gai - systemic risk; the dynamics of modern financial systems (2013)

2.1 An example of a weighted, directed, network 123.2 Systemic liquidity hoarding in a Poisson network single random idiosyncratic haircut shock; aggregate haircut shock and single rando

Systemic Risk

The Dynamics of Modern Financial Systems

Prasanna Gai

1

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In March 2000, just as the NASDAQ index was hitting its peak, I began work as

an academic consultant to a small group of staff at the Bank of Englandcharged with formulating conceptual frameworks for thinking aboutfinancialstability, led by Andy Haldane and with Prasanna Gai as a key member Theworld seemed more tranquil back then and financial crises seemed to beafflictions of emerging economies with poor governance, although the events

of the summer of 1998 associated with the LTCM crisis gave us a glimpse thatfinancial turmoil did not respect the neat taxonomy of developed andemerging economies But the prevailing sentiment back then was thatfinan-cial crises were exceptional, worthy of study in the same way that doctorsstudy a rare disease, but something that remained properly in the laboratoryrather than something that might impinge on the day to day decisions ofeconomic policymakers

Our group at the Bank of England ploughed on, nevertheless We noticedearly on that the traditional ‘domino’ models of contagion fell far short ofsounding adequate alarm bells against potential fragility The domino modelworks through cascading defaults whereby, if Bank A has borrowed fromBank B, while Bank B has borrowed from Bank C, and so on, then a shock

to Bank A’s assets that leads to default will hit Bank B, and if the hit is bigenough, Bank B’s solvency will be impaired, in which case Bank C would behit, and so on The trouble was that such ‘domino’ models almost nevergenerate systemic crises, as balance sheet interconnections in real life arerarely large enough in practice to topple banks at the second or third round

of defaults These models gave a false sense of security to policymakers andsuggested that systemic crises were remote possibilities that would not leavethe laboratory

However, the blind spot in the domino model was that banks are assumedpassive observers, and banks further down the chain stand idly by while banksstart toppling further up the chain More realistically, Bank C would berunning long before Bank A’s collapse leads to losses for Bank B It was therun on the bank, not the hit on the solvency of the bank that would lead to themost acute systemic problems This is a lesson that was driven home repeat-edly in the recent crisis Nor did the domino models take account of valuation

effects where falling prices interacted with marked-to-market constraints to setoff the ‘death spiral’ of leveraged institutions that became such a familiarfeature of the recent crisis Although we were aware of the potential impact

of these channels, we did not quite realize how potent they would prove to beonce the monetary policy frameworks at advanced economy central banksthat neglectedfinancial stability concerns allowed leverage to build up beyondbreaking point

The ideas developed by Prasanna Gai and his collaborators trace theirorigins to this early work by the team at the Bank of England They showthe richness of the conceptual frameworks for thinking about systemic riskand contagion They draw on the microeconomics of banking, quantitativerisk management, coordination games, and the theory of networks

The monograph culminates with a description of the quantitative riskassessment framework developed at the Bank of England—the Risk Assess-ment Model for Systemic Institutions (RAMSI)—that incorporated the insights

of the early research of the Bank of England team RAMSI drew on those earlylessons by integrating balance sheet-based models of banks with a networkmodel in a way that allows for feedback effect of asset sales Shocks andscenarios from a macroeconomic model are then fed through the framework

to describe how risk profiles evolve through banks’ business operations Many

of the more recent models of systemic risk being developed at central banksdraw on the insights from this early research

There is still a long way to go in incorporating systemic risk fully into thepolicymaker’s toolkit, but this monograph take us a long way down the road

we need to travel

Hyun Song ShinHughes-Rogers Professor of Economics

Princeton UniversityPrinceton

4 July 2012

In 2006, I was asked by the Bank of England to participate in a conference inNew York organized by the US National Academy of Sciences and the FederalReserve Bank of New York The aim of the conference was to bring togetheracademics, central bankers, and financial market participants to explorewhether and how financial stability analysis could benefit from the lessons

of other disciplines, such as ecology, epidemiology, and engineering At thetime, the result seemed quite unsatisfactory—in the margins of the confer-ence, some prominent economists expressed their concern at the‘black box’analyses of complex networks presented by the scientists and were sceptical ofthe benefits of any analysis not built on firm microeconomic foundations Thescientists, in turn, questioned the simplistic nature of the economists’attempts at capturing systemic risk, viewing their models as incapable ofshedding meaningful insights about the dynamics of a complex system.Systemic risk has since assumed centre-stage in public discourse as a result ofthe globalfinancial crisis of 2007–8 and, largely through the efforts of centralbankers, there has been some convergence between the economists andthe scientists This book brings together my attempts to explore the inter-disciplinary middle ground brought to light by the New York conference I try

to tackle head-on some of the challenges relating to modelling and ment in the relatively new area of financial stability analysis My targetaudience is policymakers at central banks andfinancial regulatory authorities,risk managers in financial institutions, as well as researchers and graduatestudents interested in understanding systemicfinancial crises

measure-Exploring the terrain between disciplines cannot, of course, be done inisolation and without strong support Sujit Kapadia and Kartik Anand haveplayed central roles in the development of the ideas in this book, and ourongoing collaboration continues to shape my thinking on systemic risk issues

I am most grateful to them for kindly allowing me to borrow so heavily andfreely from our joint research I would also like to acknowledge my other co-authors, Piergiorgio Alessandri, Simon Brennan, James Chapman, MatteoMarsili, Nada Mora, Claus Puhr, and Matthew Willison for their valuablecontributions in helping these ideas reach fruition

The Bank of England and the Bank of Canada have strongly supported myattempts to apply the lessons from complex adaptive systems to thefinancialsphere The material contained in this book has been profoundly influenced

by many valuable exchanges over the years with colleagues at both tions In particular, I am deeply grateful to Mark Carney, Alastair Clark, PierreDuguay, Celine Gauthier, Andrew Gracie, Toni Gravelle, Simon Hall, MoezSoussi, Paul Tucker, Tanju Yorulmazer, and Mark Zelmer for their insightfulideas, advice, and encouragement My intellectual debt to Andy Haldane,Frank Milne, and Hyun Song Shin also deserves special mention—their stead-fast guidance goes back to my student days in Canberra and Oxford, as well as

institu-in the corridors of central bankinstitu-ing I am tremendously honoured by theirendorsement of the work in this book

