Journal of Monetary Economics

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1 Journal of Monetary Economcs 58 (211) Contents lsts avalable at ScenceDrect Journal of Monetary Economcs journal homepage: Complexty, concentraton and contagon Prasanna Ga a, Andrew Haldane b, Sujt Kapada b, a Department of Economcs, Unversty of Auckland, 12 Grafton Road, Auckland 1142, New Zealand b Bank of England, Threadneedle Street, London EC2R 8AH, Unted Kngdom artcle nfo Avalable onlne 3 May 211 abstract Ths paper develops a network model of nterbank lendng n whch unsecured clams, repo actvty and shocks to the harcuts appled to collateral assume centre stage. We show how systemc lqudty crses of the knd assocated wth the nterbank market collapse of can arse wthn such a framework, wth fundng contagon spreadng wdely through the web of nterlnkages. Our model llustrates how greater complexty and concentraton n the fnancal network may amplfy ths fraglty. The analyss suggests how a range of polcy measures ncludng tougher lqudty regulaton, macro-prudental polcy, and surcharges for systemcally mportant fnancal nsttutons could make the fnancal system more reslent. & 211 Bank of England. Publshed by Elsever B.V. All rghts reserved. 1. Introducton Herbert Smon spent over half a century teachng at Carnege-Mellon Unversty. In hs classc study of The Archtecture of Complexty, he lad the foundatons for evaluatng the reslence and evoluton of complex systems (Smon, 1962). Hs analyss drew upon a wde spectrum of systems thnkng from physcal, bologcal and socal scences. Out of ths, Smon reached the powerful concluson that even complex systems tended to exhbt a basc smplcty. Systems could be arranged n a natural herarchy, comprsng nested sub-structures. Non-herarchcal structures would tend to be deselected over tme because of ther neffcency or lack of robustness relatve to smpler, herarchcal structures. Whether the complex system was bologcal, physcal, or socal, t would be a case of survval of the smplest. The modern fnancal system seems to have bucked ths evolutonary trend. In recent years, t has become much more complex, concentrated and nterconnected. 1 As much as two-thrds of the spectacular growth n banks balance sheets n the years pror to the crss reflected ncreasng clams wthn the fnancal system, rather than wth non-fnancal agents. In Smon s words, the fnancal system has become less modular, less herarchcal and thus less decomposable. In consequence, t became markedly more susceptble to systemc collapse. Ths sowed the seeds of the global fnancal crss of 27/28. In ths paper, we explore how the complexty and concentraton of fnancal lnkages can gve rse to systemc lqudty crses that threaten fnancal system reslence. In keepng wth the mult-dscplnary sprt of Smon (1962), our theoretcal framework draws upon network technques developed n epdemology and statstcal physcs to dentfy tppng ponts n complex systems, whereby a small change n the underlyng parameters or shocks can make a very large dfference Correspondng author. Tel.: þ E-mal addresses: p.ga@auckland.ac.nz (P. Ga), andy.haldane@bankofengland.co.uk (A. Haldane), sujt.kapada@bankofengland.co.uk (S. Kapada). 1 See Haldane (29). It s natural to ask why the evolutonary forces descrbed by Smon may not have forced an earler deselecton of such a complex system. One possble explanaton s polcy. Successve crses have seen a progressve wdenng and broadenng of the fnancal safety net from last resort lendng from the 19th century onwards, to depost nsurance from the 193s, to the block-buster captal and lqudty support for both banks and non-banks durng ths century (Alessandr and Haldane, 29). Untl the end of the Bretton Woods era, ncreasng mplct support went hand-n-hand wth tougher fnancal regulaton and greater restrctons on global captal flows. The 25-year postwar perod was remarkable for ts absence of systemc bankng crses. But an era of deregulaton snce the md 197s may have contrbuted to ncreasng complexty and concentraton /$ - see front matter & 211 Bank of England. Publshed by Elsever B.V. All rghts reserved. do:1.116/j.jmoneco

2 454 P. Ga et al. / Journal of Monetary Economcs 58 (211) Fg. 1. Network of large exposures between UK banks, 28. Source: FSA returns. (a) A large exposure s one that exceeds 1% of a lendng bank s elgble captal durng a perod. Elgble captal s defned as Ter 1 plus Ter 2 captal, mnus regulatory deductons. (b) Each node represents a bank n the Unted Kngdom. The sze of each node s scaled n proporton to the sum of (1) the total value of exposures to a bank, and (2) the total value of exposures of the bank to others n the network. The darkness of a lne s proportonate to the value of a sngle blateral exposure. (c) Based on 28 Q1 data. to outcomes. In lght of recent events, the paper focuses on the collapse of the nterbank market. We demonstrate both analytcally and va numercal smulatons how repo market actvty, harcut shocks, and lqudty hoardng n unsecured nterbank markets may have contrbuted to the spread of contagon and systemc collapse. 2 The network lens also offers new perspectves on a broad range of recent and proposed polcy measures amed at tacklng fnancal system rsk. Despte obvous parallels between fnancal systems and complex systems n other felds (May et al., 28; Haldane, 29; Haldane and May, 211), the use of network technques from epdemology and statstcal physcs to the study of fnancal contagon s n ts nfancy. One mportant reason for the slow up-take amongst economsts s that such network technques are typcally slent about behavoural consderatons. But they have the advantage of eclectcsm about network formaton processes and can be calbrated to match real-world networks (such as those depcted n Fg. 1). They are partcularly well suted for dealng wth heterogenety of agents, chartng the dynamc propagaton of shocks wthn the fnancal system and dentfyng the non-lneartes that characterse fnancal nstablty n a parsmonous way. 3 In what follows, we examne the nterplay between complexty, concentraton and stablty usng a smple network model of the bankng system n whch ndvdual banks are randomly lnked together by ther nterbank clams on each other. Although stylsed, the set-up s suffcently rch to permt the study of shocks that dmnsh the avalablty of nterbank loans, so-called fundng lqudty shocks. Durng the crss, ths took the form of lqudty hoardng by banks. As one bank calls n or shortens the term of ts nterbank loans, affected banks n turn do the same. The connectvty and concentraton of the players n the network play key roles n ths propagaton mechansm. We start by explorng analytcally how such fundng contagon can generate systemc lqudty crses. 4 In dong so, we artculate how tppng ponts may be embedded n the fnancal network and show how these depend on the level of lqud asset holdngs, the amount of nterbank actvty, and the sze of harcuts on banks assets. We then consder numercal smulatons whch llustrate the potental fraglty of the bankng system and shed lght on the desgn and mplementaton of polces whch mght help to address ths. The smulatons consder two possble network confguratons: one n whch the nterbank lnks n the network are dstrbuted roughly evenly across dfferent banks (Posson confguraton), and one n whch some banks n the network are much more hghly connected than the typcal bank (geometrc confguraton). The smulatons for the Posson network are a useful benchmark, whle the fat-taled geometrc confguraton s more n keepng wth real-world networks. We consder sx experments and a further four polcy exercses. Frst, to provde a baselne, we consder what happens when a random adverse harcut shock at a sngle bank forces t to start hoardng lqudty under the Posson network 2 A repo transacton entals borrowng money usng securtes as collateral. It s structured as the spot sale of a securty for cash, coupled wth an agreement to repurchase the same securty at the ntal prce plus nterest at a partcular date n the future. When the cash lent on repo trades s lower than the current market value of the securty used as collateral, the dscount s referred to as the harcut. 3 See Jackson (28) for a comprehensve study of network technques and ther applcatons to economcs. 4 We buld on Ga and Kapada (21, 211) who model default contagon and lqudty hoardng by adaptng technques advanced by Newman et al. (21) and Watts (22). May and Arnamnpathy (21) show how smlar results on the lkelhood and dynamcs of default contagon can be obtaned by assumng that all banks follow exactly average behavour, and we also draw on them n applyng ths type of (mean-feld) approxmaton to our model.

