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24 March 2010 first published online,doi:10.1098/rspa.2009.0410466 2010 Proc.R.Soc.A Prasanna Gai and Sujit Kapadia Contagion in financial networks Referenceslated-urlshttp:/rspa.royalsocietypublishing.org/content/466/2120/2401.full.html#re Article cited in:html#ref-list-1http:/rspa.royalsocietypublishing.org/content/466/2120/2401.full.This article cites 39 articles,2 of which can be accessed freeSubject collections(25 articles)statistical physics?(8 articles)graph theory?Articles on similar topics can be found in the following collectionsEmail alerting service herethe box at the top right-hand corner of the article or click Receive free email alerts when new articles cite this article-sign up in http:/rspa.royalsocietypublishing.org/subscriptions go to:Proc.R.Soc.ATo subscribe to on March 26,2014rspa.royalsocietypublishing.orgDownloaded from on March 26,2014rspa.royalsocietypublishing.orgDownloaded from Proc.R.Soc.A(2010)466,24012423doi:10.1098/rspa.2009.0410Published online 24 March 2010Contagion in financial networksBYPRASANNAGAI1,2ANDSUJITKAPADIA2,*1Crawford School of Economics and Government,Australian NationalUniversity,Canberra,ACT 0200,Australia2Bank of England,Threadneedle Street,London EC2R 8AH,UKThis paper develops an analytical model of contagion in financial networks with arbitrarystructure.We explore how the probability and potential impact of contagion is influencedby aggregate and idiosyncratic shocks,changes in network structure and asset marketliquidity.Our findings suggest that financial systems exhibit a robust-yet-fragile tendency:while the probability of contagion may be low,the effects can be extremely widespreadwhen problems occur.And we suggest why the resilience of the system in withstandingfairly large shocks prior to 2007 should not have been taken as a reliable guide to itsfuture robustness.Keywords:contagion;network models;systemic risk;liquidity risk;financial crises1.IntroductionIn modern financial systems,an intricate web of claims and obligations linksthe balance sheets of a wide variety of intermediaries,such as banks and hedgefunds,into a network structure.The advent of sophisticated financial products,such as credit default swaps and collateralized debt obligations,has heightenedthe complexity of these balance sheet connections still further.As demonstratedby the financial crisis,especially in relation to the failure of Lehman Brothers andthe rescue of American International Group(AIG),these interdependencies havecreated an environment for feedback elements to generate amplified responsesto shocks to the financial system.They have also made it difficult to assess thepotential for contagion arising from the behaviour of financial institutions underdistress or from outright default.1This paper models two key channels of contagion in financial systems by whichdefault may spread from one institution to another.The primary focus is onhow losses can potentially spread via the complex network of direct counterpartyexposures following an initial default.But,as Cifuentes et al.(2005)and Shin(2008)stress,the knock-on effects of distress at some financial institutionson asset prices can force other financial entities to write down the value of*Author for correspondence(sujit.kapadiabankofengland.co.uk).1See Rajan(2005)for a policymaker s view of the recent trends in financial development andHaldane(2009)for a discussion of the role that the structure and complexities of the financialnetwork have played in the financial turmoil of 20072009.Received 5 August 2009Accepted 9 February 2010This journal is2010 The Royal Society2401 on March 26,2014rspa.royalsocietypublishing.orgDownloaded from 2402P.Gai and S.Kapadiatheir assets,and we also model the potential for this effect to trigger furtherrounds of default.Contagion due to the direct interlinkages of interbank claimsand obligations may thus be reinforced by indirect contagion on the asset side ofthe balance sheetparticularly when the market for key financial system assetsis illiquid.The most well-known contribution to the analysis of contagion through directlinkages in financial systems is that of Allen&Gale(2000).2Using a networkstructure involving four banks,they demonstrate that the spread of contagiondepends crucially on the pattern of interconnectedness between banks.When thenetwork is complete,with all banks having exposures to each other such that theamount of interbank deposits held by any bank is evenly spread over all otherbanks,the impact of a shock is readily attenuated.Every bank takes a small hitand there is no contagion.By contrast,when the network is incomplete,withbanks only having exposures to a few counterparties,the system is more fragile.The initial impact of a shock is concentrated among neighbouring banks.Oncethese succumb,the premature liquidation of long-term assets and the associatedloss of value bring previously unaffected banks into the front line of contagion.Ina similar vein,Freixas et al.