It has been a pleasure to work with the Oxford University Press, and I wouldlike to thank Aimee Wright for her cheerful handling of the manuscriptprocess Last, but not least, my work has benefited from the stimulatingenvironment at the University of Auckland and the Australian National Uni-versity The Antipodes has been an ideal base from which to dispassionatelyassess an unfolding drama

This book is dedicated to my wife Radhika, who has patiently tolerated mymany failings with wisdom, compassion, and love And to Priyanka, whowaited

P.S.G

Auckland, July 2012

List of Figures xiii

2 The Robust-Yet-Fragile Nature of Financial Systems 9

2.1 An example of a weighted, directed, network 12

3.2 Systemic liquidity hoarding in a Poisson network (single random

idiosyncratic haircut shock; aggregate haircut shock and single random

idiosyncratic haircut shock; and single haircut shock targeted to bank

3.3 Systemic liquidity hoarding in a fat-tailed (geometric) network (single

random idiosyncratic haircut shock; single haircut shock targeted to

bank with most interbank lending links; single random idiosyncratic

3.4 Aggregate haircuts and the probability of a systemic liquidity crisis 43 3.5 Systemic liquidity hoarding in a fat-tailed network (single random

idiosyncratic haircut shock with 3.5% liquid asset holdings; single

random idiosyncratic haircut shock with 3.5% average liquid asset

5.3 Probability of a systemic crisis as a function of returns with different

6.3 Banking sector dynamics—no trading book (median, 50%, 95%, and 99%

6.5 Final quarter banking sector distributions (with trading book)—unit:

7.4 Relative losses to the domestic (light) and entire (dark) banking systems 112

7.7 Fraction of failed banks as a function of the macro shock with fire sale

7.8 Fraction of failed banks as a function of the macro shock with

3.1 Procyclicality of haircuts on term securities financing (per cent) 30

5.2 Bank i ’s balance sheet following a covered bond issue and a

6.1 Asset and liability classes on the balance sheet and associated modelling 85

Introduction

central banking has been transformed, in practice and in theory The list of assumptions that turned out to be false is lengthy: that the financial system would be self-stabilising, that managers of banks would prove competent, that financial innovation would improve risk management, that low and stable inflation would guarantee economic stability We have witnessed a bonfire of the verities

Martin Wolf, The Financial Times, 1 May 2012 1

The global financial crisis has brought home to many—particularly in thecentral banking community—the urgent need for a substantial reassessment

of the fundamental workings offinancial systems, their interactions with thereal economy, and the circumstances that tip such systems from stability toinstability The crisis revealed how rapidfinancial innovation during an era ofprolonged macroeconomic stability had resulted in a highly complex andinterconnected system that was inadequately understood by regulators Thederisory investment over the years by central banks in analytical tools forfinancial stability analysis was also cruelly exposed, as policymakers realizedthat their existing macroeconomic models were incapable of characterizingthe abrupt non-linear adjustments and spillover effects being witnessed at thesystem level

In his call for new thinking onfinancial stability, Trichet (2011) belatedlyrecognizes these failings, noting how‘the combination of complexity, inter-connectedness, payments promises in debt contracts, limits of informationand basic human behaviour—“animal spirits”—can lead to the violent feed-back and amplification mechanisms that are so typical for the transition fromstability to instability’ He suggests that economists revisit the analyticalchallenges of the crisis at a fundamental level, and in ways that eschew

1 From The Financial Times # The Financial Times Limited 2012 All rights reserved.

features of the standard economic paradigm, particularly in macroeconomics.Trichet emphasizes building models based on bottom-up simulations of indi-vidual behaviour rather than top-down maximization in order to generate thenon-linearities characteristic of systemic instability, arguing that ‘ seenfrom the perspective of public authorities experiencing a crisis, which have

to take swift and non-standard decisions in an environment of generalisednon-linearities, significant advances in this new analytical field are of theessence’.2

This book represents an attempt to understand systemicfinancial risk that isvery much in the spirit of these remarks The approach is eclectic, drawing onideas from the microeconomics of banking, quantitative risk management,coordination games, and the theory of networks As such, it reflects a strongbelief that important lessons from other disciplines can help support thecapacity of policymakers to assess and manage tail risks to the financialsystem

On the eve of crisis, most policymakers were preoccupied with ensuring thatfinancial institutions were individually stable—supervision by regulators was

‘microprudential’ in nature The subsequent realization that regulationneeded to have a‘macro’, or system-wide, element and a ‘prudential’ elementhas brought with it an emphasis on the dynamic resilience of thefinancialsector—particularly the extent to which financial firms take into account theeffects of their actions on the balance sheets of other players Butfinancialinnovation, in the form of credit risk transfer and off-balance sheet activity,has made for complex linkages betweenfinancial firms about which there islittle or no information So assessing the likelihood and possible impact of anadverse scenario—be it the idiosyncratic failure of a bank due to a rogue trader

or a sharp macroeconomic downturn influencing all banks—is less thanstraightforward, as are policy measures to combat these interdependencies.Together with the growing interconnectedness of thefinancial system, thesize, concentration, and riskiness of banks have increased markedly Financialengineering has facilitated balance sheet growth that is no longer limited bythe scale of opportunities in the real economy As King (2010) observes, thesize of the US banking sector is currently some 100 per cent of GDP, with theBank of America today accounting for the same proportion of the US bankingsystem as all of the top ten banks in 1960 Leverage ratios offinancial insti-tutions are, in some cases, over 50 times equity Banks have also run a higherdegree of maturity mismatch between their long-dated assets and short-termfunding through ever lengthening chains of transactions When the crisisbegan in 2007, suspicion about where losses would ultimately fall meant

2

For a thoughtful overview of work at the intersection of macroeconomics and finance, and the place of complexity science in modern macroeconomics, see Caballero (2010).

that confidence between financial firms began to seep away until, in the end,the evaporation of trust was complete.