3 P. Ga et al. / Journal of Monetary Economcs 58 (211) confguraton. We llustrate the frequency and contagous spread of systemc lqudty crses, dentfyng the tppng pont of the system n the process. Our second experment ntroduces an addtonal aggregate harcut shock whch affects all banks. We use t to artculate how our model speaks to the collapse of nterbank markets n the early part of the crss durng August and September 27. The thrd and fourth experments focus on the role of concentraton by assessng how the results change under a fattaled (geometrc) network confguraton, and also explore the dfferng consequences of a targeted shock whch affects the most nterconnected nterbank lender under both network confguratons. These are old exercses n epdemology and the analyss of network reslence (Anderson and May, 1991; Albert et al., 2). But the rapd emergence of nsttutons that are too bg, connected, or mportant to fal underlnes ther ncreasng mportance n fnance. The ffth and sxth experments explore dfferent dmensons of complexty what happens to fnancal vulnerablty when unsecured nterbank market actvty ncreases and ntal aggregate harcuts change. Snce harcuts appear to exhbt cyclcal behavour, tendng to be compressed n the upswng of a cycle as fnancal nsttutons become ncreasngly exuberant, the latter experment speaks to how systemc rsk n the system may change dynamcally. Our frst polcy exercse explores the consequences of mposng a unform ncrease n lqud asset holdngs on the reslence of the fnancal system. The second polcy exercse then smulates the model under an alternatve polcy rule n whch the average ncrease n lqud assets s dentcal to the frst polcy exercse but where the ncrease n lqud assets at each ndvdual bank s postvely related to ts nterbank assets. Ths allows us to assess the mpact of targetng hgher lqudty requrements on key players n the nterbank network. The thrd polcy exercse analyses how mposng harcut-dependent lqudty requrements mght affect the cyclcalty of systemc rsk. Fnally, we consder what happens f the amplfcaton mechansm underpnnng the lqudty hoardng dynamc s attenuated somewhat. Snce ths amplfcaton may be assocated wth uncertanty about the network among partcpants wthn t, ths exercse speaks to the mpact that greater network transparency could have on fnancal system reslence. The parsmony of our framework allows us to ncorporate nsghts from the recent lterature on fnancal crses. It draws on more conventonal theoretcal and emprcal approaches that vew the fnancal system as a network. 5 But t s able to assess reslence and polcy nterventons under a much more realstc network structure, whle placng lqudty effects and fundng contagon at centre stage and stll mantanng the generalty of an analytcal model. Our model also embodes the amplfcaton role of collateral n recognsng that a key rsk n repo transactons stems from shocks to the harcuts on the securtes whch serve as collateral, and complements the lterature on the behavoural foundatons of the nterbank freeze by artculatng how a strong focus on network contagon effects may help to mprove understandng of the probablty and mpact of such events. 6 The paper proceeds as follows. Secton 2 presents some emprcal motvaton for a network approach to the study of systemc rsk. Secton 3 outlnes the model and descrbes analytcally how contagon can spread across a bankng network. Secton 4 presents the results of our sx experments and four polcy exercses. Secton 5 draws out the polcy mplcatons and dscusses them n the context of the ongong debate on regulatory reform. A fnal secton concludes. 2. Stylsed evdence on fnancal system complexty and concentraton Quantfyng the complexty and concentraton of fnancal systems s dffcult. Nevertheless, data on nterbank lendng and ntra-fnancal system actvty provde some motvaton for adoptng a network approach, and for our focus on bank heterogenety, lqudty hoardng dynamcs, and non-lneartes. Fg. 1 depcts the network of blateral large exposures between the major UK banks, wth the nodes representng banks, ther sze representng each bank s overall mportance n the nterbank network, and the darkness of the lnks reflectng the value of nterbank exposures between nsttutons. The evdent complexty hghlghts the need to move beyond stylsed network representatons when analysng systemc rsk. Concentraton n the network s also clearly vsble and ths s reflected n the fat-taled nature of the underlyng dstrbuton of lnkages and loan szes mpled by Fg Evdence on the recent evoluton of concentraton and complexty motvates our focus on these ssues n relaton to the nterbank market collapse. Fg. 2 ponts to growng concentraton wthn natonal fnancal boundares, from already hgh 5 More conventonal theoretcal models nclude Allen and Gale (2), Frexas et al. (2) and Brusco and Castglones (27) see Allen and Babus (29) for a comprehensve survey. The emprcal lterature s surveyed by Upper (211). Ths tends to use regulatory data on large exposures between banks to analyse default contagon n nterbank markets apart from a bref dscusson by Furfne (23), the role of lqudty effects n the contagon process s largely gnored n ths lne of the lterature. 6 On the amplfcaton role of harcut shocks, see Brunnermeer and Pedersen (29), Adran and Shn (21a), Geanakoplos (21) and Gorton and Metrck (21). Key papers on the nterbank freeze nclude Caballero and Krshnamurthy (28), Allen et al. (29), Caballero and Smsek (21), Bolton et al. (211), Damond and Rajan (n press) and Acharya et al. (n press). Acharya and Skee (ths ssue) s another contrbuton n ths ven. Ther model explores how banks uncertanty over ther ablty to roll over ther own debt may cause them to restrct lendng n nterbank markets. Although they do not consder how such behavour may propagate through the nterbank network, ther story of a precautonary motve for lqudty hoardng s consstent wth the vew of hoardng taken n ths paper; ndeed, t may be nterpreted as provdng a behavoural foundaton for t. 7 See Boss et al. (24) and Crag and von Peter (21) for smlar evdence on the Austran and German nterbank markets, respectvely. Propertes of cross-border bankng networks are dscussed by von Peter (27), Kubelec and Sa (21) and Garratt et al. (211).