(2000)show that tiered systems with money-centrebanks,where banks on the periphery are linked to the centre but not to eachother,may also be susceptible to contagion.3The generality of insights based on simple networks with rigid structuresto real-world contagion is clearly open to debate.Moreover,while notbeing so stylized,models with endogenous network formation(e.g.Leitner2005;Castiglionesi&Navarro 2007)impose strong assumptions that lead tostark predictions on the implied network structure that do not reflect thecomplexities of real-world financial networks.And,by and large,the existingliterature fails to distinguish the probability of contagious default from itspotential spread.However,even prior to the current financial crisis,the identification of theprobability and impact of shocks to the financial system was assuming centre-stage in policy debate.Some policy institutions,for example,attempted toarticulate the probability and impact of key risks to the financial system in theirFinancial stability reports.4Moreover,the complexity of financial systems meansthat policymakers have only partial information about the true linkages betweenfinancial intermediaries.Given the speed with which shocks propagate,there is,2Other strands of the literature on financial contagion have focused on the role of liquidityconstraints(Kodres&Pritsker 2002),information asymmetries(Calvo&Mendoza 2000)andwealth constraints(Kyle&Xiong 2001).As such,their focus is less on the nexus betweennetwork structure and financial stability.Network perspectives have also been applied to othertopics in finance:for a comprehensive survey of the use of network models in finance,seeAllen&Babus(2009).3These papers assume that shocks are unexpectedan approach we follow in our analysis.Brusco&Castiglionesi(2007)model contagion in financial systems in an environment where contractsare written contingent on the realization of the liquidity shock.As in Allen&Gale(2000),they construct a simple network structure of four banks.They suggest,however,that greaterconnectivity could serve to enhance contagion risk.This is because the greater insurance providedby additional financial links may be associated with banks making more imprudent investments.And,with more links,if a bank s gamble does not pay off,its failure has wider ramifications.4See Bank of England(2007).Proc.R.Soc.A(2010)on March 26,2014rspa.royalsocietypublishing.orgDownloaded from Contagion in financial networks2403therefore,a need to develop tools that facilitate analysis of the transmission ofshocks through a given,but arbitrary,network structure.Recent events in theglobal financial system have only served to emphasize this.Our paper takes up this challenge by introducing techniques from the literatureon complex networks(Strogatz 2001)into a financial system setting.Althoughthis type of approach is frequently applied to the study of epidemiology andecology,and despite the obvious parallels between financial systems and othercomplex systems that have been highlighted by prominent authors(e.g.Mayet al.2008)and policymakers(e.g.Haldane 2009),the analytical techniques weuse have yet to be applied to economic problems and thus hold out the possibilityof novel insights.In what follows,we draw on these techniques to model contagion stemmingfrom unexpected shocks in complex financial networks with arbitrary structureand then use numerical simulations to illustrate and clarify the intuitionunderpinning our analytical results.Our framework explicitly accounts for thenature and scale of aggregate and idiosyncratic shocks and allows asset pricesto interact with balance sheets.The complex network structure and interactionsbetween financial intermediaries make for non-linear system dynamics,wherebycontagion risk can be highly sensitive to small changes in parameters.We analysethis feature of our model by isolating the probability and spread of contagion whenclaims and obligations are interlinked.In so doing,we provide an alternativeperspective on the question of whether the financial system acts as a shockabsorber or as an amplifier.We find that financial systems exhibit a robust-yet-fragile tendency:while theprobability of contagion may be low,the effects can be extremely widespreadwhen problems occur.The model also highlights how a priori indistinguishableshocks can have very different consequences for the financial system,dependingon the particular point in the network structure that the shock hits.This cautionsagainst assuming that past resilience to a particular shock will continue to applyto future shocks of a similar magnitude.And it explains why the evidence of theresilience of the financial system to fairly large shocks prior to 2007(e.g.9/11,the Dotcom crash and the collapse of Amaranth to name a few)was not a reliableguide to its future robustness.The intuition underpinning these results is straightforward.In a highlyconnected system,the counterparty losses of a failing institution can be