The Network of Counterparty Risk: Analysing Correlations in OTC Derivatives

Transcription

1 RESEARCH ARTICLE The Network of Counterparty Rsk: Analysng Correlatons n OTC Dervatves Vahan Nanumyan, Antonos Garas, Frank Schwetzer* Char of Systems Desgn, ETH Zurch, Zurch, Swtzerland * fschwetzer@ethz.ch Abstract OPEN ACCESS Ctaton: Nanumyan V, Garas A, Schwetzer F (2015) The Network of Counterparty Rsk: Analysng Correlatons n OTC Dervatves. PLoS ONE 10(9): e do: /journal.pone Edtor: Irene Sendña-Nadal, Unversdad Rey Juan Carlos, SPAIN Receved: June 16, 2015 Accepted: August 5, 2015 Publshed: September 3, 2015 Copyrght: 2015 Nanumyan et al. Ths s an open access artcle dstrbuted under the terms of the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal author and source are credted. Data Avalablty Statement: Data were obtaned from a thrd party, and are freely avalable from the followng URL: dervatves-quarterly-report.html. Fundng: Project CR12I1_ OTC Dervatves and Systemc Rsk n Fnancal Networks fnanced by the Swss Natonal Scence Foundaton. Competng Interests: The authors have declared that no competng nterests exst. Counterparty rsk denotes the rsk that a party defaults n a blateral contract. Ths rsk not only depends on the two partes nvolved, but also on the rsk from varous other contracts each of these partes holds. In rather nformal markets, such as the OTC (over-the-counter) dervatve market, nsttutons only report ther aggregated quarterly rsk exposure, but no detals about ther counterpartes. Hence, lttle s known about the dversfcaton of counterparty rsk. In ths paper, we reconstruct the weghted and tme-dependent network of counterparty rsk n the OTC dervatves market of the Unted States between 1998 and To proxy unknown blateral exposures, we frst study the co-occurrence patterns of nsttutons based on ther quarterly actvty and rankng n the offcal report. The network obtaned ths way s further analysed by a weghted k-core decomposton, to reveal a core-perphery structure. Ths allows us to compare the actvty-based rankng wth a topology-based rankng, to dentfy the most mportant nsttutons and ther mutual dependences. We also analyse correlatons n these actvtes, to show strong smlartes n the behavor of the core nsttutons. Our analyss clearly demonstrates the clusterng of counterparty rsk n a small set of about a dozen US banks. Ths not only ncreases the default rsk of the central nsttutons, but also the default rsk of perpheral nsttutons whch have contracts wth the central ones. Hence, all nsttutons ndrectly have to bear (part of) the counterparty rsk of all others, whch needs to be better reflected n the prce of OTC dervatves. Introducton After the fnancal crss of 2008 the systemc rsk resultng from OTC (over-the-counter) dervatves has become an mportant topc of publc debate and scentfc research. Dfferent from exchange-traded dervatves, OTC dervatves are traded on non-regulated markets whch have grown both n sze and mportance durng the last decade. In December 2008 the Bank for Internatonal Settlements (BIS) reported (see Semannual OTC dervatves statstcs at that total notonal amount on outstandng OTC dervatves grew up from 370,178 bn USD n June 2006 to 683,725 bn USD n June 2008,.e., t almost doubled n sze n only two years. PLOS ONE DOI: /journal.pone September 3, / 23

2 A partcular worryng feature of ths development results from the ncreasng concentraton of the counterparty rsk of OTC dervatves n the hands of only a few nsttutons. Ths trend has not changed after the fnancal crss of 2008, on the contrary the concentraton ncreased. Takng the example of the US alone, n the 4th quarter of 1998 contracts totalng 331 bn USD were sgned by 422 commercal banks and trust companes whch where not lsted n the top 25 nsttutons dealng wth OTC dervatves. Ths numbers have to be compared aganst the contracts totalng 32,668 bn USD (.e., a hundred tmes more) sgned by only the top 25 nsttutons n the OTC dervatves market. Comparng ths to the tme after the fnancal crss, the dfference became much bgger. In the 1st quarter of 2012 the 25 top ranked US nsttutons held contracts totalng 227,486 bn USD (.e., almost ten tmes more than n 1998), whereas all other nsttutons held contracts totalng only 496 bn USD (whch s almost comparable to what was held n 1998). Hence, we observe an extreme concentraton of dervatves market where the share of dervatve contracts held by the top 25 nsttutons was almost 99% n 1998 and ncreased to more than 99.5% n Ths ncreasng concentraton may also ncrease the vulnerablty of the nsttutons nvolved and can lead to cascades n case of default. Untl now, no concentraton of exposure aganst a partcular counterparty s reported by banks. The Basel Commttee on Bankng Supervson referred to ths ssue for the frst tme only n ts report of March 2013 [1]. In our paper, we address the problem n a twofold way. Based on a dataset of the 25 most actve players n the U.S. dervatve market, over a perod of 14 years, we reconstruct the network of counterparty rsk. We show that ths rsk generates an almost fully connected network of nterdependence among these players, however t s skewly dstrbuted,.e., most of the counterparty rsk s concentrated n only 10 mayor nsttutons. Ths mples two problems: n a fully connected network, t becomes much more dffcult to hedge the rsk of default, because every player s a counterparty of any other. Ths may ncrease the rsk of default cascades, whch can be amplfed by the partcularly actve counterpartes. Addtonally, the concentraton of counterparty rsk n a few nsttutons may exacerbate the problem of contagon and fnancal dstress n the whole network f those nsttutons become dstressed. OTC Dervatves The role of dervatves Dervatves are fnancal nstruments,.e., they are tradable assets. Importantly, they have no ntrnsc value. Instead, ther value depends on, or s derved at least partly from, the value of other enttes, denoted as the underlyng. These can be other assets such as commodtes, stocks, bonds, nterest rates and currences, but, dependent on the complexty of the fnancal product, the underlyng can be almost anythng that deemed to have an ntrnsc value. Ths mples that soco-psychologcal ssues such as confdence, fath or trust play an mportant role n defnng those values. Formally, dervatves are specfed as contracts between two partes. Such contracts defne how the value of the underlyng s estmated at partcular future dates and what condtons have to be fulflled for payments between these partes. Because partes do not need to own the underlyng, dervatves make for an deal nstrument to speculate about the future rsng or fallng value of underlyngs or to hedge aganst the rsk assocated wth t, provded that a counterparty s wllng to bet on ths. Tradng dervatves bascally means to fnd a counterparty for the contract. Importantly, partes can trade dervatves n two dfferent ways, n regulated markets specalzed n tradng dervatves (ETD, exchange-traded dervatves) or prvately, wthout nvolvng an exchange or other nsttutons (OTC, over-the-counter dervatves). Although OTC markets are usually well PLOS ONE DOI: /journal.pone September 3, / 23

