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Proc. R. Soc. A
doi:10.1098rspa.2009.0410
Published online
Contagion in financial networks
BY PRASANNA GAI1,2 AND SUJIT KAPADIA2,
1Crawford School of Economics and Government, Australian National University, Canberra, ACT 0200, Australia
2Bank of England, Threadneedle Street, London EC2R 8AH, UK
This paper develops an analytical model of contagion in financial networks with arbitrary structure. We explore how the probability and potential impact of contagion is influenced by aggregate and idiosyncratic shocks, changes in network structure and asset market liquidity. Our findings suggest that financial systems exhibit a robustyetfragile tendency: while the probability of contagion may be low, the effects can be extremely widespread when problems occur. And we suggest why the resilience of the system in withstanding fairly large shocks prior to 2007 should not have been taken as a reliable guide to its future robustness.
Keywords: contagion; network models; systemic risk; liquidity risk; financial crises
1. Introduction
In modern financial systems, an intricate web of claims and obligations links the balance sheets of a wide variety of intermediaries, such as banks and hedge funds, into a network structure. The advent of sophisticated financial products, such as credit default swaps and collateralized debt obligations, has heightened the complexity of these balance sheet connections still further. As demonstrated by the financial crisis, especially in relation to the failure of Lehman Brothers and the rescue of American International Group AIG, these interdependencies have created an environment for feedback elements to generate amplified responses to shocks to the financial system. They have also made it difficult to assess the potential for contagion arising from the behaviour of financial institutions under distress or from outright default.1
This paper models two key channels of contagion in financial systems by which default may spread from one institution to another. The primary focus is on how losses can potentially spread via the complex network of direct counterparty exposures following an initial default. But, as Cifuentes et al. 2005 and Shin 2008 stress, the knockon effects of distress at some financial institutions on 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 policymakers view of the recent trends in financial development and Haldane 2009 for a discussion of the role that the structure and complexities of the financial network have played in the financial turmoil of 20072009.
Received 5 August 2009
Accepted 9 February 2010 1 This journal is2010 The Royal Society
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their assets, and we also model the potential for this effect to trigger further rounds of default. Contagion due to the direct interlinkages of interbank claims and obligations may thus be reinforced by indirect contagion on the asset side of the balance sheetparticularly when the market for key financial system assets is illiquid.
The most wellknown contribution to the analysis of contagion through direct linkages in financial systems is that of AllenGale 2000.2 Using a network structure involving four banks, they demonstrate that the spread of contagion depends crucially on the pattern of interconnectedness between banks. When the network is complete, with all banks having exposures to each other such that the amount of interbank deposits held by any bank is evenly spread over all other banks, the impact of a shock is readily attenuated. Every bank takes a small hit and there is no contagion. By contrast, when the network is incomplete, with banks only having exposures to a few counterparties, the system is more fragile. The initial impact of a shock is concentrated among neighbouring banks. Once these succumb, the premature liquidation of longterm assets and the associated loss of value bring previously unaffected banks into the front line of contagion. In a similar vein, Freixas et al. 2000 show that tiered systems with moneycentre banks, where banks on the periphery are linked to the centre but not to each other, may also be susceptible to contagion.3
The generality of insights based on simple networks with rigid structures to realworld contagion is clearly open to debate. Moreover, while not being so stylized, models with endogenous network formation e.g. Leitner 2005; CastiglionesiNavarro 2007 impose strong assumptions that lead to stark predictions on the implied network structure that do not reflect the complexities of realworld financial networks. And, by and large, the existing literature fails to distinguish the probability of contagious default from its potential spread.
However, even prior to the current financial crisis, the identification of the probability and impact of shocks to the financial system was assuming centre stage in policy debate. Some policy institutions, for example, attempted to articulate the probability and impact of key risks to the financial system in their Financial stability reports.4 Moreover, the complexity of financial systems means that policymakers have only partial information about the true linkages between financial intermediaries. Given the speed with which shocks propagate, there is,
2Other strands of the literature on financial contagion have focused on the role of liquidity constraints KodresPritsker 2002, information asymmetries CalvoMendoza 2000 and wealth constraints KyleXiong 2001. As such, their focus is less on the nexus between network structure and financial stability. Network perspectives have also been applied to other topics in finance: for a comprehensive survey of the use of network models in finance, see AllenBabus 2009.
3These papers assume that shocks are unexpectedan approach we follow in our analysis. BruscoCastiglionesi 2007 model contagion in financial systems in an environment where contracts are written contingent on the realization of the liquidity shock. As in AllenGale 2000, they construct a simple network structure of four banks. They suggest, however, that greater connectivity could serve to enhance contagion risk. This is because the greater insurance provided by additional financial links may be associated with banks making more imprudent investments. And, with more links, if a banks gamble does not pay off, its failure has wider ramifications. 4See Bank of England 2007.
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Contagion in financial networks 3
therefore, a need to develop tools that facilitate analysis of the transmission of shocks through a given, but arbitrary, network structure. Recent events in the global financial system have only served to emphasize this.
Our paper takes up this challenge by introducing techniques from the literature on complex networks Strogatz 2001 into a financial system setting. Although this type of approach is frequently applied to the study of epidemiology and ecology, and despite the obvious parallels between financial systems and other complex systems that have been highlighted by prominent authors e.g. May et al. 2008 and policymakers e.g. Haldane 2009, the analytical techniques we use have yet to be applied to economic problems and thus hold out the possibility of novel insights.
In what follows, we draw on these techniques to model contagion stemming from unexpected shocks in complex financial networks with arbitrary structure and then use numerical simulations to illustrate and clarify the intuition underpinning our analytical results. Our framework explicitly accounts for the nature and scale of aggregate and idiosyncratic shocks and allows asset prices to interact with balance sheets. The complex network structure and interactions between financial intermediaries make for nonlinear system dynamics, whereby contagion risk can be highly sensitive to small changes in parameters. We analyse this feature of our model by isolating the probability and spread of contagion when claims and obligations are interlinked. In so doing, we provide an alternative perspective on the question of whether the financial system acts as a shock absorber or as an amplifier.