Such considerations, therefore, bring us to the material in this book Itscentral thesis is that an analytical understanding of systemic risk requirescoming to terms with network effects,fire sale effects, and funding liquidityrisk Default by one bank can trigger problems in others, setting off a defaultcascade Troubled institutions may opt to initiatefire sales of assets either as adefensive action or as a prelude to default Fire sales reduce valuations ofcommon assets for others in the system, and exacerbate the probability aswell as potential impact of contagion Financial firms can also withdrawfunding from counterparties as a further defence, compounding a crisis yetfurther A general insight is that these factors combine to generate fat tails inthe distribution of aggregatefinancial system losses The financial system may

be robust to most shocks, but when problems strike the effects may becatastrophic

At a practical level, these ideas have important implications for the way inwhich central banks implement frameworks for systemic risk assessment.The traditional approach has been to conduct stress tests of the bankingsystem by considering one-off effects on banks of an extreme, but plausiblescenario But the second, or higher-round, effects are critical For example,losses experienced duringfire sales and when funding liquidity risk crystal-lizes can make it more difficult for banks to extend loans to firms andhouseholds A credit crunch can result, with damaging effects on real activityand welfare An adequate framework for system-wide stress-testing mustaddress the non-linear consequences implied by the two-way relationshipbetween the financial system and the real economy, together with thesizeable externalities implied by the interconnectedness offinancial firmsfor the system as a whole

The chapters that follow tackle these issues They are grouped into threesubstantive themes:first, financial contagion and the collapse of secured andunsecured funding markets; second, the consequences of dynamic adjust-ment of bank balance sheets for systemic risk; and third, practical methodsfor quantifying systemic risk that allow for asset price and macroeconomicfeedback effects Within each group, common tools are used to tackle thequestions of interest Thus, in Chapters 2 and 3, network models help showhowfinancial connectivity can be a two-edged sword, serving to spread as well

as amplify risks to the system Chapters 4 and 5 marry ideas from the literature

on coordination games with network models to illustrate the consequences ofthe dynamic adjustment of balance sheets by banks in a system context And

in Chapters 6 and 7, the emphasis on implementation elicits a combination

of approaches that draw from previous chapters as well as quantitative riskmanagement techniques

1.1 Financial Contagion

Chapter 2 represents an exploratory attempt to use network theory to modelcontagion in thefinancial system Although the importance of network struc-ture has long been recognized in the economics literature (Allen and Gale,2000), articulating the probability and potential impact of contagion necessi-tates a move beyond simple topologies The model captures two key channels

of contagion in financial systems Losses may potentially spread via thecomplex network of direct counterparty exposures following an initial default.But the knock-on effects of distress at some financial institutions on assetprices can force otherfinancial firms to write down the value of their assets,and this effect can also trigger further rounds of default Contagion due to thedirect interlinkages of interbank claims and obligations can, thus, bereinforced, by contagion due to assetfire sales A key finding is that financialsystems display a robust-yet-fragile tendency—while the probability of conta-gion may be low, the effects are extremely widespread when problems dooccur

Chapter 3 extends the analysis of Chapter 2 to consider a more tive setting where banks rely on unsecured as well as securedfinancing In thisenvironment, the ‘haircuts’ applied to collateral assume centre-stage Themodel shows how system-wide liquidity hoarding arises—as one bank calls

representa-in or shortens the terms of its representa-interbank loans, affected banks representa-in turn do thesame The connectivity and concentration of the players in the network playimportant roles, with key nodes acting as super-spreaders of contagion

A central message is that policy measures which target systemically importantfinancial institutions are crucial to the resilience of the financial system Time-varying liquidity requirements that actively lean against the procyclical ten-dencies of thefinancial system may also help reduce systemic risk

1.2 Dynamic Adjustment of Balance Sheets

A weakness of the analysis in Chapters 2 and 3 is that the underlying topology

of interactions and the balance sheets of the intermediaries in the network arestatic The credit network does not allow for debt contracts (or links) tocontinually become established and terminate as they reach maturity More-over, the decision to foreclose is mechanistic; in reality, this decision is inher-ently strategic and resembles a coordination game In a large interbanknetwork, banks are party to many such coordination games at the sametime As lenders, they are involved in as many coordination games as

counterparties to whom they extend loans And, as borrowers, they are part ofcoordination games being played by others lending to them.

Chapter 4 responds to such issues by combining ideas from the literature onglobal games with models of network growth In so doing, it clarifies hownetwork topology interacts with the funding structure offinancial institutions

to determine system-wide crises In the model, the rate of system-wide bankfailure depends on the arrival of bad news about afinancial institution as well

as on the maturity structure of debt contracts The chapter establishes tions under which funding markets freeze, and shows how normal marketconditions can take a long time to re-establish as a result of the commonknowledge of the equilibrium Balance sheet adjustment plays a crucial role indriving such hysteresis While creditors base decisions to foreclose or refinanceloans on the basis of fundamentals, they are also influenced by the majorityopinion of other creditors Creditor optimism (or pessimism) is self-consist-ently maintained in the face of gradual changes to fundamentals But when acritical threshold is reached, opinions unravel rapidly The analysis providesclues to why the extraordinary policy measures put in place by central banks

condi-to revive global funding markets have been so prolonged

Covered bonds have renewed in popularity since the recentfinancial crisis.Despite having a venerable history dating back to Frederick the Great ofPrussia, the consequences of these instruments for systemic stability havenot been formally analysed Chapter 5 considers this matter, again drawing

on global games but also on insights from the analysis of local interactiongames Although historically limited to Europeanfinance, covered bonds areincreasingly relevant in North American and Asia-Pacific financial markets,and regulators in these countries are engaged in an active debate on thedesirable characteristics of a covered bond regime Although attention todate has been focused on the relatively asymmetric treatment of unsecuredcreditors (namely household depositors) in the event of problems at a bank,these policies also have important implications for systemic risk ex ante InEurope, there is great flexibility concerning the extent to which regulatorspermit a bank to ring-fence assets for covered bond holders In countries likeCanada and New Zealand, however, there are strict limits in place

