The Formation of Financial Networks

Transcription

1 The Formatio of Fiacial Networks Aa Babus Federal Reserve Bak of Chicago Abstract Moder bakig systems are highly itercoected. Despite various beefits, likages betwee baks carry the risk of cotagio. I this paper I ivestigate whether baks ca commit ex-ate to mutually isure each other, whe there is cotagio risk i the fiacial system. I model baks decisios to share this risk through bilateral agreemets. A fiacial etwork that allows losses to be shared amog various couterparties arises edogeously. I characterize the probability of systemic risk, defied as the evet that cotagio occurs coditioal o oe bak failig, i equilibrium iterbak etworks. I show that there exist equilibria i which cotagio does ot occur. Keywords: fiacial stability; etwork formatio; cotagio risk; JEL: C70; G1. aababus@gmail.com. I am grateful to Frakli Alle, Douglas Gale, two aoymous referees ad the editor for very useful commets ad guidace. The views i this papers are solely those of the author ad eed ot represet the views of the Federal Reserve Bak of Chicago or the Federal Reserve System. 1

2 1 Itroductio The recet turmoils i fiacial markets have revealed, oce agai, the itertwied ature of fiacial systems. I a moder fiacial world, baks ad other istitutios are liked i a variety of ways. These coectios ofte ivolve trade-offs. For istace, although baks ca solve their liquidity imbalaces by borrowig ad ledig o the iterbak market, they expose themselves, at the same time, to cotagio risk. How do baks weigh these trade-offs, ad what are the exteralities of their decisios o the fiacial system as a whole? I this paper I explore whether baks ca commit ex-ate to mutually isure each other, whe the failure of a istitutio itroduces the risk of cotagio i the fiacial system. As a form of isurace, baks ca hold mutual claims o oe aother. These mutual claims are, essetially, bilateral agreemets that allow losses to be shared amog all couterparties of a failed bak. The more bilateral agreemets a bak has, the smaller the loss that each of her couterparties icurs. Cotagio does ot take place provided each bak has suffi cietly may bilateral agreemets. I model baks decisio to share the risk of cotagio bilaterally as a etwork formatio game. Various equilibria arise. I most equilibria the fiacial system is resiliet to the demise of some baks, but ot the others. Equilibria i which there is o cotagio ca be supported as well. Moreover, I show that the welfare i equilibrium iterbak etworks is decreasig i the probability that cotagio occurs. However, more bilateral agreemets betwee baks do ot ecessarily improve welfare beyod the poit whe there is o cotagio. To study these issues, I build o the framework proposed by Alle ad Gale 000. I particular, I cosider a three-period model, where the bakig system cosists of two idetically sized regios. Baks raise deposits from cosumers who are ucertai about their liquidity prefereces, as i Diamod ad Dybvig Each regio is subject to liquidity shocks drive by cosumers liquidity eeds. Liquidity shocks are egatively correlated across the two regios. I additio, there is a small probability that oe of the baks, chose at radom, is affected by a early-withdrawal shock ad liquidated prematurely. Baks ca perfectly isure agaist liquidity shocks by exchagig iterbak deposits

3 with baks i the other regio. However, the coectios created by swappig deposits expose the system to cotagio whe the early-withdrawal shock realizes. The loss that a bak iduces whe she is affected by the early-withdrawal shock is shared across her couterparties. This implies that as baks exchage more deposits, the loss o every deposit is smaller. The model predicts a coectivity threshold above which cotagio does ot occur. Reachig this coectivity threshold may require that baks swap deposits with other baks i the same regio. I distiguish betwee a liquidity etwork, that smoothes out liquidity shocks i the bakig system, ad a solvecy etwork, that provides isurace agaist cotagio risk. The distictio betwee liquidity liks ad solvecy liks is useful to study the icetives that baks have to isure agaist cotagio. I particular, I study whether baks choose to form solvecy liks with other baks i the same regio. Whe a bak has at least as may liks as the coectivity threshold requires, the o cotagio takes place if she is affected by the early-withdrawal shock. However, her couterparties still icur a loss o their deposits. Whe decidig to form solvecy liks, baks are willig to icur a small loss o their deposits, if they ca avoid default. However, they are better off if cotagio is averted without icurrig ay cost. This implies that baks have the icetive to free-ride o others liks. Because of this, may etwork structures ca be stable i which full fiacial stability is ot ecessarily achieved. I the mai specificatio of the model, I show that at least half of the baks have swapped deposits with suffi cietly may other baks i the same regio. I other words, a systemic evet i which all baks default if oe bak is subject to a early-withdrawal shock occurs i at most half of the cases. Eve the, I fid that isurig agaist regioal fluctuatios i the fractio of early cosumers through a liquidity etwork is optimal as log as the probability of the early-withdrawal shock is suffi cietly small. The equilibria i which all baks have suffi cietly may liks ad o cotagio occurs have the highest associated welfare. This is ituitive, as baks assets are ieffi cietly liquidated if a systemic evet occurs. However, i iterbak etworks i which cotagio does ot occur, welfare is ot ecessarily icreasig with the umber of liks that baks have. This is because whe each bak has more liks, there are also more baks that icur losses o their deposits. However, sice baks have already suffi cietly may liks, 3