4 456 P. Ga et al. / Journal of Monetary Economcs 58 (211) UK US Per cent Fg. 2. Concentraton of the UK and US bankng systems. Sources: FDIC and Bank of England calculatons. (a) Largest three banks by total assets as a percentage of total bankng sector assets. (b) Data s to January 29. Fnancal corporate debt Non-fnancal corporate debt Household debt Government debt Per cent of nomnal GDP Fg. 3. Breakdown of UK Debt. Sources: ONS Blue Book and Bank of England calculatons. startng ponts. It shows the marked rse n the combned assets of the three largest banks by total assets as a percentage of total bankng system assets n both the UK and US. 8 Fnancal system complexty s lkely to go hand-n-hand wth ntra-fnancal system actvty whch tends to ncrease the length of credt chans. Shn (29) notes how the advent of securtsaton markedly ncreased the complexty of the fnancal system by lengthenng the ntermedaton chan n the lead up to the global fnancal crss. In many cases, the same securty was used repeatedly n repo markets, wth the lender usng the securty receved as collateral to borrow from others. As noted above, these transactons are subject to amplfyng dynamcs and cyclcal fluctuatons lnked to the varablty of collateral harcuts over the credt cycle. Fluctuatons of ths knd are llustrated by the dramatc rse and subsequent fall n the stock of repos and fnancal commercal paper as a percentage of broad money n the US. 9 And, as Fg. 3 shows, the growth n ntrafnancal actvty extended well beyond banks to other ntermedares, wth fnancal corporate debt (ncludng banks and nonbanks) accountng for some two-thrds of the total growth n UK debt between 23 and 27. Fg. 4 llustrates the sudden and sharp rse n the cost of unsecured nterbank borrowng that followed the onset of dffcultes at some nsttutons n August 27 and, agan n September 28, followng the collapse of Lehman Brothers. Precautonary hoardng by banks and growng counterparty rsks prompted a freeze n nterbank borrowng. Such a breakdown of the nterbank market n the US, UK and Europe was an unprecedented event and the ensung contagon 8 Kng (21) also provdes evdence on ths. 9 See Fg. 1 n Adran and Shn (21b). For evdence of cyclcalty n the underlyng harcuts, see CGFS (21), Geanakoplos (21), Gorton and Metrck (21), and Krshnamurthy (21).

5 P. Ga et al. / Journal of Monetary Economcs 58 (211) UK US Bass Ponts Jan Apr Jul Oct Jan Apr Jul Oct Jan Apr Jul Fg. 4. Three-month nterbank rates relatve to expected polcy rates. Sources: Bloomberg and Bank of England calculatons. (a) Spread of three-month Lbor to three-month overnght ndex swap (OIS) rates. (b) Fve-day movng average. placed consderable fundng pressure on banks n other jursdctons. Prces alone do not tell the full story. The quantty of fundng avalable, especally at maturtes greater than overnght, declned dramatcally. The counterpart to ths was the sharp ncrease n banks holdngs of reserves wth central banks. Ths meant that natonal fnancal systems effectvely collapsed nto a star network wth central banks at ther centre. 3. A network model of the bankng system In broad terms, our model explores the reslence of the fnancal system to lqudty shocks affectng a subset of banks under dfferent network confguratons, degrees of connectvty between fnancal nsttutons, harcut assumptons, and balance sheet characterstcs. We start by descrbng how the network of nterbank exposures and balance sheets are constructed, before dscussng how shocks to harcuts whch trgger lqudty hoardng at some nsttutons may potentally propagate across the system. We then provde some ntuton for the tppng ponts that emerge n the smulatons n Secton 4 by applyng a smplfyng assumpton to the model. Ths allows us to derve an explct condton whch dentfes whether or not the system s vulnerable to a systemc lqudty crss and how ths s affected by the parameters of the model, ncludng balance sheet characterstcs and ntal harcuts. Throughout our analyss, the only sources of randomness n our model relate to the structure of the network, whch s drawn from an exogenously set dstrbuton, and the ntal dosyncratc shocks that trgger lqudty hoardng, whch randomly affect any bank n the system. As the network fully determnes the pattern of nterbank lnkages and because we also fx each bank s total nterbank lablty poston exogenously, the value of each ndvdual nterbank lnkage s determned endogenously, and so therefore s each bank s total nterbank asset poston. So randomness n the network structure maps nto randomness n each nsttuton s total nterbank asset poston. Otherwse, the model s entrely non-stochastc: once the network and an ntal shock have been drawn, the propagaton of contagon s a purely determnstc process whch depends on the parameters of the model, all of whch are exogenous. Gven these parameters, the set-up seeks to explan whether or not an ntal shock generates more wdespread lqudty hoardng and, f so, what fracton of the system s affected by outbreaks of contagon. But despte the determnstc nature of the contagon process, the randomness over the network and ntatng shock mply uncertanty over whether contagon wll occur from an ex ante perspectve. So we also explore how the frequency of contagon s affected by changes n both the dstrbutonal assumptons that drve the network structure and n the nature of the ntatng shocks The fnancal network Table 1 provdes a summary of all notaton used n the paper. The fnancal network conssts of n fnancal ntermedares, banks for short, whch are lnked together randomly by ther unsecured clams on one another. Each bank s represented by a node on the network, and the blateral unsecured nterbank exposures of bank defne the lnks wth other banks. These lnks are drected, reflectng the fact that nterbank connectons comprse both assets and labltes. The number of ndvdual nterbank exposures vares across banks. We suppose that each bank has j lendng lnks representng ts unsecured nterbank assets (.e. money lent to counterpartes by the bank) and k borrowng lnks representng ts unsecured nterbank labltes (.e. money borrowed by the bank from counterpartes). Snce every nterbank lendng lnk for one bank s a borrowng lnk for another bank, the average number of lendng lnks across all