We find that financial systems exhibit a robustyetfragile tendency: while the probability of contagion may be low, the effects can be extremely widespread when problems occur. The model also highlights how a priori indistinguishable shocks can have very different consequences for the financial system, depending on the particular point in the network structure that the shock hits. This cautions against assuming that past resilience to a particular shock will continue to apply to future shocks of a similar magnitude. And it explains why the evidence of the resilience of the financial system to fairly large shocks prior to 2007 e.g. 911, the Dotcom crash and the collapse of Amaranth to name a few was not a reliable guide to its future robustness.
The intuition underpinning these results is straightforward. In a highly connected system, the counterparty losses of a failing institution can be more widely dispersed to, and absorbed by, other entities. So, increased connectivity and risk sharing may lower the probability of contagious default. But, conditional on the failure of one institution triggering contagious defaults, a high number of financial linkages also increases the potential for contagion to spread more widely. In particular, high connectivity increases the chances that institutions that survive the effects of the initial default will be exposed to more than one defaulting counterparty after the first round of contagion, thus making them vulnerable to a secondround 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 literature describes the behaviour of connected groups of nodes in a network and predicts the size of a susceptible cluster, i.e. the number of vulnerable nodes reached via the transmission of shocks along the links of the network. The approach
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relies on specifying all possible patterns of future transmission. Callaway et al. 2000, Newman et al. 2001 and Watts 2002 show how probabilitygenerating function techniques can identify the number of a randomly selected nodes first neighbours, second neighbours and so on. Recursive equations are constructed to consider all possible outcomes and obtain the total number of nodes that the original node is connected todirectly and indirectly. Phase transitions, which mark the thresholds for extensive contagious outbreaks, can then be identified.
In what follows, we construct a simple financial system involving entities with interlocking balance sheets and use these techniques to model the spread and probability of contagious default following an unexpected shock, analytically and numerically.5 Unlike the generic, undirected graph model of Watts 2002, our model provides an explicit characterization of balance sheets, making clear the direction of claims and obligations linking financial institutions. It also includes asset price interactions with balance sheets, allowing the effects of assetside contagion to be clearly delineated. We illustrate the robustyetfragile tendency of financial systems and analyse how contagion risk changes with capital buffers, the degree of connectivity and the liquidity of the market for failed banking assets.6
Our framework assumes that the network of interbank linkages forms randomly and exogenously: we leave aside issues related to endogenous network formation, optimal network structures and network efficiency.7 Although some realworld banking networks may exhibit coreperiphery structures and tiering see Boss et al. 2004 and Craigvon Peter in press for evidence on the Austrian and German interbank markets, respectively, the empirical evidence is limited and, given our theoretical focus, it does not seem sensible to restrict our analysis of contagion to particular network structures. In particular, our assumption that the network structure is entirely arbitrary carries the advantage that our model encompasses any structure that may emerge in the real world or as the optimal outcome of a network formation game. And it is a natural benchmark to consider.
We also model the contagion process in a relatively mechanical fashion, holding balance sheets and the size and structure of interbank linkages constant as default propagates through the system. Arguably, in normal times in developed financial systems, banks are sufficiently robust that very minor variations in their default probabilities do not affect the decision of whether or not to lend to them in
5EisenbergNoe 2001 demonstrate that, following an initial default in such a system, a unique vector that clears the obligations of all parties exists. However, they do not analyse the effects of network structure on the dynamics of contagion.
6Nier et al. 2007 also simulate the effects of unexpected shocks in financial networks, though they do not distinguish the probability of contagion from its potential spread and their results are strictly numericalthey do not consider the underlying analytics of the complex random graph network that they use. Recent work by MayArinaminpathy 2010 uses analytic meanfield approximations to offer a more complete explanation of their findings and also contrasts their results with those presented in this paper.
7See Leitner 2005, GaleKariv 2007, CastiglionesiNavarro 2007 and the survey by AllenBabus 2009 for discussion of these topics. Leitner 2005 suggests that linkages that create the threat of contagion may be optimal. The threat of contagion and the impossibility of formal commitments mean that networks develop as an ex ante optimal form of insurance, as agents are willing to bail each other out in order to prevent the collapse of the entire system. GaleKariv 2007 study the process of exchange on financial networks and show that, when networks are incomplete, substantial costs of intermediation can arise and lead to uncertainty of trade as well as market breakdowns.
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Contagion in financial networks 5
interbank markets. Meanwhile, in crises, contagion spreads very rapidly through the financial system, meaning that banks are unlikely to have time to alter their behaviour before they are affectedas such, it may be appropriate to assume that the network remains static. Note also that banks have no choice over whether they default. This precludes the type of strategic behaviour discussed by Morris 2000, JacksonYariv 2007 and GaleottiGoyal 2009, whereby nodes can choose whether or not to adopt a particular state e.g. adopting a new technology.
Our approach has some similarities to the epidemiological literature on the spread of disease in networks see AndersonMay 1991; Newman 2002; JacksonRogers 2007; or the overview by Meyers 2007. But there are two key differences. First, in epidemiological models, the susceptibility of an individual to contagion from a particular infected neighbour does not depend on the health of their other neighbours. By contrast, in our setup, contagion to a particular institution following a default is more likely to occur if another of its counterparties has also defaulted. Second, in most epidemiological models, higher connectivity simply creates more channels of contact through which infection could spread, increasing the potential for contagion. In our setting, however, greater connectivity also provides counteracting risksharing benefits as exposures are diversified across a wider set of institutions.
