Systemic Risk and Stability in Financial Networks

Transcription

1 America Ecoomic Review 2015, 105(2): 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 ( daro@mit.edu); Ozdaglar: Laboratory for Iformatio ad Decisio Systems, Massachusetts Istitute of Techology, 77 Massachusetts Aveue, 32-D630, Cambridge, MA ( asuma@mit.edu); Tahbaz-Salehi: Columbia Busiess School, Columbia Uiversity, 3022 Broadway, Uris Hall 418, New York, NY ( 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 The authors declare that they have o relevat or material fiacial iterests that relate to the research described i this paper. Go to 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

2 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.

3 566 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.

4 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).

5 568 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).

6 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.

7 570 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 } ] +,

8 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 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.

9 572 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.

10 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.

11 574 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.

12 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.

13 576 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.

14 VOL. 105 NO. 2 Acemoglu et al.: systemic risk ad stability i fiacial etworks 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.

15 578 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

16 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.

17 580 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).

18 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

19 582 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).

20 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.