6 458 P. Ga et al. / Journal of Monetary Economcs 58 (211) Table 1 Descrpton of parameters and calbraton n baselne smulaton. Parameter Descrpton Baselne calbraton n Number of banks 25 j Number of blateral unsecured nterbank lendng lnks for bank Endogenous (dependng on network) k Number of blateral unsecured nterbank borrowng lnks for bank Endogenous (dependng on network) z Average degree or connectvty Vares L IB Unsecured nterbank labltes 15% of balance sheet L R Repo labltes (.e. borrowng secured wth collateral) 2% of balance sheet L D Retal deposts Endogenous (balancng tem) K Captal 4% of balance sheet L N New unsecured nterbank borrowng rased after a shock A IB Unsecured nterbank assets Endogenous (dependng on network) A F Fxed assets (e.g. ndvdual corporate loans or mortgages) Endogenous (dependng on A IB ) A C Collateral assets (assets whch may be used as collateral n repo transactons) 1% of balance sheet A RR Reverse repo assets (.e. collateralsed lendng) 11% of balance sheet A L Unencumbered fully lqud assets 2% of balance sheet h Aggregate harcut appled to collateral used to obtan repo fundng.1 h Bank-specfc harcut appled to collateral used to obtan repo fundng e Idosyncratc lqudty shock m Fracton of banks lnked to bank whch hoard (wthdraw deposts from bank ) na l Proporton of deposts wthdrawn by hoardng banks 1 banks n the network must equal the average number of borrowng lnks. We refer to ths quantty as the average degree or connectvty of the system, and denote t by z. The structure of blateral clams and oblgatons n the fnancal network, as defned by the jont dstrbuton of lendng and borrowng lnks and ts moments (ncludng the average degree), plays a key role n determnng how shocks spread through the network. Our general modellng framework apples for any arbtrary choce of jont dstrbuton, though we adopt specfc assumptons both when analysng the theoretcal approxmaton to our model and n our numercal smulatons. 1 Ths means that our results are able to encompass all possble network structures, ncludng any mpled by a pror optmal network formaton game and real-world networks lke the UK nterbank network depcted n Fg Balance sheets and repo harcuts Fg. 5 presents the composton of ndvdual bank balance sheets n the model. The labltes of each bank are IB R composed of unsecured nterbank labltes, L ; repo labltes (.e. borrowng secured wth collateral), L ; retal deposts, D L ; and captal, K. We assume that the total unsecured nterbank lablty poston of every bank s evenly dstrbuted over each of ts borrowng lnks and s ndependent of the number of lnks the bank has. 11 IB Snce every nterbank lablty s another bank s asset, unsecured nterbank assets at each bank, A, are endogenously determned by the network lnks. So, although total unsecured nterbank assets equal total unsecured nterbank labltes n aggregate across the network, each ndvdual bank can have a surplus or defct n ther ndvdual unsecured nterbank F poston. Addtonally, banks hold four further asset classes: fxed assets (e.g. ndvdual corporate loans or mortgages), A ; C assets whch may be used as collateral n repo transactons (collateral assets), A ; reverse repo assets (.e. collateralsed lendng), A RR L ; and fully lqud assets (e.g. cash, central bank reserves, hgh-qualty government bonds), A. In our model, fully lqud assets can always be used as collateral to obtan repo fnancng f requred wthout any harcut (or alternatvely sold wthout any prce dscount),.e. borrowng can be obtaned aganst the full value of the asset. On the other hand, we assume that fxed assets and unsecured nterbank assets can never be used as collateral n repo transactons. The aggregate harcut assocated wth usng collateral assets to obtan repo fundng s denoted by h 2½,1Š. Thsharcut reflects the perceved underlyng rsk of the collateral. It s desgned to protect lenders aganst losses that they may ncur when tryng to sell collateral n the event that they are left wth t due to counterparty default. Such losses may be caused by market llqudty, changng degrees of asymmetrc nformaton, or ncreases n the probablty of default on the underlyng securty over the duraton of the loan. So changes n the lkelhood of any of these factors may affect aggregate harcuts. 12 We further allow for the possblty of a bank-specfc harcut, h, so that the maxmum amount of repo fundng that can be obtaned from collateral assets s gven by ð1 h h ÞA C. Ths dosyncratc harcut mght reflect the greater default probablty of a partcular bank f the lender perceves that there s a hgher chance t wll fal, then t mght demand 1 The framework also assumes that there s no statstcal tendency for hghly connected banks to be ether more or less lkely to be lnked to other hghly connected banks or to poorly connected banks. 11 If a bank has no borrowng lnks, L IB ¼ for that bank. 12 For a more detaled explanaton of why repo collateral nvolves harcuts and why these harcuts mght fluctuate, see Gorton and Metrck (21).