4 there is o beefit to offset this implicit cost. Thus, icreasig the coectivity of the iterbak etwork is beeficial up to the poit whe there is o cotagio i the fiacial system. The idea that a itercoected bakig system may be optimal is supported by various other studies. Leiter 005 discusses how the threat of cotagio may be part of a optimal etwork desig. His model predicts that it is optimal for some agets to bail out other agets, i order to prevet the collapse of the whole etwork. This form of isurace ca also emerge edogeously, ad I show that it is a equilibrium i a etwork formatio game. Likages betwee baks ca also be effi ciet i the model of Kah ad Satos 008 if there is suffi ciet liquidity i the fiacial system. Recetly, Acemoglu et al. 015 fid that the types of fiacial etworks that are most proe to cotagious failures deped o the umber of adverse shocks that affect the fiacial system. The ratioale for why a bak is willig to form solvecy liks with other baks i the same regio ad icur a loss o her deposits is that a early-withdrawal shock to ay bak ca have system-wide exteralities. I particular, all baks default whe a bak that has isuffi ciet liks is affected by a early-withdrawal shock. Baks are willig to pay a premium i.e. icur a loss o their deposits to avoid defaultig by cotagio. Thus, a solvecy iterbak etwork ca be iterpreted as a alterative to formal isurace markets. Moreover, the etwork formatio approach provides isights about the circumstaces i which baks are willig to purchase protectio, as well as about the premia they are willig to pay. From this perspective, the fidigs i this paper complemet solutios relyig o formal isurace arragemets previously proposed i the literature. For istace, Zawadowki 013 shows, i the cotext of OTC traded cotracts, that competitive isurace markets fail as baks fid the premia for isurig agaist couterparty default too expesive. This is because baks do ot iteralize that the default of aother bak which is ot a immediate eighbor ca evertheless affect them i subsequet default waves. Isurace is uattractive i Kyiotaki ad Moore 1997 as well, because of limited eforcemet of cotracts. Similarly, it has bee show that other formal arragemets, such as clearighouses, may icrease systemic risk either because they reduce ettig efficiecy as i Duffi e ad Zhu, 011 or they reduce dealers icetives to moitor each other as i Pirrog,

5 Startig with Alle ad Gale 000 there has bee a growig iterest i how differet etwork structures respod to the breakdow of a sigle bak i order to idetify which oes are more fragile: See, for istace, the theoretical ivestigatio of Freixas et al. 000 or Castiglioesi ad Navarro 007, ad the experimetal study of Corbae ad Duffy 008. I parallel, the empirical literature has looked for evidece of cotagious failures of fiacial istitutios resultig from mutual claims they have o oe aother ad has show such iterbak loas are ulikely to lead to sizable cotagio i developed markets Furfie, 003; Upper ad Worms, 004. Other papers are cocered with whether iterbak markets aticipate cotagio. For istace, i Dasgupta 004 cotagio arises as a equilibrium outcome coditioal o the arrival of egative iterim iformatio which leads to coordiatio problems amog depositors ad widespread rus, whereas Caballero ad Simsek 013 provide a model of market freezes whe the complexity of a fiacial etwork icreases the ucertaity about the health of tradig couterparties ad of their parters. More recetly, Alvarez ad Barlevy 014 study madatory disclosure of losses at fiacial istitutios which are exposed to cotagio via a etwork of iterbak loas. This paper is orgaized as follows. Sectio itroduces the model i its geerality. Sectio 3 describes whe cotagio ca occur ad the payoffs that baks receive. Sectio 4 provides the equilibrium aalysis. I sectio 5, I preset a extesio of the model to iclude small likig costs ad discuss welfare implicatios. Sectio 6 cocludes. The Model.1 Cosumers ad liquidity prefereces The ecoomy is divided ito sectors, each populated by a cotiuum of cosumers. Cosumers prefereces are described by a log-utility fuctio. There are three time periods t = 0, 1,. Each aget is edowed with oe uit of cosumptio good at date t = 0. Agets are ucertai about their liquidity prefereces: they ca be early cosumers, who value cosumptio oly at date 1, or they ca be late cosumers, who value cosumptio oly at date. The probability that a aget is a early cosumer is q. I assume that the law of large umbers holds i the cotiuum, which implies that, o average, the fractio of agets 5