The results show how the critical threshold for a systemic run by unsecuredcreditors depends on the extent of encumbered assets and the liquidity of thesecondary market for covered bond products Market liquidity, in turn,depends on the willingness of investors to accept those products withoutdue diligence The model highlights the key role played by the relative payoffsfrom taking on the asset, the structure of the OTC network, and the respon-siveness of investors in driving this relationship A keyfinding of the chapter

is that time-varying limits on asset encumbrance may be a useful dential tool to forestall systemic crises

macropru-1.3 Assessing Banking Sector Resilience

Policymakers are now beginning to develop quantitative models of systemicrisk that explicitly measure losses to thefinancial system that result from lowprobability scenarios These models trace shocks through bank balance sheetsand allow for macrocredit risk, network interactions between institutions, andfeedback effects arising on both the asset and liability side of the balancesheet Chapters 6 and 7 describe some of the progress made in that direction.Chapter 6 describes the quantitative risk assessment framework developed

at the Bank of England—the Risk Assessment Model for Systemic Institutions(RAMSI) The approach involves integrating balance sheet-based models ofbanks with a network model in a way that allows for the feedback effect ofasset sales Shocks and scenarios from a macroeconomic model are then fedthrough the framework to describe how risk profiles evolve through banks’business operations More recent versions of the model (Aikman et al., 2009;Gauthier et al., 2010) developed at the Bank of England and the Bank ofCanada allow for richer balance sheets and funding liquidity risk The attrac-tion for policymakers stems from the story-telling capability of RAMSI Whileother approaches to systemic risk modelling offer either rigour in terms ofmicrofoundations or consistency with market-based pricing of risk, thereduced form estimation and rules of thumb which characterize RAMSI aremore readily amenable to the stress-testing exercises now in use by regulators

A drawback of the RAMSI approach is that it relies on highly detailed,confidential, and relatively static balance sheet data to establish linkagesbetween banks and to derive the aggregate losses for the banking system as awhole In reality, however, true linkages between financial firms areunknown—either due to a lack of data or off-balance sheet activity Regulatorsalso have very limited information aboutfinancial players external to the corebanking system, such as foreign banks and hedge funds Additionally, macro-economic feedback effects are not taken into account in the framework.Chapter 7 develops a‘small-scale’ statistical model designed to overcomethese limitations Drawing from publicly available data on advanced countrybanking sectors for much of its calibration, the model shows how systemic riskcan be quantified despite information gaps Macroeconomic fluctuations,asset market liquidity, and network structure all interact simultaneously inthe framework and deliver plausible results for the aggregate loss distributions

of the financial system Even though the quantification exercise is brush, both qualitative and quantitative features of the model are suggestive.Fat tails emerge and the outcomes of‘stress tests’ do not seem too unreason-able Models such as these may also hold out the promise as useful vehicles for

broad-policy exercises, such as gauging the systemic effects of changes in capitalsurcharges.

Afinal chapter summarizes and concludes

While the book explores some aspects offinancial stability analysis, it is farfrom comprehensive Importantly, in focusing on what might be termed the

‘cross-sectional’ aspects of systemic risk, it side-steps the ‘time series’ aspects.3

The time series dimension, which has its antecedents in the Austrian School,highlights the propensity of thefinancial system to be procylical Widespreadimbalances build up in good times, as consumers andfirms increase leverage

to take advantage of favourable opportunities, often to the neglect of ing risks Unfavourable shocks can trigger reassessments of risk and anunwinding of the credit cycle, with increasingly dramatic effects the moresizeable the scale of debt in the economy The leverage-margin-cycle in manyfinancial and housing markets is typical of this story Increased optimismleads to rising prices and financial market liquidity, reductions in margins,and increased leverage Indeed, as banks have gravitated away from traditionalrelationship lending towards market-oriented transactions, the role of thischannel has gained in prominence

increas-An exhaustive treatise on systemic risk and macroprudential policy would

be very thick indeed Such is not my ambition here The book—quite ately—eschews a complete coverage of topics and key papers, restrictingattention only to those analyses that directly bear on the models beingdeveloped So while there are pointers which allow readers to pursue aspects

deliber-of the debate that they have a specific interest in, those hoping for a criticalreview and primer on the literature on systemic risk will be disappointed But

at present thefield is too fluid, and the experience of the crisis too fresh, for asatisfactory synthesis to be formed

Although the exposition lays stress on mathematical modelling, the level oftechnical analysis should be within the compass of most graduate students ineconomics orfinance, central bank research economists, and risk managers.But I hope that the overall logic and the results of the numerical simulationswill also be accessible to a broader audience Additional details of mathemat-ical methods can be found in the articles underlying the chapters, or in theappropriate references, for readers that might need them

3 Bank of England (2011) provides a comprehensive overview of the two perspectives and the recent policy debate on macroprudential instruments.

May et al (2008: 893) 1

In modernfinancial 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 prod-ucts, such as credit default swaps and collateralized debt obligations, hasheightened the complexity of these balance sheet connections still further

As the events following the collapse of Lehman Brothers serve to emphasize,these interdependencies create an environment for feedback elements togenerate amplified responses to shocks to the financial system They alsomake it difficult to assess the potential for contagion arising from the distress(or outright default) of afinancial institution.2

Identification of the probability and potential impact of shocks to thefinancial system has, unsurprisingly, assumed great prominence in the policydebate on systemic risk Even before the globalfinancial crisis, several centralbanks were seeking to explore ways of articulating key risks in their FinancialStability Reports But the paucity of information about the true linkagesbetweenfinancial intermediaries means that policymakers have been limited

This chapter is based on material originally published in Gai and Kapadia (2010), Contagion in financial networks, Proceedings of the Royal Society A, 466, 2401–23 DOI: 10.1098/rspa.2009.0410.

