar to that of the core-perphery structure. The algorthm to generate the scale-free network as well as the results can be found n Appendx B. 4. Results The results n ths secton are organzed nto subsectons n whch we sequentally ntroduce each asset type. Secton 4.1 begns wth only drect exposures through nterbank lendng. Secton 4.2 adds a common asset owned by all banks and Secton 4.3 adds a partally overlappng ownershp asset to drect exposures. Fnally, Secton 4.4 adds the combnaton of a common asset and a partally overlappng ownershp asset to drect exposures. When a common asset s present we present contagon results from both the ntal falure of a random core node and the ntal falure of a random perphery node. In the absence of a common asset, contagons do not result from the ntal falure of any perphery nodes and only the results from an ntal core node falure are gven (n the cases where the ntal falure of perphery nodes does not result n a contagon the overall contagon probablty (resultng from the ntal falure of any node) can be determned by multplyng the probablty of contagon from a core node falure by the proporton of nodes that are core nodes, or 25/N, whch s equal, on average, to.69). In addton, we restrct our focus to the core-perphery network structure, as t s more representatve of the US bankng system. The results for the scale-free network structure are extremely smlar and for the complete results for the scalefree network structure see Appendx B Drect Exposures Only. As a baselne case we begn by consderng the possblty of a fnancal contagon arsng only through drect exposures to counterpartes through nterbank lendng. The approach was frst developed n [] and then expanded n [21, 22]. Network characterstcs such as the number of nodes, the total assets of each node, and the captalbufferofeachnodearecalbratedfromdataonthe US fnancal system and nterbank exposures are generated wth a core-perphery network structure (see Sectons 3.2 and 3.3 for further detals). Our man results are that, wth thecalbrateddatafortheusfnancalsystemandonly

9 Complexty (a) Contagon probablty (core node) (b) Contagon extent (core node) Fgure 4: Contagons wth Drect Exposures Only. Probablty (a) and extent (b) of a fnancal contagon for a core-perphery network wth drect exposures only, condtoned on a random core bank falng. drect exposures, fnancal contagons orgnate only from the falure of core banks and that there was a sgnfcant ncrease n the probablty and extent of a fnancal contagon begnnng n 7, pror to the US fnancal crss. Wth only drect exposures, fnancal contagons begn only wth the ntal falure of a core node or the falure of one of the largest 25 banks. Due to the concentraton of lendng at core nodes and ther greater connectedness, the falure ofasnglecorenodewllhavegreaterandwder-reachng mplcatons than the falure of a perphery node. Fgure 4 shows the smulated probablty of contagon (a) and average extent of contagon (b) based on the ntal falure of a random core node. Overall between and 15 the probablty of a fnancal contagon resultng from the falure of a random core node generally declned (from 25.4% to 16.4%). A sgnfcant excepton was n 7, when the probablty rose from19.7%to21.4%,and8,whentroseto24.6%,pror to the begnnng of the US fnancal crss (the US fnancal crses are often marked as begnnng wth the bankruptcy of LehmanBrothersnSeptember8.Theeffectn7salso present and more pronounced n the scale-free model, where theprobabltyofcontagonrosefrom17.6%to19.5%n7). Ths rse n the probablty of contagon s accompaned by a more than trplng n the average contagon extent n 7, from 11.8% to 37.3%. Vewed ex post, these results can be nterpreted as a sgnfcant warnng sgn n 7 of thecomngfnancalcrss.theresultsfromthescale-free network structure (see Appendx B) are smlar. Whle the rse n probablty and extent of a fnancal contagon n 7 and 8 are consstent wth the fnancal crss, the result that an ntal falure of a perphery node never leads to a fnancal contagon suggests that drect exposures through nterbank lendng alone may not provde a rch enough set of exposures to explan more than a small class of extreme fnancal contagons. To address ths, n the next sectons we consder addtonal types of assets and exposures Common Asset. In ths secton we add a common asset, n a manner consstent wth [4], to the exstng drect exposures. Ths common asset provdes an addtonal exposure commontoeverybank,whchncreasesthechanceofa fnancal contagon. Consstent wth the results of [4], we fnd that the addton of a common asset to drect exposures sgnfcantly ncreases both the probablty and