6 Regio A Regio B Probability State/Sector ϕ/ S 1 p H p H... p H p L p L... p L 1 ϕ/ S p L p L... p L p H p H... p H q + χ q... q q q... q ϕ S q q + χ... q q q... q q q... q q q... q + χ Table 1: Distributio of shocks i the ecoomy at date 1 that value cosumptio at date 1 is q. However, each sector experieces fluctuatios of early withdrawals. With probability 1/, i each sector there is either a high proportio, p H, or a low proportio, p L, of early cosumers, so that q = p H+p L. I particular, the ecoomy cosists of two regios, A = {1,,..., } ad B = { + 1, +,..., }, such that fluctuatios i the fractio of early cosumers are perfectly correlated withi each regio ad egatively correlated across regios. That is, whe sectors i regio A receive a high fractio, sectors i regio B receive a low fractio, ad the other way aroud. Aggregate early-withdrawal shocks ca affect the ecoomy with a small, but positive, probability ϕ. I this case, the average fractio of early cosumers is higher tha q. For tractability, I assume that exactly oe of the sectors receives a fractio q + χ of early cosumers, whereas the others receive a fractio q. Each sector is equally likely to experiece a early-withdrawal shock. The ucertaity i the liquidity prefereces of cosumers at date 1 is summarized i Table 1. 1 At date 0 each of the sectors is ex-ate idetical. All the ucertaity is resolved at date 1, whe the state of the world is realized ad commoly kow. At date, the fractio of late cosumers i each regio will be 1 p where the value of p is kow at date 1, as either p H, p L, q or q + χ. 1 Each realizatio of the aggregate early-withdawal shock i which sector k receives a fractio q + χ of early cosumers represets a state S k that occurs with probability ϕ. I abuse otatio ad refer to the set of states { Sk }k as state S that occurs with probability ϕ. 6

7 . Baks, ivestmet opportuities ad iterbak deposits I each sector i there is a competitive represetative bak. Agets deposit their edowmet i their sector s bak. I exchage, they receive a deposit cotract that promises a fiite amout of cosumptio depedig o the date they choose to withdraw their deposits, ad that is, possibly, cotiget o the state of the world. I particular, the deposit cotract specifies that if they withdraw at date 1, they receive C S 1, ad if they withdraw at date, they receive Ci S CS, where S {S1, S, S}. Baks have two ivestmet opportuities: a liquid asset with a retur of 1 after oe period, or a illiquid asset that pays a retur of r < 1 after oe period, or R > 1 after two periods. Let x i ad y i be the per capita amouts that a bak i ivests i the liquid ad illiquid asset, respectively. I additio, baks ca deposit fuds at other baks i exchage for the same deposit cotract offered to cosumers. That is, for each uit that bak i deposits at bak j, she is promised C1j S if withdrawig at t = 1 ad CS j if withdrawig at time t =. Let z ij deote the amout that bak i deposits at bak j, ad z i deote the total amout of iterbak deposits that bak i holds. Iterbak deposits coect the baks i a etwork g. I particular, if bak i holds deposits at bak j, they are cosidered to have a lik ij ad to be eighbors i the etwork g. The set of eighbors of a bak i i the etwork g is N i g = {j A B ij g for ay j i}. A set of cotracts C S, CS i i A B ad portfolio allocatios x i, y i, z i i A B is feasible if it respects the feasibility costraits at date 0, 1, ad, as follows. At date 0, each bak i s portfolio must satisfy the followig feasibility costrait x i + y i + j N i g z ij = 1 + j i N j g I other words, the amout that a bak i receives from depositors, 1, ad other baks, j, i Njg z ji, ca be ivested i the liquid asset, the illiquid asset, or as deposits at other baks. The feasibility costrait at date 1 requires that the paymets to the early cosumers ad to baks that withdraw at date 1 equal the cash iflows from the liquid asset ad the deposits withdraw from other bak at date 1, i state S 1 ad S. I state S 1, baks z ji. 7

8 i regio A withdraw deposits from baks i regio A ad B, whereas baks i regio B withdraw deposits oly from other baks i regio B. Similarly, i state S, baks i regio B withdraw deposits from baks i regio A ad B, whereas baks i regio A withdraw deposits oly from other baks i regio A. The feasibility costrait at date requires that the paymets to the late cosumers ad to baks that withdraw at date equals the cash iflows from the illiquid asset ad from the deposits withdraw from other baks at date, i state S 1 ad S. I state S 1, baks i regio B withdraw deposits from baks i regio A. Similarly, i state S, baks i regio A withdraw deposits from baks i regio B. The feasibility costraits are, essetially, budget costraits. The coditio that cotracts ad portfolio allocatios respect the feasibility costraits i states S 1 ad S, simply rules out that defaults occur i either of these states. However, I allow for the possibility that defaults occur i state S, as baks are ot required to have a balaced budget i this state..3 Iterbak etworks Various etworks of iterbak deposits ca be cosidered. Figure 1 illustrates several patters of coectios betwee baks. I a liquidity etwork each bak has coectios oly with baks i the other regio. I a symmetric etwork each bak has the same umber of liks. A symmetric liquidity etwork is show i Figure 1a, ad a symmetric etwork is show i Figure 1b. Figure 1c represets a etwork i which each bak has the same umber of coectios with baks i the other regio, but a differet umber of coectios with baks i the same regio. The liks that baks have with other baks i the same regio represet a solvecy etwork. I a complete etwork each baks has coectios with all other baks, as show i Figure 1d. I will use the followig otatio throughout the paper. Let g l,ηi represet etworks where each bak i has l liquidity liks with baks i the other regio ad η i solvecy liks with baks i the same regio. If all baks have same the umber, η, of coectios with baks i the same regio, the the etwork is symmetric ad deoted g l,η. For istace g, represets the complete etwork, while g,0 represets a symmetric liquidity etwork i which each bak has liks with all the bak i the other regio, ad o liks with baks 8