7 P. Ga et al. / Journal of Monetary Economcs 58 (211) Assets Labltes Fxed Assets (A F ) Retal Deposts (L D ) Collateral Assets (A C ) Reverse Repo (A RR ) Unsecured Interbank Assets (A IB ) Lqud Assets (A L ) Repo (L R ) Unsecured Interbank Labltes (L IB ) Captal (K) Fg. 5. Stylsed balance sheet. a hgher harcut as extra protecton both because t s more lkely to end up wth the collateral n practce, and because there may be some legal rsk n accessng the collateral n a tmely fashon. C Reverse repo transactons are secured wth collateral that commands the same aggregate harcut as on A. Ths mples that the amount of collateral that bank receves on ts reverse repo assets s gven by A RR =ð1 hþ. We allow for ths collateral to be fully rehypothecated (.e. reused n another unrelated transacton) to obtan repo fundng wth the same aggregate harcut, h. The maxmum amount of repo fundng that can be obtaned from rehypothecatng collateral obtaned n reverse repo transactons s then gven by ½ð1 h h Þ=ð1 hþša RR Lqudty shortages, replenshment and propagaton Gven the balance sheet and harcut assumptons above, we now seek the condton under whch bank wll reman lqud n each perod. To smplfy our analyss, we preclude the possblty of systematc retal depost nflows or outflows and assume that banks cannot rase fresh equty. We also assume that the central bank never takes collateral at more generous terms than the market n ts lqudty operatons. A bank s lqud f the total amount of collateral t has avalable to obtan repo fundng (whch ncludes ts fully lqud assets) plus any new unsecured nterbank borrowng, L N, s suffcent to exceed the amount of exstng repo fundng t has and meet any dosyncratc lqudty shock, e, or loss of nterbank fundng that t mght experence. Lqudty shocks or shocks to aggregate or dosyncratc harcuts thus have the potental to trgger a lqudty shortage at the bank. If a bank faces such a lqudty shortage, t needs to take defensve acton to avod defaultng on requred payments. In our model, we assume that the bank tres n the frst nstance to rase any resources needed by wthdrawng (or, equvalently, refusng to roll over) unsecured nterbank assets, A IB, from counterpartes t was lendng to n the nterbank network,.e. t hoards lqudty. 13 Clearly, there are other avenues open to the bank. For example, t may try to lqudate ts fxed assets, A F, n a fre sale. Alternatvely, t mght seek to rase the nterest rate t s prepared to offer on new nterbank borrowng. 14 As the recent crss demonstrates, however, lqudty hoardng was wdely observed, whereas there was less evdence of wdespread fre sales or of banks payng up sgnfcantly n nterbank fundng markets. Ths s largely because banks typcally vewed such actons as unattractve, last resort measures, gven ther large drect, costs n terms of proftablty and mportant adverse stgma effects due to ther hgh vsblty. By contrast, hoardng lqudty has fewer drect costs and snce unsecured nterbank transactons are over-the-counter, adverse stgma effects can be kept to a mnmum. Indeed, n contrast to other optons, the bank may not need to make any actve decson f t chooses not to roll over nterbank loans Note that hoardng n ths context s drven purely by the bank s own lqudty needs; concerns over the solvency of ts counterparty play no role n ts decson. In the model presented here, we assume that such hoardng behavour represents a genune dran on the lqudty of the entre bankng system, for example, endng up as ncreased reserves at central banks. As dscussed by Ga and Kapada (211) n a smlar model whch abstracts from repo market actvty, hoardng behavour can also be nterpreted as a swtch from lendng at long maturtes (e.g. for three months or one year) to lendng at much shorter maturtes (e.g. overnght). 14 The bank could also try to cut lendng to the real economy, A F. Whle ths was certanly observed durng the crss, we abstract from ths possblty as t s only lkely to generate lqudty slowly. But the potental for ths type of credt crunch reacton s one of the key channels through whch systemc lqudty crses can become so costly for socety. 15 Damond and Rajan (n press) observe that banks may be hestant to enter nto a fre sale because the alternatve of holdng on to the asset may be more benefcal. Adverse selecton n the sprt of Stgltz and Wess (1981) ponts to the fact that banks may be unwllng to pay up sgnfcantly n

8 46 P. Ga et al. / Journal of Monetary Economcs 58 (211) Once we recognse lqudty hoardng, network effects take centre stage. In partcular, lqudty shortages can propagate through the system va the network of nterbank lnkages. Suppose that a fracton, m, of banks connected to bank n the network hoard lqudty from t, wthdrawng a porton of ther deposts held at bank. Further, let us suppose that, on average, these hoardng banks wthdraw a fracton l of the deposts that they hold at bank. Under these assumptons, bank loses lm L IB of ts labltes due to lqudty hoardng by ts counterpartes n the network. Therefore, takng nto account the potental for lost deposts due to lqudty hoardng and the harcut assumptons, ts overall lqudty condton can be formally expressed as A L þð1 h h ÞA C þ ð1 h h Þ ð1 hþ A RR þl N LR lm LIB e 4 where the frst four terms represent ts avalable lqudty and the last three terms represent fundng t needs to cover wth collateral, and fundng outflows. The value of l s a key determnant of the strength of amplfcaton n the model n partcular, the hgher the value of l, the larger the shocks that can ht banks further down the chan of contagon. The extreme case of l ¼ 1 corresponds to full wthdrawal, wth lendng banks wthdrawng ther entre depost rrespectve of ther own lqudty shortfall (ther outstandng shortage of lqudty after all collateral and lqud assets have been used). At the other extreme, f banks lqudty shortfalls were the only determnant of the amount of hoardng, then l would be fully endogenous wthn the model. In practce, l s lkely to le between these two extremes. In our baselne numercal smulatons, however, we set l ¼ 1 to generate the sharpest possble results. Immedate full wthdrawal may seem unlkely because contractual oblgatons may prevent banks from wthdrawng ther entre depost straght away. On the other hand, f a bank has a lqudty shortfall because t has lost some porton of ts deposts from a hoardng counterparty, t may consder t to be only a matter of tme before the full amount s lost, for example when contractual oblgatons expre. So, even f current wthdrawal s only partal, a forward-lookng bank may choose to act mmedately as f t had lost ts entre depost, n order to lmt the prospect of t sufferng lqudty problems n the future. As such, assumng l ¼ 1 effectvely captures a rch set of dynamcs whch may operate through forward-lookng expectatons. The propagaton of contagon also depends on the extent to whch wthdrawals are concentrated on partcular counterpartes or whether they are more evenly dstrbuted. We assume the latter: banks rase any resources needed by wthdrawng fundng equally and proportonately from all of ther counterpartes. Ths assumpton seems plausble as mmedately accessble deposts are only lkely to be avalable n relatvely small amounts from each counterparty. ð1þ 3.4. Contagon dynamcs and tppng ponts Eq. (1) makes clear that the decson by a sngle bank to hoard lqudty makes t harder for banks that were prevously borrowng from t to meet ther own lqudty condton wthout resortng to hoardng themselves. In partcular, as each successve bank suffers a lqudty shortfall, ts hoardng has the potental to trgger a lqudty shortage at other banks to whch t s connected by nterbank lendng. Ths process wll only de out f ether no neghbours to newly dstressed banks become dstressed themselves or when every bank n the network s n dstress. So hoardng can potentally spread across the system, wth the structure and connectvty of the unsecured nterbank network playng a crtcal role n determnng the evoluton of contagon. 