extent of fnancal contagons. Unlke the prevous case n Secton 4.1 wth drect exposures only, durng some years the probablty of contagon from the ntal falure of perphery node s postve. Fgure 5 shows the smulated probablty of contagon from the falure of a random core node (a) and a random perphery node (b) as well as the average extent of contagon for each type of ntal falure ((c) and (d), resp.). Included neachfgurenredforcomparsonaretheresultsfrom Secton 4.1 for drect exposures only. Wth the addton of a common asset, the probablty thatthefalureofacorenodetrggersacontagonsboth extremely hgh and hghly volatle (Fgure 5(a)), especally begnnng n year 8 and through the years of the US fnancal crss. The probablty of contagon drops from 82%nto47%n6beforejumpngto99%n 8andtheneventuallydroppngto39%n15.Whle ths probablty may appear extremely hgh, recall that t s condtoned on the falure of a core node or one of the 25 largest banks. The overall probablty of contagon resultng from the ntal falure of any random node (core or perphery) for year s just 5.9%.

10 1 Complexty Common asset Drect exposures only (a) Contagon probablty (core node) Common asset Drect exposures only (b) Contagon probablty (perphery node) Common asset Drect exposures only (c) Contagon extent (core node) Common asset (d) Contagon extent (perphery node) Fgure 5: Contagons wth a Common Asset. Probablty of a contagon ((a) and (b)) and average extent of contagon ((c) and (d)) for a coreperphery network wth a common asset and drect exposures (black sold crcles) and drect exposures only (red empty crcles) condtoned on a random core bank falng ((a) and (c)) and condtoned on a random perphery bank falng ((b) and (d)). Contagon extent shown only when average contagon frequency s greater than 1%. The addton of a common asset also greatly ncreases the average extent of a fnancal contagon stemmng from the falure of a core node for most years (Fgure 5(c)). Between years to 6 and 12 to 15 the contagon extent from a core node falure s 15 to 25 percentage ponts hgher wth the addton of a common asset. Durng 7 to 11, the years leadng up to and durng the fnancal crss, the average extent s the same wth and wthout a common asset, due to a large ncrease n the number of smaller contagons, whch s dscussed further below. In addton to sgnfcantly ncreasng the probablty and extent of a contagon resultng from the falure of a core node, the addton of a common asset ntroduces the possblty of a contagon from a perphery node, although only durng the mmedate years of the fnancal crss. Pror to 8 and after 1 the probablty of a contagon from a perphery node s zero (Fgure 5(b)). In 8 ths probablty rses to 17.2%, then to 98.7% n 9 before fallng to 5.9% n 1. Combned wth the results for the core node falures, n 9 the probablty of a contagon resultng from the falure of any

11 Complexty 11 sngle node (core or perphery) was 98.9%, or a near certanty. However, ths near certanty of a contagon s offset by an extremely low contagon extent for perphery node falures (Fgure 5(d)). In 9, the average extent of a contagon resultng from the falure perphery node was only 6.3%. The extremely hgh probablty of contagon n 9 for both core and perphery nodes s beng prmarly drven by the declne n captal buffers we observe n the data among banks wth the lowest captal buffers. In 9 the 5% of banks wth the lowest captal buffers (see Fgure 2 and Table 1) had captal buffers at or below 4.1%. Snce the exogenous deprecaton of the common asset can be nterpreted as a four-percentage pont loss n total assets ths deprecaton alone results n the falure of a sgnfcant number of nodes. Durng most years the ntal deprecaton of the common asset results n between and 3 falures drectly (.e., before thefalureofarandomnodeandthespreadoftheshock through drect exposures). However, n 9 due to low captal buffers ths deprecaton of the common asset drectly results n the falure of 17 nodes (4.59% of the network). Combned wth our mposed falure of one node, only one addtonal node needs to fal through nterbank exposures n ths year to meet the defnton of a contagon. As a result of the declne n captal buffers a fnancal contagon s almost guaranteed wth a common asset n 9; however, the extent of such a contagon may be relatvely small. Ths result also explans the convergence of contagon extent we see for core node falures n 8, 9, and 1. Whle for most years the average contagon extent from a core node falure wth a common asset s 15 to 25 percentage ponts hgher than wth drect exposures only, between 8 and 1 ths extent drops and converges to that of the