Another strand of related literature e.g. DavisLo 2001; FreyBackhaus 2003; Giesecke 2004; GieseckeWeber 2004; CossinSchellhorn 2007; Egloff et al. 2007 considers default correlation and credit contagion among firms, often using reducedform credit risk models. In contrast to these papers, clearly specified bank balance sheets are central to our approach, with bilateral linkages precisely defined with reference to these. And our differing modelling strategy, which focuses on the transmission of contagion along these links, reflects the greater structure embedded in our network setup.
The structure of the paper is as follows. Section 2 describes the structure of the financial network, the transmission process for contagion and analytical results characterizing a default cascade. Section 3 uses numerical simulations to study the effects of failures of individual institutions and to articulate the likelihood and extent of contagion. Section 4 considers the impact of liquidity effects on system stability. Section 5 discusses points of contact with the empirical literature on interbank contagion being pursued by central banks. The final section concludes.
2. The model
a Network structure
Consider a financial network in which n financial intermediaries, banks for short, are randomly linked together by their claims on each other. In the language of graph theory, each bank represents a node on the graph and the interbank exposures of bank i define the links with other banks. These links are directed and weighted, reflecting the fact that interbank exposures comprise assets as well as liabilities and that the size of these exposures is important for contagion analysis. Figure 1 shows an example of a directed, weighted financial network in which there are five banks, with darker lines corresponding to higher value links.
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Figure 1. A weighted, directed network with five nodes.
A crucial property of graphs such as those in figure 1 is their degree distribution. In a directed graph, each node has two degrees, an indegree, the number of links that point into the node, and an outdegree, which is the number pointing out. Incoming links to a node or bank reflect the interbank assetsexposures of that bank, i.e. monies owed to the bank by a counterparty. Outgoing links from a bank, by contrast, correspond to its interbank liabilities. In what follows, the joint distribution of in and outdegree governs the potential for the spread of shocks through the network.
For reasons outlined above, our analysis takes this joint degree distribution, and hence the structure of the links in the network, as being entirely arbitrary, though a specific distributional assumption is made in our numerical simulations in 3. This implies that the network is entirely random in all respects other than its degree distribution. In particular, there is no statistical correlation between nodes and mixing between nodes is proportionate i.e. there is no statistical tendency for highly connected nodes to be particularly connected with other highly connected nodes or with poorly connected nodes.
Suppose that the total assets of each bank consist of interbank assets, AIB, i
and illiquid external assets, such as mortgages, AMi . Further, let us assume that
the total interbank asset position of every bank is evenly distributed over each of
its incoming links and is independent of the number of links the bank has if a
bank has no incoming links, AIB0 for that bank. Although these assumptions i
are stylized, they provide a useful benchmark that emphasizes the possible benefits of diversification and allows us to highlight the distinction between risk sharing and risk spreading within the financial network. In particular, they allow us to show that widespread contagion is possible even when risk sharing in the system is maximized. We consider the implications of relaxing these assumptions in 2e.
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Contagion in financial networks 7
Since every interbank asset is another banks liability, interbank liabilities, LIB, i
are endogenously determined. Apart from interbank liabilities, we assume that the only other component of a banks liabilities is exogenously given customer deposits, Di. The condition for bank i to be solvent is therefore
1fAIB qAM LIB Di 0, 2.1 iii
where f is the fraction of banks with obligations to bank i that have defaulted, and q is the resale price of the illiquid asset.8 The value of q may be less than 1 in the event of asset sales by banks in default, but equals 1 if there are no fire sales. We make a zero recovery assumption, namely that, when a linked bank defaults, bank i loses all of its interbank assets held against that bank.9 The solvency condition can also be expressed as
fKi1qAMi , forAIB0, 2.2 AIB i
i
where KiAIBAMLIBDi is the banks capital buffer, i.e. the difference iii
between the book value of its assets and liabilities.10
To model the dynamics of contagion, we suppose that all banks in the network
are initially solvent and that the network is perturbed at time t1 by the initial default of a single bank. Although purely idiosyncratic shocks are rare, the crystallization of operational risk e.g. fraud has led to the failure of financial institutions in the past e.g. Barings Bank. Alternatively, bank failure may result from an aggregate shock that has particularly adverse consequences for one institution: this can be captured in the model through a general erosion in the stock of illiquid assets or, equivalently, capital buffers across all banks, combined with a major loss for one particular institution.
Let ji denote the number of incoming links for bank i the indegree. Since linked banks each lose a fraction 1ji of their interbank assets when a single counterparty defaults, it is clear from equation 2.2 that the only way default can spread is if there is a neighbouring bank for which
Ki1qAMi 1. 2.3 AIB ji
i
We define banks that are exposed in this sense to the default of a single neighbour as vulnerable and other banks as safe. The vulnerability of a bank clearly depends
8A regulatory requirement for banks to maintain capital above a certain level at all times
could easily be incorporated into the model by rewriting the solvency condition to require that
1fAIBqAMLIBDi exceeds a positive constant. This would not fundamentally alter the iii
analysis.
9This assumption is likely to be realistic in the midst of a crisis: in the immediate aftermath of a default, the recovery rate and the timing of recovery will be highly uncertain and banks funders are likely to assume the worstcase scenario. Nevertheless, in our numerical simulations, we show that our results are robust to relaxing this assumption.
10Formally, this capital buffer is taken to be a random variablethe underlying source of its variability may be viewed as being generated by variability in Di, drawn from its appropriate distribution. For notational simplicity, we do not explicitly denote this dependence of Ki on Di in the subsequent expressions.