morewidely dispersed to,and absorbed by,other entities.So,increased connectivityand risk sharing may lower the probability of contagious default.But,conditionalon the failure of one institution triggering contagious defaults,a high numberof financial linkages also increases the potential for contagion to spread morewidely.In particular,high connectivity increases the chances that institutionsthat survive the effects of the initial default will be exposed to more than onedefaulting counterparty after the first round of contagion,thus making themvulnerable to a second-round default.The effects of any crises that do occur can,therefore,be extremely widespread.Our model draws on the mathematics of complex networks(see Strogatz(2001)and Newman(2003)for authoritative and accessible surveys).This literaturedescribes the behaviour of connected groups of nodes in a network and predictsthe size of a susceptible cluster,i.e.the number of vulnerable nodes reachedvia the transmission of shocks along the links of the network.The approachProc.R.Soc.A(2010)on March 26,2014rspa.royalsocietypublishing.orgDownloaded from 2404P.Gai and S.Kapadiarelies on specifying all possible patterns of future transmission.Callaway et al.(2000),Newman et al.(2001)and Watts(2002)show how probability-generatingfunction techniques can identify the number of a randomly selected node s firstneighbours,second neighbours and so on.Recursive equations are constructedto consider all possible outcomes and obtain the total number of nodes that theoriginal node is connected todirectly and indirectly.Phase transitions,whichmark the threshold(s)for extensive contagious outbreaks,can then be identified.In what follows,we construct a simple financial system involving entities withinterlocking balance sheets and use these techniques to model the spread andprobability of contagious default following an unexpected shock,analytically andnumerically.5Unlike the generic,undirected graph model of Watts(2002),ourmodel provides an explicit characterization of balance sheets,making clear thedirection of claims and obligations linking financial institutions.It also includesasset price interactions with balance sheets,allowing the effects of asset-sidecontagion to be clearly delineated.We illustrate the robust-yet-fragile tendency offinancial systems and analyse how contagion risk changes with capital buffers,thedegree of connectivity and the liquidity of the market for failed banking assets.6Our framework assumes that the network of interbank linkages forms randomlyand exogenously:we leave aside issues related to endogenous network formation,optimal network structures and network efficiency.7Although some real-worldbanking networks may exhibit coreperiphery structures and tiering(see Bosset al.(2004)and Craig&von Peter(in press)for evidence on the Austrian andGerman interbank markets,respectively),the empirical evidence is limited and,given our theoretical focus,it does not seem sensible to restrict our analysis ofcontagion to particular network structures.In particular,our assumption thatthe network structure is entirely arbitrary carries the advantage that our modelencompasses any structure that may emerge in the real world or as the optimaloutcome of a network formation game.And it is a natural benchmark to consider.We also model the contagion process in a relatively mechanical fashion,holdingbalance sheets and the size and structure of interbank linkages constant as defaultpropagates through the system.Arguably,in normal times in developed financialsystems,banks are sufficiently robust that very minor variations in their defaultprobabilities do not affect the decision of whether or not to lend to them in5Eisenberg&Noe(2001)demonstrate that,following an initial default in such a system,a uniquevector that clears the obligations of all parties exists.However,they do not analyse the effects ofnetwork structure on the dynamics of contagion.6Nier et al.(2007)also simulate the effects of unexpected shocks in financial networks,thoughthey do not distinguish the probability of contagion from its potential spread and their results arestrictly numericalthey do not consider the underlying analytics of the complex(random graph)network that they use.Recent work by May&Arinaminpathy(2010)uses analytic mean-fieldapproximations to offer a more complete explanation of their findings and also contrasts theirresults with those presented in this paper.7See Leitner(2005),Gale&Kariv(2007),Castiglionesi&Navarro(2007)and the survey byAllen&Babus(2009)for discussion of these topics.Leitner(2005)suggests that linkages thatcreate the threat of contagion may be optimal.The threat of contagion and the impossibility offormal commitments mean that networks develop as an ex ante optimal form of insurance,asagents are willing to bail each other out in order to prevent the collapse of the entire system.Gale&Kariv(2007)study the process of exchange on financial networks and show that,whennetworks are incomplete,su