Systemic Risk and Stability in Financial Networks

America Ecoomic Review 2015, 105(2): 564 608 http://dx.doi.org/10.1257/aer.20130456 Systemic Risk ad Stability i Fiacial Networks By Daro Acemoglu, Asuma Ozdaglar, ad Alireza Tahbaz-Salehi * This paper argues that the extet of fiacial cotagio exhibits a form of phase trasitio: as log as the magitude of egative shocks affectig fiacial istitutios are sufficietly small, a more desely coected fiacial etwork (correspodig to a more diversified patter of iterbak liabilities) ehaces fiacial stability. However, beyod a certai poit, dese itercoectios serve as a mechaism for the propagatio of shocks, leadig to a more fragile fiacial system. Our results thus highlight that the same factors that cotribute to resiliece uder certai coditios may fuctio as sigificat sources of systemic risk uder others. (JEL D85, E44, G21, G28, L14) Sice the global fiacial crisis of 2008, the view that the architecture of the fiacial system plays a cetral role i shapig systemic risk has become covetioal wisdom. The itertwied ature of the fiacial markets has ot oly bee proffered as a explaatio for the spread of risk throughout the system (see, e.g., Plosser 2009 ad Yelle 2013), but also motivated may of the policy actios both durig ad i the aftermath of the crisis. 1 Such views have eve bee icorporated ito the ew regulatory frameworks developed sice. 2 Yet, the exact role played by the fiacial system s architecture i creatig systemic risk remais, at best, imperfectly uderstood. The curret state of ucertaity about the ature ad causes of systemic risk is reflected i the potetially coflictig views o the relatioship betwee the structure of the fiacial etwork ad the extet of fiacial cotagio. Pioeerig works by * Acemoglu: Departmet of Ecoomics, Massachusetts Istitute of Techology, 77 Massachusetts Aveue, E18-269D, Cambridge, MA 02142 (e-mail: daro@mit.edu); Ozdaglar: Laboratory for Iformatio ad Decisio Systems, Massachusetts Istitute of Techology, 77 Massachusetts Aveue, 32-D630, Cambridge, MA 02139-4307 (e-mail: asuma@mit.edu); Tahbaz-Salehi: Columbia Busiess School, Columbia Uiversity, 3022 Broadway, Uris Hall 418, New York, NY 10027 (e-mail: alirezat@columbia.edu). We thak five aoymous referees for very helpful remarks ad suggestios. We are grateful to Ali Jadbabaie for extesive commets ad coversatios. We also thak Ferado Alvarez, Gadi Barlevy, Oza Cadoga, Adrew Clause, Marco Di Maggio, Paul Glasserma, Be Golub, Gary Gorto, Sajeev Goyal, Jeifer La O, Joh Moore, Fracesco Nava, Marti Oehmke, Jo Pogach, Jea-Charles Rochet, Alp Simsek, Ali Shourideh, Larry Wall, ad umerous semiar ad coferece participats. Acemoglu ad Ozdaglar gratefully ackowledge fiacial support from the Army Research Office, Grat MURI W911NF-12-1-0509. The authors declare that they have o relevat or material fiacial iterests that relate to the research described i this paper. Go to http://dx.doi.org/10.1257/aer.20130456 to visit the article page for additioal materials ad author disclosure statemet(s). 1 For a accout of the policy actios durig the crisis, see Sorki (2009). 2 A example of recet policy chages motivated by this perspective is the provisio o sigle couterparty exposure limits i the Dodd-Frak Act, which attempts to prevet the distress at a istitutio from spreadig to the rest of the system by limitig each firm s exposure to ay sigle couterparty. 564

VOL. 105 NO. 2 Acemoglu et al.: systemic risk ad stability i fiacial etworks 565 Alle ad Gale (2000) ad Freixas, Parigi, ad Rochet (2000) suggested that a more itercoected architecture ehaces the resiliece of the system to the isolvecy of ay idividual bak. Alle ad Gale, for example, argue that i a more desely itercoected fiacial etwork, the losses of a distressed bak are divided amog more creditors, reducig the impact of egative shocks to idividual istitutios o the rest of the system. I cotrast to this view, however, others have suggested that dese itercoectios may fuctio as a destabilizig force, pavig the way for systemic failures. For example, Vivier-Lirimot (2006) argues that as the umber of a bak s couterparties grows, the likelihood of a systemic collapse icreases. This perspective is shared by Blume et al. (2011, 2013) who model iterbak cotagio as a epidemic. I view of the coflictig perspectives oted above, this paper provides a framework for studyig the etwork s role as a shock propagatio ad amplificatio mechaism. Though stylized, our model is motivated by a fiacial system i which differet istitutios are liked to oe aother via usecured debt cotracts ad hece are susceptible to couterparty risk. Our setup eables us to provide a umber of theoretical results that highlight the implicatios of the etwork s structure o the extet of fiacial cotagio ad systemic risk. 3 More cocretely, we focus o a ecoomy cosistig of fiacial istitutios that lasts for three periods. I the iitial period, baks borrow fuds from oe aother to ivest i projects that yield returs i both the itermediate ad fial periods. The liability structure that emerges from such iterbak loas determies the fiacial etwork. I additio to its commitmets to other fiacial istitutios, each bak also has to make other paymets with respect to claims that are seior to those of other baks. These claims may correspod to paymets due to retail depositors or other types of commitmets such as wages, taxes, or claims by other seior creditors. We assume that the returs i the fial period are ot pledgeable, so all debts have to be repaid i the itermediate period. Thus, a bak whose short-term returs are below a certai level may have to liquidate its project prematurely (i.e., before the fial period returs are realized). If the proceeds from liquidatios are isufficiet to pay all its debts, the bak defaults. Depedig o the structure of the fiacial etwork, this may the trigger a cascade of failures: the default of a bak o its debt may cause the default of its creditor baks o their ow couterparties, ad so o. The mai focus of the paper is to study the extet of fiacial cotagio as a fuctio of the structure of iterbak liabilities. By geeralizig the results of Eiseberg ad Noe (2001), we first show that, regardless of the structure of the fiacial etwork, a paymet equilibrium cosistig of a mutually cosistet collectio of asset liquidatios ad repaymets o iterbak loas always exists ad is geerically uique. We the characterize the role of the structure of the fiacial etwork o the resiliece of the system. To start with, we restrict our attetio to regular fiacial etworks i which the total claims ad liabilities of all baks are equal. Such a ormalizatio guaratees that ay variatio i the fragility of the system is due to the 3 The stylized ature of our model otwithstadig, we refer to our etwork as a fiacial etwork ad to its comprisig etities as fiacial istitutios or baks for ease of termiology.

566 THE AMERICAN ECONOMIC REVIEW february 2015 fiacial etwork s structure rather tha ay heterogeeity i size or leverage amog baks. Our first set of results shows that whe the magitude of egative shocks is below a certai threshold, a result similar to those of Alle ad Gale (2000) ad Freixas, Parigi, ad Rochet (2000) holds: a more diversified patter of iterbak liabilities leads to a less fragile fiacial system. I particular, the complete fiacial etwork, i which the liabilities of each istitutio are equally held by all other baks, is the cofiguratio least proe to cotagious defaults. At the opposite ed of the spectrum, the rig etwork a cofiguratio i which all liabilities of a bak are held by a sigle couterparty is the most fragile of all fiacial etwork structures. The ituitio uderlyig these results is simple: a more diversified patter of iterbak liabilities guaratees that the burde of ay potetial losses is shared amog more couterparties. Hece, i the presece of relatively small shocks, the excess liquidity of the o-distressed baks ca be efficietly utilized i forestallig further defaults. Our ext set of results shows that as the magitude or the umber of egative shocks crosses certai thresholds, the types of fiacial etworks that are most proe to cotagious failures chage dramatically. I particular, a more itercoected etwork structure is o loger a guaratee for stability. Rather, i the presece of large shocks, highly diversified ledig patters facilitate fiacial cotagio ad create a more fragile system. O the other had, weakly coected fiacial etworks i which differet subsets of baks have miimal claims o oe aother are sigificatly less fragile. 4 The ituitio uderlyig such a sharp phase trasitio is that, with large egative shocks, the excess liquidity of the bakig system may o loger be sufficiet for absorbig the losses. Uder such a sceario, a less diversified ledig patter guaratees that the losses are shared with the seior creditors of the distressed baks, protectig the rest of the system. Our results thus cofirm a cojecture of Haldae (2009, pp. 9 10), the Executive Director for Fiacial Stability at the Bak of Eglad, who suggested that highly itercoected fiacial etworks may be robust-yet-fragile i the sese that withi a certai rage, coectios serve as shock-absorbers [ad] coectivity egeders robustess. However, beyod that rage, itercoectios start to serve as a mechaism for the propagatio of shocks, the system [flips to] the wrog side of the kife-edge, ad fragility prevails. More broadly, our results highlight that the same features that make a fiacial system more resiliet uder certai coditios may fuctio as sources of systemic risk ad istability uder others. I additio to illustratig the role of the etwork structure o the stability of the fiacial system, we itroduce a ew otio of distace over the fiacial etwork, called the harmoic distace, which captures the susceptibility of each bak to the distress at ay other. We show that, i the presece of large shocks, all baks whose harmoic distaces to a distressed bak are below a certai threshold default. This characterizatio shows that, i cotrast to what is ofte presumed i the empirical literature, various off-the-shelf (ad popular) measures of etwork cetrality such as eigevector or Boacich cetralities may ot be the right otios for idetifyig 4 Such weakly coected fiacial etworks are somewhat remiiscet of the old-style uit bakig system, i which baks withi a regio are oly weakly coected to the rest of the fiacial system, eve though itra-regio ties might be strog.

VOL. 105 NO. 2 Acemoglu et al.: systemic risk ad stability i fiacial etworks 567 systemically importat fiacial istitutios. Rather, if the iterbak iteractios exhibit o-liearities similar to those iduced by the presece of usecured debt cotracts, the it is the bak closest to all others accordig to our harmoic distace measure that may be too-itercoected-to-fail. Related Literature. Our paper is part of a recet but growig literature that focuses o the role of the architecture of the fiacial system as a amplificatio mechaism. Kiyotaki ad Moore (1997); Alle ad Gale (2000); ad Freixas, Parigi, ad Rochet (2000) provided some of the first formal models of cotagio over etworks. Usig a multi-regio versio of Diamod ad Dybvig s (1983) model, Alle ad Gale, for example, show that the iterbak relatios that emerge to pool regio-specific shocks may at the same time create fragility i respose to uaticipated shocks. 5 Dasgupta (2004) studies how the cross-holdigs of deposits motivated by imperfectly correlated regioal liquidity shocks ca lead to cotagious breakdows. Shi (2008, 2009), o the other had, costructs a accoutig framework of the fiacial system as a etwork of iterliked balace sheets. He shows that securitizatio eables credit expasio through greater leverage of the fiacial system as a whole, drives dow ledig stadards, ad hece icreases fragility. More recetly, Alle, Babus, ad Carletti (2012) have argued that the patter of asset commoalities betwee differet baks determies the extet of iformatio cotagio ad hece, the likelihood of systemic crises. Also related is the work of Castiglioesi, Feriozzi, ad Lorezoi (2012), who show that a higher degree of fiacial itegratio leads to more stable iterbak iterest rates i ormal times, but to larger iterest rate spikes durig crises. Noe of the above papers, however, provides a comprehesive aalysis of the relatioship betwee the structure of the fiacial etwork ad the likelihood of systemic failures due to cotagio of couterparty risk. 6 Our paper is also related to several recet, idepedet works, such as Elliott, Golub, ad Jackso (2014) ad Cabrales, Gottardi, ad Vega-Redodo (2014), that study the broad questio of propagatio of shocks i a etwork of firms with fiacial iterdepedecies. These papers, however, focus o a cotagio mechaism differet from ours. I particular, they study whether ad how cross-holdigs of differet orgaizatios shares or assets may lead to cascadig failures. Elliott, Golub, ad Jackso (2014) cosider a model with cross-owership of equity shares ad show that i the presece of bakruptcy costs, a firm s default may iduce losses o all firms owig its equity, triggerig a chai reactio. O the other had, Cabrales, Gottardi, ad Vega-Redodo (2014) study how securitizatio modeled 5 Alle ad Gale also ote that compared to a four-bak rig etwork, a pairwise-coected (ad thus overall discoected) etwork ca be less proe to fiacial cotagio origiatig from a sigle shock. Their paper, however, does ot cotai ay of our results o the cetral role played by the size of the shocks i the fragility of the system ad the phase trasitio of highly itercoected etworks. 6 Other related cotributios iclude Rochet ad Tirole (1996); Cifuetes, Ferrucci, ad Shi (2005); Leiter (2005); Nier et al. (2007); Rotemberg (2011); Zawadowski (2011); Battisto et al. (2012); Gofma (2011, 2014); Caballero ad Simsek (2013); Georg (2013); Cohe-Cole, Patacchii, ad Zeou (2013); Debee et al. (2014); Di Maggio ad Tahbaz-Salehi (2014); ad Amii, Cot, ad Mica (2013). For a more detailed discussio of the literature, see the survey by Alle ad Babus (2009). A more recet ad smaller literature focuses o the formatio of fiacial etworks. Examples iclude Babus (2013); Zawadowski (2013); ad Farboodi (2014); as well as the workig paper versio of the curret work (Acemoglu, Ozdaglar, ad Tahbaz-Salehi 2013). For empirical evidece o iterbak cotagio, see Iyer ad Peydró (2011).