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on its indegree. Specifically, recalling that the capital buffer is taken to be a random variable see footnote 10, a bank with indegree j is vulnerable with
probability
Ki 1qAMi 1
yjP AIBj j1. 2.4
i
Further, the probability of a bank having indegree j, outdegree k and being vulnerable is yjpjk , where pjk is the joint degree distribution of in and outdegree. The model structure described by equations 2.12.4 captures several features of interest in systemic risk analysis. First, as noted above, the nature and scale of adverse aggregate or macroeconomic events can be interpreted as a negative shock to the stock of illiquid assets, AMi , or, equivalently, to the capital buffer, Ki. Second, idiosyncratic shocks can be modelled by assuming the exogenous default of a bank. Third, the structural characteristics of the financial system are described by the distribution of interbank linkages, pjk. And, finally, liquidity effects associated with the potential knockon effects of default on asset prices are captured by allowing q to vary. To keep matters simple, we initially fix
q1, returning later to endogenize it.
b Generating functions and the transmission of shocks
In sufficiently large networks, for contagion to spread beyond the first neighbours of the initially defaulting bank, those neighbours must themselves have outgoing links i.e. liabilities to other vulnerable banks.11 We therefore define the probabilitygenerating function for the joint degree distribution of a vulnerable bank as
yj pjk xj yk. 2.5
The generating function contains all the same information that is contained in the degree distribution, pjk , and the vulnerability distribution, yj , but in a form that allows us to work with sums of independent draws from different probability distributions. Specifically, for our purposes, it generates all the moments of the degree distribution of only those banks that are vulnerable.12
Since every interbank asset of one bank is an interbank liability of another,
Gx,y
j,k
every outgoing link for one node is an incoming link for another node. This
means that the average indegree in the network, 1nji jpjk , must i j,k
11If the number of nodes, n, is sufficiently large, banks are highly unlikely to be exposed to more than one failed bank after the first round of contagion, meaning that safe banks will never fail in the second round. This assumption clearly breaks down either when n is small or when contagion spreads more widely. However, the logic of this section still holds in both cases: in the former, the exact solutions derived for large n will only approximate reality this is confirmed by the numerical results in 3; in the latter, the exact solutions will apply but the extent of contagion will be affected, as discussed further in 2d.
12For a description of the key properties of probabilitygenerating functions, including those that are used in this paper, see GrimmettWelsh 1986; for a more advanced treatment, see Wilf 1994.
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Contagion in financial networks 9
equal the average outdegree, 1nki kpjk . We refer to this quantity as i j,k
the average degree and denote it by
zjpjkkpjk . 2.6 j,k j,k
From Gx,y, we can define a singleargument generating function, G0y, for the number of links leaving a randomly chosen vulnerable bank. This is given by
Note that
G0yG1, y
yj pjk yk.
G1, 1G01
yj pjk,
2.7
2.8
j,k
so that G01 yields the fraction of banks that are vulnerable.
We can also define a second singleargument generating function, G1y, for the number of links leaving a bank reached by following a randomly chosen incoming link. Because we are interested in the propagation of shocks from one bank to another, we require the degree distribution, yjrjk , of a vulnerable bank that is
a random neighbour of our initially chosen bank.
At this point, it is important to note that this is not the same as the degree
distribution of vulnerable banks on the network as a whole. This is because a bank with a higher indegree has a greater number of links pointing towards it, meaning that there is a higher chance that any given outgoing link will terminate at it, in precise proportion to its indegree. Therefore, the larger the indegree of a bank, the more likely it is to be a neighbour of our initially chosen bank, with the probability of choosing it being proportional to jpjk.13 The generating function for the number of links leaving a vulnerable neighbour of a randomly chosen vulnerable bank is thus given by
j,k
yjjpjkyk k j,k
G1yyjrjkyjp . 2.9 j,k j,k jk
Now suppose that we follow a randomly chosen outgoing link from a vulnerable bank to its end and then to every other vulnerable bank reachable from that end. We refer to this set of banks as the outgoing vulnerable cluster at the end of a randomly chosen outgoing link from a vulnerable bank. Because it captures links between vulnerable banks, the size and distribution of the vulnerable cluster characterize how default spreads across the financial network following an initial failure.
13This point is discussed in more detail in the context of undirected graphs by Feld 1991, Newman et al. 2001 and Newman 2003.
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svvvv…
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v
Figure 2. Transmission of contagion implied by equation 2.10.
As figure 2 illustrates, each vulnerable cluster represented by a square in the figure can take many different forms see also Newman 2003. We can follow a randomly chosen outgoing link and find a single bank at its end with no further outgoing connections emanating from it. This bank may be safe s or vulnerable v. Or we may find a vulnerable bank with one, two or more links emanating from it to further clusters. At this point, we assume that the links emanating from the defaulting node are treelike and contain no cycles or closed loops. This is solely to make an exact solution possible: the thrust of the argument goes through without this restriction and we do not apply it when conducting our numerical simulations in 3.
Let H1y be the generating function for the probability of reaching an outgoing vulnerable cluster of given size in terms of numbers of vulnerable banks by following a random outgoing link from a vulnerable bank. As shown in figure 2, the total probability of all possible forms can be represented selfconsistently as the sum of probabilities of hitting a safe bank, hitting only a single vulnerable bank, hitting a single vulnerable bank connected to one other cluster, two other clusters and so on. Each cluster that may be arrived at is independent. Therefore, H1y satisfies the following selfconsistency condition:
yjrjkH1yk , 2.10
where the leading factor of y accounts for the one vertex at the end of the initial edge and we have used the fact that if a generating function generates the probability distribution of some property of an object, then the sum of that property over m independent such objects is distributed according to the mth power of the generating function. By using equation 2.9 and noting that G11 represents the probability that a random neighbour of a vulnerable bank is vulnerable, we may write equation 2.10 in implicit form as
H1y1G11yG1H1y. 2.11
It remains to establish the distribution of outgoing vulnerable cluster sizes to which a randomly chosen bank belongs. There are two possibilities that can arise. First, a randomly chosen bank may be safe. Second, it may have indegree j and outdegree k , and be vulnerable, the probability of which is yjpjk . In this second
H1yPrreach safe banky
j,k
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Contagion in financial networks 11
case, each outgoing link leads to a vulnerable cluster whose size is drawn from the distribution generated by H1y. So, the size of the vulnerable cluster to which a randomly chosen bank belongs is generated by
vjpjkH1yk
1G01yG0H1y. 2.12
And, in principle, we can calculate the complete distribution of vulnerable cluster sizes by solving equation 2.11 for H1y and substituting the result into equation 2.12.