16 To llustrate the dynamcs of contagon, recall that bank has k borrowng lnks. So f a sngle counterparty to bank hoards, m ¼ 1=k snce nterbank labltes are evenly dstrbuted across counterpartes. Suppose we randomly perturb the network by assumng that a sngle bank suffers a harcut or dosyncratc lqudty shock whch s suffcently large to cause t to start hoardng lqudty. Then, by substtutng for m n Eq. (1) and rearrangng, we can see that for contagon to spread beyond the frst bank, there must be at least one neghbourng bank for whch A L þð1 h h ÞA C þ½ð1 h h Þ=ð1 hþša RR þl N LR e ll IB o 1 ð2þ k If ths condton holds, then contagon starts to spread. In partcular, a second bank s forced nto hoardng lqudty whch may, n turn, create lqudty shortfalls at other banks n the network, and so on. And the broader contagon dynamcs can obvously also ental the possblty of banks beng exposed to multple hoardng counterpartes, n whch case smlar equatons derved from (1) determne the spread of lqudty hoardng across the network. (footnote contnued) nterbank markets as there s a rsk that wholesale lenders may vew t as a sgnal of underlyng dffcultes. Acharya and Skee (ths ssue) provde behavoural foundatons for why banks may choose to hoard lqudty for precautonary reasons. 16 As ponted out by Lo (ths ssue), a bank s lqudty trgger s unlkely to be as determnstc as (1). Hs suggeston to ntroduce a probablstc lqudty trgger, whereby the probablty of needng to take defensve acton ncreases as the lqudty poston of the bank deterorates, would certanly enrch the framework. In partcular, t would ntroduce an explct role for uncertanty and would also allow for heghtened volatlty to trgger lqudty events. But the man effect would be to ntroduce a greater dsperson n outcomes for gven underlyng parameters rather than fundamentally alterng our man results.

9 P. Ga et al. / Journal of Monetary Economcs 58 (211) Before smulatng ths process, we can gan further ntutve nsght nto the nature of contagon by makng some further smplfyng assumptons whch allow us to obtan clean analytcal results. Specfcally, rather than takng the network to be randomly generated, let us suppose that each bank s connected to exactly z other banks as both a lender and borrower (whch mples that j ¼ k ¼ z for all banks). 17 Let us also suppose that there are no dosyncratc harcuts or shocks (so h ¼ e ¼ ) and that all banks have dentcal balance sheets, allowng us to drop all subscrpts. Gven that every nterbank asset s another bank s nterbank lablty, these assumptons also mply that L IB ¼ A IB for all banks. Then, assumng full wthdrawal (l ¼ 1) and the absence of any new unsecured nterbank fundng (L N ¼ ), we can rewrte (2) as A IB zo A L þð1 hþa C þa RR L R ð3þ Ths expresson s dentcal for every bank n the network under these stark assumptons. It yelds a tppng pont condton, determnng when contagon may break out across the fnancal network. In partcular, f (3) s satsfed, then provded that z s greater than or equal to 1 so that there s suffcent network connectvty, any lqudty hoardng by a sngle bank wll cause all neghbourng banks to become dstressed and start hoardng. Because neghbours of neghbours face the same lqudty condton, hoardng behavour wll cascade through the entre network. By contrast, f (3) s volated, an ntal case of lqudty hoardng by one bank wll have no systemc consequences at all. As such, the tppng pont condton clearly llustrates how very small changes n the underlyng parameters of the model can lead to fundamentally dfferent outcomes. 18 Eq. (3) also clarfes the condtons under whch systemc lqudty crses may occur. In partcular, t hghlghts the way n whch low lqud asset holdngs, large adverse aggregate harcut shocks, a hgh amount of repo borrowng, and a hgh level of unsecured nterbank lendng are all lkely to ncrease the susceptblty of the system to a wdespread lqudty crss. Our smulatons demonstrate that these results are borne out under more general assumptons. 4. Model smulatons We now draw upon numercal smulatons to offer further nsght nto the role of concentraton and complexty n contrbutng to systemc lqudty crses, and to assess a range of possble fnancal stablty polcy nterventons. Table 1 summarses the baselne smulaton parameters. The system comprses 25 banks. In the ntal step n each realsaton, the network of unsecured nterbank lnkages between banks s drawn randomly from an underlyng dstrbuton whch s assumed to characterse the structure of the network. In what follows, we focus on two characterstc network structures: one n whch the lnks connectng banks are dstrbuted roughly unformly (Posson); and one n whch the underlyng dstrbuton descrbng lnkages results n some banks n the network beng much more hghly connected than the typcal bank (geometrc). And when drawng the network, we allow for the possblty that two banks can be lnked to each other va both lendng and borrowng lnks no nettng of exposures s assumed. Although the model apples to fully heterogeneous banks, for the purpose of llustraton we take the lablty sde of the balance sheet of all banks to be dentcally composed of a captal buffer of 4% of the balance sheet, unsecured nterbank labltes (15%), repo labltes (determned as descrbed below) and retal deposts. Snce each bank s nterbank labltes are evenly dstrbuted over each of ther borrowng lnks, nterbank assets are determned endogenously wthn the (random) network structure and thus vary across banks. In addton to nterbank assets, the bank has lqud assets (2% of the balance sheet), reverse repos and collateral assets. The asset sde s further topped up by fxed assets untl the total asset poston equals the total lablty poston. For the repo aspects of the balance sheet, we assume that, n the ntal state, all collateral assets and assets receved as part of reverse repo transactons are used as collateral to obtan repo fundng so that L R ¼ð1 h h ÞA C þ ð1 h h Þ A RR ð1 hþ ð4þ We suppose that reverse repo assets are 11% of the balance sheet, collateral assets are 1%, the ntal aggregate harcut, h, s.1, and there are no dosyncratc harcuts. Ths mples that repo labltes comprse 2% of the balance sheet ntally. To keep the number of experments manageable, we assume throughout that banks can never rase any new deposts n the unsecured nterbank market (.e. L N ¼ for all banks). 19 Apart from n one of the polcy experments, full wthdrawal by hoardng banks (l ¼ 1) s assumed throughout. In most of what follows, we vary the average connectvty between banks, z, drawng 1 realsatons of the network for each value, and then shock the network n dfferent ways accordng to the experment n queston. The dynamcs of contagon 17 In formal terms, ths represents a mean-feld approxmaton to the model. 18 Ga and Kapada (211) show that although a precse threshold n the form of Eq. (3) cannot normally be dentfed, t s possble to prove the exstence of such tppng ponts (or, more formally, phase transtons) n ths class of model wthout takng a mean-feld approxmaton and whle stll mantanng heterogeneous balance sheets (though the l ¼ 1 assumpton s stll necessary n ther formal analyss). 19 As Ga and Kapada (211) show, although allowng for replacement of lost nterbank deposts can sgnfcantly reduce the lkelhood of systemc lqudty crses, t generally does not otherwse change the fundamental propertes of the model and ts fundng contagon dynamc n partcular, the tppng pont s stll present and contagon can stll spread to the entre system on some occasons.