3 organzed, they are less formal. In partcular, there s no central authorty whch would regulate the condtons of the dervatve contracts or would control the fulfllment of these condtons. OTC dervatves are usually preferred over the exchange traded ones because taxes and other expenses are lower and they are much more flexble, meanng that the counterpartes can agree on very specfc or unusual condtons as opposed to the lmted set of dervatve types desgned and operated by an exchange. As a trade-off for flexblty and the possblty of hgher earnngs OTC dervatves bear sgnfcant addtonal rsks as compared to the exchange traded ones. Rsk nvolved n OTC dervatves Dervatves are generally used to hedge rsks, but dervatves themselves are a source of rsk. These are credt rsk and market rsk, along wth lqudty, operatonal and legal rsks [2]. In case of OTC dervatves, credt rsk s the man source of rsk because of the usual absence of a clearng house that guarantees the fulfllment of oblgatons between partes. Thus, the two contractng partes are exposed to counterparty default rsk,.e., the rsk that a counterparty wll undergo dstress, or even default pror to expraton of the contract and thus wll not make the current and future payments. In contrast to lendng rsk, to whch only the party whch lends s exposed, both sdes nvolved n OTC contract are exposed to counterparty rsk. To have some sort of mtgaton, the partes nvolved n OTC dervatves are usually banks whch act on ther own behalf or on behalf of ther clents. There are dfferent ways to mtgate counterparty rsk n case of default. For example, usng close-out nettng agreements allows that all contracts are netted, elmnatng the possblty of selectve executon of contracts [3]. For blateral close-out nettng, whch mostly apples to OTC dervatves markets, the two partes agree to net wth one another,.e., to set off gans and losses from all of ther blateral contracts. Ths dffers from the case of multlateral close-out nettng whch mostly apples to ETD,.e., to markets where all partes oblgatons are netted together. In both cases, nettng s only a procedure to follow after a default and thus does not address the emergence of counterparty rsk. It s obvous that nettng decreases credt exposure, as t takes nto account only the net oblgatons, thus reducng both operatonal and settlement rsk and operatonal costs. In order to know the rsk, the present value of contracts, pror to ther contracted termnaton, has to be determned. Outstandng contracts are marked to market, takng nto account the replacement costs,.e., the loss suffered by the non-defaultng party n replacng the relevant contract. Ths assessment of credt exposure at a sngle pont n tme s denoted as current credt exposure (CCE). However, dervatve contracts usually have consderable lfetmes and are very often characterzed by fast and large changes n credt exposure. Therefore, the potental future exposure (PFE) s used to estmate the possble CCE ncrease over a fxed tme frame. These estmates are, of course, predctons that depend on the choce of fnancal models and correspondng confdence level. The total credt exposure (TCE) s then measured as the sum of CCE and PFE, followng the Basel I framework. In Secton Correlatons n rsk we wll use the TCE values reported by fnancal nsttutons to estmate correlatons n ther rsk. Whereas nettng agreements work n the absence of clearng houses, recent developments try to mtgate counterparty rsk by means of central counterparty clearng houses (CCPs) [4]. In the presence of a CCP a blateral contract between two counterpartes s substtuted by two contracts, so that the CCP stands between the two contractng partes. Ths allows for more transparency and for multlateral nettng, whch can facltate the reducton of both counterparty and systemc rsk. Although nvolvement of a CCP was prevously requred n contracts for credt default swaps (CDS) [5], a specal class of dervatves, ts broader utlzaton can be seen as a reacton to the fnancal crss of PLOS ONE DOI: /journal.pone September 3, / 23

4 However, regulatons requrng CCPs n all standardzed types of OTC dervatves are ether new, e.g. the US Dodd-Frank Act from 2010, or are stll beng developed. Therefore, ther mpact on OTC dervatves markets s not well known yet, both emprcally and theoretcally. [6] recently attempted to shed some lght on the possble systemc effects from CCPs. They performed a theoretcal nvestgaton of cascadng effects and systemc rsk n dfferent fnancal networks wth one or two CCPs. One may argue that not consderng the role of CCPs n OTC dervatves networks s a lmtaton of ths paper. But one should bear n mnd that we analyse data rangng from 1998 to 2012,.e., most of the tme CCP were not requred, and not reflected, n the OTC data. To keep our methods consstent for the whole tme perod, we neglect the possble (but not documented) presence of CCPs. Moreover, even today t s not known whether the wde adopton of CCPs wll succeed n makng the OTC dervatves network entrely transparent. So our methods to nfer undscovered and potentally dangerous lnks of the network may stll be needed n the future. Clusterng of counterparty rsk In ths paper, we dscuss a partcular rsk nvolved n OTC dervatves, namely the clusterng of counterparty rsk. Whle counterparty rsk tself s already dffcult to estmate, t becomes even more tedous for a party to fnd out about the addtonal rsk that a counterparty bears because of t s nvolvement n other OTC dervatves. The problem s llustrated n Fg 1. It shows nne nsttutons that have n total ten dfferent OTC contracts. The wdth of the lnks shall ndcate the volume of these contracts,.e., the three nsttutons 1, 2, 3 n the center (ndcated by the dashed lne) form a fully connected cluster of strongly engaged nsttutons. What s ther mplct mpact on those nsttutons outsde the center? Each of these has only one contract wth one of the major nsttutons n the center and s lkely not aware of the whole structure of the network of OTC dervatves. There s a two-step scenaro to ncrease the rsk of the dfferent nsttutons: () Transfer of rsk from the outer nsttutons to the central counterparty: Insttuton 4 s probably not aware that ts counterparty 1 also has contracts wth nsttutons 5 and 6. If one of these outer nsttutons defaults, ths puts an addtonal rsk for nsttuton 1 to default, whch s lkely not accounted for n the OTC contract between 4 and 1. Addtonally, nsttutons 4 and 5 also Fg 1. Schematc llustraton of the exposure clusterng. do: /journal.pone g001 PLOS ONE DOI: /journal.pone September 3, / 23