c Phase transitions
Although it is not usually possible to find a closedform expression for the complete distribution of cluster sizes in a network, we can obtain closed form expressions for the moments of its distribution from equations 2.11 and 2.12. In particular, the average vulnerable cluster size, S, is given by
SH01. 2.13 Noting that H1y is a standard generating function so that H111, it follows
H0yPrbank safey
from equation 2.12 that
H01G0H11G0 H11H11
G01G0 1H11. And we know from equation 2.11 that
H1 G11 . 1 1G11
So, substituting equation 2.15 into 2.14 yields
SG01G0 1G11.
1G11
2.14
2.15
2.16
j,k
From equation 2.16, it is apparent that the points that mark the phase transitions at which the average vulnerable cluster size diverges are given by
or, equivalently, by
j,k
G1 11, jkvj pjk z,
2.17 2.18
where we have used equations 2.6 and 2.9.
The term G11 is the average outdegree of a vulnerable first neighbour,
counting only those links that end up at another vulnerable bank. If this quantity is less than 1, all vulnerable clusters are small and contagion dies out quickly since the number of vulnerable banks reached declines. But if G1 1 is greater than 1, a giant vulnerable clustera vulnerable cluster whose size scales linearly with the size of the whole networkexists and occupies a finite fraction of the network. In
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this case, systemwide contagion is possible: with positive probability, a random initial default at one bank can lead to the spread of default across the entire vulnerable portion of the financial network.
As the average degree, z, increases, typical in and outdegrees increase, so that more of the mass of pjk is at higher values for j and k. This increases the lefthand side of equation 2.18 monotonically through the jk term but reduces it through the vj term as vj is lower for higher j from equation 2.4. So, equations 2.17 and 2.18 will either have two solutions or none at all. In the first case, there are two phase transitions and a continuous window of intermediate values of z for which contagion is possible. For values of z that lie outside the window and below
jkpjk term is too small and the network is
the lower phase transition, the
j,k
insufficiently connected for contagion to spread consider what would happen in
a network with no links; for values of z outside the window and above the upper phase transition, the vj term is too small and contagion cannot spread because there are too many safe banks.
dThe probability and spread of contagion
From a system stability perspective, we are primarily interested in contagion within the giant vulnerable cluster. This only emerges for intermediate values of z, and only when the initially defaulting bank is either in the giant vulnerable cluster or directly adjacent to it. The likelihood of contagion is, therefore, directly linked to the size of the vulnerable cluster within the window.14 Intuitively, near both the lower and upper phase transitions, the probability of contagion must be close to zero since the size of the vulnerable cluster is either curtailed by limited connectivity or by the presence of a high fraction of safe banks. The probability of contagion is thus nonmonotonic in z: initially, the riskspreading effects stemming from a more connected system will increase the size of the vulnerable cluster and the probability of contagion; eventually, however, risksharing effects that serve to reduce the number of vulnerable banks dominate, and the probability of contagion falls.15
At the minimum, the conditional spread of contagion i.e. conditional on contagion breaking out must correspond to the size of the giant vulnerable cluster. But once contagion has spread through the entire vulnerable cluster, the assumption that banks are adjacent to no more than one failed bank breaks down. So, safe banks may be susceptible to default and contagion can spread well beyond the vulnerable cluster to affect the entire connected component of the network. Near the lower phase transition, z is sufficiently low that nearly all banks are likely to be vulnerable. Therefore, in this region, the size of the giant vulnerable cluster corresponds closely to the size of the connected component of
14Note that this is not given by equation 2.16 since this equation is derived on the assumption that there are no cycles connecting subclusters. This will not hold in the giant vulnerable cluster. 15In the special case of a uniform Poisson random graph in which each possible link is present with independent probability p, an analytical solution for the size of the giant vulnerable cluster can be obtained using techniques discussed in Watts 2002 and Newman 2003. Since this does not account for the possibility of contagion being triggered by nodes directly adjacent to the vulnerable cluster, it does not represent an analytical solution for the probability of contagion. However, it highlights that the size of the giant vulnerable cluster, and hence the probability of contagion, is nonmonotonic in z.
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Contagion in financial networks 13
the network, meaning that the fraction of the network affected by episodes of contagion is roughly similar to the probability that contagion breaks out. But these quantities diverge as z increases, and, near the upper phase transition, the system will exhibit a robustyetfragile tendency, with episodes of contagion occurring rarely, but spreading very widely when they do take place.
From equation 2.18, the size of the contagion window is larger if, for a given j, the probability that a bank is vulnerable, vj, is larger. Greater levels of vulnerability also increase the size of the giant vulnerable cluster and, hence, the probability of contagion within the range of intermediate z values. Therefore, it is clear from equation 2.4 that an adverse shock that erodes capital buffers will both increase the probability of contagion and extend the range of z for which contagious outbreaks are possible.
e Relaxing the diversification assumptions
In our presentation of the model, we assumed that the total interbank asset position of each bank was independent of the number of incoming links to that bank and that these assets were evenly distributed over each link. In reality, we might expect a bank with a higher number of incoming links to have a larger total interbank asset position. Intuitively, this would curtail the risksharing benefits of greater connectivity because the greater absolute exposure associated with a higher number of links would partially offset the positive effects from greater diversification. But, as long as the total interbank asset position increases less than proportionately with the number of links, all of our main results continue to apply. In particular, vj will still decrease in z, though at a slower rate. As a result, equation 2.18 will continue to generate two solutions, though in an extended range of cases. The contagion window will thus be wider. On the other hand, if the total interbank asset position increases more than proportionately with the number of links, vj will increase in z and greater connectivity will unambiguously increase contagion risk. This latter case does not seem a particularly plausible description of reality.