9 A B 1 5 A B a A B b A B c 4 d 8 Figure 1: This figure illustrates various patters of coectios betwee baks. Pael a shows a symmetric liquidity etwork. Pael b illustrates a symmetric iterbak etwork. Pael c illustrates a iterbak etwork i which each bak has the same umber of liks with baks i the other regio, but a differet umber of liks with baks i the same regio. Pael d illustrates a complete etwork. i the same regio. I the aalysis i the followig sectio I focus o the case i which each bak has l coectios with baks i the other regio. The mai results, aside of Propositio 5, are derived for l =. 3 Iterbak Cotagio 3.1 Optimal risk-sharig without aggregate ucertaity The optimal risk sharig problem is well uderstood if there is o aggregate ucertaity about the average fractio of early cosumers i.e. whe ϕ = 0. Alle ad Gale 000 characterize the optimal risk sharig allocatio as the solutio to a plaig problem. They show that the optimal deposit cotract C1, C is state-idepedet, ad maximizes the ex-ate expected utility of cosumers. I the case whe cosumers have log-prefereces it follows straightforwardly that C 1, C = 1, R. 1 9

10 Moreover, the optimal portfolio allocatio requires that each bak i ivests a amout x i the liquid asset i order to pay the early cosumers ad a amout y = 1 x i the illiquid asset i order to pay the late cosumers x, y = qc 1, 1 qc /R. As Alle ad Gale 000 show, a iterbak system ca decetralize the plaer s solutio, if each bak deposits with baks i the other regio a total amout z = p H q = q p L, at date 0. The plaer s solutio ca be implemeted by ay symmetric liquidity etwork, g l,0, where each bak has liquidity liks with l {1,,..., } baks i the other regio, ad the amout of deposits exchaged betwee ay two baks is z l at date 0. Moreover, ay etwork g l,ηi i which ay two baks that have a lik exchage z l as deposits at date 0 ca also decetralize the plaer solutio. This is because deposits exchaged with baks i the same regio mutually cacel out i either state S 1 or S, as they do ot provide isurace agaist regioal liquidity fluctuatios. At the same time, although a bak ca deposit more tha z with baks i the other regio, i a iterbak etwork g l,ηi isurace agaist liquidity shocks ca be achieved oly if baks that have a liquidity shortage withdraw a et amout of z l from the baks that have a liquidity surplus. It is straightforward to see that the cotract 1 ad the portfolio respect the feasibility costraits itroduced i Sectio.. Moreover, o defaults occur whe ϕ = Defaults ad the cotagio mechaism Next, I cosider the case of ϕ > 0, whe each bak icurs a early-withdrawal shock with probability 1/ i state S. I show that defaults ad cotagio ca occur i a give iterbak etwork, g l,ηi, i which baks offer the deposit cotract 1, hold the portfolio, ad isure a amout z /l as bilateral deposits with other baks. I assume that each bak exchages z /l deposits with both baks i the other regio, as well as baks i the Exchagig iterbak deposits ex-ate i order to isure agaist liquidity shocks may be see as ucovetioal. Acharya et al. 01 emphasize some of the problems that occurs whe liquidity trasfers occur ex-post. For istace, surplus baks may strategically uder-provide ledig to iduce ieffi ciet sales of assets from eedy baks. 10

11 same regio. Although it is feasible to cosider that baks i the same regio exchage a differet amout as deposits, this assumptio simplifies our aalysis without losig ay isights. A bak that eeds to repay q + χ C 1 to the early cosumers i state S does ot have suffi ciet liquidity at date 1, as the proceeds from the liquid asset are x = qc 1. Hece, the bak must liquidate either some of its iterbak deposits ad/or the illiquid asset. As i Alle ad Gale 000, I assume that the costliest i terms of early liquidatio is the illiquid asset, followed by iterbak deposits: C C 1 < R r. 3 This implies that the bak liquidates deposits i other regios before it liquidates the illiquid asset. I state S, liquidatig iterbak deposits, although beeficial, as I describe below, does ot geerate liquidity. Thus, the bak must liquidate at least part of the illiquid asset i order to meet withdrawals from early cosumers. A bak that liquidates the illiquid asset prematurely, affects egatively the cosumptio of late withdrawers. I fact, if too much of the illiquid asset is liquidated early, the cosumptio of late cosumers may be reduced to a level below C1. I this case, the late cosumers gai more by imitatig the early cosumers ad withdrawig their ivestmet from the bak at date 1. This iduces a ru o the bak. The maximum amout of illiquid asset that ca be liquidated without causig a ru is give by or, substitutig from, b y 1 qc 1, 4 R b = 1 q C C 1. R For the remaider of the paper, I assume that χ > r b C1. 5 I other words, the amout that ca be obtaied at date 1 by liquidatig the log asset without causig a ru, r b, is ot suffi ciet to repay the additioal fractio, χ, of early depositors. Thus, a bak which offers the deposit cotract 1, ad holds the portfolio caot repay C 1 to depositors that withdraw at date 1, if she icurs a early-withdrawal 11