5 have a contract whch s lkely not known to nsttuton 1. Thus, the default of ether 4or5 ncreases the rsk for the remanng one, whch ndrectly ncreases the rsk for nsttuton 1 [7]. () Increase of rsk between central nsttutons: Because the center nsttutons form a fully connected cluster, f one of these undergoes dstress or even defaults ths mmedately affects the other two core nsttutons. Ths n turn affects the outer nsttutons. In concluson, because of the strong couplng of the center nsttutons, whch we call clusterng of counterparty rsk here, all nsttutons ndrectly have to bear (part of) the counterparty rsk of all other nsttutons n the network. Ths should be prced n ther OTC dervatves, but effectvely t s not because that would mply to know (a) all the lnks and (b) all ther weghts or, n plan words, all the OTC contracts made. But, as explaned above, the exstence of OTC dervatves s precsely because such nformaton should not be made publcly avalable. As we wll see from the data, all publc nformaton only refers to the total amount of OTC dervatves for each nsttuton, but not to ther counterparty network. Ths sets the stage for our paper. Even n the absence of offcal nformaton about the network of counterparty rsk, we want to derve some nsghts nto ts structure, from a dataset descrbed n the followng. Specfcally, we want to derve a proxy for the structure of ths weghted, and tme dependent, network. Further, we want to estmate correlatons between OTC dervatves,.e., nfer on possble counterpartes from the co-movement of the engagement of nsttutons. The Network of OTC Dervatves Actvtes and Ranks In order to reconstruct the network of counterparty rsk from the avalable dataset, we need to ntroduce a few varables that are later to be mapped to specfc data. Frst of all, we dentfy each nsttuton n the dataset by an ndex =1,..., N, where N = 61,.e., the total number of dstnct nsttutons. Note that the dataset for each quarter only lsts the 25 best ranked nsttutons, whch are not necessarly the same for each quarter (see also Fg 2). Thus, durng the whole perod of 14 years, 61 dfferent nsttutons appeared n the dataset. At each tme step t, where t s dscrete and measured n quarters, up to T = 57, nsttutons and j can act as counterpartes,.e., they have contracts of total volume x j (t). Importantly, the dataset nether lsts the counterpartes j nor the volume of ther contracts, x j (t). It lsts, however, the quarterly actvty of each nsttuton, a ðtþ ¼ P N x j¼1 jðtþ,.e., the aggregated volume, gven n column 5 of Table A n S1 Appendx Thus, the am of our paper s to reconstruct the network of dependences from ths aggregated data. Note that, f an nsttuton was not actve n a partcular quarter,.e., not lsted n the dataset for that perod, ts actvty s set to zero. To gve an example, Fg 3 shows the actvty of two banks that are consstently engaged n OTC dervatves n every quarter. Impressvely enough, ther actvtes dffer n about two orders of magntude and further show a dfferent busness strategy over tme. Whle the quarterly actvty of Keybank remans almost constant over 12 years, the actvty of Bank of Amerca grew exponentally durng the same perod of tme, clearly shown n the lnear slope n the logarthmc plot. Only n 2012, after the fnancal crss, ths nvolvement was slghtly reduced. Based on the quarterly actvtes, a (t), we can assgn each nsttuton a rank r (t) r[a (t)] wth r dscrete and r 2 {1, 2,...N} such that r[a (t)] < r[a j (t)] f a (t) > a j (t) for any par, j 2 N. I.e., rank 1 corresponds to the nsttuton wth the hghest actvty value at tme t, rank 2 to the one wth the second hghest actvty, and so forth. If an nsttuton was not actve n a gven perod, ts rank s set to zero. Because the rank r consders the poston relatve to other nsttutons, t can change even f the actvty of an nsttuton remaned constant over a certan perod. Fg 2 gves an overvew of how often the nsttutons were present n the rankng up to 25 n any of the quarters, wth ther ranks color coded. Ths matrx already ndcates that there are PLOS ONE DOI: /journal.pone September 3, / 23

6 Fg 2. Tme seres of the fnancal nsttutons appearng among the 25 top ranked between 1998 and Color codes the rank: the darker the color the better the rank (rank 1 consdered the hghest), whte ndcates the absence n the rankng. do: /journal.pone g002 remarkable fluctuatons n the ranks of most of the nsttutons, except for a group of about 10 nsttutons. Fg 4 gves a more detaled pcture by plottng the ranks of ths group over tme. We observe that there exsts a smaller core group (of about 7 members) wth consstently low ranks, whch can be well separated from a second group wth hgher, and more fluctuatng, ranks. Ths can be also observed by lookng at the ranks R r[a ] resultng from the aggregated actvtes A ¼ P T a t¼1 ðtþ. Plottng the nverse functon A(R) shown n Fg 5, we observe a rather skew dstrbuton of the aggregated actvtes wth respect to the rank, wth a skewness PLOS ONE DOI: /journal.pone September 3, / 23