Assuming an uneven distribution of interbank assets over incoming links would not change any of our fundamental results. In particular, vj would still decrease in z, maintaining the possibility of two solutions to equation 2.18. But an uneven distribution of exposures would make banks vulnerable to the default of particular counterparties for higher values of z than would otherwise be the case. As a result, the contagion window will be wider.
3. Numerical simulations
a Methodology
To illustrate our results, we calibrate the model and simulate it numerically. Although the findings apply to random graphs with arbitrary degree distributions, we assume a uniform Poisson random graph in which each possible directed link in the graph is present with independent probability p. In other words, the network is constructed by looping over all possible directed links and choosing each one to be present with probability pnote that this algorithm does not preclude the possibility of cycles in the generated network and thus encompasses
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all of the structures considered by AllenGale 2000. The Poisson random graph was chosen for simplicity given the primary focus of this section in empirically confirming our theoretical results; conducting the simulation analysis under different joint degree distributions would be a useful extension but is left for future work.
Consistent with bankruptcy law, we do not net interbank positions, so it is possible for two banks to be linked with each other in both directions. The average degree, z, is allowed to vary in each simulation. And although our model applies to networks of fully heterogeneous financial intermediaries, we take the capital buffers and asset positions on banks balance sheets to be identical.16
As a benchmark, we consider a network of 1000 banks. Clearly, the number of financial intermediaries in a system depends on how the system is defined and what counts as a financial intermediary. But several countries have banking networks of this size, and a figure of 1000 intermediaries also seems reasonable if we are considering a global financial system involving investment banks, hedge funds and other players.
The initial assets of each bank are chosen so that they comprise 80 per cent external nonbank assets and 20 per cent interbank assetsthe 20 per cent share of interbank assets is broadly consistent with the figures for developed countries reported by Upper 2007. Banks capital buffers are set at 4 per cent of total nonriskweighted assets, a figure calibrated from data contained in the 2005 published accounts of a range of large, international financial institutions. Since each banks interbank assets are evenly distributed over its incoming links, interbank liabilities are determined endogenously within the network structure. And the liability side of the balance sheet is topped up by customer deposits until the total liability position equals the total asset position.
In the experiments that follow, we draw 1000 realizations of the network for each value of z. In each of these draws, we shock one bank at random, wiping out all of its external assetsthis type of idiosyncratic shock may be interpreted as a fraud shock. The failed bank defaults on all of its interbank liabilities. As a result, neighbouring banks may also default if their capital buffer is insufficient to cover their loss on interbank assets. Any neighbouring banks that fail are also assumed to default on all of their interbank liabilities, and the iterative process continues until no new banks are pushed into default.
Since we are only interested in the likelihood and conditional spread of system wide contagion, we wish to exclude very small outbreaks of default outside the giant vulnerable cluster from our analysis. So, when calculating the probability and conditional spread of contagion, we only count episodes in which over 5 per cent of the system defaults. As well as being analytically consistent on the basis of numerical simulations, a 5 per cent failure rate seems a suitable lower bound for defining a systemic financial crisis.
16 With heterogeneous banks, the critical Ki AIB ratio, which determines vulnerability in i
equation 2.4, would vary across banks. In his undirected framework, Watts 2002 shows that, when thresholds such as this are allowed to vary, the qualitative theoretical results continue to apply but the contagion window is wider. Intuitively, with heterogeneity, some banks remain vulnerable even when relatively well connected because they have low capital buffers relative to their interbank asset position. Therefore, incorporating bank heterogeneity into our numerical simulations would simply widen the contagion window. See also Iori et al. 2006 for a discussion of how bank heterogeneity may increase contagion risk.
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Contagion in financial networks 15
1.0
0.8
0.6
0.4
0.2
contagion window
0 1 2 3 4 5 6 7 8 9 10 average degree i.e. connectivity
Figure 3. The benchmark case. Grey circle, extent of contagion; cross, frequency of contagion.
b Results
Figure 3 summarizes the benchmark case. In this and all subsequent diagrams, the extent of contagion measures the fraction of banks that default, conditional on contagion over the 5 per cent threshold breaking out.
The benchmark simulation confirms the results and intuition of 2c,d. Contagion only occurs within a certain window of z. Within this range, the probability of contagion is nonmonotonic in connectivity, peaking at approximately 0.8 when z is between 3 and 4. As noted above, the conditional spread of contagion as a fraction of network size is approximately the same as the frequency of contagion near the lower phase transitionin this region, contagion breaks out when any bank in, or adjacent to, the giant vulnerable cluster is shocked and spreads across the entire cluster, which roughly corresponds to the entire connected component of the network.
For higher values of z, however, a large proportion of banks in the network fail when contagion breaks out. Of particular interest are the points near the upper phase transition: when z8, contagion never occurs more than five times in 1000 draws; but in each case where it does break out, every bank in the network fails. This highlights that a priori indistinguishable shocks to the network can have vastly different consequences for contagion.
In figure 4, we compare our benchmark results with the limiting case, since our analytical results only strictly apply in the limit as n. Watts 2002 notes that numerical results in random graph models approximate analytical solutions in the vicinity of n10 000. Figure 4 demonstrates that a smaller number of nodes in the benchmark simulation does not fundamentally affect the results: the contagion window is widened slightly, but the qualitative results of the analytical model remain intact.