drect exposures (see Fgure 5(c)). Ths convergence of contagon extent s an artfact of the ntroducton of a large number of small contagons durng these years (see Fgure 5(a)). As n [21], the separaton of contagons nto mld contagons (when 5 3% of nodes fal) and moderate contagons (when over 3% of nodes fal) would contnue to show sgnfcantly hgher contagon extents from core node falures wth the common asset compared to drect exposures only. The addton of a common asset results n a sgnfcantly larger contagon probablty and extent for all years. However, one potental crtcsm of ncludng a common asset s that t has an extremely homogenzng effect on exposures. Because the asset s truly common to all banks, any deprecaton n the asset wll necessarly weaken the balance sheet of all banks. In addton, wthout a separate market and prcng structure for ths common asset any deprecaton must be external and arbtrary. In the next secton we ntroduce a partally overlappng asset or a common asset whch only some banks chose to hold (.e., t s not common to all banks). In addton, we nterpret ths asset as a fxed ownershp portfolo of other banks, whch allows us to endogenously devalue ths asset followng the falure of a bank based on ts weght n the portfolo of ownershp Partally Overlappng Ownershp Asset. In ths secton we return to the case of drect exposures (.e., wthout a common asset) and then add a partally overlappng ownershp asset. Unlke the common asset n Secton 4.2, ths asset wll be held by only some banks (hence partally overlappng ). We also nterpret ths asset as reflectng a fxed portfolo of ownershp n other banks. For a dscusson of ths ownershp observed n the data see Secton 2.2 and of how we mplement ths ownershp n the model see Secton 3.2. One of the advantages of nterpretng ths asset as ownershp s that t provdes a natural way to devalue the asset followng the falure of a bank.wththefalureofabank,thevalueofownershpnthat bank falls to zero and the ownershp asset devalues accordng toshareofthatbankheldntheportfolo. Wth the addton of ths partally overlappng ownershp asset, our man results are a small ncrease n the probablty of a contagon and a large ncrease n the extent of a contagon stemmng from the falure of a core node. As n the case of drect exposures only, contagons do not arse from the falure of a perphery node and fgures for an ntal perphery falure are omtted. Fgure 6 shows the smulated probablty of contagon (a) and average extent of contagon (b) based on thefalureofarandomcorenode.includedneachfguren red for comparson are the results from Secton 4.1 for drect exposures only. The addton of the partally overlappng ownershp asset has a small but postve effect on the probablty of a contagon stemmng from the falure of a core node (Fgure 6(a)). Unlke the case of the common asset n Secton 4.2, the partal ownershp asset nether consstently deprecates for all falures nor unformly transmts the ths shock to all other banks.forcorenodesthatareweghtedlttleornotatalln the portfolo, the effect of ther falure wll be mnmal. For example, n 12, whle 91% of the ownershp portfolo was n core banks, three-quarters of ths ownershp was concentrated n just four of these banks: Bank of Amerca (33%), Wells Fargo (18%), SunTrust Banks (1%), and Ctbank (8%). The effect of a falure of any core bank outsde of these four banks would be smlar to the case wth drect exposures only (.e., wthout the partally overlappng ownershp asset). Because perphery banks are mnmally represented n the ownershp asset, the probablty of a contagon from the falure of a perphery bank remans at zero, as n Secton 4.1. Whle the effect of a partally overlappng ownershp asset on the probablty of a fnancal contagon may be small, the effect on the average extent of a contagon s relatvely large. Ths s because the ownershp portfolo s heavly concentrated n a small number of core node banks. For example, n 12 the largest shares of ownershp by the portfolo are n Bank of Amerca (33%), Wells Fargo (18%), SunTrust Banks (1%), and Ctbank (8%). In the event that one of these banks fals, a contagon due solely to drect exposures s already hghly lkely and the partally overlappng ownershp asset serves as a channel to transmt the contagon to owner-banks who may not have otherwse been affected solely through drect (and ndrect) exposures. Ultmately, the partally overlappng ownershp asset has the effect of ncreasng the mportance of a select few core banks and worsenng the extent of a contagon should one of these banks fal. In the next secton we combne ths partally overlappng ownershp asset wth the common asset.