12 shock i state S. I this case, the bak defaults, ad its portfolio of assets is liquidated at the curret value ad distributed equally amog creditors. Suppose that bak k is affected by the early-withdrawal shock i state S, whe the etwork of iterbak deposits is g l,ηi. differet returs upo liquidatio i period 1. The three assets i the bak k s portfolio yield First, the liquid asset pays a retur of 1. Secod, the illiquid asset, pays a retur of r < 1 if liquidated early. Ad lastly, the iterbak deposits held at a bak j yield a retur, C1j d C 1. O the liability side, a bak has to pay its depositors, ormalized to 1 ad at the same time to repay its iterbak creditors that add up to l + η k z l. This yields at date 1 a ew retur per uit of good deposited i bak k equal to C d 1k g l,η i = x + ry + j N k g z 1 + l + η k z l l Cd 1j g. 6 The retur that bak k pays o early withdrawals, C d 1k, depeds o the etwork g l,η i iterbak deposits, as it is described i Sectio 3.3. However, whe uecessary, I suppress the depedecy i the otatio ad take as implicit that C1k d g l,η i = C1k d for ay k. Followig the default of bak k, subsequet defaults are possible. of I particular, if C1k d < C 1, the a bak j that has deposits at bak k icurs a loss of value o its deposits, or a loss give default heceforth, LGD. The LGD that bak j icurs i a etwork g l,ηi whe bak k has bee liquidated is give by or, substitutig i 6, LGD jk g l,ηi = z l C 1 C1k d, 7 LGD jk g l,ηi = z l 1 q C1 r l + η k z l l N k g l,ηi z l 1 + l + η k z l C 1 C1l d. 8 A positive LGD triggers the early liquidatio of the illiquid asset to meet early withdrawals. If bak j eeds to liquidate a amout of the illiquid asset higher tha b, she fails, as explaied above. Whe a bak fails by cotagio, its portfolio of assets is also liquidated at the curret value ad distributed equally amog creditors. 3 I cotrast, if 3 This explais why bak k, that has iitially icured the early-withdrawal shock, may reiceve a retur o its iterbak deposits lower tha C 1, as reflected i 6. 1

13 the amout of the illiquid asset liquidated at date 1 is below the threshold b, bak j does ot default ad returs C 1 for early withdrawals. Nevertheless, it will be costly for the late cosumers, as their cosumptio is ow reduced to Cj d < C. Thus, b give by 4 represets a cotagio threshold, as it is the maximum amout of illiquid asset that a bak ca liquidate without causig a ru. Two distict implicatios follow from 8. First, the loss give default LGD jk is z icreasig i the amout of deposits, l, exchaged betwee the two baks. I other words, the more liquidity liks, l, each bak has with baks i the other regio, the smaller is the loss give default. Secod, the more liks bak k has with baks that are able to repay C 1 for deposits, the smaller is the loss LGD jk it iduces to a eighbor j. This effect is idepedet of the amout of deposits exchaged betwee baks, ad it arises, for istace, whe keepig the umber of liks, l, with baks i the other regio costat ad icreasig the umber of solvecy liks, η k, with baks i the same regio. There are two chaels that explai this secod implicatio. Everythig else equal, icreasig the umber of liks with baks that are able to repay a retur for deposits of C 1 at date 1, mechaically decreases the loss that bak k iduces to its eighbors. This is because, the deomiator i 8 icreases, while the umerator remais the same. I other words, the loss that bak k iduces whe it fails is redistributed across more couterparties. At the same time, a positive spiral that further reduces the loss-give-default may arise. For istace, the LGD jk icurred by eighbor j may decrease suffi cietly, such that bak j does ot default by cotagio ad is also able to repay bak k a retur C 1 per-uit of deposits. The, the loss that k iduces to its eighbors decreases eve further, reachig its miimum whe oe of the eighbor baks defaults LGDjk mi η k = z 1 q C1 r, 9 l 1 + l + η k z l for ay j N k g l,ηi. Thus, the cotract 1 ad portfolio ivolve a trade-off. Because defaults ivolve the liquidatio of the illiquid asset, there is a utility loss i state S. A differet arragemet, i which baks ivest more i the liquid asset ad isure less through the iterbak system may be desirable. A bak that has cash reserves, ca avoid a ru if it icurs a earlywithdrawal shock. At the same time, she holds fewer deposits with other baks i the 13