568 THE AMERICAN ECONOMIC REVIEW february 2015 as exchage of assets amog firms may lead to the istability of the fiacial system as a whole. Our work, i cotrast, focuses o the likelihood of systemic failures due to cotagio of couterparty risk. Focusig o fiacial cotagio through direct cotractual likages, Alvarez ad Barlevy (2014) use a model similar to ours to study the welfare implicatios of a policy of madatory disclosure of iformatio i the presece of couterparty risk. Glasserma ad Youg (2015) also rely o a similar model, but rather tha characterizig the fragility of the system as a fuctio of the fiacial etwork s structure, they provide a etwork-idepedet boud o the probability of fiacial cotagio. 7 Our paper is also related to Eboli (2013), who studies the extet of cotagio i some classes of etworks. I cotrast to our paper, his focus is o the idetermiacy of iterbak paymets i the presece of cyclical etaglemet of assets ad liabilities. 8 Gai, Haldae, ad Kapadia (2011) also study a etwork model of iterbak ledig with usecured claims. Usig umerical simulatios, they show how greater complexity ad cocetratio i the fiacial etwork may amplify the fragility of the system. The role of the etworks as shock propagatio ad amplificatio mechaisms has also bee studied i the cotext of productio relatios i the real ecoomy. Focusig o the iput-output likages betwee differet sectors, Acemoglu et al. (2012) ad Acemoglu, Ozdaglar, ad Tahbaz-Salehi (2014) show that i the presece of liear (or log-liear) ecoomic iteractios, the volatility of aggregate output ad the likelihood of large ecoomic dowturs are idepedet of the sparseess or deseess of coectios, but rather deped o the extet of asymmetry i differet etities itercoectivity. The cotrast betwee the isights o propagatio of shocks i productio ecoomies with (log) liear iteractios ad those i the presece of default (due to debt-like fiacial istrumets) preseted i this paper highlights that the role of etworks i cotagio crucially depeds o the ature of ecoomic iteractios betwee differet etities that costitute the etwork. Outlie of the Paper. The rest of the paper is orgaized as follows. Our model is preseted i Sectio I. I Sectio II, we defie our solutio cocept ad show that a paymet equilibrium always exists ad is geerically uique. Sectio III cotais our results o the relatioship betwee the extet of fiacial cotagio ad the etwork structure. Sectio IV cocludes. A discussio o the properties of the harmoic distace ad the proofs are preseted i the Appedix, while a olie Appedix cotais several omitted proofs. 7 I additio to focusig o differet questios, the liability structures of the fiacial istitutios are also differet i the two papers. I particular, due to the absece of the outside seior claims, the model of Glasserma ad Youg (2015) imposes a implicit upper boud o the size of the egative shocks, essetially limitig the extet of cotagio. 8 A differet strad of literature studies the possibility of idirect spillovers i the fiacial markets. I particular, rather tha takig place through direct cotractual relatios as i our paper, the amplificatio mechaisms studied i this literature work through the edogeous resposes of various market participats. Examples iclude Holmström ad Tirole (1998); Bruermeier ad Pederse (2005); Lorezoi (2008); ad Krishamurthy (2010). For a recet survey, see Bruermeier ad Oehmke (2013).

VOL. 105 NO. 2 Acemoglu et al.: systemic risk ad stability i fiacial etworks 569 I. Model A. Fiacial Istitutios Cosider a sigle-good ecoomy, cosistig of risk-eutral baks idexed by N = {1, 2,, }. The ecoomy lasts for three periods, t = 0, 1, 2. At the iitial period, each bak i is edowed with k i uits of capital that it ca either hoard as cash (deoted by c i ), led to other baks, or ivest i a project that yields returs i the itermediate ad fial periods. More specifically, bak i s project yields a radom retur of z i at t = 1, ad if held to maturity, a fixed, o-pledgeable log-term retur of A at t = 2. The bak ca (partially) liquidate its project at t = 1, but ca oly recover a fractio ζ < 1 of the project s full value. This assumptio is motivated by the fact that rapid liquidatio of real ad fiacial assets o baks balace sheets may be costly. 9 Iterbak ledig takes place through stadard debt cotracts siged at t = 0. Let k ij deote the amout of capital borrowed by bak j from bak i. The face value of j s debt to i is thus equal to y ij = R ij k ij, where R ij is the correspodig iterest rate. 10 I additio to its liabilities to other baks, each bak must also meet a outside obligatio of magitude v > 0 at t = 1, which is assumed to have seiority relative to its other liabilities. These more seior commitmets may be claims by the bak s retail depositors, wages due to its workers, taxes due to the govermet, or secured claims by o-bak fiacial istitutios such as moey market fuds. 11 The sum of liabilities of bak i is thus equal to y i + v, where y i = j i y ji. 12 Give the assumptio that log-term returs are ot pledgeable, all debts have to be cleared at t = 1. If bak j is uable to meet its t = 1 liabilities i full, it has to liquidate its project prematurely (i part or i full), where the proceeds are distributed amog its creditors. We assume that all juior creditors that is, the other baks are of equal seiority. Hece, if bak j ca meet its seior liabilities, v, but defaults o its debt to the juior creditors, they are repaid i proportio to the face value of the cotracts. O the other had, if j caot meet its more seior outside liabilities, its juior creditors receive othig. B. The Fiacial Network The ledig decisios of the baks ad the resultig couterparty relatios ca be represeted by a iterbak etwork. I particular, we defie the fiacial etwork correspodig to the bilateral debt cotracts i the ecoomy as a weighted, 9 This ca be either due to iefficiet abadomet of ogoig projects or due to the fact that rapid liquidatio of fiacial assets may happe at depressed prices (e.g., i fire sales). Furthermore, durig bakruptcy, the liabilities of the istitutio may be froze ad its creditors may ot immediately receive paymet, leadig to a effectively small ζ. 10 I this versio of the paper, we take the iterbak ledig decisios ad the correspodig iterest rates as give, ad do ot formally treat the baks actios at t = 0. This stage of the game is studied i detail i the workig paper versio of this article Acemoglu, Ozdaglar, ad Tahbaz-Salehi (2013). 11 Moey market fuds, for example, are amog the most major creditors i the repo market, i which ledig is collateralized, makig them de facto more seior tha all other creditors (Bolto ad Oehmke forthcomig). 12 This formulatio also allows for liabilities to outside fiacial istitutios that have the same level of seiority as iterbak loas by simply settig oe of the baks, say bak, to have claims but o iside-the-etwork liabilities, i.e., y i = 0 for all i.

570 THE AMERICAN ECONOMIC REVIEW february 2015 directed graph o vertices, where each vertex correspods to a bak ad a directed edge from vertex j to vertex i is preset if bak i is a creditor of bak j. The weight assiged to this edge is equal to y ij, the face value of the cotract betwee the two baks. Throughout the paper, we deote a fiacial etwork with the collectio of iterbak liabilities { y ij }. We say a fiacial etwork is symmetric if y ij = y ji for all pairs of baks i ad j. O the other had, a fiacial etwork is said to be regular if all baks have idetical iterbak claims ad liabilities; i.e., j i y ij = j i y ji = y for some y ad all baks i. Paels A ad B of Figure 1 illustrate two regular fiacial etworks, kow as the rig ad the complete etworks, respectively. The rig fiacial etwork represets a cofiguratio i which bak i > 1 is the sole creditor of bak i 1 ad bak 1 is the sole creditor of bak ; that is, y i, i 1 = y 1, = y. Hece, for a give value of y, the rig etwork is the regular fiacial etwork with the sparsest coectios. I cotrast, i the complete etwork, the liabilities of each bak are held equally by all others; that is, y ij = y/( 1) for all i j, implyig that the iterbak coectios i such a etwork are maximally dese. Fially, we defie, Defiitio 1: The fiacial etwork { y ij } is a γ -covex combiatio of fiacial etworks { y ij } ad { y ij } if there exists γ [0, 1] such that y ij = (1 γ) y ij + γ y ij for all baks i ad j. Thus, for example, a fiacial etwork that is a γ -covex combiatio of the rig ad the complete fiacial etworks exhibits a itermediate degree of desity of coectios: as γ icreases, the fiacial etwork approaches the complete fiacial etwork. II. Paymet Equilibrium The ability of a bak to fulfill its promises to its creditors depeds o the resources it has available to meet those liabilities, which iclude ot oly the returs o its ivestmet ad the cash at had, but also the realized value of repaymets by the bak s debtors. I this sectio, we show that a mutually cosistet collectio of repaymets o iterbak loas ad asset liquidatios always exists ad is geerically uique. Let x js deote the repaymet by bak s o its debt to bak j at t = 1. By defiitio, x js [0, y js ]. The total cash flow of bak j whe it does ot liquidate its project is thus equal to h j = c j + z j + s j x js, where c j is the cash carried over by the bak from the iitial period. If h j is larger tha the bak s total liabilities, v + y j, the the bak is capable of meetig its liabilities i full, ad as a result, x ij = y ij for all i j. If, o the other had, h j < v + y j, the bak eeds to start liquidatig its project i order to avoid default. Give that liquidatio is costly, the bak liquidates its project up to the poit where it ca cover the shortfall v + y j h j, or otherwise i its etirety to pay back its creditors as much as possible. Mathematically, the bak s liquidatio decisio, l j [0, A], is give by (1) l j = [ mi { 1_ ζ (v + y j h j ), A } ] +,

VOL. 105 NO. 2 Acemoglu et al.: systemic risk ad stability i fiacial etworks 571 Pael A. The rig fiacial etwork 1 y Pael B. The complete fiacial etwork y 1 2 3 Figure 1. The Rig ad the Complete Fiacial Networks where [ ] + stads for max {, 0} ad guaratees that the bak does ot liquidate its project if it ca meet its liabilities with a combiatio of the cash it holds, the shortterm retur o its project, ad the repaymet by its debtor baks. If the bak caot pay its debts i full eve with the full liquidatio of its project, it defaults ad its creditors are repaid accordig to their seiority. If h j + ζa is less tha v, the bak defaults o its seior liabilities ad its juior creditors receive othig; that is, x ij = 0. O the other had, if h j + ζa (v, v + y j ), seior liabilities are paid i full ad the juior creditors are repaid i proportio to the face value of their cotracts. Thus, the t = 1 paymet of bak j to a creditor bak i is equal to y ij (2) x ij = y j [ mi { y j, h j + ζ l j v} ] +, where recall that h j = c j + z j + s j x js deotes the fuds available to the bak i the absece of ay liquidatio ad l j is its liquidatio decisio give by (1). Thus, equatios (1) ad (2) together determie the liquidatio decisio ad the debt repaymets of bak j as a fuctio of its debtors repaymets o their ow liabilities. Defiitio 2: For a give realizatio of the projects short-term returs ad the cash available to the baks, the collectio ({ x ij }, { l i }) of iterbak debt repaymets ad liquidatio decisios is a paymet equilibrium of the fiacial etwork if (1) ad (2) are satisfied for all i ad j simultaeously. A paymet equilibrium is thus a collectio of mutually cosistet iterbak paymets ad liquidatios at t = 1. The otio of paymet equilibrium i our model is a geeralizatio of the otio of a clearig vector itroduced by Eiseberg ad Noe (2001) ad utilized by Shi (2008, 2009). I cotrast to these papers, baks i our model ot oly have fiacial liabilities of differet seiorities, but also ca obtai extra proceeds by (partially or completely) liquidatig their log-term projects.

572 THE AMERICAN ECONOMIC REVIEW february 2015 Via equatios (1) ad (2), the paymet equilibrium captures the possibility of fiacial cotagio i the fiacial system. I particular, give the iterdepedece of iterbak paymets across the etwork, a (sufficietly large) egative shock to a bak ot oly leads to that bak s default, but may also iitiate a cascade of failures, spreadig to its creditors, its creditors creditors, ad so o. The ext propositio shows that, regardless of the structure of the fiacial etwork, the paymet equilibrium always exists ad is uiquely determied over a geeric set of parameter values ad shock realizatios. 13 Propositio 1: For ay give fiacial etwork, cash holdigs, ad realizatio of shocks, a paymet equilibrium always exists ad is geerically uique. Fially, for ay give fiacial etwork ad the correspodig paymet equilibrium, we defie the (utilitaria) social surplus i the ecoomy as the sum of the returs to all agets; that is, u = ( π i + T i ), i=1 where T i v is the trasfer from bak i to its seior creditors ad π i is the bak s profit. III. Fiacial Cotagio As discussed above, the iterdepedece of iterbak paymets over the etwork implies that distress at a sigle bak may iduce a cascade of defaults throughout the fiacial system. I this sectio, we study how the structure of the fiacial etwork determies the extet of cotagio. For most of our aalysis, we focus o regular fiacial etworks i which the total claims ad liabilities of all baks are equal. Such a ormalizatio guaratees that ay variatio i the fragility of the system is simply due to how iterbak liabilities are distributed, while abstractig away from effects that are drive by other features of the fiacial etwork, such as size or leverage heterogeeity across baks. 14 To simplify the aalysis ad the expositio of our results, we also assume that the short-term returs o the baks ivestmets are i.i.d. ad ca oly take two values z i {a, a ϵ}, where a > v is the retur i the busiess as usual regime ad ϵ (a v + ζa, a) correspods to the magitude of a egative shock. The upper boud o ϵ simply implies that the retur of the project is always positive, whereas the lower boud guaratees that abset ay paymets by other baks, a distressed 13 As we show i the proof of Propositio 1, i ay coected fiacial etwork, the paymet equilibrium is uique as log as j=1 ( z j + c j ) v ζa. I the o-geeric case i which j=1 ( z j + c j ) = v ζ A, there may exist a cotiuum of paymet equilibria, i almost all of which baks default due to coordiatio failures. For example, if the ecoomy cosists of two baks with c 1 = c 2 = v, bilateral cotracts of face values y 12 = y 21, o shocks ad o proceeds from liquidatio (that is, ζ = 0 ), the defaults ca occur if baks do ot pay oe aother, eve though both are solvet. See Alvarez ad Barlevy (2014) for a similar characterizatio i fiacial etworks with some weak form of symmetry. 14 For example, Acemoglu et al. (2012) show that asymmetry i the degree of itercoectivity of differet idustries as iput suppliers i the real ecoomy plays a crucial role i the propagatio of shocks.