12 12 Complexty Ownershp asset Drect exposures only (a) Contagon probablty (core node) Ownershp asset Drect exposures only (b) Contagon extent (core node) Fgure 6: Contagons wth a Partally Overlappng Ownershp Asset. Probablty of a contagon (a) and average extent of contagon (b) for a core-perphery network wth a partally overlappng ownershp asset and drect exposures (black sold crcles) and drect exposures only (red empty crcles) condtoned on a random core bank falng Common and Partally Overlappng Ownershp Assets. In ths secton we add both the partally overlappng ownershp and common asset to drect exposures. Fgure 7 shows the smulated probablty of contagon (a) and average extent of contagon (c) based on the falure of a random core node as well as the probablty of contagon (b) and average extent of contagon (d) based on the falure of a random perphery node. Included n each fgure n red for comparson are the results from Secton 4.2 for drect exposures wth a common asset. The addton of the partally overlappng ownershp asset wth the common asset and drect exposures has smlar effects to those outlned n Secton 4.3. There s a small ncrease n the probablty of contagon from core nodes n the years pror to and after the fnancal crss (Fgure 7(a)) and there s a substantal ncrease n the extent of contagon (Fgure 7(b)). There s no sgnfcant effect on contagons resultng from perphery node falures. The lack of a sgnfcant nteracton between the common asset and partally overlappng ownershp asset s not surprsng gven that the frst dsproportonately affects contagons from perphery node falures whle the second dsproportonately affects contagons from core node falures. Ths s prmarly the case due to the observed makeup of the ownershp asset portfolo whch heavly reflects core node banks (e.g., n 15 over 92% of the ownershp portfolo was n core banks). Whle the partally overlappng ownershp asset s held by many perphery banks, falures from perphery banks havealmted,fany,effectonthevalueofthsasset.if, hypothetcally, the ownershp asset was constructed prmarly of perphery banks the results would be qute dfferent. 5. Concluson Usng hstorcal data on the US bankng ndustry from to 15 we calbrate a core-perphery fnancal network model and characterze the probablty and average extent of a fnancal contagon over tme from the random falure of a core or perphery node. The fnancal network s composed of a network of heterogeneous drect exposures, calbrated to data where possble. In addton to drect exposures we consder two other types of assets. The frst s a common asset as n [2] and for the second we ntroduce a partally overlappng ownershp asset to capture the growng equty ownershp of banks by banks that we observe n the data. Ths partally overlappng ownershp asset s held by a subset of banks and devalues endogenously based on the falure of banks wthn the system. The results show that wth drect exposures only, n 7 and 8, pror to the US fnancal crss, the probablty of contagon resultng from the falure of a core bank rose from 19.7% to 24.6% (a 24% ncrease) and the average extent of a potental contagon more than trpled. The addton of a common asset greatly ncreases the probablty of a contagon n all years, brngng t close to 1% for core banks between 8and1andcloseto1%n9forperphery banks, prmarly due to the declne n captal buffers among certan banks n 9. Addng a common asset also more than doubles the average extent of contagon from a core bank falure. The addton of a partally overlappng ownershp asset only slghtly ncreases the probablty of contagon from a core bank falure (by about 5%) but doubles the extent of a contagon on average across all years. The combnaton of