14 system ad icurs a lower loss-give-default. This way, baks ca avoid default i state S, at the expese of providig a lower utility to cosumers i states S 1 ad S. I the remaider of the sectio I discuss the payoffs that baks expect to receive i each state of the world 4, uder the assumptio that each bak offers the deposit cotract 1 ad holds the portfolio. I sectio 4, I come back to this issue ad show that baks fid it optimal to offer the deposit cotract 1 ad to hold the portfolio provided ϕ is suffi cietly small. For ow, I start by describig the payoffs i state S. 3.3 Expected payoffs Cosider as before that i state S bak k is affected by the early-withdrawal shock, ad suppose that the umber of coectios that bak k has with baks is i the same regio, η k, satisfies the followig iequality z 1 q C1 r l 1 + l + η k z l r b. 10 The left had side of the iequality represets the loss-give default that a eighbor of bak k receives i a etwork g l,ηi, provided all k s eighbor baks repay a retur C 1 for the deposits they have received from bak k. At the same time, repayig C 1 is ideed cosistet with the loss-give-default that a eighbor j of bak k receives i the etwork g l,ηi, as the maximum amout of the illiquid asset that ca be liquidated without causig a ru, b, is larger tha LGDmi jk g l,η i r. This implies that i the etwork g l,ηi each bak returs for early withdrawals C1, except for bak k which returs C1k d η k = C1 1 q C1 r l + η k z l Moreover, each of the [ l + η k + 1] baks that do ot have a coectio with bak k returs C for late withdrawals. However, each of the l + η k baks that have a coectio with bak k must liquidate a amout of LGDmi jk g l,η i r from the illiquid asset ad returs for late withdrawals C d j η k = C z l R r C 1 C 1 + l + η k z l, 1 4 Baks are perfectly competitive ad make zero-profits. However, because baks maximize the expected utility of cosumers, I abuse termiology ad use baks payoffs to refer to baks cosumers payoffs. 14

15 for ay j N k g l,ηi, with C d j η k C 1. I cotrast, if iequality 10 does ot hold, the ay eighbor of bak k defaults by cotagio. This is because eve if all l + η k eighbors of bak k repay C1, they still eed to liquidate too much of the illiquid asset. Clearly, this implies that it is impossible for ay of them to repay C 1 to start with, ad the realized loss-give-default is eve higher. Whe iequality 10 does ot hold, the payoffs that baks receive deped o the etire etwork structure. The procedure to fidig the solutio ivolves a sequece of steps. First, fid the retur that bak k ad each of its eighbors j N k g l,ηi repay for early withdrawals. For this, solve the system of l + η k + 1 equatios implied by 6, uder the assumptio that the remaiig baks that are ot eighbors of k do ot default ad are able to repay C 1. Secod, verify that these baks are ideed able to repay C 1, give the losses-give-default implied at the first step. If all baks that are ot eighbors of k are able to repay C1, the the solutio is the oe foud at the first step. Otherwise, if a subset of m of these baks are ot able to repay C 1, solve the system of l + η k m equatios implied by 6, uder the assumptio that the remaiig baks do ot default ad are able to repay C1. Verify that this is ideed cosistet with the solutio foud. Otherwise, cotiue the procedure util all baks default. This solutios cocept is similar to the algorithm that Elliott, Golub ad Jackso 013 propose to characterize waves of defaults i a etwork of liabilities which is a geeralizatio of the algorithm i Eiseberg ad Noe, 001. Although a solutio for a geeral etwork g l,ηi is diffi cult to characterize, the followig propositio describes the payoffs that baks receive whe l =, for ay umber of liks η i that a bak i has with baks i the same regio. Moreover, for the remaider of the paper I assume as well that l =, ad relax this assumptio whe I discuss icomplete liquidity etworks i Sectio 5. 15

16 Propositio 1 Cosider ay iterbak etwork g,ηi i which each bak offers the deposit cotract 1 ad holds the portfolio. Let η be the smallest positive iteger that satisfies the iequality z 1 q C1 r r b η z If the bak that is subject to the early-withdrawal shock i state S has less tha η coectios, the each bak returs per uit of deposit at date 1 C1 d = C1 1 q C1 r. 14 The proof for Propositio 1 follows i two steps. First, I show that if the bak that is subject to the early-withdrawal shock has less tha η coectios, the all the other 1 baks fail by cotagio. Importatly, this is idepedet of how may coectios each bak has with baks i the same regio, as the result holds for ay etwork g,ηi. Secod, I fid the vector of returs C d that is a fixed poit of the system of equatios implied by 6. i A B Propositio 1 easily geeralizes to ay etwork g l,ηi i which all baks default i state S. I particular, let ηl be the smallest iteger for which iequality 10 holds. Cosider parameters such that if the bak subject to the early-withdrawal shock has less tha η l coectios, the all the other 1 baks fail by cotagio. The all baks retur C d 1 per uit of deposit as give by 14. At this stage I ca characterize the payoffs that each bak i expects to receive i a etwork g,ηi. I both states S 1 ad S, each bak has with probability half either a high fractio, p H, of early cosumers or a low fractio, p L, of early cosumers. Hece, cosumers expect to receive i each of these states qu C q u C. I state S, cosumers expected utility depeds o how may baks have at least η coectios with baks i the same regio ad o whether bak i, itself has at least η coectios with baks i the same regio. Let Hg,ηi = { j A B η j η } be the set of baks that have at least η coectios, ad let h = Hg,ηi be the umber of baks that have at least at least η coectios. This implies that there are h baks that each iduces the default of the etire system, whe affected by the early-withdrawal shock. Moreover, 16