7 Fg 3. The total dervatves notonal amount of two banks whch constantly appear durng the whole perod from 1998 to The dfference of order of magntude motvates to take nto account the ranks of nsttutons when buldng ther network. The lnear regresson slope for log(a BoA ) for the perod 1999/Q3 2011/Q3 (bolder lne) s , whch corresponds to yearly growth rato (a t (t+1)/a (t)) equal to do: /journal.pone g003 value γ = and a Gn coeffcent [8]g = Moreover, the plot suggests that the aggregated actvty A follows a log-normal dstrbuton wth respect to the rank R: AðRÞ ¼ 1 p Rs ffffffffff ð ln R mþ2 exp ; R 1 ð1þ 2p 2s 2 where μ = s the mean value and σ = the standard devaton of the dstrbuton. To further compare the emprcal wth the log-normal dstrbuton, Fg A n S1 Appendx shows the Q Q plot and gves the results of the two-sample Kolmogorov-Smrnov test. The nset of Fg 5 presents the cumulatve dstrbuton PðR < YÞ ¼ P Y R¼1 AðRÞ. It ndcates that about 95% of the total actvty results from the seven frst ranked nsttutons, whle the 15 frst ranked nsttutons cover more than 99% of the total actvty. It may be temptng to restrct the analyss to only these 15 nsttutons. However, the aggregated actvtes do not allow to draw conclusons about the concentraton of actvtes n certan tme perods or a change of PLOS ONE DOI: /journal.pone September 3, / 23

8 Fg 4. Changes of the ranks r (t) of a set of banks, wth the number showng ther dstance to the core of the weghted network based on the cooccurrence and actvty of fnancal nsttutons ntroduced n Temporal and aggregated networks. do: /journal.pone g004 strategy n choosng counterpartes, before and after the fnancal crss. Therefore, we wll present more detals on the temporal actvtes n Secton Temporal and aggregated networks. The avalable data also allows us to analyse the composton of the actvtes a (t) wth respect to exchange traded dervatves (ETD) and OTC dervatves. I.e., the value of the total dervatves s splt nto a ðtþ ¼a ETD ðtþþa OTC ðtþ and A ¼ A ETD þ A OTC ðtþ tell that OTC dervatves make up for the vast, respectvely. Already the sheer numbers of the a (t) and a OTC amount of contracts. I.e., we should not assume that the ranks r (t)orr obtaned from both ETD and OTC dervatves are dfferent from those ranks that would result from only consderng the values of a OTC ðtþ or A OTC. To test ths hypothess, Fg B n the S1 Appendx provdes a Q Q plot to compare both values. We see that up to rank 15 there s no dfference n the ranks obtaned by these two measures, whereas between ranks 15 and 50 the dfference n ranks would be 1 or 2. Only for ranks above 50, the dfferences become remarkable. So t s reasonable to use the ranks r (t) and R n the further evaluaton. However, when analysng the counterparty rsk n dervatve contracts, we wll make a dstncton between the (less rsky) ETD and the more rsky OTC dervatves. In fact, as Fg 6 ndcates, the mportance of OTC dervatves as compared to the ETD vastly dffers across nsttutons. The rato A OTC =A ETD s below 10 for about 1/3 of all nsttutons, whch mples that 10% or more of the actvtes s n ETD. However, lookng at the 15 best ranked nsttutons, we see for most of them the ETD busness accounts for only 2%-5% of ther actvty. So PLOS ONE DOI: /journal.pone September 3, / 23

9 Fg 5. Dstrbuton of the aggregated actvty A over the rank R obtaned from the whole reportng perod. (nset) Cumulatve sum PðR < YÞ ¼ P Y R¼1 AðRÞ. The celng of the dstrbuton, whch s the capacty of the market over the whole perod of tme s shown by the grey lne, whle the orange lne shows the correspondng 95% percentle. do: /journal.pone g005 Fg 6. Rato A OTC =A ETD versus ranks R based on the total actvty A. do: /journal.pone g006 PLOS ONE DOI: /journal.pone September 3, / 23

10 agan, t s reasonable to proxy actvtes related to OTC dervatves by the total actvtes but whenever possble, we wll take nto account the real values for OTC dervatves. Temporal and aggregated networks In order to estmate the lnk structure of the network of counterparty rsk, we frst look nto the co-occurrence of any two nsttutons among the 25 best ranked nsttutons n each gven quarter. I.e., we defne a lnk as l j (t) = 1 f for both nsttutons 1 {r (t), r j (t)} 25 and l j (t)= 0, otherwse. Ther co-occurrence does not necessarly mply that the two nsttutons are counterpartes of an OTC dervatve. A ranked nsttuton could do all ts OTC contracts wth the many nsttutons that have ranks too hgh (.e., actvtes too low), to be lsted n ths dataset. Practcally, however, ths cannot be the case because, as the OCC reports verfy, already 99% of all OTC dervatves are held by the 25 best ranked nsttutons. So, the not lsted ones would make only for 1%, whch cannot explan the large actvtes of any of the 25 best ranked nsttutons. Consequently, t s reasonable to assume that has at least one contract wth any of the other 24 nsttutons, and the best ranked nsttutons have lkely more than one. The co-occurrence network certanly overestmates the busness relatons based on OTC contracts because t s bascally a fully connected network between the 25 best ranked nsttutons. Further, the co-occurrence may change n each quarter. Therefore, as the next step, t s reasonable to assgn weghts for the lnks between any two nsttutons based on the number of quarters, they co-appear n the dataset. I.e., we defne weghts as w j ¼ 1 T X T t¼1 l j ðtþ ð2þ to normalze them to the avalable tme perod. A node that has lnks wth hgh weghts to ts neghbors certanly represents an mportant nsttuton n the OTC dervatves market. We use the weghts to defne the mportance of an nsttuton as W ¼ P N w j¼1 j. In the followng network fgures, the sze of the nodes s scaled to the normalzed mportance, W / W. Ths allows us now, based on the aggregated values, to draw n Fg 7 a frst approxmaton of the network of counterparty rsk. Whle ths fgure clearly shows the mportant nsttutons wth respect to ther co-occurrence, t neglects another mportant nformaton, namely ther rankng whch s a proxy of ther relatve actvty. Imagne nsttuton wth a steady but relatvely low actvty over tme, just enough for frequently appearng n the network, whle nsttuton j may have a much hgher actvty, but durng a shorter perod of tme, resultng n a better, but less frequent rankng. As a result, nsttuton wll be over-presented n the network drawn n Fg 7, whle nsttuton j wll be under-represented. Such actvty dfferences are prevalent n the dataset as the nvestgatons n Secton Actvtes and Ranks show. In the example shown n Fg 3, the actvty of Keybank was two to three orders of magntude lower than the actvty of Bank of Amerca. But because KeyBank was present n the top 25 lst durng the whole tme perod, t ganed a smlar poston n the network n Fg 7 as gants such as Bank of Amerca or Ctbank. Therefore, to further mprove our estmaton of the network of counterparty rsk, we take nto account the overall actvty of an nsttuton by usng ther ranks to assgn weghts to the lnks of co-occurrence. I.e., nstead of l j (t) = 1, we use ( ) 1 l j ðtþ ¼ mn r ðtþ ; 1 ð3þ r j ðtþ The ratonale behnd s to bnd the weght of a lnk to the actvty of the less actve nsttuton. PLOS ONE DOI: /journal.pone September 3, / 23