VOL. 105 NO. 2 Acemoglu et al.: systemic risk ad stability i fiacial etworks 573 bak that is, a bak directly hit by the egative shock would ot be able to pay its seior creditors. Fially, i what follows we assume that all baks hold the same amout of cash, which we ormalize to zero. Propositio 2: Coditioal o the realizatio of p egative shocks, the social surplus i the ecoomy is equal to u = (a + A) pϵ (1 ζ ) l i. As expected, the social surplus is decreasig i the extet of liquidatio i the correspodig paymet equilibrium. I particular, i the case that proceeds from liquidatio are trivial, that is, ζ = 0, the social surplus is simply determied by the umber of bak failures, that is, 15 i=1 u = a pϵ + ( # defaults)a. Uder this assumptio, it is atural to measure the performace of a fiacial etwork i terms of the umber of baks i default. Defiitio 3: Cosider two regular fiacial etworks { y ij } ad { y ij }. Coditioal o the realizatio of p egative shocks, (i) { y ij } is more stable tha { y ij } if E p u E p u, where E p is the expectatio coditioal o the realizatio of p egative shocks. (ii) { y ij } is more resiliet tha { y ij } if mi u mi u, where the miimum is take over all possible realizatios of p egative shocks. Stability ad resiliece capture the expected ad worst-case performaces of the fiacial etwork i the presece of p egative shocks, respectively. Clearly, both measures of performace ot oly deped o the umber ( p ) ad the size ( ϵ ) of the realized shocks, but also o the structure of the fiacial etwork. To illustrate the relatio betwee the extet of fiacial cotagio ad the etwork structure i the most trasparet maer, we iitially assume that exactly oe bak is hit with a egative shock ad that the proceeds from liquidatios are trivial, i.e., p = 1 ad ζ = 0. We relax these assumptios i Sectios IIID ad IIIE. A. Aggregate Iterbak Liabilities Our first result formalizes the ofte-made claim that the size of iterbak liabilities i the fiacial system is liked to the likelihood of fiacial cotagio. I particular, it shows that icreasig all pairwise fiacial liabilities by the same factor 15 Here ζ = 0 stads for ζ 0, sice i the limit where ζ = 0 there is o ecoomic reaso for liquidatio ad i fact, equatio (1) is ot well-defied.

574 THE AMERICAN ECONOMIC REVIEW february 2015 leads to a more fragile system, regardless of the structure of the origial fiacial etwork. Propositio 3: For a give regular fiacial etwork { y ij }, let y ij = β y ij for all i j ad some costat β > 1. The, fiacial etwork { y ij } is less stable ad resiliet tha { y ij }. I other words, a icrease i iterbak ledig comes at a cost i terms of fiacial stability. The ituitio for this result is simple: larger liabilities raise the exposure of each bak to the potetial distress at its couterparties, hece facilitatig cotagio. B. Small Shock Regime We ow characterize the fragility of differet fiacial etworks whe the size of the egative shock is less tha a critical threshold. Propositio 4: Let ϵ = (a v) ad suppose that ϵ < ϵ. The, there exists y such that for y > y, (i) The rig etwork is the least resiliet ad least stable fiacial etwork. (ii) The complete etwork is the most resiliet ad most stable fiacial etwork. (iii) The γ -covex combiatio of the rig ad complete etworks becomes (weakly) more stable ad resiliet as γ icreases. The above propositio thus establishes that as log as the size of the egative shock is below the critical threshold ϵ, the rig is the fiacial etwork most proe to fiacial cotagio, whereas the complete etwork is the least fragile. Furthermore, a more equal distributio of iterbak liabilities leads to less fragility. Propositio 4 is thus i lie with, ad geeralizes, the observatios made by Alle ad Gale (2000) ad Freixas, Parigi, ad Rochet (2000). The uderlyig ituitio is that a more diversified patter of iterbak liabilities implies that the burde of ay potetial losses is shared amog more baks, creatig a more robust fiacial system. I particular, i the extreme case of the complete fiacial etwork, the losses of a distressed bak are divided amog as may creditors as possible, guarateeig that the excess liquidity i the fiacial system ca fully absorb the trasmitted losses. O the other had, i the rig fiacial etwork, the losses of the distressed bak rather tha beig divided up betwee multiple couterparties are fully trasferred to its immediate creditor, leadig to the creditor s possible default. The coditio that ϵ < ϵ meas that the size of the egative shock is less tha the total excess liquidity available to the fiacial etwork as a whole. 16 Propositio 4 16 Recall that i the absece of ay shock, a v is the liquidity available to each bak after meetig its liabilities to the seior creditors outside the etwork.

VOL. 105 NO. 2 Acemoglu et al.: systemic risk ad stability i fiacial etworks 575 also requires that iterbak liabilities (ad claims) are above a certai threshold y, which is atural give that for small values of y, o cotagio would occur, regardless of the structure of fiacial etwork. The extreme fragility of the rig fiacial etwork established by Propositio 4 is i cotrast with the results of Acemoglu et al. (2012) ad Acemoglu, Ozdaglar, ad Tahbaz-Salehi (2014), who show that if the iteractios over the etwork are liear (or log liear), the rig is as stable as ay other regular etwork structure. This cotrast reflects the fact that, with liear iteractios, egative ad positive shocks cacel each other out i exactly the same way, idepedetly of the structure of the etwork. However, the ofte oliear ature of fiacial iteractios (captured i our model by the presece of usecured debt cotracts) implies that the effects of egative ad positive shocks are ot ecessarily symmetric. Stability ad resiliece are thus achieved by miimizig the impact of distress at ay give bak o the rest of the system. The rig fiacial etwork is highly fragile precisely because the adverse effects of a egative shock to ay bak are fully trasmitted to the bak s immediate creditor, triggerig maximal fiacial cotagio. I cotrast, a more diversified patter of iterbak liabilities reduces the impact of a bak s distress o ay sigle couterparty. Our ext result shows that this ituitio exteds to a broad set of etwork structures. We first itroduce a class of trasformatios that lead to a more diversified patter of iterbak liabilities. Defiitio 4: For two give subsets of baks M ad S, the fiacial etwork { y ij } is a (M, S, P) -majorizatio of the regular fiacial etwork { y ij } if y ij = p ik y kj if i S, j S k S, { y ij if i, j M where P is a doubly stochastic matrix of the appropriate size. 17 Followig such a trasformatio, the liabilities of baks i M to oe aother remai uchaged, while the liabilities of baks i S to baks i the complemet of S (deoted by S c ) become more evely distributed. This is due to the fact that premultiplicatio of a submatrix of liabilities by a doubly stochastic matrix P correspods to a mixig of those liabilities. Note also that Defiitio 4 does ot put ay restrictios o the liabilities of baks i S c to those i S or o those of baks i M c to oe aother beyod the fact that the resultig fiacial etwork { y ij } has to remai regular. Thus, the (M, S, P) -majorizatio of a fiacial etwork is ot ecessarily uique. Note further that this trasformatio is distict from a γ -covex combiatio with the complete etwork, accordig to which the liabilities of all baks become more equally distributed. 17 A square matrix is said to be doubly stochastic if it is elemet-wise oegative ad each of whose rows ad colums add up to 1.

576 THE AMERICAN ECONOMIC REVIEW february 2015 Propositio 5: Suppose that ϵ < ϵ ad y > y. For a give fiacial etwork, let D deote the set of baks i default ad suppose that the distressed bak ca meet its liabilities to its seior creditors. The, ay (D, D, P) -majorizatio of the fiacial etwork does ot icrease the umber of defaults. 18 The ituitio behid this result is similar to that of Propositio 4: a trasformatio of a fiacial etwork that spreads the fiacial liabilities of baks i default to the rest of the system guaratees that the excess liquidity available to the o-distressed baks are utilized more effectively. I the presece of small eough shocks, this ca ever lead to more defaults. A immediate corollary to this propositio exteds Propositio 4 to the γ -covex combiatio of a give etwork with the complete etwork. Corollary 1: Suppose that ϵ < ϵ ad y > y. If there is o cotagio i a fiacial etwork, the there is o cotagio i ay γ -covex combiatio of that etwork ad the complete etwork. Our results thus far show that as log as ϵ < ϵ, a more uiform distributio of iterbak liabilities, formalized by the otios of covex combiatios ad majorizatio trasformatios, ca ever icrease the fragility of a already stable fiacial etwork. Our ext example, however, illustrates that ot all trasformatios that equalize iterbak liabilities lead to a less fragile system. Example. Cosider the fiacial etwork depicted i Figure 2, i which iterbak liabilities are give by y i, i+1 = y i+1, i = qy if i odd { (1 q)y if i eve, where 1/2 q < 1 ad y > y ; i.e, the fiacial etwork cosists of pairs of itercoected baks located o a rig-like structure, with weaker iter-pair liabilities. The liabilities of a give bak i to baks i 1 ad i + 1 become more equalized as q approaches 1/2. Now suppose that bak 1 is hit with a egative shock of size ϵ = (3 + ω)(a v) for some small ω > 0. If q = 1/2, the, by symmetry, baks 1, 2, ad caot meet their liabilities i full, ad i particular, baks 2 ad default due to a small shortfall of size ω(a v). However, give that is just at the verge of solvecy, icreasig q slightly above 1/2 guaratees that bak o loger defaults, as a larger fractio of the losses would ow be trasferred to bak 2. More specifically, oe ca show that if q = (1 + ω)/2, the oly baks 1 ad 2 default, whereas all other baks ca meet their liabilities i full. To summarize, though Propositios 4(iii) ad 5 ad Corollary 1 show that, i the presece of small shocks, γ -covex combiatios with the complete etwork ad various majorizatio trasformatios do ot icrease the fragility of the fiacial 18 We would like to thak a aoymous referee for suggestig this result.

VOL. 105 NO. 2 Acemoglu et al.: systemic risk ad stability i fiacial etworks 577 2 qy qy 1 (1 q)y (1 q)y 3 Figure 2. The Liabilities of Each Bak to Its Two Couterparties Become More Equalized as q Approaches 1/2 system, the same logic does ot apply to all trasformatios that equalize iterbak liabilities. For istace, i the precedig example, lower values of q which make the liabilities of a give bak to its two couterparties more equal, may evertheless icrease fragility by trasferrig resources away from the bak that relies o them for survival. C. Large Shock Regime Propositios 4 ad 5, alog with Corollary 1, show that as log as the magitude of the egative shock is below the threshold ϵ, a more equal distributio of iterbak liabilities leads to less fragility. I particular, the complete etwork is the most stable ad resiliet fiacial etwork: except for the bak that is directly hit with the egative shock, o other bak defaults. Our ext set of results, however, shows that whe the magitude of the shock is above the critical threshold ϵ, this picture chages dramatically. We start with the followig defiitio: Defiitio 5: A regular fiacial etwork is δ -coected if there exists a collectio of baks S N such that max { y ij, y ji } δy for all i S ad j S. I other words, i a δ -coected fiacial etwork, the fractio of liabilities of baks iside ad outside of S to oe aother is o more tha δ [0, 1]. Hece, for small values of δ, the baks i S have weak ties i terms of both claims ad liabilities to the rest of the fiacial etwork. We have the followig result: Propositio 6: Suppose that ϵ > ϵ ad y > y. The, (i) The complete ad the rig etworks are the least stable ad least resiliet fiacial etworks.

578 THE AMERICAN ECONOMIC REVIEW february 2015 (ii) For small eough values of δ, ay δ -coected fiacial etwork is strictly more stable ad resiliet tha the rig ad complete fiacial etworks. Thus, whe the magitude of the egative shock crosses the critical threshold ϵ, the complete etwork exhibits a form of phase trasitio: it flips from beig the most to the least stable ad resiliet etwork, achievig the same level of fragility as the rig etwork. I particular, whe ϵ > ϵ, all baks i the complete etwork default. The ituitio behid this result is simple: sice all baks i the complete etwork are creditors of the distressed bak, the adverse effects of the egative shock are directly trasmitted to them. Thus, whe the size of the egative shock is large eough, all baks icludig those origially uaffected by the egative shock default. Not all fiacial systems, however, are as fragile i the presece of large shocks. I fact, as part (ii) shows, for small eough values of δ, ay δ -coected fiacial etwork is strictly more stable ad resiliet tha both the complete ad the rig etworks. The presece of such weakly coected compoets i the etwork guaratees that the losses rather tha beig trasmitted to all other baks are bore i part by the distressed bak s seior creditors. Take together, Propositios 4 ad 6 illustrate the robust-yet-fragile property of highly itercoected fiacial etworks cojectured by Haldae (2009). They show that more desely itercoected fiacial etworks, epitomized by the complete etwork, are more stable ad resiliet i respose to a rage of shocks. However, oce we move outside this rage, these dese itercoectios act as a chael through which shocks to a subset of the fiacial istitutios trasmit to the etire system, creatig a vehicle for istability ad systemic risk. The ituitio behid such a phase trasitio is related to the presece of two types of shock absorbers i our model, each of which is capable of reducig the extet of cotagio i the etwork. The first absorber is the excess liquidity, a v > 0, of the o-distressed baks at t = 1 : the impact of a shock is atteuated oce it reaches baks with excess liquidity. This mechaism is utilized more effectively whe the fiacial etwork is more complete, a observatio i lie with the results of Alle ad Gale (2000) ad Freixas, Parigi, ad Rochet (2000). However, the claim v of seior creditors of the distressed bak also fuctios as a shock absorptio mechaism. Rather tha trasmittig the shocks to other baks i the system, the seior creditors ca be forced to bear (some of) the losses, ad hece limit the extet of cotagio. I cotrast to the first mechaism, this shock absorptio mechaism is best utilized i weakly coected fiacial etworks ad is the least effective i the complete etwork. Thus, whe the shock is so large that it caot be fully absorbed by the excess liquidity i the system which is exactly whe ϵ > ϵ fiacial etworks that sigificatly utilize the secod absorber are less fragile. Usig Defiitio 4, we ca exted this ituitio to a broader class of fiacial etworks: Propositio 7: Suppose that y > y ad suppose that bak j is hit with a egative shock ϵ > ϵ. Let D deote the set of baks other tha j that default ad let P = (1 γ)i + γ / ( 1)11 for some γ [0, 1]. The, ay

VOL. 105 NO. 2 Acemoglu et al.: systemic risk ad stability i fiacial etworks 579 (D, { j}, P) -majorizatio of the fiacial etwork does ot decrease the umber of defaults. The above result, which is a large shock couterpart to Propositio 5, shows that i cotrast to the small shock regime, a trasformatio that leads to a more uiform distributio of the distressed bak s liabilities does ot reduce but may icrease the extet of cotagio. The remaider of this subsectio provides a characterizatio of the set of baks that default i a geeral fiacial etwork ad shows that the ituitio o the role of itercoectivity i the fragility of the system remais valid for a broad set of etwork structures. We first defie a ew otio of distace over the fiacial etwork. Defiitio 6: The harmoic distace from bak i to bak j is (3) m ij = 1 + ( y ik y ) m kj, k j with the covetio that m ii = 0 for all i. 19 The harmoic distace from bak i to bak j depeds ot oly o how far each of its immediate debtors are from j, but also o the itesity of their liabilities to i. Such a defiitio implies that the harmoic distace betwee ay pair of baks ca be cosiderably differet from the shortest-path, geodesic distace defied over the fiacial etwork. I particular, the more liability chais (direct or idirect) exist betwee baks i ad j, the closer the two baks are to oe aother. Propositio 8: Suppose that bak j, hit with the egative shock, defaults o its seior liabilities. The, there exists m such that, (i) If m ij < m, the bak i defaults. (ii) If all baks i the fiacial etwork default, the m ij < m for all i. This result implies that baks that are closer to the distressed bak i the sese of the harmoic distace are more vulerable to default. Cosequetly, a fiacial etwork i which the pairwise harmoic distaces betwee ay pairs of baks are smaller is less stable ad resiliet i the presece of large shocks. 20 Propositio 8 thus geeralizes Propositio 6. I particular, oe ca verify that the harmoic distace betwee ay pair of baks is miimized i the complete fiacial etwork as predicted by Propositio 6(i). 21 O the other had, i a δ -coected etwork (for sufficietly small δ ), there always exists a pair of baks whose pairwise harmoic 19 Strictly speakig, the harmoic distace is a quasi-metric, as it does ot satisfy the symmetry axiom (that is, i geeral, m ij m ji ). Nevertheless, for ease of referece, we simply refer to m ij as the distace from bak i to bak j. For a discussio o the properties of the harmoic distace, see Appedix A. 20 Note that ϵ > ϵ implies that the bak hit by the egative shock defaults o its seior creditors. For a proof, see Lemma B6 i the Appedix. 21 See Appedix A.