13 Complexty Common & ownershp assets Common asset only (a) Contagon probablty (core node) Common & ownershp assets Common asset only (b) Contagon probablty (perphery node) Common & ownershp assets Common asset only (c) Contagon extent (core node) Common & ownershp assets Common asset only (d) Contagon extent (perphery node) Fgure 7: Contagons wth a Partally Overlappng Ownershp and Common Asset.Probablty of acontagon ((a)and (b))and average extent of contagon ((c) and (d)) for a core-perphery network wth common and partally overlappng ownershp assets and drect exposures (black sold crcles) and common asset and drect exposures only (red empty crcles) condtoned on a random core bank falng ((a) and (c)) and condtoned on a perphery bank falng ((b) and (d)). Contagon extent shown only when average contagon frequency s greater than 1%. both a common asset and a partally overlappng ownershp asset ncreases both the probablty and average extent of a contagon, but there does not appear to be any sgnfcant nteracton or amplfcaton. Our data show that many key fnancal network characterstcs, such as number of nodes, total assets, and captal buffers, change sgnfcantly over tme and these changes have a large effect on the probablty and average extent of a potental fnancal contagon. As a result, researchers may want to consder data from more than a sngle or small number of years when characterzng a fnancal network for contagon analyss. A comparson between results from core-perphery networks and scalefree networks (see Appendx B) shows that the dfferences n probablty and average extent of a fnancal contagon across the two network structures are smaller than the

14 14 Complexty Captal buffer Basel ter 1 captal rato Fgure 8: Comparson of medan captal buffers and medan Basel ter 1 captal ratos. dfferences across years due to changes n fnancal network characterstcs. Whle we nterpret equty ownershp as nvestment n a sngle portfolo asset, such as an ndex fund of bankng stock, a complete network of ownershp exposures may also be nterpreted as observatonally equvalent to a second network,smlartothatofnterbanklendng.insuchacase, ownershp can be captured by approprately amplfyng the network of nterbank exposures. However, there are scenaros n whch equty ownershp s more approprately treated as a separate asset from debt. For example, ownershp may nclude the ownershp of other frms outsde the network of banks (such as nsurance or other fnancal frms) or may reflect more complex types of ownershp. In other stuatons, the order of lqudaton n a bankruptcy may be relevant, wth nterbank debt clams beng satsfed pror to, or more wholly than, nterbank equty clams. Fnally, by ts nature, debt typcally requres mutual consent to the transacton, whle equty does not. There may be a scenaro n whch one bank delberately chooses not to borrow from another bank but s unable to prevent ths other bank from buyng ther equty. The dstncton between debt and equty become more relevant to the addton of strategc behavor. In the future we would lke to expand ths research to look at the effect of partally overlappng ownershp assets wth other characterstcs. For example, f the portfolo was more heavly weghted toward perphery banks then ths may ncrease the probablty of contagon more sgnfcantly and amplfy shocks further once a contagon begns. We would also lke to expand the ownershp portfolo to nclude other sourcesofownershp,suchasdebtownershp,aswellasthe ownershp of other frms n related fnancal areas, such as nondepostory banks and nsurance frms. Another way to expand ths research s to add a dynamc component to the model. For example, f banks n the fnancal network respond to a frst round shock strategcally, ther response may affect the contagon results. Reference [2] dscusses a scenaro where banks try to rebalance ther portfolotoreachthertargetleveragelevelandtheyconclude that ths rebalancng of portfolo destablzes the system. In contrast, [21] shows that f banks rebalance ther portfolo by reducng nvestment n potentally weakened banks, ths rebalancng stablzes the system to