1 Reprinted by permission from Macmillan Publishers Ltd: Nature, R May, S Levin, and

G Sugihara, Complex systems: ecology for bankers, 451, 893–5, # 2008.

2 See Haldane (2009) for a general discussion of the role that the structure and complexities of the financial network played in the financial turmoil of 2007–8 For a comprehensive overview of the use of network models in finance, see Allen and Babus (2009).

in their ability to trace the transmission of shocks through what is a dense cat’scradle offinancial exposures.

In this chapter, we describe how contagion stemming from unexpectedshocks can spread in a financial network of arbitrary structure The modelhighlights two key channels by which default spreads from one institution toanother First, losses can potentially spread through the network of directcounterparty exposures following an initial default And second, as empha-sized by Cifuentes et al (2005), the knock-on effects of distress at somefinan-cial institutions on asset prices can force otherfinancial entities to write downthe value of their assets, potentially triggering further rounds of default.Contagion due to the direct interlinkages of interbank claims and obligations

is thus reinforced by indirect contagion on the asset-side of the bank balancesheet—particularly when the market for key financial system assets is illiquid.3

The approach taken in this chapter is inspired by recent work on complexnetworks (in particular Newman et al., 2001; and Watts, 2002).4This literaturedescribes the behaviour of connected groups of nodes in a network andpredicts the size of a susceptible cluster, i.e the number of vulnerable nodesreached via the transmission of shocks along the links of the network Theanalysis relies on specifying all possible patterns of future transmission Prob-ability generating function techniques allow identification of the number ofrandomly selected nodes’ first neighbours, second neighbours, and so on.Recursive equations are then constructed to consider all possible outcomesand obtain the total number of nodes that the original node is connected to—directly and indirectly Phase transitions, or tipping points, that mark the thresh-old(s) for extensive contagious outbreaks are then identified

When applied to afinancial system setting involving entities with locking balance sheets, the method allows the probability and spread ofcontagious defaults to be modelled analytically and numerically UnlikeWatts (2002) who models cascades in undirected networks, the direction ofclaims and obligations linkingfinancial institutions is taken into account sothat the balance sheet of the bank is carefully characterized Moreover, themodel allows asset-side contagion to be clearly delineated in the financialnetwork since it allows balance sheets to interact with asset prices Thecomplex network structure and interactions between intermediaries makefor non-linear system dynamics, in which contagion risk is highly sensitive

inter-to small changes in parameters Financial systems display a robust-yet-fragile

3

Chapters 3 and 4 consider liability-side or ‘funding’ contagion in more depth.

4 See Strogatz (2001) and Newman (2003) for accessible surveys.

tendency—while the probability of contagion may be low, the effects can beextremely widespread when problems occur.

The intuition is straightforward In a highly connected system, the party losses of a failing institution can be more widely dispersed to, andabsorbed by, other entities So increased connectivity and risk-sharing maylower the probability of contagious default But, conditional on the failure ofone institution triggering contagious defaults, a high number of financiallinkages also increases the potential for contagion to spread more widely Inparticular, high connectivity increases the chances that institutions that sur-vive the effects of the initial default will be exposed to more than one default-ing counterparty after the first round of contagion, thus making themvulnerable to a second round default The effects of any crises that do occurcan, therefore, be extremely widespread

counter-The model framework in this chapter assumes that the network of interbanklinkages forms randomly and exogenously, leaving aside issues related toendogenous network formation and optimal network structures Randomgraphs serve as a metaphor for the complexity of real-worldfinancial systems.The analysis is, thus, not restricted to particular network structures, eventhough some empirical work (e.g Boss et al., 2004; Craig and von Peter,2010) points to core–periphery structures in European interbank marketsand fat-tailed distributions of linkages and loan sizes Indeed, the assumptionthat the network structure is entirely arbitrary carries the advantage that themodel encompasses any structure which may emerge in the real world or asthe optimal outcome of a network formation game As such, it is a naturalbenchmark to consider

The contagion process described below is relatively mechanical Balancesheets and the size and structure of interbank linkages are held constant asdefault propagates through the system Arguably, in normal times and indeveloped financial systems, banks are sufficiently robust that very minorvariations in their default probabilities do not affect the decision of whether

or not to lend to them in interbank markets Meanwhile, in crises, contagionspreads very rapidly through thefinancial system, leaving banks little time toalter their behaviour before they are affected So regarding the network asstatic is a usefulfirst approximation Chapter 4 explores some consequences ofrelaxing this assumption.5

5 Banks also do not have a choice over whether they default This precludes the type of strategic behaviour discussed by Morris (2000), Jackson and Yariv (2007), and Galeotti and Goyal (2009) in which nodes can choose whether or not to adopt a particular state (e.g adopting a new technology).

2.1 A Simple Model of Contagion

2.1.1 Network structure

Consider afinancial network in which n financial intermediaries, ‘banks’ forshort, are randomly linked together by their claims on each other In thelanguage of graph theory, each bank represents a node on the graph and theinterbank exposures of bank i define the links with other banks These links aredirected and weighted, reflecting the fact that interbank exposures compriseassets as well as liabilities, and that the size of these exposures matters for theanalysis of contagion Figure 2.1 shows an example of a directed, weighted,financial network of five banks, with darker lines corresponding to highervalue links

A crucial property of graphs such as those in Figure 2.1 is their degreedistribution In a directed graph, each node has an in-degree (the number oflinks that point into the node) and an out-degree (the number of links pointingout of the node) Incoming links to a bank reflect the interbank assets of thatbank, i.e monies owed to the bank by a counterparty By contrast, the out-going links correspond to the bank’s interbank liabilities Critically, the jointdistribution of the in- and out-degrees governs the potential for the spread ofshocks through the network