580 THE AMERICAN ECONOMIC REVIEW february 2015 distace is greater tha m, esurig that the etwork is strictly more stable ad resiliet tha the complete fiacial etwork, thus establishig Propositio 6(ii) as a corollary. Propositio 8 also highlights that i a give fiacial etwork, the bak that is closest to all others i the sese of harmoic distace is the most systemically importat fiacial istitutio: a shock to such a bak would lead to the maximal umber of defaults. This observatio cotrasts with much of the recet empirical literature that relies o off-the-shelf measures of etwork cetrality such as eigevector or Boacich cetralities for idetifyig systemically importat fiacial istitutios. 22 Such stadard etwork cetrality measures would be appropriate if iterbak iteractios are liear. I cotrast, Propositio 8 shows that if iterbak iteractios exhibit oliearities similar to those iduced by the presece of debt cotracts, it is the harmoic distaces of other baks to a fiacial istitutio that determie its importace from a systemic perspective. 23 Our last result i this subsectio relates the iterbak harmoic distaces to a ituitive structural property of the fiacial etwork. Defiitio 7: The bottleeck parameter of a fiacial etwork is ϕ = mi ( y ij / y) S N S S c. i S j S Roughly speakig, ϕ quatifies how the fiacial etwork ca be partitioed ito two roughly equally-sized compoets, while miimizig the extet of itercoectivity betwee the two. 24 I particular, for a give partitio of the fiacial etwork ito two subsets of baks, S ad S c, the quatity i S, j S y ij is equal to the total liabilities of baks i S c to those i S (see Figure 3). The bottleeck parameter thus measures the miimal extet of itercoectivity betwee the baks i ay partitio (S, S c ), while esurig that either set is sigificatly smaller tha the other. Thus, a highly itercoected fiacial etwork, such as the complete etwork, exhibits a large bottleeck parameter, whereas ϕ = 0 for ay discoected etwork. We have the followig result: Lemma 1: For ay symmetric fiacial etwork, (4) 1 max 2ϕ m ij 16 i j ϕ 2, where ϕ is the correspodig bottleeck parameter. 22 See, for example, Bech, Chapma, ad Garratt (2010); Bech ad Atalay (2010); Akram ad Christopherse (2010); Bisias et al. (2012); ad Craig, Fecht, ad Tümer-Alka (2013). 23 I their umerical simulatios, Soramäki ad Cook (2013) make a similar observatio ad propose a measure of relative importace of baks that is related to our otio of harmoic distace. 24 The bottleeck parameter is closely related to the otios of coductace ad Cheeger costat i spectral graph theory. For a discussio, see Chapters 2 ad 6 of Chug (1997).

VOL. 105 NO. 2 Acemoglu et al.: systemic risk ad stability i fiacial etworks 581 S S c Figure 3. A Partitio of the Fiacial Network ito Subsets S ad S c Note: i S, j S y ij is equal to the sum of iterbak liabilities over the dashed lie ad measures the aggregate liabilities of baks i S c to those i S ad vice versa. The above lemma thus provides bouds o the maximum harmoic distace betwee ay pairs of baks i the fiacial etwork i terms of the etwork s bottleeck parameter. More importatly, it shows that the relatioship betwee the extet of iterbak coectivity ad the fiacial etwork s fragility discussed after Propositio 6 holds for a broad set of etwork structures. I particular, the iterbak harmoic distaces are smaller whe the fiacial etwork is more itercoected, guarateeig more defaults i the presece of a large shock. The followig corollary to Propositio 8 ad Lemma 1 formalizes this observatio: Corollary 2: Suppose that ϵ > ϵ. The, there exist costats ϕ > ϕ such that for ay symmetric fiacial etwork, (i) If ϕ > ϕ, the all baks default. (ii) If ϕ < ϕ, the at least oe bak does ot default. We ed this discussio by demostratig the implicatios of the above results by meas of a few examples. First, cosider the complete fiacial etwork. It is clear that for ay partitio (S, S c ) of the set of baks, i S, j S y ij y = 1 S S c, ad as a result, ϕ comp = 1/( 1). O the other had, choosig S = {i} i ay arbitrary fiacial etwork guaratees that ϕ 1/ ( 1). Therefore, the complete etwork has the largest bottleeck parameter amog all regular fiacial etworks. Corollary 2 thus implies that if a large shock leads to the default of all baks i ay fiacial etwork, it would also do so i the complete etwork, as predicted by Propositio 6. At the other ed of the spectrum, i a δ -coected fiacial etwork, there exists a partitio (S, S c ) of the set of baks for which max { y ij, y ji } δy for all i S ad j S c. It is the immediate to verify that the bottleeck parameter of ay such etwork satisfies ϕ δ. Hece, for small eough values of δ ad i the presece

582 THE AMERICAN ECONOMIC REVIEW february 2015 of large shocks, the fiacial etwork is strictly more stable ad resiliet tha the complete etwork, agai i lie with the predictios of Propositio 6. Fially, give a regular fiacial etwork with iterbak liabilities { y ij } ad bottleeck parameter ϕ, let y ij (γ) = (1 γ) y ij + γ y ij comp deote the iterbak liabilities i the γ -covex combiatio of the former with the complete etwork. Oe ca show that the correspodig bottleeck parameter satisfies 25 ϕ(γ) = (1 γ)ϕ + γ ϕ comp. I view of the observatio that the complete etwork has the greatest bottleeck parameter across all fiacial etworks, the above equality implies that ϕ(γ) is icreasig i γ, establishig the followig couterpart to Corollary 1: Corollary 3: Suppose that ϵ > ϵ, ad cosider a symmetric fiacial etwork for which ϕ > ϕ. The, the γ -covex combiatio of the etwork ad the complete etwork is o more stable or resiliet for all γ. This corollary implies that, i cotrast to our results for the small shock regime, a more diversified patter of iterbak liabilities caot prevet the systemic collapse of the etwork i the presece of large shocks. D. Multiple Shocks The isights o the relatioship betwee the extet of cotagio ad the structure of the fiacial etwork studied so far geeralize to the case of multiple egative shocks. Propositio 9: Let p deote the umber of egative shocks ad let ϵ p = (a v)/p. There exist costats y p > y ˆ p > 0, such that (i) If ϵ < ϵ p ad y > y p, the the complete etwork is the most stable ad resiliet fiacial etwork, whereas the rig etwork is the least resiliet. (ii) If ϵ > ϵ p ad y > y p, the the complete ad the rig fiacial etworks are the least stable ad resiliet fiacial etworks. Furthermore, if p < 1, the there exists a δ -coected fiacial etwork that is strictly more stable tha the complete ad rig fiacial etworks. (iii) If ϵ > ϵ p ad y ( y ˆ p, y p ), the the complete etwork is the least stable ad resiliet fiacial etwork. Furthermore, the rig etwork is strictly more stable tha the complete fiacial etwork. 25 For ay subset of baks S, we have i S, j S y ij (γ) = (1 γ) i S, j S y ij + γ S S c /( 1).

VOL. 105 NO. 2 Acemoglu et al.: systemic risk ad stability i fiacial etworks 583 Parts (i) ad (ii) geeralize the isights of Propositios 4 ad 6 to the case of multiple shocks. The key ew observatio is that the critical threshold ϵ p that defies the boudary of the small ad large shock regimes is a decreasig fuctio of p. Cosequetly, the umber of egative shocks plays a role similar to that of the size of the shocks. More specifically, as log as the magitude ad the umber of egative shocks affectig fiacial istitutios are sufficietly small, more complete iterbak claims ehace the stability of the fiacial system. The uderlyig ituitio is idetical to that behid Propositio 4: the more itercoected the fiacial etwork is, the better the excess liquidity of o-distressed baks is utilized i absorbig the shocks. O the other had, if the magitude or the umber of shocks is large eough so that the excess liquidity i the fiacial system is ot sufficiet for absorbig the losses, fiacial itercoectios serve as a propagatio mechaism, creatig a more fragile fiacial system. Furthermore, as i Propositio 6, weakly coected etworks esure that the losses are shared with the seior creditors of the distressed baks, protectig the rest of the system. Part (iii) of Propositio 9 cotais a ew result. It shows that i the presece of multiple shocks, the claims of the seior creditors i the rig fiacial etwork are used more effectively as a shock absorptio mechaism tha i the complete fiacial etwork. I particular, the closer the distressed baks i the rig fiacial etwork are to oe aother, the larger the loss their seior creditors are collectively forced to bear. This limits the extet of cotagio i the etwork. 26 As a fial remark, we ote that a multi-shock couterpart to Propositio 8 ca also be established. I particular, if m ij < m for all i ad j, the all baks i the fiacial etwork default at the face of p shocks of size ϵ > ϵ p. E. No-Trivial Liquidatio Proceeds Our results thus far were restricted to the case i which the proceeds from liquidatios are trivial, i.e., ζ = 0. The ext propositio shows that our mai results remai valid eve whe liquidatio recovers a positive fractio ζ > 0 of a project s returs (while cotiuig to assume that projects ca be partially liquidated, a assumptio we relax at the ed of this subsectio). Propositio 10: Suppose that baks ca partially liquidate their projects at t = 1. Let ϵ (ζ) = (a v) + ζa ad ϵ (ζ) = (a v) + ζa. The, there exists y (ζ) such that for y > y (ζ) : (i) If ϵ < ϵ (ζ), the the complete ad the rig fiacial etworks are, respectively, the most ad the least stable ad resiliet fiacial etworks. (ii) If ϵ > ϵ (ζ), the the complete ad the rig fiacial etworks are the least stable ad resiliet etworks, while ay δ -coected etwork for small eough δ is strictly more stable ad resiliet. 26 I a related cotext, Alvarez ad Barlevy (2014) show that the aggregate equity of the bakig system with a rig etwork structure depeds o the locatio of the shocks. Also see Barlevy ad Nagaraja (2013) for a iterestig coectio betwee the problem of cotagio i the rig fiacial etwork ad the so-called circle-coverig problem.

584 THE AMERICAN ECONOMIC REVIEW february 2015 (iii) If ϵ (ζ) < ϵ < ϵ (ζ), the the complete etwork is strictly more stable ad resiliet tha the rig etwork. Furthermore, if ϵ > ϵ (ζ) + ζa, the there exists a δ -coected etwork which is strictly more stable ad resiliet tha the complete etwork. Part (i) correspods to the small shock regime, i which the complete etwork outperforms all other regular fiacial etworks ad the rig etwork is the most fragile of all. Part (ii), o the other had, correspods to our large shock regime results: for large eough shocks, the complete etwork becomes as fragile as the rig fiacial etwork, 27 whereas the presece of weakly coected compoets i the fiacial system guaratees that the losses are shared with the distressed bak s seior creditors, protectig the rest of the etwork. Propositio 10 also establishes a itermediate regime, i which the complete etwork lies strictly betwee the rig ad δ -coected fiacial etworks i terms of stability ad resiliece. As part (iii) shows, the threshold ϵ (ζ) at which the complete etwork becomes the most fragile fiacial etwork o loger coicides with the threshold ϵ (ζ) + ζa at which it starts uderperformig δ -coected etworks. I other words, eve though both the small ad large shock regimes exist regardless of the value of ζ, the phase trasitio betwee the two becomes smoother as ζ icreases. Note that, as expected, the two thresholds coicide whe ζ = 0. The emergece of the itermediate regime i Propositio 10 relies ot oly o the fact that ζ > 0, but also o the assumptio that baks ca partially liquidate their projects at t = 1. I cotrast, if baks are forced to liquidate their projects i full (e.g., because it is difficult to liquidate a fractio of a ogoig real project), the sharp phase trasitio result from the earlier subsectios is restored, eve whe ζ is strictly positive. I particular, i this case, the complete etwork flips from beig the most stable to the most fragile fiacial etwork oce the size of the shock crosses some threshold ϵ (ζ). The ext propositio formalizes this statemet. Propositio 11: Suppose that partial liquidatio at t = 1 is ot feasible. Let ϵ (ζ) ad y (ζ) be as defied i Propositio 10, ad suppose that y > y (ζ). (i) If ϵ < ϵ (ζ), the the complete etwork is the most stable ad resiliet fiacial etwork, whereas the rig etwork is the least stable ad resiliet fiacial etwork. (ii) If ϵ > ϵ (ζ), the the complete ad the rig etworks are the least stable ad resiliet fiacial etworks. Moreover, for small eough values of δ, ay δ -coected fiacial etwork is strictly more stable ad resiliet tha the rig ad complete fiacial etworks. 27 As i Propositio 9, the rig fiacial etwork may be more stable tha the complete fiacial etwork i the presece of multiple shocks.