17 let H i g,ηi = { j N i g,ηi η j η } be the set of eighbors of bak i that have at least η coectios. This implies that wheever a bak j H i g,ηi is affected by the earlywithdrawal shock, bak i returs to its late cosumers C d i ηj as give by 1 for l =. I additio, the retur that bak i pays i case it is affected by the early-withdrawal shock depeds o how may coectios it has. Thus, if i Hg,ηi, the it returs C d η i as give by 11 for l =. Otherwise, it returs C d 1 The expected payoff of a give bak i is as follows as give by 14. π i g,ηi = 1 ϕ [q l C1 + 1 q l C] 15 [ h +ϕ l C1 d + 1 q lc j H i g,ηi q l Ci d ηj + 1 ] q j Hg,ηi \H i g,ηi lc q l C, if i / Hg,ηi, ad π i g,ηi = 1 ϕ [q l C1 + 1 q l C] 16 [ h +ϕ l C1 d + 1 q lc j H i g,ηi q l Ci d ηj + 1 q j Hg,ηi \H i g,ηi lc q l C + 1 l C d η i ], if i Hg,ηi. 4 Edogeous Solvecy Networks Whe there is a risk of cotagio i the fiacial system, baks ca take actios to isure agaist it. The decisios that each bak i must cosider at date 0 i order to maximize the expected utility of cosumers, cosist of a deposit cotract C S, CS i for early ad late withdrawals, a portfolio of liquid ad illiquid assets ad iterbak deposits x i, y i, zi, a set of liquidity liks with baks i the other regio, ad a set of solvecy liks with baks i the same regio. I particular, cosider the followig timig of evets at date 0. At stage 1, each bak chooses a deposit cotract ad a portfolio allocatio. At stage, each bak chooses a set of liks with baks i the other regio, specifyig for each lik the amout of iterbak deposits she wats to isure with the respective couterparty. At stage 3, 17

18 each bak chooses a set of liks with baks i the same regio, specifyig as well, for each lik, the amout of iterbak deposits it wats to isure with the respective couterparty. At each stage, each bak takes the decisios at previous stages, as well, the decisios of other at the curret stage as give. Furthermore, at each stage baks uderstad the cosequeces of their curret decisios o the choices to be made at future stages. I makig choices, at each stage baks could reaso backwards ad choose the deposit the cotract that maximizes the expected utility of cosumers. The diffi culty with this approach is that decisios at stage ad 3 ivolve both a set of liks, as well as a amout of deposits for each lik. This prevets agets from takig decisios uilaterally, or eve bilaterally. etwork g l,0 For istace, cosider a symmetric liquidity where each bak has l liks with baks i the other regio, ad ay two baks that have a lik exchage a amout z l as deposits. Suppose that at stage 1, baks offer the deposit cotract C 1, C, hold the portfolio x, y ad isure a amout z = p H q as iterbak deposits. The, a bak that is cosiderig decreasig the umber of liks with baks i the other regio from l to l 1, must re-adjust the amout of deposits it exchages with at least oe other bak from z l to z l, i order to respect the feasibility costraits described i Sectio.. However, the re-adjustmet caot be doe uilaterally as her couterparty has a excess of icomig deposits, uless she re-adjusts the deposits she exchages with other couterparties i order to respect her feasibility costraits. The approach I take istead is similar to the solutio method i Dasgupta 004. First, I aalyze the equilibrium etworks o the cotiuatio path iduced by the deposit cotract 1 ad the portfolio, whe each bak has coectios with baks i the other regio ad ay two baks that have a lik exchage z i deposits. Secod, I show that each bak s best respose is to offer the deposit cotract 1 ad the portfolio whe all other baks offer the deposit cotract 1 ad the portfolio, at least whe the probability of state S is suffi cietly small. 5 5 A complimetary approach is to cosider that baks first choose a set of liks ad the choose a deposit cotract ad a portfolio allocatio. Acemoglu et al. 015 explore this route usig a solutio strategy similar to mie. I particular, they solve for the equilibrium iterbak ledig decisios ad iterest rates, give that baks are coected i a exogeous etwork which restricts the amout they ca borrow from 18

19 To aalyze the equilibrium etworks, I model the iteractio betwee baks i the same regio as a game i which baks choose with whom to form liks. Such a game is called a etwork formatio game. The formatio of a lik requires the coset of both parties ivolved, sice baks mutually exchage deposits. However, the severace of a lik ca be doe uilaterally, as deposits ca be withdraw o demad by either of the baks ivolved i a lik i this case, deposits will be restituted to both baks, although oly oe party exercises the claim. To idetify equilibrium etworks, I use the cocept of pairwise stability itroduced by Jackso ad Wolisky Defiitio 1 A iterbak etwork g is pairwise stable if i for ay pair of baks i ad j that are liked i the iterbak etwork g, either of them has a icetive to uilaterally sever their lik ij. That is, the expected profit each of them receives from deviatig to the iterbak etwork g ij is ot larger tha the expected profit that each of them obtais i the iterbak etwork g π i g ij π i g ad π j g ij π j g; ii for ay two baks i ad j that are ot liked i the iterbak etwork g, at least oe of them has o icetive to form the lik ij. That is, the expected profit that at least oe of them receives from deviatig to the iterbak etwork g + ij is ot larger tha the expected profit that it obtais i the iterbak etwork g if π i g + ij > π i g the π j g + ij < π j g. The result below describes whether formig or severig liks is profitable for ay give pair of baks i ad j, depedig o the umber of liks that they have i a etwork g,ηi. The derivatios are show i Appedix B. Propositio The margial payoff s of formig ad severig a lik, respectively, for bak i i a etwork g,ηi are characterized by the followig properties: each other. They study the whether equilibrium outcomes are optimal or ot, depedig o the etwork structure. 6 This cocept has bee applied by Zawadowski 013 to aalyze the stability of bilateral isurace cotracts i the cotext of OTC markets. More recetly, Farboodi 014 uses a extesio of the cocept that allows for group deviatios to explai itermediatio i iterbak markets. 19