11 Fg 7. Weghted network based on the co-occurrence of fnancal nsttutons n the top 25 rankng, aggregated over all quarter years. The sze of a node ncreases wth ts mportance W, the wdth of the lnks ncreases wth ther weghts w j, where l j 2 {0,1} (.e., do not depend on the ranks). The lnks are colored accordng to the non-normalzed correlaton coeffcent (defned n Secton Correlatons n actvtes) between actvtes n OTC dervatves of the two banks. do: /journal.pone g007 To elucdate ths, let us assume that nsttuton s a bg player wth rank r (t) = 2 at tme t, whle j s a less mportant nsttuton wth rank r j (t) = 21. Because both nsttutons co-appear n the same quarter, each of them has lnks to all other nsttutons lsted n the same tme perod,.e., 24 lnks. For the less mportant nsttuton j, 20 of these lnks get assgned a weght PLOS ONE DOI: /journal.pone September 3, / 23

12 of 1/20, namely those lnks to nsttutons wth better ranks. But there are 4 lnks to nsttutons wth an actvty less than j and therefore wth hgher ranks. Those lnks get assgned the weghts 1/22, 1/23, 1/24, 1/25. I.e., for each nsttuton, lnks to less actve counterpartes have less weght, whle lnks to more actve counterpartes have the maxmum weght that could occur gven the rank of that nsttuton. Lkewse, for nsttuton only one lnk, namely the lnk to the hghest ranked nsttuton, gets a weght 1/2, whereas the 23 lnks to all other nsttutons become less and less mportant as 1/3, 1/4,..., 1/25. The resultng network s shown as an anmaton (at the tme of wrtng only supported n Adobe products) n Fg D n S1 Appendx. At each tme step ths s a fully connected network, but the weghts of the lnks, as well as the mportance of the nsttutons, change durng every tmestep. The anmaton ncely elucdates the emergence of new key players n the OTC dervatves markets before and after the crss, as well as the changed preferences n choosng counterpartes. To allow a comparson wth Fg 7, we aggregate the weghts of the lnks over tme accordng to Eq (2), to take nto account both co-occurrence and actvty, and calculate the mportance of an nsttuton as before, W ¼ P N j¼1 w j. The resultng weghted network s then shown n Fg 8, whch should be compared to Fg 7. The most obvous dfference s a less dense core, bult up by a smaller number of mportant nsttutons, n Fg 8. Tracng partcular nsttutons, e.g. Unon Bank, we see that ther poston becomes less nfluental. But the core of the network,.e., the set of the ten most mportant nsttutons, remans the same and shall be nvestgated n the followng. Core-perphery structure So far, we have used the followng nformaton to descrbe counterparty relatons: () Aggregated measures derved from the aggregated co-occurrence l j n the rankng of the 25 top players n the OTC market, n partcular the weghts w j and the mportance W. The results are concluded n the network of Fg 7. () Temporal measures derved from the rankng r (t), n partcular the temporal co-occurrence l j (t). The results are concluded n the anmated network of Fg D n S1 Appendx, wth the tme-aggregated network shown n Fg 8. Whle the latter can be seen as the most refned network of counterparty rsk, the characterzaton of both nodes and lnks s stll based on the actvty a (t) of the respectve nsttuton,.e., t s derved from a sngle scalar measure. So, the queston s whether the reconstructon of the aggregated temporal network would allow us to add another dmenson to characterze nsttutons, based on topologcal nformaton. Already a vsual nspecton of Fgs 7 and 8 verfes that the network s rather heterogeneous wth respect to ts densty. We can easly detect a core of larger (.e., more actve) and more densely connected nodes whch can be dstngushed from a perphery of nodes that are smaller (.e., less actve) and less densely connected. In fact, perpheral nodes are mostly connected towards the core and much less to other perpheral nodes. The core of the network s depcted n Fg 9 and gves a good mpresson of the fully connected network, albet wth lnks of dfferent weghts. Whether nsttutons can be found n the core or n the perphery of the network certanly relates to ther mportance n the OTC market. In order to quantfy the topologcal nformaton encoded n the network structure, we use the weghted K core analyss, whch s an establshed method to assgn an mportance value to nodes. In the frst step, for the tme aggregated network shown n Fg 8, each node gets assgned a weghted degree ^k [9]: "! X # ^k k b aþb 1 ¼ w j ; k a j PLOS ONE DOI: /journal.pone September 3, / 23

13 Fg 8. Weghted network based on the co-occurrence and actvty of fnancal nsttutons n the top 25 rankng, aggregated over all quarter years. The codng of sze and color of nodes and lnks are the same as n Fg 7, but the w j and W are calculated from the l j as gven by Eq (3).e., dependent on the ranks. The tme resolved network s shown n Fg D n S1 Appendx. The aggregated network should be compared wth Fg 7 where actvtes are not taken nto account. do: /journal.pone g008 where k s the degree of node,.e., ts number of lnks to neghborng nodes, and P k j w j s the sum over all ts lnk weghts as defned n Eq (2) wth the weghted l j gven by Eq (3). The exponents α and β are used to weght the two dfferent contrbutons,.e., number of lnks versus weght of lnks. In our analyss we used α = 0 and β = 1,.e., we focused only on the weghts snce the network s almost fully connected and the node degree does not gve us any nformaton. In the second step, we follow a prunng procedure to recursvely remove all nodes wth degree ^k K from the network, where K =1,2,...I.e., frst all nodes wth ^k 1 are removed, PLOS ONE DOI: /journal.pone September 3, / 23