We will suppose that the joint degree distribution, and hence the structure

of the links in the network, is entirely arbitrary This implies that the network

is entirely random in all respects other than its degree distribution We also donot assume any statistical correlation between nodes and we treat mixingbetween nodes as proportionate Thus, there is no statistical tendency for

Figure 2.1 An example of a weighted, directed, network

highly connected nodes to be particularly connected to other highly nected nodes or with poorly connected nodes Later on in the chapter, wemake a specific distributional assumption in our numerical simulations inorder to illustrate the main results.

con-Let the total assets of each bank consist of interbank assets, AIB

i , and illiquidassets such as mortgages, AM

i Suppose also that the total interbank assetposition of every bank is evenly distributed over each of its incoming linksand is independent of the number of links the bank has These stylizedassumptions emphasize the benefits of diversification and allows us to high-light the distinction between risk sharing and risk spreading within thefinan-cial network In particular, we shall see that widespread contagion is possibleeven when risk sharing in the system is maximized

Since every interbank asset is another bank’s liability, interbank liabilities,

LIB

i , are endogenously determined Apart from interbank liabilities, we assumethat the only other component of a bank’s liabilities is exogenously givencustomer deposits, Di The condition for bank i to be solvent is therefore

ð1  fÞAIB

i þ qAM

i  LIB

i  Di> 0 ð2:1Þwhere f is the fraction of banks with obligations to bank i that have defaulted,and q is the resale price of the illiquid asset The value of q may be less than 1 inthe event of asset sales by banks in default, but equals 1 if there are no‘firesales’ We make a zero recovery assumption, namely that, when a linked bankdefaults, bank i loses all of its interbank assets held against that bank Thisassumption is plausible in the midst of a crisis—in the immediate aftermath of

a default, the recovery rate and timing of recovery are both highly uncertainand those funding the banks are likely to assume the worst But the assump-tion can be relaxed without altering the main results Notice also that thesolvency condition of the bank can be expressed as

f<Ki ð1  qÞAM

i

AIB i

6 Formally, this capital buffer is taken to be a random variable—the underlying source of its variability may be viewed as being generated by the variability in D i , drawn from its appropriate distribution For notational simplicity, we do not explicitly denote this dependence of K i on D i in the subsequent expressions.

1994) leading to the failure of the financial institution Alternatively, bankfailure may result from an aggregate shock that has particularly adverse con-sequences for one institution—this can be captured in the model through ageneral erosion in the stock of illiquid assets or, equivalently, capital buffersacross all banks, combined with a major loss for one particular institution.Let jidenote the number of incoming links for bank i Since linked bankseach lose a fraction 1/ji, of their interbank assets when a single counterpartydefaults, equation (2.2) implies that the only way default can spread is if there

is a neighbouring bank for which

Ki ð1  qÞAM

i

AIB i

< 1

We define banks that are exposed in this sense to the default of a singleneighbour as vulnerable and other banks as safe The vulnerability of a bankclearly depends on its in-degree And since the capital buffer is a randomvariable, a bank with in-degree j is vulnerable with probability

uj¼ Pr Ki ð1  qÞAM

i

AIB i

<1j

Furthermore, the probability of a bank having in-degree j, out-degree k, andbeing vulnerable is uj pjk, where pjkis the joint degree distribution of in- andout-degree

The model structure described by equations (2.1)—(2.4) captures severalfeatures of interest in systemic risk analysis First, the nature and scale ofadverse aggregate or macroeconomic events can be interpreted as a negativeshock to the stock of illiquid assets, AiM, or, equivalently, to the capital buffer,

Ki Second, idiosyncratic shocks can be modelled by assuming the exogenousdefault of a bank Third, the structural characteristics of thefinancial systemare described by the distribution of interbank linkages, pjk And,finally, liquid-ity effects associated with the potential knock-on effects of default on assetprices are captured by allowing q to vary We initiallyfix q = 1, returning later

to consider the implications of liquidity risk

2.1.2 Contagion dynamics

To track the transmission of a shock to a randomly chosen bank, we make use

of generating function techniques to describe the distributions of links andcluster sizes in thefinancial network.7

In particular, we would like to know (a)

7

See Grimmett and Welsh (1986) for a description of the key properties of probability generating functions.

the distribution of the number of links leaving a vulnerable bank chosen atrandom; (b) the distribution of links leaving a vulnerable neighbour reached

by following a randomly chosen link; (c) the distribution of the cluster sizes ofvulnerable banks reached by following a random link; and (d) the distribution

of the size of the vulnerable cluster to which a randomly chosen bank belongs.For contagion to spread beyond thefirst neighbours of the initially default-ing bank in a sufficiently large network, these neighbours must themselveshave outgoing links (i.e liabilities) to other vulnerable banks.8We thereforedefine the probability generating function for the joint degree distribution of avulnerable bank as

Gðx; yÞ ¼X

j;k

uj pjk xj yk: ð2:5Þ

The generating function contains all the same information that is contained

in the degree distribution, pjk, and the vulnerability distribution, uj, but in aform that allows us to work with sums of independent draws from differentprobability distributions Specifically, it generates all the moments of thedegree distribution of only those banks that are vulnerable

Since every interbank asset of one bank is an interbank liability of another,every outgoing link for one node is an incoming link for another node So theaverage in-degree in the network must equal the average out-degree We refer

to this quantity as the average degree and denote it by

z¼X

j;kjpjk¼X

j;kkpjk:From G(x, y), define a single-argument generating function, G0(y), for thenumber of links leaving a randomly chosen vulnerable bank This is given by

8 If 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 safe banks will never fail in the second round This assumption clearly breaks down when n is small or when contagion spreads more widely But the logic of the model holds in both cases In the former, the exact solutions derived for large n will only approximate reality (and is confirmed by the numerical simulations below); in the latter, the exact solutions apply but the extent of contagion will be affected.