future shocks. Appendx A. Basel Accords The defnton of captal buffers used n ths paper and common n the lterature, begnnng wth [] and earler, s of net-worth (total labltes mnus total assets) dvded by total assets. Ths defnton s the most lteral defnton of solvency but does not take nto consderaton the rskness of dfferent asset classes. Another nterpretaton of captal buffers that does consder the rskness of assets comes from the Basel Accords or the nternatonal recommendatons on bankng regulatons. Basel I was developed n 1988 by the Basel Commttee on Bankng Supervson (BCBS) and was adopted n law by the Untes States and other G-1 countres n Snce then the Basel Accord has gone through revsons (Basel II n 4 and Basel III n 1). In the Unted States Basel II became effectve on Aprl 1, 8, but wth some rules ntally delayed or waved due the 7/8 fnancal crses. The Basel III recommendatons were approved on July 9, 13, n the Unted States. Under the Basel Accords the regulatory captal buffer s measured n part by the ter 1 captal rato or the rato of a bank s core equty captal to ts total rsk-weghted assets. Whle the ter 1 captal rato more accurately measures solvency rskness than actual solvency, t may be useful to compare ths measure wth the more tradtonal defnton of captal buffers. Fgure 8 compares the medan captal buffer (as defned n ths paper) wth the ter 1 captal rato under the Basel Accord. In all years the ter 1 captal rato s sgnfcantly hgher than the captal buffer, reflectng that t accounts for the rskness of relatvely safe or rsk-free assets on the balance sheet. Overall the trend we see n the data for ter 1 captal ratos s smlar to the trend n captal buffers, other than a sharper declne durng the years leadng up to the US fnancal crss. Another reason we do not use the ter 1 captal rato n our model s that partcpaton n the Basel Accord was voluntary pror to 8 (n % of our sample reported ter 1 captal buffers under the Basel Accords,

15 Complexty Common and ownershp assets Common asset Drect exposures only Common and ownershp assets Common asset Drect exposures only (a) Contagon probablty (core node) (b) Contagon probablty (perphery node) Common and ownershp assets Common asset Drect exposures only Common and ownershp assets Common asset (c) Contagon extent (core node) (d) Contagon extent (perphery node) Fgure 9: Results for the Scale-Free Network Structure. Probablty of a contagon ((a) and (b)) and average extent of contagon ((c) and (d)) for a scale-free network wth common and partally overlappng ownershp assets and drect exposures (black sold crcles), common asset and drect exposures only (red empty crcles), and drect exposures only (blue sold trangles), condtoned on a random core bank falng ((a) and (c)) and condtoned on a perphery bank falng ((b) and (d)). Contagon extent shown only when average contagon frequency s greater than 1%.

16 16 Complexty compared to only 62.3% n 3 and 6.6% n ), and the ruleswerefurtherchangedn1. B. Results from Scale-Free Network Structure For the purpose of comparng our results for the coreperphery network structure wth the commonly used scale-free structure we use Barabás-Albert model to construct a scale-free network, whch reflects the preferental attachment characterstc of scale-free networks. The algorthm we use here s a drected verson of Barabás-Albert model from Bollobás et al. [23]. Assume that α, β, γ, δ n,andδ out are nonnegatve real numbers such that α+β+γ = 1.Startngwthanntal graph G = G(t ),weformg(t + 1) from G(t) accordng to the followng steps: (1) Wth probablty α, add a new vertex V, together wth an edge from V to an exstng vertex w, where w s chosen accordng to d n +δ n (P(w = w ) = (d n (w )+δ n )/(t + δ n n(t))),whered n (w ) represents the ncomng degree for node and n(t) represents the number of vertces n the graph at tme t. (2) Wth probablty β, add an edge from an exstng