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1.0
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0.4
0.2
0
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Figure 4. Benchmark and analytical solutions compared. Cross, frequency of contagion 1000 banks; open circle, extent of contagion 1000 banks; grey line, frequency of contagion 10000 banks, smoothed fit; black line, extent of contagion 10 000 banks, smoothed fit.
Figure 5 considers the effects of varying banks capital buffers. As expected, an erosion of capital buffers both widens the contagion window and increases the probability of contagion for fixed values of z.17 For small values of z, the extent of contagion is also slightly greater when capital buffers are lower but, in all cases, it reaches one for sufficiently high values of z. When the capital buffer is increased to 5 per cent, however, this occurs well after the peak probability of contagion. This neatly illustrates how increased connectivity can simultaneously reduce the probability of contagion but increase its spread conditional on it breaking out.
Figure 6 illustrates how changes in the average degree and capital buffers jointly affect the expected number of defaults in the system. Since this diagram does not isolate the probability of contagion from its potential spread, rare but highimpact events appear in the benign flat region as the expected number of defaults in these cases is low. Figure 6 serves to highlight another nonlinear feature of the system: when capital buffers are eroded to critical levels, the level of contagion risk can increase extremely rapidly.
Finally, in figure 7, we relax the zero recovery assumption. Instead, we assume that, when a bank fails, its default in the interbank market equals its asset shortfall i.e. its outstanding loss after its capital buffer is absorbed plus half of any remaining interbank liabilities, where the additional amount is interpreted as reflecting bankruptcy costs that are lost outside the system.18 As we might expect,
17Reduced capital buffers may also increase the likelihood of an initial default. Therefore, they may contribute to an increased probability of contagion from this perspective as well.
18Since interbank assets make up 20 per cent of each banks total asset position, interbank liabilities must, on average, make up 20 per cent of total liabilities. Therefore, for the average bank, if we take insured customer deposits as senior, the maximum bankruptcy cost under this assumption is 10 per cent of total assetsliabilities, which accords with the empirical estimates of bankruptcy costs in the banking sector reported by James 1991.
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Contagion in financial networks 17
1.0
0.8
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0.4
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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 average degree i.e. connectivity
Figure 5. Varying the capital buffer. Plus, frequency of contagion 3 capital buffer; diamond, extent of contagion 3 capital buffer; cross, frequency of contagion 4 capital buffer; filled circle, extent of contagion 4 capital buffer; open triangle, frequency of contagion 5 capital buffer; filled triangle, extent of contagion 5 capital buffer.
1000 800 600
400 200
0
20 15
2
4
10 5
6
80
Figure 6. Connectivity, capital buffers and the expected number of defaults.
this reduces the likelihood of contagion because fewer banks are vulnerable when the recovery rate can be positive. But it is also evident that relaxing the zero recovery assumption does not fundamentally affect our broad results.
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capital buffer
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1.0 0.8 0.6 0.4 0.2
0
1 2 3 4 5 6 7 8 9 10
average degree i.e. connectivity
Figure 7. Relaxing the zero recovery rate assumption. Cross, frequency of contagion benchmark case; filled circle, extent of contagion benchmark case; plus, frequency of contagion positive recovery rate; diamond, extent of contagion positive recovery rate.
c Interpretation and discussion
Contagious crises occur infrequently in developed countries, suggesting that financial systems are located near to the upper phase transition of our model. The findings of Soramaki et al. 2007, who report average degrees in financial systems of 15, are consistent with this. But recent events have demonstrated that financial systems are prone to occasional systemwide breakdown, with policymakers intervening to limit the extent of contagion.
Our framework implies that financial systems exhibit a robustyetfragile tendency. Although the likelihood of contagion may be low, its impact can be extremely widespread. Moreover, even if contagion from idiosyncratic shocks never occurs when banks have relatively high capital buffers, figure 5 highlights that if an adverse aggregate shock, such as the macroeconomic downturn we are currently experiencing, erodes capital buffers, the system could be susceptible to contagion risk.
A priori indistinguishable shocks also have vastly different consequences in our model. Although the system may be robust to most shocks of a given size, if it is hit by a similarly sized shock at a particular pressure point, possibly reflecting a structural weakness, the ensuing financial instability could be significant. This result cautions against assuming that the resilience of a financial system to large shocks at some point in the past will continue to apply to future shocks of a similar magnitude.
4. Liquidity risk
We now incorporate liquidity effects into our analysis. When a bank fails, financial markets may have a limited capacity to absorb the illiquid external assets that are sold. As a result, the asset price may be depressed. Following SchnabelShin
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Figure 8. Liquidity effects and contagion. Cross, frequency of contagion; filled circle, extent of contagion; plus, frequency of contagion with liquidity effects; diamond, extent of contagion with liquidity effects.
2004 and Cifuentes et al. 2005, suppose that the price of the illiquid asset, q, is given by
qeax , 4.1
where x0 is the fraction of system illiquid assets that have been sold in the market if assets are not being sold in the market, q1. We calibrate a so that the asset price falls by 10 per cent when onetenth of system assets have been sold.
We integrate this pricing equation into our numerical simulations. Specifically, when a bank defaults, all of its external assets are sold in the market, reducing the asset price according to equation 4.1. We assume that, when the asset price falls, the external assets of all other banks are markedtomarket to reflect the new asset price. From equation 2.4, it is clear that this will reduce banks capital buffers and has the potential to make some banks vulnerable, possibly ultimately tipping them into default.
The incorporation of market liquidity risk introduces a second potential source of contagion into the model from the asset side of banks balance sheets. Note, however, that this liquidity risk only materializes upon default. Realistically, asset prices are likely to be depressed by asset sales before any bank defaults. So, accounting only for the postdefault impact probably understates the true effects of liquidity risk.