10 462 P. Ga et al. / Journal of Monetary Economcs 58 (211) follow the process descrbed n Secton 3.3, wth Eq. (1) at the centre of the propagaton dynamcs. For each realsaton, we follow these dynamcs teratvely untl no new banks are forced nto hoardng lqudty or untl every bank s hoardng. We count as systemc those epsodes n whch at least 1% of banks are forced nto hoardng lqudty. Our results dentfy the frequency of systemc lqudty crses and ther mpact n terms of the average fracton of the system affected n each systemc outbreak (.e. how wdely contagon spreads, condtonal on t spreadng to at least 1% of banks n the system) Experment 1: a stylsed systemc lqudty crss In our frst experment, we assume that the lnks n the network are spread roughly unformly, wth each possble drected lnk n the network beng present wth ndependent probablty p (a Posson network). We hold aggregate harcuts constant but shock the system by assumng that a sngle bank receves a very large adverse dosyncratc harcut shock whch causes t to start hoardng lqudty. Fg. 6 (baselne) presents the results. Contagon occurs for values of z between and 2 and ts probablty s non-monotonc n connectvty, at frst ncreasng before fallng. But when contagon breaks out, t nvarably spreads to the entre network. These results accord well wth the analytcal approxmaton at the end of Secton 3.4, whch s not too surprsng because ths approxmaton s most reasonable for a Posson network. Specfcally, gven the parameters chosen n the baselne smulaton, Eq. (3) suggests that contagon wll occur for zo7:5 but not for z47:5. From Fg. 6 (baselne), t s evdent that z¼7.5 s the pont around whch the probablty of contagon starts to fall from close to one. The reason t remans postve for hgher values of z s due to the randomness of the network structure whch means that contagon can stll break out under certan confguratons. And the reason contagon s not always certan for smaller values of z s that the ntal shock may ht a bank whch ether has no nterbank assets, and s therefore unable to trgger any contagon by hoardng lqudty, or s n an solated subset of the network Experment 2: addng aggregate harcut shocks Our second experment repeats the frst but also ncorporates an aggregate harcut shock that ncreases the harcut from.1 to.2. Ths acts lke a bank run n dranng lqud assets from every repo borrower n the system as they are all requred to post more collateral to obtan the same amount of repo fundng. It s clear from (3) that ths shfts the tppng pont to around z¼15. Ths s borne out by the results presented n Fg. 6 (wth aggregate harcut shock) Average degree (.e. connectvty) Frequency of systemc hoardng (Posson baselne) Extent of systemc hoardng (Posson baselne) Frequency of systemc hoardng (Posson wth aggregate harcut shock) Extent of systemc hoardng (Posson wth aggregate harcut shock) Frequency of systemc hoardng (Posson wth targeted shock) Extent of systemc hoardng (Posson wth targeted shock) Fg. 6. Systemc lqudty hoardng (Posson network: sngle random dosyncratc harcut shock; aggregate harcut shock and sngle random dosyncratc harcut shock; and sngle harcut shock targeted to bank wth most nterbank lendng lnks).

11 P. Ga et al. / Journal of Monetary Economcs 58 (211) Average degree (.e. connectvty) Frequency of systemc hoardng (geometrc baselne) Extent of systemc hoardng (geometrc baselne) Frequency of systemc hoardng (geometrc wth targeted shock) Extent of systemc hoardng (geometrc wth targeted shock) Frequency of systemc hoardng (geometrc wth 25% nterbank labltes) Extent of systemc hoardng (geometrc wth 25% nterbank labltes) Fg. 7. Systemc lqudty hoardng (geometrc network: sngle random dosyncratc harcut shock; sngle harcut shock targeted to bank wth most nterbank lendng lnks; sngle random dosyncratc harcut shock wth 25% unsecured nterbank labltes). In some respects, ths type of experment mmcs the behavour of nterbank markets n the early part of the crss durng August and September 27. In lght of both bad news and greater uncertanty on subprme mortgages and other types of collateral whch were beng used to back repo and other secured forms of fundng, aggregate harcuts ncreased. As a result, some banks found themselves short of lqudty. Ths was especally true of banks whch were forced to take back assets from off-balance sheet vehcles for whch lqudty had dred up. In response to ther own fundng lqudty stress, some banks started to hoard n the unsecured nterbank market. As hghlghted by Lo (ths ssue), uncertanty over the value of assets may also have exacerbated hoardng behavour. The result was paralyss n the nterbank market as reflected by the sharp ncrease n spreads depcted n Fg. 4. Ths wll have ncreased counterparty rsk whch may have ntensfed problems for some nsttutons. Although ths framework abstracts from the counterparty rsk dmenson, t makes clear how a seemngly small shock to a lmted set of assets whch are beng used as collateral can lead to a collapse n both secured and unsecured nterbank markets Experment 3: systemc lqudty crses n a concentrated network Real-world fnancal networks do not appear to be partcularly unform; nstead they appear to exhbt fat tals, wth a small number of key players who are very hghly connected both n terms of the number of nterbank relatonshps they have and n terms of the overall value of those relatonshps. Ths reflects the underlyng concentraton n the bankng sector. To explore the mpact of concentraton on our results, we repeat our frst smulaton exercse but draw the network from a (geometrc) dstrbuton whch embeds fat tals by allowng some banks to have substantally more connectons than the typcal bank. 21 Fg. 7 (baselne) presents the results. Contagon s less lkely and less severe for low values of z than under the Posson dstrbuton. Ths reflects the well-known result that fat-taled networks tend to be more robust to random shocks (Anderson and May, 1991; Albert et al., 2). On the other hand, t s also clear that contagon can occur 2 In the model, hoardng entals wthdrawal of funds whle, n realty, much of the hoardng behavour early n the crss nvolved banks dramatcally reducng the maturty of ther lendng. But, as noted above, the broad framework can be nterpreted as speakng to ths behavour as well. 21 In the verson mplemented, we draw the number of borrowng and lendng lnks separately from the same dstrbuton, mplyng that there s no correlaton between the number of counterpartes a bank lends to and borrows from. To construct the network, we also need to ensure that the total number of borrowng lnks drawn equals the total number of lendng lnks. We follow the algorthm outlned by Newman et al. (21) to acheve ths.