14 The Network of Counterparty Rsk Fg 9. The core of the aggregated weghted temporal network presented n Fg 8. For the codng see the legend n Fg 8. do: /journal.pone g009 whch may leave the network wth other nodes that now have ^k 1 smply because some of ther neghbors were removed. So the procedure contnues wth removng these nodes, too, unless no nodes wth ^k 1 are left. Then all nodes removed durng ths step get assgned to a core K = 1, and the procedure contnues to successvely remove all nodes wth degree ^k 2 and assgn them to a core K = 2, etc. The procedure stops at a certan hgh core value, K, when all nodes are removed. The hgher the K-core a node s assgned to, the more t belongs to the PLOS ONE DOI: /journal.pone September 3, / 23

15 core of the network and the more mportant t s, from a topologcal perspectve. Evdently, nodes assgned to a core wth low K value are much less ntegrated n the network. Ths does not refer smply to the number of neghbors, but also to non-local propertes such as the number of neghbors of ther neghbors, because the K-core decomposton also takes these nto account. That means, the K-core a node s assgned to reflects s poston n the network much better than smple measures such as the degree (.e., the number of neghbors), alone. The results of the weghted K-core analyss are shown n the left sde of Fg C n S1 Appendx, where the K value s normalzed to 1. Based on ther K value, nsttutons can be ranked such that the hgher the K value (.e., the better the ntegraton n the network), the better the rank. Ths topologcal rankng does not necessarly concdes wth the rankng R obtaned from the aggregated actvty A whch s shown on the rght sde of Fg C n S1 Appendx, for comparson. Ths ndcates that structural measures based on the network topology ndeed provde nformaton dfferent from the temporal measures based on the market actvtes of the nsttutons. But, comparng the left and the rght sdes wth respect to the color codng, we observe that only n a few cases nsttutons have consderably dfferent levels of mportance dependent on the measurement. It would be worth lookng at these n a case-by-case study, to fnd out whch mportance measure better reflects ther overall performance n the fnancal market. We would lke to note that, for consstency, we have used the rankng obtaned from the weghted K-core analyss to sort the dfferent nsttutons n all the fgures. Correlatons Correlaton measures So far, we have analysed the co-occurrence of fnancal nsttutons n the set of the 25 best ranked nsttutons, weghted by ther ranks. These ranks were based on ther actvtes,.e., total dervatves. As a result, we could reconstruct the weghted network of counterparty rsk whch also reflects the mportance of the nodes. Ths network was reconstructed (a) on a tme resoluton of one quarter year, to show the dynamcs of the network (Fg D n S1 Appendx), and (b) on the tme aggregated level (Fg 8). To further analyse the mutual dependence between the best ranked nsttutons, we now calculate dfferent correlatons. The network of counterparty rsk has revealed how the co-occurrence changes over tme. But wll the OTC dervatves of nsttuton ncrease, or decrease, f the same measure of nsttuton j ncreases? Answerng ths queston allows some more refned conclusons about the dependence between these nsttutons. The smplest measure s the Pearson correlaton coeffcent ρ, whch ponts to a lnear dependence between two varables. As explaned above, for each nsttuton we have a dataset a = {a (1), a (2),..., a (T)} avalable whch contans up to T entres about ts quarterly actvty a (t) measured by means of ts total dervatves. We recall that some of these entres are zero whenever nsttuton was not lsted among the best 25 ranked. Let us defne the mean value and the standard devaton of each of these samples as: sffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff a ¼ 1 X T 1 X T a T ðtþ ; s a ¼ ½a T 1 ðtþ a Š 2 : ð4þ t¼1 The Pearson correlaton coeffcent wth respect to the varable a s then defned as r a j ¼ 1 X " # T a ðtþ a aj ðtþ a j : ð5þ T 1 s a t¼1 s a j Values of ρ can be between -1 and +1. The latter ndcates that the relaton between actvtes a t¼1 PLOS ONE DOI: /journal.pone September 3, / 23

16 and a j can be perfectly descrbed by a lnear relatonshp, where a ncreases as a j ncreases. -1, on the other hand, ndcates a perfect lnear relatonshp where a decreases as a j ncreases, and vce versa. Zero would ndcate that there are no lnear dependences detected n the data. Eq (5) also shows that, n case of a postve correlaton, f a ðkþ > ^a then also a j ðkþ > ^a j for most of the tme, and f a ðkþ < ^a then also a j ðkþ < ^a j for most of the tme,.e., the actvtes of both nsttutons are mostly above (or below) ther respectve average, at the same tme. Correlatons n actvtes We frst dscuss the results for the most actve nsttutons,.e., those appearng among the 25 best ranked nsttutons wth respect to ther total dervatves n every quarter. Interestngly, ths apples only to 8 out of the 61 lsted nsttutons. Fg 10 shows the correlaton matrx for these nsttutons, ther actvtes proxed by the total notonal amount of dervatve contracts as lsted n column 5 of Table A n S1 Appendx. Fg 10. Correlaton matrx of the reported total dervatves of the nsttutons appearng n top 25 commercal banks, savngs assocatons or trust companes n dervatves durng the whole perod from 1998 to do: /journal.pone g010 PLOS ONE DOI: /journal.pone September 3, / 23