Because we are interested in the propagation of shocks from one bank toanother, we require the degree distribution, uj rjk, of a vulnerable bank that

is a random neighbour of our initially chosen bank Notice that this is not thesame as the degree distribution of vulnerable banks on the network as a whole.This is because a bank with a higher in-degree has a greater number of linkspointing towards it, meaning that there is a higher chance that any givenoutgoing link will terminate at it, in precise proportion to its in-degree So thelarger the in-degree of a bank, the more likely it is to be a neighbour of theinitially chosen bank, with the probability of choosing it being proportional

to jpjk.9The generating function for the number of links leaving a vulnerableneighbour of a randomly chosen vulnerable bank is thus given by

As Figure 2.2 shows, each vulnerable cluster (represented by a square) cantake many different forms We could follow a randomly chosen outgoing linkand find a single bank at its end with no further outgoing connectionsemanating from it This bank may be safe (s) or vulnerable (v) Or we couldfind a vulnerable bank with one, two, or more links emanating from it tofurther clusters We assume that the links emanating from the defaulting node

are tree-like, containing no cycles or closed loops, to make an exact solutionpossible.10

Let H1(y) be the generating function for the probability of reaching anoutgoing vulnerable cluster of given size (in terms of numbers of vulnerablebanks) by following a random outgoing link from a vulnerable bank InFigure 2.2, the total probability of all possible forms can be represented self-consistently as the sum of probabilities of hitting a safe bank, hitting only asingle vulnerable bank, hitting a single vulnerable bank connected to oneother cluster, two other clusters and so on Each cluster that may be arrived

at is independent Therefore H1(y) satisfies the self-consistency condition:

H1ðyÞ ¼ Pr ½reach safe bank þ yX

j;k

uj rjk ½H1ðyÞk; ð2:9Þ

where the leading factor of y accounts for the one vertex at the end of theinitial edge and we have used the fact that if a generating function generatesthe probability distribution of some property of an object, then the sum ofthat property over m independent such objects is distributed according to themth power of the generating function Using equation (2.8) and noting that

G1(1) represents the probability that a random neighbour of a vulnerable bank

is vulnerable, equation (2.9) can be written in implicit form as

H1ðyÞ ¼ 1  G1ðyÞ þ yG1ðH1ðyÞÞ ð2:10Þ

It remains to establish the distribution of outgoing vulnerable cluster sizes towhich a randomly chosen bank belongs There are two possibilities that canarise First, a randomly chosen bank may be safe Second, it may have in-degree j and out-degree k and be vulnerable, the probability of which is uj pjkInthis second case, each outgoing link leads to a vulnerable cluster whose size isdrawn from the distribution generated by H1(y) So the size of the vulnerablecluster to which a randomly chosen bank belongs is generated by

H0ðyÞ ¼ Pr ½bank safe þ yX

uj pjk ½H1ðyÞk¼ 1  G0ð1Þ þ yG0½H1ðyÞ: ð2:11ÞAnd, in principle, we can calculate the complete distribution of vulnerablecluster sizes by solving equation (2.10) for H1(y) and substituting the resultinto equation (2.11)

financial network The average vulnerable cluster size can be deduced fromequations (2.10) and (2.11), namely

G01ð1Þ is greater than 1, a ‘giant’ vulnerable cluster—a vulnerable cluster whosesize scales linearly with the size of the whole network—exists and occupies afinite fraction of the network In this case, system-wide contagion is possible.With positive probability, a random initial default at one bank can lead to thespread of default across the entire vulnerable portion of thefinancial system

As the average degree, z, increases, typical in- and out-degrees increase sothat more of the mass of pjkis at higher values for j and k This increases theleft-hand side of equation (2.17) monotonically through the j.k term, butreduces it through the ujterm as ujis lower for higher j from equation (2.4)

So equations (2.16) and (2.17) will either have two solutions or none at all Inthe former instance, 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 the lower phase transition, theP

j;kj k  pjkterm is too small and the network is insufficiently connected forcontagion to spread; for values of z outside the window and above the upperphase transition, the vjterm is too small and contagion cannot spread becausethere are too many safe banks

2.1.4 The probability and spread of contagion

We focus attention on contagion within the giant vulnerable cluster Thisonly emerges for intermediate values of z, and only when the initially default-ing bank is either in the giant vulnerable cluster or directly adjacent to it Thelikelihood of contagion is, therefore, directly linked to the size of the vulner-able cluster within the window Intuitively, near both the lower and upperphase transitions, the probability of contagion must be close to zero since thesize of the vulnerable cluster is either restricted by limited connectivity or bythe presence of a high fraction of safe banks The probability of contagion is,thus, non-monotonic in z Initially, the risk-spreading effects stemming from

a more connected system will increase the size of the vulnerable cluster andthe probability of contagion Eventually, however, risk-sharing effects thatserve to reduce the number of vulnerable banks dominate, and the probability

of contagion falls

The conditional spread of contagion (i.e conditional on contagion breakingout) also corresponds to the size of the giant vulnerable cluster But oncecontagion has spread through the entire vulnerable cluster, the assumptionthat banks are adjacent to no more than one failed bank breaks down Thus,

‘safe’ banks may be susceptible to default and contagion can spread wellbeyond the vulnerable cluster to affect the entire connected component ofthe network Near the lower phase transition, z is sufficiently low that nearlyall banks are likely to be vulnerable So in this region, the size of the giantvulnerable cluster corresponds closely to the size of the connected component

of 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 transi-tion, the system exhibits a robust-yet-fragile tendency—episodes of contagionoccur rarely, but spread very widely when they do take place

2.1.5 Diversification

In the presentation of the model so far, the assumptions made emphasize thediversification benefits of financial connectivity We have assumed that thetotal interbank asset position of each bank is independent of the number ofincoming links to that bank and that these assets were evenly distributed over

each link Such assumptions deliberately serve to accentuate the role of thefinancial system as a ‘shock absorber’.