Figure 8 illustrates the effects of incorporating liquidity risk into the model. As we might expect, liquidity effects magnify the extent of contagion when it breaks out. The contagion window also widens.
As shown, liquidity effects do not drastically alter the main results of our model. But this should not be taken to mean that liquidity effects are unimportant. In part, the limited effect of liquidity risk reflects the already high spread of contagion embedded in the benchmark scenario. As demonstrated by
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MayArinaminpathy 2010 in a similar setup, liquidity effects can be more material if recovery rates upon interbank default are relatively high. And, to the extent that liquidity risk materializes before any bank defaults, it can be viewed as having the potential to erode capital buffers and increase the likelihood of an initial default.
5. Relationship with the empirical literature
There is a large empirical literature that uses counterfactual simulations to assess the danger of contagion in a range of national banking systems see Upper 2007 for a comprehensive survey. This literature has largely tended to use actual or estimated data on interbank lending to simulate the effects of the failure of an individual bank on financial stability.19 The evidence of contagion risk from idiosyncratic shocks is mixed. Furfine 2003 and Wells 2004 report relatively limited scope for contagion in the US and UK banking systems. By contrast, UpperWorms 2004 and van LelyveldLiedorp 2006 suggest that contagion risk may be somewhat higher in Germany and The Netherlands. Meanwhile, Mistrullis 2007 results for the Italian banking system echo the findings of this paper: he finds that, while only a relatively low fraction of banks can trigger contagion, large parts of the system are affected in worstcase scenarios. Moreover, he shows that, when moving from an analysis of actual bilateral exposures which form an incomplete network to a complete structure estimated using maximum entropy techniques, the probability of contagion from a random, idiosyncratic bank failure is reduced but its spread is sometimes widened.
Contagion owing to aggregate shocks is examined by Elsinger et al. 2006, who combine a model of interbank lending in the Austrian banking system with models of market and credit risk. They take draws from a distribution of risk factors and compute the effects on banks solvency, calculating the probability and the severity of contagion. Their findings also echo the results reported in our paper. While contagious failures are relatively rare, if contagion does occur, it affects a large part of the banking system.
Counterfactual simulations have also been used to assess how changes in the structure of interbank loan markets affect the risk of contagion. But these results do not show a clear relationship. Mistrulli 2005 and DegryseNguyen 2007 consider how contagion risk has evolved in Italy and Belgium as their banking structures have shifted away from a comparatively complete graph structure towards one with multiple moneycentre banks. Their findings suggest that, while this shift appears to have reduced contagion risk in Belgium, the possibility of contagion risk in Italy appears to have increased.
As noted by Upper 2007, existing empirical studies are plagued by data problems, and the extent to which reported interbank exposures reflect true linkages is unclear: generally, interbank exposures are reported only on a particular day once a quarter and exclude a range of items, including intraday exposures. As such, they underestimate the true scale of financial connectivity. Moreover, national supervisory authorities do not generally receive information on the exposures of foreign banks to domestic institutions, making it difficult to model the risk of global contagion in the increasingly international financial
19A parallel literature explores contagion risk in payment systems see Angelini et al. 1996. Proc. R. Soc. A
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Contagion in financial networks 21
system. And studies attempting to analyse the effects of changes in network structure on contagion risk are constrained by short time series for the relevant data series.
6. Conclusion
In this paper, we develop a model of contagion in arbitrary financial networks that speaks to concerns about the widespread transmission of shocks in an era of rapid financial globalization and in the wake of a major systemic financial crisis. Our model applies broadly to systems of agents linked together by their financial claims on each other, including through interbank markets and payment systems. While high connectivity may reduce the probability of contagion, it can also increase its spread when problems occur. Adverse aggregate shocks and liquidity risk also amplify the likelihood and extent of contagion.
Our results suggest that financial systems may exhibit a robustyetfragile tendency. They also highlight how a priori indistinguishable shocks can have vastly different consequences, which helps explain why the evidence of the resilience of the system to fairly large shocks prior to 2007 was not a reliable guide to its future robustness.
The approach provides a first step towards modelling contagion risk when true linkages are unknown. It would be useful to extend the simulation analysis by relaxing the assumption that the defaulting bank is randomly selected and, along the lines of Albert et al. 2000, considering the implications of targeted failure affecting big or highly connected interbank borrowers. This would be particularly interesting in a setup in which the joint degree distribution was calibrated to match observed data. Added realism could also be incorporated into the model by using real balance sheets for each bank or endogenizing the formation of the network. Extending the model in this direction could help guide the empirical modelling of contagion risk and is left for future work.
This paper represents the views of the authors and should not be thought to represent those of the Bank of England or Monetary Policy Committee members. We thank Emma Mattingley, Nick Moore, Barry Willis and, particularly, Jason Dowson for excellent research assistance. We are also grateful to Kartik Anand, Fabio Castiglionesi, Geoff Coppins, Avinash Dixit, John Driffill, Sanjeev Goyal, Andy Haldane, Simon Hall, Matteo Marsili, Robert May, Marcus Miller, Emma Murphy, Filipa Sa, Nancy Stokey, Merxe Tudela, Jing Yang, three anonymous referees and seminar participants at the Bank of England, the University of Oxford, the University of Warwick research workshop and conference on World Economy and Global Finance Warwick, 1115 July 2007, the UniCredit Group Conference on Banking and Finance: Span and Scope of Banks, Stability and Regulation Naples, 1718 December 2007, the 2008 Royal Economic Society Annual Conference Warwick, 1719 March 2008 and the 2008 Southern Workshop in Macroeconomics Auckland, 2830 March 2008 for helpful comments and suggestions.
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