17 There are two observatons to be made: () the correlatons between any two of these nsttutons are always postve and often even close to 1, () Keybank s a notceable excepton. Ths can be explaned by the combnaton of two effects: The frst one s the vastly growng market n OTC dervatve durng the observaton perod whch resulted n the growth of OTC dervatves for these core nsttutons. Thus, the observed correlatons could, n prncple, be caused by the underlyng market dynamcs rather than by the mutual nteracton. However, takng nto account that the 10 best ranked nsttutons already account for 95% of the OTC dervatves market, there s lttle room for the assumpton that ther growth s based on OTC dervatve contracts wth nsttutons that do not belong to the core of 10, or to the 25 best ranked nsttutons. In concluson, these eght nsttutons ncreased ther OTC dervatves actvtes by repeatedly choosng the same core nsttutons as counterpartes. The low correlatons for Keybank could result both from the absence of growth (see Fg 3), whle all others were growng, and from choosng counterpartes from outsde the set of core banks. If we wsh to extend ths correlaton analyss to the whole set of 61 nsttutons, t would generate a number of artfacts whch should be avoded. We dscuss them here, frst, to motvate our own approach presented afterward. As already shown n Fg 2, most of these nsttutons were not present n the rankng of the best 25, for some longer or shorter perod. So, one could lmt the correlaton analyss to those quarters where the two nsttutons were ndeed present n the rankng. I.e., f nsttuton appeared at tmes t 1, t 2, t 3, t 4 and whle nsttuton j appeared at tmes t 2, t 4, t 5, t 6, the correlaton coeffcent for them s computed usng only the observatons at tmes t 2 and t 4 where both were present. The Pearson correlaton coeffcent based on parwse avalable observatons wth respect to the varable a s then defned as r a j ¼ 1 X # ½T \ T j Š 1 t2t \T j a ðtþ a s a " # aj ðtþ a j s a j ð6þ T and T j are subsets of {1, 2,..., T}, comprsng the tme steps when the nsttutons and j appeared n the rankng among the top 25, and #[T ] and #[T ] are the numbers of these tme steps. T \T j then defnes the subset of tmesteps where both nsttutons and j appeared together, and #[T \T j ] gves the respectve number of those tme steps. Consequently, the average actvty a and the standard devaton s a are also calculated only for the subset T : a ¼ 1 X a # ½T Š ðtþ ; s a ¼ t2t sffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 1 X ½a # ½T Š 1 ðtþ a Š 2 t2t The results of ths analyss are shown n Fg E n S1 Appendx. We observe that, n addton to the strong correlatons n the core of those nsttutons always present, there are a lot of strongly ant-correlated actvtes (ndcated by rch red) among the low ranked nsttutons whch need to be nterpreted, both wth respect to the correlaton and to the magntude. We start wth the latter. Defnng the Pearson correlaton coeffcent accordng to Eq (6) has the drawback that the correlaton coeffcents for dfferent nsttutons are no longer normalzed to the same number of observatons, T, as n Eq (5) and thus cannot be compared. Precsely, the correlatons between Bank of Amerca and Ctbank, whch were both present n the rankng for T =57 quarters wll get the same weght as the correlatons between Ctbank Nevada and Chase Manhattan Bank USA whch were present together only two tmes. The second drawback results from the tme lapse between the co-appearance. Whle the tmes t 4 and t 6 n the above example may stll be relatvely close, the nterval between t 4 and t 56 ð7þ PLOS ONE DOI: /journal.pone September 3, / 23

18 would be much longer and, because of the unknown ntermedate values, nterpretatons about the correlated move of both nsttutons become hghly speculatve. In contrast to the above example, n whch the two ntermedares appear only n a few quarters, but yet co-appear twce, some pars of ntermedares whch are mportant both by means of long term presence and good rankngs, never appeared together, for example Goldman Sachs and Bank of New York, and, as a consequence, the Pearson correlaton coeffcent s not even defned for them, whch s yet another drawback. One could argue that these drawbacks dsappear f we smply keep the normalzaton T,as n Eqs (4) and (5), and nstead assgn an actvty a (t) = 0 whenever an nsttuton s not present n the rankng. Whle there s no evdence that the actvty was ndeed zero, the error produced ths way s certanly small because of the very skew dstrbuton of actvtes shown n Fg 5, and both the mean and the standard devaton of the actvty are not substantally affected. But t becomes a problem when there s ndeed no data because the nsttuton does not exst n certan quarters, e.g. because of mergers and acqustons, as n the case of Chase Manhattan Bank and JPMorgan Chase Bank. Addtonally, by proceedng lke ths we would generate another artfact, namely generatng artfcal correlatons between those nsttutons that are often not n the rankngs and, n the worst case, never co-appear. It s n fact the absence of data that generates ther correlatons, artfcally. Takng agan the example of Goldman Sachs and Bank of New York, these two nsttutons would then appear ant-correlated whle, n fact, no correlaton was defned for them. Thus, solvng the above mentoned drawbacks ths way would generate yet a dfferent one. Consequently, we wll go wth the correlatons defned on the parwse co-appearance, Eq (6), but we compensate for the dfferent normalzaton by multplyng the correlaton coeffcents r a wth the weghts w j j defned n Eq (2) wth l j = 1, whch s the relatve number of coappearances. Ths mples that the correlatons between two nsttutons that rarely coappeared n the rankng are scaled down. Precsely, after ths correcton, the weghts w j defne the bounds of the values of the correlaton coeffcents, whch are dfferent for each par of nsttuton, namely [ w j,+w j ] nstead of [ 1, +1]. These weghted correlaton coeffcents shall be nterpreted dfferently from the conventonal correlaton coeffcents n that a close-tozero coeffcent no longer means that the varables are uncorrelated, but that there s no sgnfcant correlaton because of the low weght. The resultng correlaton matrx s shown n Fg 11. Compared to the non-scaled Fg E n S1 Appendx, both the correlated and the ant-correlated actvtes loose mportance for nsttutons wth hgher ranks, because the co-appearance n the rankng s rather sparse. But stll, t s obvous that the correlated actvtes are concentrated n the core, whle the ant-correlated actvtes can be mostly found n the perphery. Keepng n mnd the exponental growth of the dervatve volume of some of the key players, as shown n Fg 3, t means that the OTC market acted rather heterogeneous. Most banks wth hgh ranks,.e., key players, ncreased ther actvtes n a growng market. Banks wth lower ranks, such as Frst Natonal Bank of Chcago or RBS Ctzens, have ether reduced ther overall OTC exposure or have concentrated ther actvtes towards only mayor nsttutons, avodng other low ranked nsttutons. Correlatons n rsk So far, we have only analysed correlatons n actvtes,.e., the correlated ncrease or decrease n OTC dervatves volumes between any two nsttutons. We found that the correlated behavor was the domnatng one whch, together wth a vastly growng OTC market, mples that most nsttutons ncreased ther nvolvement. The queston remans what ths would mean for the rsk of the counterpartes, PLOS ONE DOI: /journal.pone September 3, / 23