VOL. 105 NO. 2 Acemoglu et al.: systemic risk ad stability i fiacial etworks 585 F. Size Heterogeeity As metioed earlier, our aalysis has so far focused o regular fiacial etworks i order to delieate the role of etwork structure o the fiacial system s fragility, while abstractig from the impact of asymmetries i size ad leverage across baks. Focusig o the large shock regime, the ext result illustrates that our key isights cotiue to hold eve i the presece of such asymmetries. Suppose that all assets ad liabilities of bak i are scaled by a costat θ i > 0. I particular, bak i s liabilities to its seior creditors ad all other baks are equal to θ i v ad y i = θ i y, whereas its short-term ad log-term returs are give by θ i z i ad θ i A. As before, we assume that oly oe egative shock is realized ad that ζ = 0. Propositio 12: Suppose that bak j is hit with a egative shock ϵ > (a v) k=1 ( θ k / θ j ). The, (i) Bak j defaults o its seior liabilities; (ii) All other baks also default if ad oly if m ˆ ij < θ i m for all baks i, where y (5) m ˆ ij = θ i + ( ik y i ) m ˆ kj. k j I lie with our earlier results, part (i) shows that a large eough shock guaratees that the seior creditors will bear some of the losses. Naturally, the correspodig threshold ow depeds o the relative size of the distressed bak: the greater the size of the bak, the smaller the threshold at which the seior creditors start to suffer. The secod part of Propositio 12 is the couterpart to Propositio 8, establishig that the risk of systemic failures depeds o the size-adjusted harmoic distaces of other baks from the distressed bak j. Comparig (5) with (3) shows that the susceptibility of bak i to default ot oly depeds o the itesity of the liabilities alog the chais that coect j to i, but also o the size of all the itermediary baks that exist betwee the two. Ideed, if the fiacial etwork is coected, a icrease i the relative size of ay bak k i, j makes bak i more robust to a shock to bak j. This is a simple cosequece of the fact that ay such icrease would raise i s distace from j. The effect of bak i s size o its ow fragility is more subtle, as a greater θ i icreases both the distace m ˆ ij of bak i from the distressed bak j as well as the threshold θ i m. The two effects, however, are ot proportioal: whereas θ i m icreases liearly, m ˆ ij is sub-liear i θ i. Cosequetly, icreasig the relative size of a bak makes it more vulerable to cotagious defaults. Fially, we remark that eve though our results i this sectio were illustrated for a eviromet i which shocks ca take oly two values, similar results ca be obtaied for more geeral shock distributios. IV. Cocludig Remarks The recet fiacial crisis has rekidled iterest i the relatioship betwee the structure of the fiacial etwork ad systemic risk. Two polar views o this

586 THE AMERICAN ECONOMIC REVIEW february 2015 relatioship have bee suggested i the academic literature ad the policy world. The first maitais that the icompleteess of the fiacial etwork is a source of istability, as idividual baks are overly exposed to the liabilities of a hadful of fiacial istitutios. Accordig to this argumet, a more complete fiacial etwork which limits the exposure of baks to ay sigle couterparty would be less proe to systemic failures. The secod view, i stark cotrast, hypothesizes that it is the highly itercoected ature of the fiacial system that cotributes to its fragility, as it facilitates the spread of fiacial distress ad solvecy problems from oe istitutio to the rest i a epidemic-like fashio. This paper provides a tractable theoretical framework for the study of the ecoomic forces shapig the relatioship betwee the structure of the fiacial etwork ad systemic risk. We show that as log as the magitude (or the umber) of egative shocks is below a critical threshold, a more diversified patter of iterbak liabilities leads to less fragility. I particular, all else equal, the sparsely coected rig fiacial etwork (correspodig to a credit chai) is the most fragile of all cofiguratios, whereas the highly itercoected complete fiacial etwork is the cofiguratio least proe to cotagio. I lie with the observatios made by Alle ad Gale (2000), our results establish that, i more complete etworks, the losses of a distressed bak are passed to a larger umber of couterparties, guarateeig a more efficiet use of the excess liquidity i the system i forestallig defaults. We also show, however, that whe egative shocks are larger tha a certai threshold, the secod view o the relatioship betwee the structure of the fiacial etwork ad the extet of cotagio prevails. Now, completeess is o loger a guaratee for stability. Rather, i the presece of large shocks, fiacial etworks i which baks are oly weakly coected to oe aother are less proe to systemic failures. Such a phase trasitio is due to the fact that, i additio to the excess liquidity withi the fiacial etwork, the seior liabilities of baks ca also act as shock absorbers. Weak itercoectios guaratee that the more seior creditors of a distressed bak bear most of the losses, ad hece protect the rest of the system agaist cascadig defaults. Our model thus formalizes the robust-yet-fragile property of itercoected fiacial etworks cojectured by Haldae (2009). More broadly, our results highlight the possibility that the same features that make a fiacial etwork structure more stable uder certai coditios may fuctio as sigificat sources of systemic risk ad istability uder other coditios. Our results idicate that the idetificatio of systemically importat fiacial istitutios i the iterbak etwork requires some care. I particular, some of the existig empirical aalyses that rely o off-the-shelf ad well-kow measures of etwork cetrality may be misleadig, as such measures are relevat oly if the iterbak iteractios are liear. I cotrast, our aalysis shows that, if the fiacial iteractios exhibit oliearities similar to those iduced by usecured debt cotracts, the systemic importace of a fiacial istitutio is captured via its harmoic distace to other baks, suggestig that this ew otio of etwork distace should feature i theoretically-motivated policy aalyses. Our model also highlights several possible aveues for policy itervetios. From a ex ate perspective, a atural objective is to icrease the stability ad resiliece of the fiacial system by regulatig the extet ad ature of iterbak likages. The importat isight of our aalysis is that such itervetios have to be iformed

VOL. 105 NO. 2 Acemoglu et al.: systemic risk ad stability i fiacial etworks 587 by the expected size of the egative shocks. For istace, imposig limits o the exposure betwee pairs of fiacial istitutios motivated by the possibility of small shocks may be couterproductive if the shocks are i fact large. I additio, our characterizatio results i terms of the harmoic distace ca serve as a guidelie for ex post policy itervetios oce a shock is realized. I particular, our aalysis suggests that ijectig additioal fuds to or bailig out systemically importat fiacial istitutios that have a large impact o other etities withi the etwork would cotai the extet of cotagio (though such itervetios may iduce moral hazard-type cocers ex ate). Aother importat dimesio of policy aalysis is the discussio of whether observed fiacial etworks are likely to be (costraied) efficiet. This is studied i some detail i the workig paper versio of our work (Acemoglu, Ozdaglar, ad Tahbaz-Salehi 2013). There we show that, i the presece of coveats that make the iterbak iterest rates cotiget o the borrowers ledig decisios, the equilibrium patter of iterbak liabilities is efficiet as log as the ecoomy cosists of two or three baks. However, i the presece of more tha three baks, a specific (ad ew) type of exterality arises: give that the cotracts offered to a bak do ot coditio o the asset positios of the baks to whom it leds, the borrowers do ot iteralize the effect of their decisios o their creditors creditors. We show that this may lead to the formatio of iefficiet etworks, embeddig excessive couterparty risk from the viewpoit of social efficiecy, both i the small ad the large shock regimes. These results suggest that there may be room for welfare-improvig govermet itervetios at the etwork formatio stage. We view our paper as a first step i the directio of a systematic aalysis of the broader implicatios of the fiacial etwork architecture. Several importat issues remai ope to future research. First, this paper focused o the implicatios of a give etwork structure o the fragility of the fiacial system. A systematic aalysis of the edogeous formatio of fiacial etworks ad their efficiecy ad policy implicatios, alog the lies of Acemoglu, Ozdaglar, ad Tahbaz-Salehi (2013) briefly discussed above, is a obvious area for future research. Secod, our focus was o a specific form of etwork iteractios amog fiacial istitutios; amely, the spread of couterparty risk via usecured debt cotracts. I practice, however, there are other importat types of fiacial iterdepedecies. I particular, (i) the fire sales of some assets by a bak may create distress o other istitutios that hold similar assets; ad (ii) withdrawal of liquidity by a bak (for example, by ot rollig over a repo agreemet or icreasig the haircut o the collateral) may lead to a chai reactio playig out over the fiacial etwork. How the ature of these differet types of fiacial iterdepedecies determie the relatioship betwee uderlyig etwork structure ad systemic risk remais a ope questio for future research. Third, our model purposefully abstracted from several importat istitutioal details of the bakig system. For example, it eschewed other forms of iterbak ledig (such as repurchase agreemets) as well as the differeces betwee deposit-takig istitutios, ivestmet baks, ad other specialized fiacial istitutios such as hedge fuds. It also abstracted from the complex maturity structure of iterbak liabilities which ca create differet types of cotagio owig to a mismatch i

588 THE AMERICAN ECONOMIC REVIEW february 2015 the maturity of the assets ad liabilities of a fiacial istitutio. Icorporatig these importat istitutioal realities is aother obvious area of ivestigatio. Last but ot least, a systematic empirical ivestigatio of these ad other types of etwork iteractios i fiacial markets is a importat area for research. Appedix A: The Harmoic Distace The harmoic distace defied i (3) provides a measure of proximity betwee a pair of baks i the fiacial etwork. Accordig to this otio, two baks are closer to oe aother the more direct or idirect liability chais exist betwee them ad the higher the face value of those liabilities are. This Appedix studies some of the basic properties of the harmoic distace. The first key observatio is that the otio of harmoic distace is closely related to a discrete-state Markov chai defied over the etwork. I particular, let Q be a matrix whose (i, j) elemet is equal to the fractio of bak j s liabilities to i ; that is, q ij = y ij / y. By costructio, Q is a (row ad colum) stochastic matrix, ad hece ca be iterpreted as the trasitio probability matrix of a Markov chai with the uiform statioary distributio. For this Markov chai, defie the mea hittig time from i to j as the expected umber of time steps it takes the chai to hit state j coditioal o startig from state i. We have the followig result: Propositio A1: The harmoic distace from bak i to j is equal to the mea hittig time of the Markov chai from state i to state j. Thus, the harmoic distace provides a ituitive measure of proximity betwee ay pair of baks: the loger it takes o average for the Markov chai to reach state j from state i, the larger the harmoic distace betwee the two correspodig baks i the fiacial etwork. The above observatio eables us to idetify the properties of the harmoic distace by relyig o kow results from the theory of Markov chais. For istace, give that expected hittig times i a Markov chai are o-symmetric, it is immediate that i geeral, m ij m ji, eve whe the fiacial etwork is symmetric. This observatio thus implies that the harmoic distace is ot a otio of distace i its strictest sese. Nevertheless, it satisfies a weaker form of symmetry: Propositio A2: Suppose that the fiacial etwork is symmetric. For ay three baks i, j, ad k, (A1) m ij + m jk + m ki = m ik + m kj + m ji. The followig is a immediate implicatio of the above result (Lovász 1996): Corollary A1: If the fiacial etwork is symmetric, the there exists a orderig of baks such that if i is preceded by j, the m ij m ji. The above orderig ca be obtaied by fixig a arbitrary referece bak k ad orderig the rest of the baks accordig to the value of m ik m ki. Such a orderig,

VOL. 105 NO. 2 Acemoglu et al.: systemic risk ad stability i fiacial etworks 589 however, is ot ecessarily uique due to the possibility of ties. Nevertheless, by Propositio A2, pairs of baks for which m ij = m ji form a equivalece class, implyig that there exists a well-defied orderig of the equivalece classes, idepedet of the referece bak k (Lovász 1996). More specifically, there exists a partitio ( S 1,, S r ) of the set of baks such that (i) m ij = m ji for all i, j S t ; ad (ii) if t < t, the m ij < m ji for all i S t ad j S t. Therefore, i view of Propositio 8, baks i the higher equivalece classes are systemically more importat, as a large shock to them would lead to loger chais of defaults. Yet, at the same time, such baks are ot at much risk of cotagious defaults if aother bak is hit with a egative shock. I cotrast, baks i the lower equivalece classes are proe to default due to cotagio eve though a egative shock to them would ot lead to a large cascade of failures. Propositio A3 (Triagle Iequality): For ay triple of baks i, j, ad k, m ij + m jk m ik. Furthermore, the iequality is tight if ad oly if all liability chais from k to i pass through bak j. The sum of harmoic distaces of a give bak from all others is a ivariat structural property of the fiacial etwork that does ot deped o the idetity of the bak: Propositio A4: j i m ij = j k m kj for all pairs of baks i ad k. We ed this discussio by showig that the maximum pairwise harmoic distace betwee ay two baks is miimized i the complete fiacial etwork. First, ote that give the full symmetry i the complete fiacial etwork, m ij is the same for all distict pairs of baks i ad j. Thus, equatio (3) immediately implies that m ij = 1 for i j. O the other had, summig both sides of (3) over j i i a arbitrary fiacial etwork implies j i m ij = ( 1) + q ik m kj j i k j = ( 1) + q ik m kj q ik m ki. k i j k k i Give that the quatity j k m kj does ot deped o k, the above equality simplifies to k i q ik m ki = 1, implyig that max k i m ki 1. Thus, the maximum pairwise harmoic distace is miimized i the complete fiacial etwork.

590 THE AMERICAN ECONOMIC REVIEW february 2015 Appedix B: Proofs This Appedix cotais the proofs of Propositios 1 8, Corollary 1, ad Lemma 1. The rest of the proofs are provided i the olie Appedix. Notatio. Let Q R be the matrix whose (i, j) elemet is equal to the fractio of bak j s liabilities to i ; that is, q ij = y ij / y j. We let y = [ y 1,, y ] deote the vector of the baks total liabilities to oe aother ad use l = [ l 1,, l ] to deote the vector of baks liquidatio decisios. Thus, rewritig equatios (1) ad (2) i matrix otatio implies that a paymet equilibrium is simply a pair of vectors (x, l) that simultaeously solve (B1) x = [ mi {Qx + e + ζl, y } ] + (B2) l = mi _ 1 [ { ζ (y Qx e), A1 }] +, where e j = c j + z j v ad x j = i j x ij is the total debt repaymet of bak j to the rest of the baks. Throughout the proofs, we use 1 to deote a vector (of appropriate size) whose elemets are equal to 1. Lemma B1: Suppose that β > 0. The, A. Prelimiary Lemmas [mi {α, β} ] + [mi { α ˆ, β} ] + α α ˆ. Furthermore, the iequality is tight oly if either α = α ˆ or α, α ˆ [0, β]. Lemma B2: Suppose that (x, l) is a paymet equilibrium of the fiacial etwork. The, x satisfies (B3) x = [mi {Qx + e + ζa1, y} ] +. Coversely, if x R satisfies (B3), the there exists l [0, A] such that (x, l) is a paymet equilibrium. Proof: First, suppose that the pair (x, l) is a paymet equilibrium of the fiacial etwork. Thus, by defiitio, ζl = [mi {y (Qx + e), ζa1}] +, which implies that Therefore, Qx + e + ζl = max {Qx + e, mi {y, Qx + e + ζa1} }. mi{qx + e + ζl, y} = mi { y, max { Qx + e, mi {y, Qx + e + ζa1 } } }.