But when these assumptions are relaxed, thefinancial system can serve toamplify shocks In reality, we might expect a bank with a high number ofincoming links to have a larger total interbank asset position Intuitively, thiscurtails the risk-sharing benefits of higher connectivity because the greaterabsolute exposure associated with a higher number of links would (partially)offset the positive effects from greater diversification But as long as the totalinterbank asset position increases less than proportionately with the number

of links, the main insights of the model continue to hold In particular, uj

will still decrease in z, albeit at a slower rate As a result, equation (2.17) willcontinue to generate two solutions, though in an extended range of cases Thecontagion window will thus be wider On the other hand, if the total inter-bank asset position increases more than proportionately with the number oflinks, vjwill increase in z and greater connectivity unambiguously increasescontagion risk But this latter case does not seem particularly plausible

An uneven distribution of interbank assets over incoming links would alsonot change the results fundamentally In particular, ujwould still decrease in z,maintaining the possibility of two solutions to equation (2.17) But an unevendistribution of exposures would make banks vulnerable to the default ofparticular counterparties for higher values of z than would otherwise be thecase So the contagion window will be wider.11

2.2 Model Simulations

2.2.1 The benchmark case

To illustrate our results, we calibrate the model and simulate it numerically.Although the modelfindings apply to random graphs with arbitrary degreedistributions, we assume a Poisson random graph—in which each possibledirected link in the graph is present with independent probability p—forconvenience The implications of‘fat-tailed’ distributions are considered inChapter 3

We consider a network of 1,000 banks in the simulation exercise Althoughthis number is somewhat arbitrary, several countries have banking systems ofthis size, and afigure of 1,000 intermediaries seems reasonable in the context

of a financial system involving hedge funds, investment banks, and otherplayers Consistent with bankruptcy law, we do not net interbank positionsand it is possible for two banks to be linked with each other in both directions

11

In recent work, Amini et al (2010) study the implications of relaxing this assumption in greater detail.

And, although the model can be applied to networks of fully heterogeneousfinancial intermediaries, we take the capital buffers and asset positions onbanks’ balance sheets to be identical We return to a more detailed treatment

of heterogeneous institutions and capital buffers in Chapter 7

The initial assets of each bank are chosen to comprise 80 per cent external(non-bank) assets and 20 per cent interbank assets,figures broadly consistentfor banks in advanced economies (see Upper, 2011) Banks’ capital buffers areset at 4 per cent of total (non-risk-weighted) assets, afigure calibrated fromdata contained in the published 2005 accounts of a range of large financialinstitutions Since the interbank assets of a bank are evenly distributed over itsincoming links, interbank liabilities are endogenously determined within thenetwork structure The liability side of the balance sheet is topped up bycustomer deposits until the total liability position equals the total assetposition

The average degree, z, is allowed to vary in each simulation and we draw1,000 realizations of the network for each value of z In each draw, we shockone bank at random, wiping out all of its external assets.12 The failed bankdefaults on all of its interbank liabilities and, as a result, neighbouring banksmay also default if their capital buffer is insufficient to cover losses on inter-bank assets Any neighbouring banks that fail are also assumed to default onall of their interbank liabilities, and the iterative process continues until nonew banks are pushed into default

Since we are only interested in the likelihood and potential spread ofsystem-wide contagion, we exclude very small outbreaks of default outsidethe giant vulnerable cluster from the analysis So when calculating the prob-ability and conditional spread, we only count episodes in which over 5 percent of the system defaults Again this is somewhat arbitrary, but it seems to be

a plausible and suitable lower bound for defining a systemic financial crisis.Figure 2.3 illustrates the frequency and extent of contagion in the benchmarkcase

The benchmark simulation confirms the insights of the preceding analysis.Contagion only occurs within a certain window of z Within this range, theprobability of contagion is non-monotonic in connectivity, peaking atapproximately 0.8 when z is between 3 and 4 As anticipated, the conditionalspread of contagion as a fraction of network size is approximately the same asthe frequency of contagion near the lower phase transition—in this region,contagion breaks out when shocks strike any bank in, or adjacent to, the giantvulnerable cluster and spread across the entire cluster This roughly corres-ponds to the entire connected component of the network

12 This type of idiosyncratic shock can be interpreted as a fraud shock.

For higher values of z, however, a large proportion of banks in the networkfail when contagion breaks out Of particular interest are points near the upperphase transition When z > 8, contagion never occurs more thanfive times in1,000 draws; but in each case where it does break out, every bank in thenetwork fails So a priori indistinguishable shocks to the network can havevastly different consequences for contagion This cautions against assumingthat past resilience to a particular shock will continue to apply to future shocks

of a similar magnitude And it suggests why the resilience of the globalfinancial system to fairly large shocks prior to the crisis of 2007/8 (e.g 9/11,the Dotcom crash, and the collapse of the hedge fund Amaranth) should nothave been regarded as a reliable guide to its future resilience

Even if contagion from idiosyncratic shocks never occurs when banks haverelatively high capital buffers, an adverse aggregate shock (such as a macro-economic downturn) may erode capital buffers and expose the financialsystem to contagion risk Figure 2.4 considers this possibility As expected,

an erosion of capital buffers both widens the contagion window and increasesthe probability of contagion forfixed values of z For small values of z, theextent of contagion is also slightly greater when capital buffers are lower but,

in all cases, it reaches unity for sufficiently high values of z When the capital

Figure 2.3 Contagion in the benchmark case

buffer is increased to 5 per cent, however, this occurs well after the peakprobability of contagion This illustrates how increased connectivity can sim-ultaneously reduce the probability of contagion but increase its spread, condi-tional on it breaking out.

2.2.2 Fire sale of assets

A number of recent papers (e.g Diamond and Rajan, 2011; Stein, 2012) havestressed the role of assetfire sales as an important amplifying mechanism inthe recent crisis Our model can be readily adapted to illustrate their concernsand highlight how indirect contagion through asset prices supplements adirect default cascade When a bank fails,financial markets may have a limitedcapacity to absorb the illiquid external assets that are sold in the ensuingfiresale As a result, the asset price may be depressed Following Cifuentes et al.(2005), suppose that the price of the illiquid asset, q, is given by

Figure 2.4 Varying the capital buffer