19 Fg 11. Non-normalzed Pearson correlaton coeffcents ρ j (based on parwse avalable data), scaled by w j wth l j =1. do: /journal.pone g011 We already mentoned n Secton Rsk nvolved n OTC dervatves that credt rsk s the man source of rsk for bankng nsttutons. To estmate the total credt exposure (TCE), we sum up ther current credt exposure (CCE) and ther potental future exposure (PFE) as explaned n Secton Rsk nvolved n OTC dervatves. Ths data has been made avalable n Table 4 of the OCC reports for each quarter year (see Table B n S1 Appendx) and s used for our subsequent correlaton analyss. Table 4 lsts, for each of the 25 frst ranked nsttutons, the blaterally netted current credt exposure and the blaterally netted potental future exposure and the sum of both, TCE = CCE+PFE, as reported by the nsttutons themselves. Lookng at Q1 of 2012, we frst notce that, for the hgh ranked nsttutons (accordng to ther actvty n OTC dervatves), the potental future exposure exceeds consderably the current exposure, whch s generally not the case for the lower ranked nsttutons. The queston whether ths observaton s related to the fnancal crss of 2008 s addressed further below. We can now defne a correlaton coeffcent r TCE j based on Eq (6) by just replacng the values of the actvtes a (t) by TCE (t), and for r CCE j accordngly. Followng the argumentaton above, we weght these correlatons agan by the weghts w j. The results are shown n Fg 12. Both fgure parts ndcate that, at least for the subset of banks whch are the closest to the core accordng to the core-perphery decomposton, the credt exposures are hghly postvely correlated. Ths ndcates that the core of the network conssts of nsttutons whch are very strongly nterdependent. Ths can become a reason for systemc nstablty, as the credt exposures and the connected rsks cannot be well dversfed. PLOS ONE DOI: /journal.pone September 3, / 23

20 Fg 12. Non-normalzed Pearson correlaton coeffcents r CCE j and r TCE j (based on parwse avalable data), scaled by w j wth l j = 1 between (a) blaterally netted current credt exposures and (b) total credt exposures of the banks. do: /journal.pone g012 The correlaton pattern for the rsk resembles the one found for the actvtes, Fg 11.We have to note, however, that a large correlaton coeffcent r a j s a good ndcator of a long-term actvty between nsttutons and j, but a large correlaton coeffcent r TCE j does not allow us to derve such a concluson. Up to ths pont the analyss was based on the whole avalable perod of tme ( ). It s nterestng to repeat the correlaton analyss of rsk for the tme before and after the fnancal crss, separately. We avod to dscuss the precse mappng of before and after and have chosen Q4 of 2008 to dvde the tme seres nto two perods. In Q4 of 2008 Goldman Sachs entered the rankng of the OCC, for the frst tme, rght after the collapse of Lehman Brothers on 15 September, The results of our analyss before and after Q4 of 2008 are shown n Fg 13. Comparng the two parts of the fgure, we make two observatons: () All lsted banks follow a smlar behavor before and after the crss. But after the crss the correlatons became more homogeneous and non-negatve even between low-to-low ranked and low-to-hgh ranked nsttutons. () Except only few banks, the key players n the core dd not change. Therefore, the OTC dervatves market structurally remaned the same despte ts vast growth. Conclusons Our nvestgaton reveals the hdden network structure behnd the OTC market n the Unted States, and the network evoluton from 1998 to For ths, we use publcly avalable data PLOS ONE DOI: /journal.pone September 3, / 23

21 Fg 13. Correlatons n total credt exposure (a) before and (b) after the fnancal crss of 2008 Q4. do: /journal.pone g013 from the [10] reports, whch contans aggregated numbers about the actvtes of fnancal nsttutons, measured by the volume of ther dfferent dervatves. We focus on two dfferent aspects: () co-occurrence patterns of nsttutons, whch take nto account ther ranks and actvtes to reconstruct the network of counterparty rsk. Ths network was further analysed usng of a weghted k-core method, to reveal ts core-perphery structure. Ths allowed us to compare the topology-based rankng wth the actvty-based rankng, and to dentfy the most mportant nsttutons and ther mutual relatons. () correlaton patters, to reveal dependences n actvtes, and the subsequent counterparty rsks of any two nsttutons. Our fndngs, namely an emergence of a pronounced core and the hgher correlatons n credt exposure assocated wth t, hnt at ncreasng counterparty and systemc rsk n OTC dervatves market. One could argue that the lst of the few top nsttutons wth the hghest counterparty rsk s not really surprsng and fnancal experts would have known ths anyway. But the pont of our nvestgaton s to present a formal, yet smple approach, to decompose ther known aggregated actvtes nto unknown blateral exposures. Only ths allows us to reveal the hdden network, and to estmate the systemc rsk. Counterparty rsk s not just the sum of ndvdual rsks, but can be amplfed over the network of dependences. Precsely, the falure of sngle nsttutons, even n the perphery of the OTC network, can lead to the collapse of the whole system because of dstress and load dstrbuted over the network [11]. Such consderatons do not only enhance our understandng of systemc rsk, they also allow to develop more refned rsk measures, and a more realstc prcng of OTC contracts. Ths network perspectve s mssng n exstng nvestgatons [12, 13] on systemc rsk n OTC PLOS ONE DOI: /journal.pone September 3, / 23