VOL. 105 NO. 2 Acemoglu et al.: systemic risk ad stability i fiacial etworks 591 Simplifyig the expressio o the right-had side above leads to mi{qx + e + ζl, y} = mi {y, Qx + e + ζa1}, ad hece, x = [mi {y, Qx + e + ζa1}] +, which is the same as (B3). To prove the coverse, suppose that x R satisfies (B3) ad let l = (1/ζ) [mi {y (Qx + e), ζa1}] +. By costructio, (B2) is satisfied. Thus, to prove that (x, l) is ideed a paymet equilibrium, it is sufficiet to show that the pair (x, l) also satisfies (B1). It is immediate that ad therefore, Qx + e + ζl = max {Qx + e, mi{y, Qx + e + ζa1} }, [mi{qx + e + ζl, y} ] + = [ mi { y, max {Qx + e, mi{y, Qx + e + ζa1} } } ] + = [mi{y, Qx + e + ζa1} ] + = x, where the last equality is simply a cosequece of the assumptio that x satisfies (B3). B. Proof of Propositio 1 Existece. I view of Lemma B2, it is sufficiet to show that there exists x R + that satisfies x = [ mi {Q x + e + ζa1, y}] +. Defie the mappig Φ : H H as Φ(x) = [mi {Qx + e + ζa1, y} ] +, where H = i=0 [0, y i ]. This mappig cotiuously maps a covex ad compact subset of the Euclidea space to itself, ad hece, by the Brouwer fixed poit theorem, there exists x H such that Φ( x ) = x. Lemma B2 the implies that the pair ( x, l ) is a paymet equilibrium of the fiacial etwork, where l = (1/ζ) [mi{y (Q x + e), ζa1}] +. I particular, the collectio of pairwise iterbak paymet { x ij } defied as x ij = q ij x j alogside liquidatio decisios l i satisfy the collectio of equatios (1) ad (2) for all i ad j simultaeously. Geeric Uiqueess. Without loss of geerality, we restrict our attetio to a coected fiacial etwork. 28 Suppose that the fiacial etwork has two distict 28 If the fiacial etwork is ot coected, the proof ca be applied to each coected compoet separately.

592 THE AMERICAN ECONOMIC REVIEW february 2015 paymet equilibria, deoted by (x, l) ad ( x ˆ, l ˆ ) such that x x. 29 By Lemma B2, both x ad x ˆ satisfy (B3). Therefore, for ay give bak i, (B4) x i x i = [mi {(Qx ) i + e i + ζa, y i }] + [mi {(Q x ) i + e i + ζa, y i }] + (Qx) i (Q x ˆ ) i, where the iequality is a cosequece of Lemma B1. Summig both sides of the above iequality over all baks i leads to (B5) x x ˆ 1 Q(x x ˆ ) 1 (B6) Q 1 x x ˆ 1 = x x ˆ 1, where the last equality is due to the fact that the colum sums of Q are equal to oe. Cosequetly, iequalities (B4) (B6) are all tight simultaeously. I particular, give that (B4) is tight, ad i view of Lemma B1, for ay give bak i either or (Qx) i = (Q x ˆ ) i (B7) 0 (Qx) i + e i + ζa, (Q x ˆ ) i + e i + ζa y i. Deotig the set of baks that satisfy (B7) by B, it is immediate that, for all i B, ad therefore, x i = e i + ζa + (Qx) i x ˆ i = e i + ζa + (Q x ˆ ) i, (Qx) i (Q x ˆ ) i = x i x ˆ i i B. O the other had, i view of the fact that (Qx) i = (Q x ˆ ) i for all i B, we have Q(x x ˆ ) = x B x ˆ B, [ 0 ] 29 The assumptio that the two paymet equilibria are distict requires that x x. Note that if x = x, the (B2) immediately implies that l = l.

VOL. 105 NO. 2 Acemoglu et al.: systemic risk ad stability i fiacial etworks 593 ad hece, (B8) Q(x x ˆ ) 1 = x B x ˆ B 1. Therefore, (B5) holds as a equality oly if x i = x ˆ i for all i B, as it would otherwise violate (B8). Hece, (B9) Q BB ( x B x ˆ B ) = x B x ˆ B, where Q BB is the submatrix of Q correspodig to the baks i B. O the other had, the fact that the fiacial etwork is coected implies that Q, ad hece, Q BB are irreducible matrices. 30 Give that x x ˆ, equality (B9) caot hold uless B c is empty. As a cosequece, x i = e i + ζa + (Qx) i for all i, ad hece, x i = ζa + e i + q ij x j, i=1 i=1 i=1 j=1 which implies i=1 e i = ζa, a equality that holds oly for a o-geeric set of parameters z 1,, z. Thus, the paymet equilibria of the ecoomy are geerically uique. C. Proof of Propositio 2 Deote the set of baks that default o their seior debt by f, the set of baks that default but ca pay their debts to the seior creditors by d, ad the set baks that do ot default by s. For ay bak i f, we have whereas for i d, π i + T i = z i + ζ l i + x ij, π i + T i = v. O the other had, for ay bak i s which does ot default, we have j i π i + T i = A l i + ζ l i + z i y + x ij, j i where ζ l i is the proceeds that bak i obtais from liquidatig its project (if ay). 30 A matrix Q is said to be reducible, if for some permutatio matrix P, the matrix P QP is block upper-triagular. If a square matrix is ot reducible, it is said to be irreducible. If Q is a oegative irreducible matrix with uit colum sums, the all eigevalues of ay square submatrix of Q, say Q, lie withi the uit circle, implyig that equatio Q x = x has o o-trivial solutios. For more o this, see e.g., Berma ad Plemmos (1979).

594 THE AMERICAN ECONOMIC REVIEW february 2015 Summig the above three equalities over all baks implies that the social surplus i the ecoomy is equal to u = s(a y) + vd + ( z i + ζ l i ) + x ij l i i d i d i s = s(a y) + ( z i + ζ l i ) + y ij l i, i=1 i s j i where with some abuse of otatio, we deote the size of sets s, d, ad f with s, d, ad f, respectively. The secod equality is a cosequece of the fact that for i d we have, j i ( x ji x ij ) = z i + ζ l i v. Further simplifyig the above equality thus implies which completes the proof. j i u = z i + sa + ζ l i l i i=1 i=1 i s i s = (a + A) pϵ (1 ζ ) l i, D. Proof of Propositio 3 Lemma B3: Suppose that o bak defaults o its seior liabilities. The, baks i sets s ad d ca meet their liabilities ad default, respectively, if ad oly if (B10) (I Q dd ) 1 e d < 0 (B11) Q sd (I Q dd ) 1 e d + e s 0. Lemma B4: Suppose that the distressed bak j defaults o its seior liabilities. If d deotes the set of all other baks that default, the (B12) (I Q dd ) 1 [(a v)1 y Q dj ] < 0. Furthermore, if (B12) is satisfied for a subset of baks d, the all baks i d default. Proof: To prove the first statemet, suppose that d ad s deote the set of baks that default (excludig bak j ) ad ca meet their liabilities i full, respectively. By defiitio, which implies i=1 x d = Q dd x d + y Q ds 1 + (a v)1, x d = (I Q dd ) 1 [ y Q ds 1 + (a v)1].

VOL. 105 NO. 2 Acemoglu et al.: systemic risk ad stability i fiacial etworks 595 O the other had, give that Q is a stochastic matrix, we have Q ds 1 + Q dd 1 + Q dj = 1. Therefore, for all baks i d to default, it is ecessary that (I Q dd ) 1 [(a v)1 y Q dj ] < 0, which proves the first statemet. To prove the secod statemet, suppose that for a give subset of baks d, iequality (B12) is satisfied. Replacig for Q dj = 1 Q dd 1 Q ds 1 implies that (I Q dd ) 1 [(a v)1 + y Q ds ] < y1. I other words, eve if o other bak outside of d (ad the origially distressed bak j ) defaults, still o bak i d ca meet its liabilities i full. Therefore, regardless of the state of other baks, all baks i d default. Proof of Propositio 3: We prove this propositio for three separate cases, depedig o whether or ot the distressed bak, say bak f, defaults o its seior liabilities i the two fiacial etworks { y ij } ad { y ij }, where y ij = β y ij for all i j ad some costat β > 1. I each case, we show that the set of baks that default i { y ij } is a subset of the set of defaultig baks i { y ij }. Case (i): First, suppose that the distressed bak f does ot default o its seior liabilities i the fiacial etwork { y ij }. Let d ad s deote the set of baks that default ad ca meet their liabilities i full, ad deote the correspodig paymet equilibrium with x = ( x d, y1). It is immediate that x d = y1 + (I Q dd ) 1 e d. Furthermore, Lemma B3 implies that iequalities (B10) ad (B11) must be satisfied. Now cosider the fiacial etwork { y ij }. We verify that x = ( x d, y 1) is a paymet equilibrium of the ew fiacial etwork, where y = βy ad x d = y 1 + (I Q dd ) 1 e d. Note that x d = x d + ( y y)1 > 0, which implies that all baks i d, icludig the distressed bak f, ca meet their seior liabilities. Moreover, give (B10), it is immediate that x d < y 1. Fially, iequality (B11) ad the fact that Q is a stochastic matrix imply that Q sd x d + y Q ss 1 + e s y 1. Hece, x solves x = [mi{q x + e, y 1}] +, implyig that it ideed is a paymet equilibrium of { y ij }. Cosequetly, the set of baks that default i the two fiacial etworks are idetical. Case (ii): Next suppose that the distressed bak defaults o its seior liabilities i both fiacial etworks. By Lemma B4, it is immediate that (I Q dd ) 1 [(a v)1 y Q df ] < 0,

596 THE AMERICAN ECONOMIC REVIEW february 2015 where s deotes the set of baks that ca meet their obligatios i full ad d deotes the set of baks, excludig bak f, that default. Now cosider the fiacial etwork { y ij }, for which y ij = β y ij for all i j ad β > 1. Recall that (I Q dd ) 1 is a iverse M-matrix ad hece, is elemet-wise oegative. Therefore, it is immediate that (I Q dd ) 1 [(a v)1 y Q d f ] < 0, where y = β y is the total liabilities of each bak i { y ij }. Furthermore, by assumptio, bak f defaults o its seior liabilities. Thus, the secod part of Lemma B4 implies that the set of defaultig baks i { y ij } cotais the set of defaultig baks i { y ij }. Case (iii): Fially, suppose that bak f defaults o its seior liabilities i { y ij } but ca fully meet them i { y ij }. I other words, by icreasig all bilateral liabilities by a factor β, the total resources available to bak f icreases from a umber smaller tha v to a umber greater tha v. Thus, by cotiuity, there exists b (1, β) such that i the fiacial etwork { y ij (b)} the liquid resources available to bak j is exactly equal to v, where { y ij (b)} is defied as the fiacial etwork i which y ij (b) = b y ij for all i j. Let β 0 be the smallest such b. By case (ii) above, the set of defaultig baks i the fiacial etwork { y ij ( β 0 )} cotais the set of defaultig baks i { y ij } = { y ij (1)}. O the other had, give that bak f ca just meet its obligatios to its seior creditors, case (i) above implies that the set of defaultig baks i { y ij } = { y ij (β)} coicides with the set of defaultig baks i { y ij ( β 0 )}. Thus, to summarize, the set of baks i default does ot shrik if all liabilities are icreased by a factor β. E. Two Auxiliary Lemmas Lemma B5: Suppose that ζ = 0. The umber of bak defaults satisfies p # (defaults) < pϵ a v, where p is the umber of realizatios of egative shocks i the etwork. Proof: Give that the total iterbak liabilities of each bak are equal to its total iterbak claims ad that v > a ϵ, ay bak that is hit with a egative shock defaults. Hece, the lower boud is trivial. To obtai the upper boud, ote that for ay bak i that defaults but ca meet its seior liabilities i full, we have z i + x ij = v + x ji. j i j i

VOL. 105 NO. 2 Acemoglu et al.: systemic risk ad stability i fiacial etworks 597 Deotig the set of such baks by d ad summig over all i d implies (B13) i d z i + x ij = vd + x ji. i d j i i d j i O the other had, for ay bak i that defaults o its seior liabilities (if such a bak exists), we have x ij + z i < v. j i Summig over the set of all such baks, f, implies (B14) i f x ij + z i < vf. j i i f Addig (B13) ad (B14) leads to pϵ (a v) #(defaults) ( y ij x ij ), j s i s where s is the set of baks that do ot default. By defiitio, the right-had side of the above equality is strictly positive, provig that the umber of defaults is strictly smaller tha pϵ / (a v). Lemma B6: If ϵ < ϵ p, the at least oe bak does ot default, where ϵ p = (a v) / p. O the other had, if ϵ > ϵ p, the at least oe bak defaults o its seior creditors. Proof: Suppose that ϵ < ϵ p ad that all baks default. Therefore, for all baks i. Summig over i implies z i + x ij v + x ji, j i j i a pϵ v, which cotradicts the assumptio that ϵ < ϵ p. To prove the secod statemet, suppose that ϵ > ϵ p ad that o bak defaults o its seior liabilities. Thus, z i + x ij v + x ji, j i j i for all baks i. Summig over i implies a pϵ v, leadig to a cotradictio.