12 464 P. Ga et al. / Journal of Monetary Economcs 58 (211) for much hgher values of z, albet rarely. So, for a broad range of connectvty, hgher concentraton n the network makes the system more susceptble to a systemc lqudty crss Experment 4: the mpact of targeted shocks n concentrated and less concentrated networks Experment 3 masks what s perhaps the key dfference between the fraglty of concentrated and less concentrated networks. Thus far, we have assumed that when the ntal dosyncratc harcut shock occurs, t hts any bank n the network at random. Now suppose nstead that the ntal shock hts the bank wth the largest number of unsecured nterbank lendng relatonshps. Fgs. 6 and 7 (wth targeted shock) show the results for each type of network. When the shock s targeted at the most connected nterbank lender, contagon occurs more frequently n both cases. But for the less concentrated (Posson) network, a targeted shock only makes a relatvely small dfference to the results. By contrast, a targeted shock under the concentrated (geometrc) network has catastrophc consequences, makng contagon a near certanty for a very wde range of z. In the Posson network, the most connected bank s not that much more connected than the typcal bank. But under the fat-taled (geometrc) network, the most connected bank s lkely to be connected to a very large porton of the other banks n the network, so f t becomes dstressed, t has the potental to spread contagon very wdely. Ths s agan consstent wth the results of Anderson and May (1991) and Albert et al. (2), who both demonstrate how fat-taled networks are partcularly susceptble to shocks targeted at key partcpants. It s clear that banks who are heavly nvolved n repo actvty are more lkely to face lqudty shortages from aggregate harcut shocks. So the most dangerous banks for the stablty of the network are those whch are both heavly nvolved n repo actvty and bg lenders n the unsecured nterbank market the former because t makes them hghly susceptble to the ntal shock, the latter because they propagate the shock wdely. Because the banks that are heavly nvolved n repo actvty are typcally the same large, complex fnancal nsttutons that are bg players n the unsecured nterbank market, we can mmedately see how the structure of the modern fnancal system may be partcularly prone to systemc collapse. It also demonstrates why the seemngly small shocks of early/md 27 could have had such catastrophc systemc consequences as they affected banks that were both hghly susceptble to lqudty dsturbances and central to the structure of the network Experment 5: the mpact of greater complexty Fnancal system complexty has no sngle defnton and s dffcult to measure. But t seems lkely that, on any defnton, t wll be ncreased by ntra-fnancal system actvty, both n unsecured and secured markets. Therefore, n ths experment, we consder a random shock n a concentrated network (as n Experment 3) but now suppose that unsecured nterbank labltes comprse 25% rather than 15% of the balance sheet. As can be seen n Fg. 7 (wth 25% nterbank labltes), contagon occurs more frequently than n the baselne. Intutvely, ths s because the overall rse n nterbank labltes ncreases the lkelhood of larger fundng wthdrawals whch cannot be absorbed by lqud assets Experment 6: cyclcalty n harcuts and the lkelhood of systemc lqudty crses It s perhaps more nterestng to consder complexty n a dynamc settng. If unsecured nterbank lendng were to ncrease over the economc cycle, then the system would become ncreasngly vulnerable to shocks. And compresson n aggregate harcuts, whch often occurs durng the upswng of a cycle as the fnancal system becomes ncreasngly exuberant, may nfluence the amount of repo market actvty by allowng more secured fundng to be generated from a fxed amount of collateral. 22 Rehypothecaton of collateral may also ncrease n upswngs, servng to ncrease the money multpler and expand balance sheets. All of ths may alter the vulnerablty of the fnancal system over tme. We provde an ndcatve smulaton of such tme-varyng rsks n our model by varyng the ntal aggregate harcut from.25 to (whch, n turn, affects the ntal amount of repo fundng va Eq. (4)). We then assess the effect of ths on the frequency of systemc crses n response to a combned shock n whch the aggregate harcut jumps to a stressed level, taken to be.25, and a sngle bank suffers a suffcently large dosyncratc shock that t s forced to start hoardng. Fg. 8 (damonds) presents the results for a concentrated network wth average connectvty (z) fxed at 5. It s clear that rsk n the system ncreases as the ntal aggregate harcut falls, pontng towards the potental for systemc rsk to ncrease as the credt cycle evolves. The non-lnearty s due to the tppng pont embedded n the model whch mples that around crtcal thresholds, the system suddenly becomes much more vulnerable to collapse Polcy Exercse 1: tougher lqudty requrements We now turn to a seres of polcy experments. Eq. (3) mples that an ncrease n lqud asset holdngs should make the system drectly less susceptble to systemc lqudty crses. So, n our frst polcy exercse, we repeat Experment 3 22 For more on the lnks between harcuts and repo actvty over the economc cycle, see Adran and Shn (21a) or CGFS (21).