vertex V to an exstng vertex w, wherev and w are chosen ndependently, V s chosen accordng to d out + δ out,andws chosen accordng to d n +δ n. (3) Wth probablty γ, add a new vertex w,togetherwth an edge from an exstng vertex V to w, wherev s chosen accordng to d out +δ out. To smplfy comparsons wth the core-perphery network structure, we use the termnology core node to refer to the 25 largest nodes (by total assets) n the scale-free model and perphery node to refer to other nodes. Fgure 9 presents for the scale-free network that are comparable to those of the core-perphery structure presented n Sectons 4.1, 4.2, and 4.4. Both network structures yeld extremely smlar results. Conflcts of Interest The authors declare that there are no conflcts of nterest regardng the publcaton of ths artcle. References [1] M. Chnazz and G. Fagolo, Systemc rsk, contagon, and fnancal networks: a survey, SSRN Electronc Journal, p.57, 15. [2] F. Caccol, M. Shrestha, C. Moore, and J. D. Farmer, Stablty analyss of fnancal contagon due to overlappng portfolos, Journal of Bankng & Fnance,vol.46,no.1,pp ,14. [3] A. Sensoy, Frm sze, ownershp structure, and systematc lqudty rsk: the case of an emergng market, Journal of Fnancal Stablty,vol.31,pp.62 8,17. [4] F. Caccol, J. D. Farmer, N. Fot, and D. Rockmore, Overlappng portfolos, contagon, and fnancal stablty, Journal of Economc Dynamcs & Control,vol.51,pp.5 63,15. [5] S.Poledna,J.L.Molna-Borboa,S.Martínez-Jaramllo, M. van der Lej, and S. Thurner, The mult-layer network nature of systemc rsk and ts mplcatons for the costs of fnancal crses, Journal of Fnancal Stablty, vol., pp. 7 81, 15. [6]M.Boss,H.Elsnger,M.Summer,andS.Thurner, Network topology of the nterbank market, Quanttatve Fnance, vol. 4, no. 6, pp , 4. [7] K. Soramäk, M. L. Bech, J. Arnold, R. J. Glass, and W. E. Beyeler, The topology of nterbank payment flows, Physca A: Statstcal Mechancs and ts Applcatons,vol.379,no.1,pp , 7. [8]R.Cont,A.Moussa,andE.B.Santos, Networkstructure and systemc rsk n bankng systems, Handbook of Systemc Rsk, pp , 13, Network structure and systemc rsk n bankng systems, Handbook of Systemc Rsk. [9] D. n t Veld and I. van Lelyveld, Fndng the core: network structure n nterbank markets, Journal of Bankng & Fnance, vol. 49, pp. 27 4, 14. [1] D. Frcke and T. Lux, Core perphery structure n the overnght money market: evdence from the e-mid tradng platform, Computatonal Economcs, vol. 45, no. 3, pp , 14. [11] B. Crag and G. Von Peter, Interbank terng and money center banks, Journal of Fnancal Intermedaton, vol.23,no.3,pp , 14. [12]S.Langfeld,Z.Lu,andT.Ota, MappngtheUKnterbank system, Journal of Bankng & Fnance, vol. 45, no. 1, pp , 14. [13]T.C.Slva,S.R.S.deSouza,andB.M.Tabak, Network structure analyss of the Brazlan nterbank market, Emergng Markets Revew,vol.26,pp ,16. [14] M. P. Sols-Montes, The structure of the Mexcan nterbank market, Banco de Méxco,13. [15] T. Lux, Emergence of a core-perphery structure n a smple dynamc model of the nterbank market, Journal of Economc Dynamcs & Control,vol.52,pp.A11 A23,15. [16] M. van der Lej, D. n t Veld, and C. Hommes, The formaton of a core perphery structure n heterogeneous fnancal networks, De Nederlandsche Bank Workng Paper No [17] A. Galeott, C. Ghglno, and S. Goyal, Fnancal lnkages, portfolo choce and systemc rsk, 16, CAM [18] R. McCord and E. S. Prescott, The fnancal crss, the collapse of bank entry, and changes n the sze dstrbuton of banks, FRB Rchmond Economc Quarterly,vol.1,pp.23 5,14. [19] M. Kowalk, T. Davg, C. S. Morrs, and K. Regehr, Bank consoldaton and merger actvty followng the crss, Economc Revew-Federal Reserve Bank of Kansas Cty,pp.31 49,15. [] P. Ga and S. Kapada, Contagon n fnancal networks, Proceedngs of the Royal Socety A Mathematcal, Physcal and Engneerng Scences,vol.466,no.21,pp ,1. [21] Y. Guan and M. 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