598 THE AMERICAN ECONOMIC REVIEW february 2015 F. Proof of Propositio 4 Proof of Part (i): Without loss of geerality assume that bak 1 is hit with the egative shock. By Lemma B6, bak does ot default as it is the bak furthest away from the distressed bak. Moreover, as log as y > y = ( 1)(a v), bak 1 ad cosequetly all baks ca meet their seior liabilities i full. This is due to the fact that y + a ϵ > v. Give that baks i default form a coected chai, say of legth τ, the repaymet of the last bak i default to its sole creditor satisfies x τ+1, τ = y + τ (a v) ϵ. O the other had, give that bak τ + 1 does ot default, we have a + x τ+1, τ y + v. As a result, τ ϵ / (a v) 1, implyig that the umber of defaults reaches the upper boud established i Lemma B5. Hece, the rig etwork is the least stable ad least resiliet fiacial etwork. Proof of Part (ii): By Lemma B6, as log as ϵ < ϵ, there exists at least oe bak that does ot default. Give the full symmetry i the complete etwork, the 1 baks that are ot hit with the egative shock ca thus meet their liabilities i full. Hece, the complete fiacial etwork is the most stable ad most resiliet regular fiacial etwork. Proof of Part (iii): Cosider the fiacial etwork costructed as the γ -covex combiatio of the rig ad the complete fiacial etworks. Without loss of geerality, we assume that bak 1 is hit with the egative shock. Defie γ d to be the value at which baks 1 through d 1 default while bak d is at the verge of default. At this value of γ, we have γ (B15) x 1 = ( d 1) [( x 1 + + x d ) x 1 + ( d)y] + (1 γ d )y + (a v ϵ) γ (B16) x i = ( d 1) [( x 1 + + x d ) x i + ( d)y] for 2 i d. Hece, + (1 γ d ) x i 1 + (a v ) γ Δ 2 = x 2 x 1 = (1 γ d ) Δ 1 ( d 1) Δ 2 + ϵ γ Δ i = x i x i 1 = (1 γ d ) Δ i 1 ( d 1) Δ i,

VOL. 105 NO. 2 Acemoglu et al.: systemic risk ad stability i fiacial etworks 599 where Δ 1 = x 1 y. Thus, for all 2 i d, Δ i = β i 1 ( Δ 1 + ϵ / (1 γ d )), where for otatioal simplicity we have defied β = ( 1 + γ d 1 ) 1 (1 γ d ). Give that x d = y, the terms Δ i must add up to zero, that is, d 1 β d Δ i = ( 1 β ) Δ 1 β d 1 1 + ( 1 β ) ( βϵ 1 γ d ) = 0, i=1 which immediately implies Therefore, ad as a result, Δ 1 = 1 β d 1 ( 1 β d ) ( βϵ 1 γ d ). i x i = y + Δ s = y + β d 1 β i 1 s=1 ( 1 β d ) ( βϵ 1 γ d ), d 1 x i = (d 1)y + d β d 1 (1 β) (1 β d ) i=1 [ (1 γ d )(1 β)(1 β d ) βϵ. ] O the other had, from (B15) ad (B16), we have d 1 i=1 x i = (d 1)y + d(a v) ϵ γ d ( d)/( 1). Equatig the above two equalities thus leads to (a v)/ϵ 1 = β d 1 (1 β) [ 1 β d ( d). ] Therefore, the value γ d at which d 1 baks default ad bak d is at the verge of default must satisfy the above equality. For a fixed value of d, the right-had side is icreasig i β (ad hece, decreasig i γ d ). 31 As a cosequece, i order for the right-had side to remai equal to the costat o the left-had side, d has to decrease as γ icreases. I other words, there will be weakly less defaults for higher values of γ. 31 This ca be easily verified by oticig that β d 1 (1 β) = (1 β d ) (1 + β 1 + + β (d 1) 1 ).

600 THE AMERICAN ECONOMIC REVIEW february 2015 G. Proof of Propositio 5 First, cosider the origial fiacial etwork { y ij } ad label the set of baks that default ad ca meet their liabilities i full by d ad s, respectively. Lemma B3 implies that iequalities (B10) ad (B11) are satisfied. Next, cosider the fiacial etwork { y ij }, which is a (D, D, P) -majorizatio of { y ij } for some doubly stochastic matrix P. By costructio, Q dd = Q dd, which immediately implies that the trasformed etwork { y ij } satisfies (B10). Thus, by Lemma B3, the proof is complete oce we show that { y ij } also satisfies iequality (B11). We have Q sd (I Q 1 dd ) e d + e s = P Q sd (I Q dd ) 1 e d + e s = P [ Q sd (I Q dd ) 1 e d + e s ], where we are usig the fact that P is a doubly-stochastic matrix. Give that { y ij } satisfies (B11) (ad usig the fact that P is a o-egative matrix), it is the immediate that the right-had side of the above equality is oegative. H. Proof of Corollary 1 Cosider the regular fiacial etwork { y ij }. By assumptio, oly the distressed bak, say bak j, defaults. This meas that bak j is paid i full by its creditors, implyig that x j > 0. To see this, ote that x j = 0 implies that x j = [mi{y + a v ϵ, y}] + = 0, which ca arise oly if y + a v ϵ 0. That, however, would be i cotradictio with the assumptios that ϵ < ϵ ad y > y. The fact that x j > 0 implies that bak j ca meet its liabilities to the seior creditors i full. Thus, by Propositio 5, ay ({ j}, { j}, P) -majorizatio of the fiacial etwork would lead to weakly less defaults. O the other had, ote that i the absece of fiacial cotagio, the γ -covex combiatio of the etwork with the complete etwork is essetially a ({ j}, { j}, P) -majorizatio of the origial etwork, with the doubly stochastic matrix P give by P = (1 γ)i + γ / ( 1)11. I. Proof of Propositio 6 Proof of Part (i): First cosider the complete fiacial etwork. By Lemma B6, the distressed bak defaults o its seior liabilities. Suppose that the remaiig 1 baks do ot default ad that they ca all meet their liabilities i full. 32 This would be the case oly if y ( 2) + (a v) y, 1 32 Note that give the symmetric structure of the complete etwork ad the uiqueess of paymet equilibrium, either all other baks default together or they all meet their liabilities fully at the same time.

VOL. 105 NO. 2 Acemoglu et al.: systemic risk ad stability i fiacial etworks 601 where we are usig the fact that the distressed bak does ot pay aythig to its juior creditors. The above iequality, however, cotradicts the assumptio that y > y. Hece, all baks default, implyig that the complete etwork is the least resiliet ad the least stable fiacial etwork. Now cosider the rig fiacial etwork ad assume without loss of geerality that bak 1 is hit with the egative shock. Oce agai by Lemma B6, bak 1 defaults o its seior liabilities, ad hece does ot pay aythig to its juior creditor, bak 2. Thus, if some bak does ot default, the legth of the default cascade, τ, must satisfy τ(a v) > y, which i light of the assumptio that y > y guaratees that τ > 1, leadig to a cotradictio. Hece, all baks default, implyig that the rig fiacial etwork is the least stable ad resiliet fiacial etwork. Proof of Part (ii): Cosider a δ -coected fiacial etwork where δ < a y v. By defiitio, there exists a partitio (S, S c ) of the set of baks such that y ij δy for all i S ad j S c. Cosequetly, j S y ij δy S c for all baks i S, ad therefore, y ij y δy S c j S y (a v). Therefore, if the egative shock hits a bak i S c, all baks i S ca still meet their liabilities i full. This is a cosequece of the fact that a v + j S y ij y for all i S, which guaratees that i the uique paymet equilibrium of the fiacial etwork, o bak i S defaults. A similar argumet shows that whe the shock hits a bak i S, all baks i S c ca meet their liabilities i full. Thus, the fiacial etwork is strictly more stable ad resiliet tha both the complete ad the rig etworks, i which all baks default. J. Proof of Propositio 7 Cosider the origial fiacial etwork { y ij }. Give that ϵ > ϵ, Lemma B6 implies that the distressed bak j defaults o its seior liabilities. Thus, by the first part of Lemma B4, iequality (B12) is satisfied, where d deotes the set of all other baks that default. Next, cosider the fiacial etwork { y ij } obtaied by the (D, { j}, P) -majorizatio of { y ij }. By costructio, Q dd = Q dd ad q ij = (1 γ) q ij + γ /( 1)

602 THE AMERICAN ECONOMIC REVIEW february 2015 for ay bak i j. Cosequetly, (I Q 1 dd ) [(a v)1 y Q dj ] = (1 γ) (I Q dd ) 1 [(a v)1 y Q dj ] + γ(a v y/( 1)) (I Q dd ) 1 1. The first part of Lemma B4 guaratees that the first term o the right-had side above is egative. As for the secod term, ote that (I Q dd ) 1 is a iverse M-matrix, ad hece, is elemet-wise oegative. 33 Furthermore, by assumptio, y > y = ( 1)(a v), implyig that the secod term is also egative. Therefore, by the secod part of Lemma B4, all baks i d default followig the majorizatio trasformatio. K. Proof of Propositio 8 Proof of Part (i): Let m = y / (a v). We prove the result by showig that if a bak i does ot default, the m ij m. Give that the distressed bak j defaults o its seior liabilities, it is immediate that x j = 0. Let x d R d deote the subvector of the equilibrium paymet vector x correspodig to the baks i default, excludig the origially distressed bak j. Similarly, let x s R s deote the subvector correspodig to the baks that ca meet their liabilities i full. 34 From the defiitio of a paymet equilibrium, it is immediate that ad as a result, x d = Q dd x d + y Q ds 1 + (a v)1, (B17) x d = (I Q dd ) 1 ( y Q ds 1 + (a v)1). Furthermore, give that baks idexed s ca meet their liabilities i full, it must be the case that Q sd x d + y Q ss 1 + (a v)1 y1. Substitutig for x d from (B17) implies (B18) Q sd (I Q dd ) 1 1 + 1 m [I Q ss Q sd (I Q dd ) 1 Q ds )] 1. 33 A square matrix X is a M-matrix if there exist a oegative square matrix B ad a costat r > ρ(b) such that X = ri B, where ρ(b) is the spectral radius of B. By (Plemmos 1977, Theorem 2), the iverse of ay o-sigular M-matrix is elemet-wise oegative. For more o M-matrices ad their properties, see Berma ad Plemmos (1979). 34 With some abuse of otatio, we use d ad s to ot oly deote the set of defaultig ad solvet baks, but also the size of the two sets, respectively. Hece, s + d + 1 =.

VOL. 105 NO. 2 Acemoglu et al.: systemic risk ad stability i fiacial etworks 603 O the other had, by the defiitio of the harmoic distace (3), we have (B19) m dj = 1 + Q dd m dj + Q ds m sj (B20) m sj = 1 + Q sd m dj + Q ss m sj, where m dj R d ad m sj R s are vectors that capture the harmoic distaces of the defaultig ad solvet baks to the distressed bak j, respectively. Solvig for m dj i (B19) ad replacig it i (B20) leads to m sj = 1 + Q sd (I Q dd ) 1 1 + Q sd (I Q dd ) 1 Q ds m sj + Q ss m sj m [I Q ss Q sd (I Q dd ) 1 Q ds )] 1 + Q sd (I Q dd ) 1 Q ds m sj + Q ss m sj, where the iequality is a cosequece of (B18). Rearragig the terms thus implies where matrix C R s s is give by C m sj m (C1), C = [I Q ss Q sd (I Q dd ) 1 Q ds )]. The proof is thus complete oce we show that C 1 is a elemet-wise oegative matrix, as it would immediately imply that m sj m 1. To establish this, first ote that C is the Schur complemet of the o-sigular, M-matrix G defied as G = I Q ss Q sd [ Q ds I Q dd ]. By Berma ad Plemmos (1979, exercise 5.8, p. 159), the Schur complemet of ay o-sigular M-matrix is itself a o-sigular M-matrix. 35 Thus, Theorem 2 of Plemmos (1977) guaratees that C 1 is elemet-wise oegative, completig the proof. Proof of Part (ii): Suppose that all baks default if bak j is hit with the egative shock, which meas that x i < y for all baks i j. O the other had, from the defiitio of a paymet equilibrium it is immediate that x i = a v + q ik x k, k j for all i j. Dividig both sides of the above equatio by a v ad comparig it with the defiitio of the harmoic distace over the fiacial etwork (3) implies that x i / (a v) = m ij. 36 Thus, the fact that x i < y guaratees that m ij < m. 35 For a proof, see Theorem 1 of Carlso ad Markham (1979). 36 Note that we are usig the fact that for ay j, equatio (3) has a uique solutio.

604 THE AMERICAN ECONOMIC REVIEW february 2015 L. Proof of Lemma 1 Lemma B7: For ay regular fiacial etwork, 1_ 8 2 ϕ 2 1 λ 2 2ϕ, where λ 2 is the secod largest eigevalue of Q ad ϕ is the bottleeck parameter. Proof: Let Φ be the coductace of the graph, defied as Φ = mi Q(S, S c ) S N mi { S, S c }, where Q(S, S c ) = i S, j S q ij. It is easy to verify that which implies that ( / 2) mi { S, S c } S S c mi { S, S c }, (B21) Φ / ϕ 2Φ /. O the other had, by Theorem 2 of Siclair (1992), (B22) Φ 2 / 2 1 λ 2 2Φ. Combiig (B21) ad (B22) completes the proof. Proof of of Lemma 1: We first prove the lower boud. Let M = [ m ij ] deote the matrix of pairwise harmoic distaces betwee the baks, ad defie T = 11 + QM M. For ay pair of baks i j, we have t ij = 1 + q ik m kj m ij = 0, k=1 where the secod equality is a cosequece of the defiitio of the harmoic distace i (3). This meas that T is a diagoal matrix. Furthermore, 1 T = 1, which implies that all diagoal elemets of T are equal to, ad as a cosequece, (B23) (I Q)M = 11 I. Solvig the above equatio for the matrix of pairwise harmoic distaces M implies that (B24) M = (11 diag(z) Z),

VOL. 105 NO. 2 Acemoglu et al.: systemic risk ad stability i fiacial etworks 605 where Z = (I Q + (1/)11 ) 1 ad diag(x) is a diagoal matrix whose elemets are the diagoal etries of X. 37 Therefore, M1 = (11 diag(z)1 Z1) = (trace(z) 1)1, where the secod equality is a cosequece of the fact that Z1 = 1. Cosequetly, for ay bak i, (B25) 1_ m ij = trace(z) 1 = 1, j i k=2 1 λ k i which λ k is the k -th largest eigevalue of Q. The secod equality above relies o the fact that the eigevalues of Z are simply the reciprocal of the ozero eigevalues of I Q. Thus, (B25) implies max m ij 1 1. i j 1 λ k 1 λ 2 k=2 Lemma B7 ow guaratees that max i j m ij 1/(2ϕ) establishig the lower boud i (4). We ext establish the upper boud. 38 Note that sice Q is symmetric with 1 as its top eigevalue, it ca be writte i spectral form as Q = _ 1 11 + λ k w k w k, where { w 2,, w } are eigevectors of Q correspodig to eigevalues { λ 2,, λ } 1 with legths ormalized to oe. Cosequetly, Z = k=2 w 1 λ k k w k. Equatio (B24) the implies k=2 m ij = 1 ( w 1 λ k kj 2 w ki w kj ), k=2 for all pairs of baks i ad j, ad hece, m ij + m ji = 1 ( w 1 λ k ki w kj ) 2 k=2 ( w 1 λ 2 ki 2 + w kj 2 2 w ki w kj ) = 2. 1 λ 2 37 Sice I Q is ot ivertible, equatio (B23) has ifiitely may solutios. However, the restrictio that m ii = 0 for all i uiquely determies the matrix M of pairwise harmoic distaces. 38 The proof of the upper boud follows steps similar to those i, ad geeralizes, Lovász (1996, Theorem 3.1). k=1

606 THE AMERICAN ECONOMIC REVIEW february 2015 The last equality above is a cosequece of the observatio that the collectio of vectors { w 1,, w } form a orthoormal basis, ad hece, it must be the case that 2 k=1 w ki = 1 for all baks i ad that k=1 w ki w kj = 0 for all i j. 39 Give that the above iequality holds for ay pairs of baks i j, it is immediate that (B26) max m ij 2. i j 1 λ 2 Combiig (B26) with the lower boud i Lemma B7 completes the proof. REFERENCES Acemoglu, Daro, Vasco M. Carvalho, Asuma Ozdaglar, ad Alireza Tahbaz-Salehi. 2012. The Network Origis of Aggregate Fluctuatios. Ecoometrica 80 (5): 1977 2016. Acemoglu, Daro, Asuma Ozdaglar, ad Alireza Tahbaz-Salehi. 2013. Systemic Risk ad Stability i Fiacial Networks. Natioal Bureau of Ecoomic Research Workig Paper 18727. Acemoglu, Daro, Asuma Ozdaglar, ad Alireza Tahbaz-Salehi. 2014. Microecoomic Origis of Macroecoomic Tail Risks. http://www.columbia.edu/~at2761/largedowturs.pdf (accessed December 12, 2014). Akram, Q. Farooq, ad Casper Christopherse. 2010. Iterbak Overight Iterest Rates Gais from Systemic Importace. Norges Bak Workig Paper 11/2010. Alle, Frakli, ad Aa Babus. 2009. Networks i Fiace. I The Network Challege: Strategy, Profit, ad Risk i a Iterliked World, edited by Paul R. Kleidorfer ad Yoram (Jerry) Wid, 367 82. Upper Saddle River, NJ: Wharto School Publishig. Alle, Frakli, Aa Babus, ad Elea Carletti. 2012. Asset Commoality, Debt Maturity ad Systemic Risk. Joural of Fiacial Ecoomics 104 (3): 519 34. Alle, Frakli, ad Douglas Gale. 2000. Fiacial Cotagio. Joural of Political Ecoomy 108 (1): 1 33. Alvarez, Ferado, ad Gadi Barlevy. 2014. Madatory Disclosure ad Fiacial Cotagio. Federal Reserve Bak of Chicago Workig Paper No. WP-2014-4. Amii, Hamed, Rama Cot, ad Adreea Mica. 2013. Resiliece to Cotagio i Fiacial Networks. Mathematical Fiace. doi:10.111/mafi.12051. Babus, Aa. 2013. The Formatio of Fiacial Networks. Tiberge Istitute Discussio Paper 2006-093/2. Barlevy, Gadi, ad H. N. Nagaraja. 2013. Properties of the Vacacy Statistic i the Discrete Circle Coverig Problem. Federal Reserve Bak of Chicago Workig Paper No. WP-2013-05. Battisto, Stefao, Domeico Delli Gatti, Mauro Gallegati, Bruce Greewald, ad Joseph E. Stiglitz. 2012. Default Cascades: Whe Does Risk Diversificatio Icrease Stability? Joural of Fiacial Stability 8 (3): 138 49. Bech, Morte L., ad Eghi Atalay. 2010. The Topology of the Federal Fuds Market. Physica A: Statistical Mechaics ad its Applicatios 389 (22): 5223 46. Bech, Morte L., James T. E. Chapma, ad Rodey J. Garratt. 2010. Which Bak Is the Cetral Bak? Joural of Moetary Ecoomics 57 (3): 352 63. Berma, Abraham, ad Robert J. Plemmos. 1979. Noegative Matrices i the Mathematical Scieces. New York: Academic Press. Bisias, Dimitrios, Mark Flood, Adrew W. Lo, ad Stavros Valavais. 2012. A Survey of Systemic Risk Aalytics. Aual Review of Fiacial Ecoomics 4 (1): 255 96. Blume, Lawrece, David Easley, Jo Kleiberg, Robert Kleiberg, ad Éva Tardos. 2011. Which Networks are Least Susceptible to Cascadig Failures? I Proceedigs of the 2011 52d IEEE Aual Symposium o Foudatios of Computer Sciece, 393 402. Washigto, DC: IEEE Computer Society. 39 By Hor ad Johso (1985, Theorem 2.1.4), if the rows of a square matrix X form a orthoormal basis, so do its colums.

VOL. 105 NO. 2 Acemoglu et al.: systemic risk ad stability i fiacial etworks 607 Blume, Lawrece, David Easley, Jo Kleiberg, Robert Kleiberg, ad Éva Tardos. 2013. Network Formatio i the Presece of Cotagious Risk. ACM Trasactios o Ecoomics ad Computatio 1 (2). Bolto, Patrick, ad Marti Oehmke. Forthcomig. Should Derivatives Be Privileged i Bakruptcy? Joural of Fiace. Bruermeier, Markus K., ad Marti Oehmke. 2013. Bubbles, Fiacial Crises, ad Systemic Risk. I Hadbook of the Ecoomics of Fiace. Vol. 2B, Fiacial Markets ad Asset Pricig, edited by George M. Costatiides, Milto Harris, ad Reé M. Stulz, 1221 88. Oxford: Elsevier. Bruermeier, Markus K., ad Lasse Heje Pederse. 2005. Predatory Tradig. Joural of Fiace 60 (4): 1825 63. Caballero, Ricardo J., ad Alp Simsek. 2013. Fire Sales i a Model of Complexity. Joural of Fiace 68 (6): 2549 87. Cabrales, Atoio, Piero Gottardi, ad Ferado Vega-Redodo. 2014. Risk-Sharig ad Cotagio i Networks. CESifo Group Muich CESifo Workig Paper No. 4715. Carlso, David, ad Thomas L. Markham. 1979. Schur Complemets of Diagoally Domiat Matrices. Czechoslovak Mathematical Joural 29 (2): 246 51. Castiglioesi, Fabio, Fabio Feriozzi, ad Guido Lorezoi. 2012. Fiacial Itegratio ad Liquidity Crises. http://www.cireqmotreal.com/wp-cotet/uploads/2013/03/castiglioesi.pdf (accessed December 12, 2014). Chug, Fa R. K. 1997. Spectral Graph Theory. CBMS Regioal Coferece Series i Mathematics, No. 92. Providece, RI: America Mathematical Society. Cifuetes, Rodrigo, Gialuigi Ferrucci, ad Hyu Sog Shi. 2005. Liquidity Risk ad Cotagio. Joural of the Europea Ecoomic Associatio 3 (2 3): 556 66. Cohe-Cole, Etha, Eleoora Patacchii, ad Yves Zeou. 2013. Systemic Risk ad Network Formatio i the Iterbak Market. CAREFIN Research Paper No. 25/2010. Craig, Be R., Falko Fecht, ad Güseli Tümer-Alka. 2013. The Role of Iterbak Relatioships ad Liquidity Needs. Deutsche Budesbak Discussio Paper 54/2013. Dasgupta, Amil. 2004. Fiacial Cotagio through Capital Coectios: A Model of the Origi ad Spread of Bak Paics. Joural of the Europea Ecoomic Associatio 2 (6): 1049 84. Debee, Edward, Christia Julliard, Ye Li, ad Kathy Yua. 2014. Network Risk ad Key Players: A Structural Aalysis of Iterbak Liquidity. Fiacial Markets Group (FMG) Discussio Paper dp734. Di Maggio, Marco, ad Alireza Tahbaz-Salehi. 2014. Fiacial Itermediatio Networks. Columbia Busiess School Research Paper 14-40. Diamod, Douglas W., ad Philip H. Dybvig. 1983. Bak Rus, Deposit Isurace, ad Liquidity. Joural of Political Ecoomy 91 (3): 401 19. Eboli, Mario. 2013. A Flow Network Aalysis of Direct Balace-Sheet Cotagio i Fiacial Networks. Kiel Workig Paper 1862. Eiseberg, Larry, ad Thomas H. Noe. 2001. Systemic Risk i Fiacial Systems. Maagemet Sciece 47 (2): 236 49. Elliot, Matthew, Bejami Golub, ad Matthew O. Jackso. 2014. Fiacial Networks ad Cotagio. America Ecoomic Review 104 (10): 3115 53. Farboodi, Maryam. 2014. Itermediatio ad Volutary Exposure to Couterparty Risk. http:// home.uchicago.edu/~/farboodi/maryamfarboodijmp.pdf (accessed December 12, 2014). Freixas, Xavier, Bruo M. Parigi, ad Jea-Charles Rochet. 2000. Systemic Risk, Iterbak Relatios, ad Liquidity Provisio by the Cetral Bak. Joural of Moey, Credit, ad Bakig 32 (3): 611 38. Gai, Prasaa, Adrew Haldae, ad Sujit Kapadia. 2011. Complexity, Cocetratio ad Cotagio. Joural of Moetary Ecoomics 58 (5): 453 70. Georg, Co-Pierre. 2013. The Effect of the Iterbak Network Structure o Cotagio ad Commo Shocks. Joural of Bakig ad Fiace 37 (7): 2216 28. Glasserma, Paul, ad H. Peyto Youg. 2015. How Likely is Cotagio i Fiacial Networks? Joural of Bakig & Fiace (50): 383 99. Gofma, Michael. 2011. A Network-Based Aalysis of Over-the-Couter Markets. http://ssr.com/ abstract=1681151. Gofma, Michael. 2014. Efficiecy ad Stability of a Fiacial Architecture with Too-Itercoected-to-Fail Istitutios. http://www.imf.org/exteral/p/semiars/eg/2014/itercoect/pdf/ gofma.pdf. Haldae, Adrew G. 2009. Rethikig the Fiacial Network. Speech preseted at the Fiacial Studet Associatio, Amsterdam. http://www.bakofeglad.co.uk/archive/documets/historicpubs/ speeches/2009/speech386.pdf (accessed December 12, 2014).

608 THE AMERICAN ECONOMIC REVIEW february 2015 Holmström, Begt, ad Jea Tirole. 1998. Private ad Public Supply of Liquidity. Joural of Political Ecoomy 106 (1): 1 40. Hor, Roger A., ad Charles R. Johso. 1985. Matrix Aalysis. New York: Cambridge Uiversity Press. Iyer, Rajkamal, ad José-Luis Peydró. 2011. Iterbak Cotagio at Work: Evidece from a Natural Experimet. Review of Fiacial Studies 24 (4): 1337 77. Kiyotaki, Nobuhiro, ad Joh Moore. 1997. Credit Chais. Upublished. https://www.priceto. edu/~kiyotaki/papers/creditchais.pdf (accessed December 12, 2014). Krishamurthy, Arvid. 2010. Amplificatio Mechaisms i Liquidity Crises. America Ecoomic Joural: Macroecoomics 2 (3): 1 30. Leiter, Yaro. 2005. Fiacial Networks: Cotagio, Commitmet, ad Private Sector Bailouts. Joural of Fiace 60 (6): 2925 53. Lorezoi, Guido. 2008. Iefficiet Credit Booms. Review of Ecoomic Studies 75 (3): 809 33. Lovász, László. 1996. Radom Walks o Graphs: A Survey. I Combiatorics, Paul Edr ős is Eighty. Vol. 2, edited by D. Miklós, T. Szőyi, ad V. T. Sós, 353 398. Budapest, Hugary: Jáos Bolyai Mathematical Society. Nier, Erled, Jig Yag, Taju Yorulmazer, ad Amadeo Aletor. 2007. Network Models ad Fiacial Stability. Joural of Ecoomic Dyamics ad Cotrol 31 (6): 2033 60. Plemmos, R. J. 1977. M-Matrix Characterizatios.I Nosigular M-Matrices. Liear Algebra ad its Applicatios 18 (2): 175 88. Plosser, Charles I. 2009. Redesigig Fiacial System Regulatio. Speech preseted at Restorig Fiacial Stability: How to Repair a Failed System New York Uiversity Coferece, New York. Rochet, Jea-Charles, ad Jea Tirole. 1996. Iterbak Ledig ad Systemic Risk. Joural of Moey, Credit, ad Bakig 28 (4): 733 62. Rotemberg, Julio J. 2011. Miimal Settlemet Assets i Ecoomies with Itercoected Fiacial Obligatios. Joural of Moey, Credit, ad Bakig 43 (1): 81 108. Shi, Hyu Sog. 2008. Risk ad Liquidity i a System Cotext. Joural of Fiacial Itermediatio 17 (3): 315 29. Shi, Hyu Sog. 2009. Securitisatio ad Fiacial Stability. Ecoomic Joural 119 (536): 309 32. Siclair, Alistair. 1992. Improved Bouds for Mixig Rates of Markov Chais ad Multicommodity Flow. Combiatorics, Probability ad Computig 1 (4): 351 70. Soramäki, Kimmo, ad Samatha Cook. 2013. SikRak: A Algorithm for Idetifyig Systemically Importat Baks i Paymet Systems. Ecoomics: The Ope-Access, Ope-Assessmet E-Joural. Sorki, Adrew Ross. 2009. Too Big to Fail: The Iside Story of How Wall Street ad Washigto Fought to Save the Fiacial System ad Themselves. New York: Vikig. Vivier-Lirimot, Sébastie. 2006. Cotagio i Iterbak Debt Networks. http://eveemets.uivlille3.fr/recherche/jemb/programme/papiers/vivierlirimot_lille06.pdf. Yelle, Jaet L. 2013. Itercoectedess ad Systemic Risk: Lessos from the Fiacial Crisis ad Policy Implicatios. Speech preseted at The America Ecoomic Associatio/America Fiace Associatio joit lucheo, Sa Diego, CA. Zawadowski, Adam. 2011. Iterwove Ledig, Ucertaity, ad Liquidity Hoardig. Bosto Uiversity School of Maagemet Research Paper No. 2011-13. Zawadowski, Adam. 2013. Etagled Fiacial Systems. Review of Fiacial Studies 26 (5): 1291 323.