20 Property 1: π i g,ηi + ij < π i g,ηi if η i η or η i η for ay η j η. Property : π i g,ηi ij > π i g,ηi if η i η + 1 or η i η 1 for ay η j η + 1. Property 3: π i g,ηi + ij > π i g,ηi if η i η 1 for ay η j η 1, or for ay η i if η j = η 1. Property 4: π i g,ηi ij < π i g,ηi if η i η for ay η j η, or for ay η i if η j = η. Property 5: π i g,ηi + ij < π i g,ηi, if η i = η 1 ad η j η, for ay r < z + η z +1+ η 1. Property 6: π i g,ηi ij > π i g,ηi, if η i = η ad η j η + 1, for ay r < z + η z +1+ η 1. Property 7: π i g,ηi + ij = π i g,ηi if η i η ad η j η. Property 8: π i g,ηi ij = π i g,ηi if η i η 1 ad η j η 1. The mai trade-offs that baks take ito accout whe formig or severig liks are as follows. O the oe had, there is a implicit cost of havig a lik with a bak that has at least η + 1 liks or formig a lik with a bak that has η liks. If she icurs a early-withdrawals shock, her eighbors, though they do ot default, are ot able to retur C to the late cosumers. This effect is captured by Properties 1. O the other had, baks are willig to sacrifice some utility from the late cosumers if they ca avoid defaultig by cotagio. Thus, baks fid it beeficial to maitai a lik with a bak that has η liks or form a lik with a bak that has η 1 liks. This effect is captured by Properties 3 4. The trade-off betwee the cost ad beefit of likig is more complex i two cases, which are characterized by Properties 5 6. Whe bak i has η i = η 1 liks, formig a ew a lik with a bak j that has η j η liks implies that bak i is able to repay C d η i + 1, as opposed to C1 d, if she is affected by the early-withdrawal shock. However, bak i ca retur oly Ci d η j + 1 for late withdrawals, as opposed to C, if bak j is affected by the early-withdrawal shock. Similarly, whe bak i has η i = η liks, severig a a lik with a bak j that has η j η + 1 liks implies that bak i returs oly C d 1, as opposed to C d η i, if she is affected by the early-withdrawal shock. However, bak i is 0

21 able to retur C for late withdrawals, as opposed to Cd i η j, if bak j is affected by the early-withdrawal shock. Whe r is suffi cietly small, the utility gaied whe bak i is affected by the early-withdrawal shock is ot suffi ciet to compesate for the utility lost whe bak j is affected by the liquidity shock. As r icreases, whether bak i fids it profitable to form or severe a lik with bak j that has at least η liks is ambiguous, ad it depeds o exactly how may liks bak j has. I particular, the more liks bak j has, the more attractive is for bak i to form or maitai a lik with bak j. I ay case, this does ot hider the characterizatio of stable etworks i terms of how may baks have at least η coectios. Lastly, there is o utility gai or loss from formig a ew lik whe baks have less tha η liks, or severig a existig lik whe baks have less tha η 1 liks. This effect is captured by Properties 7 8. Typically, i may models of etwork formatio ad otherwise, multiple equilibria arise whe agets are idifferet betwee takig ad ot takig a actio. The umber of equilibria is particularly high i models of etwork formatio simply because of the dimesioality of the problem. Not surprisigly, the curret model features multiple equilibria as well. For istace, give Properties 7 8 described above, ay etwork g,ηi i which a bak i has η i η is pairwise stable. Clearly, there is a cocer whether these equilibria are robust, sice a bak may be tempted to switch to a payoff-equivalet lik profile. Geerally, elimiatig cases where players are idifferet results i a sigificat reductio of equilibria. Several solutios have bee explored i the literature. For istace, Bala ad Goyal 000 impose a stroger equilibrium cocept, which requires that each aget gets a strictly higher payoff from his curret likig strategy tha he would with ay other strategy, whereas Hojma ad Szeidl 008 impose coditios o the payoff fuctio to reduce the set of equilibrium etworks. A useful bechmark that miimizes free-ridig for the curret model is to assume that if two baks are idifferet betwee formig ad ot formig the lik, the they form the lik. Similarly, if two baks are idifferet betwee keepig ad severig a lik, the they keep the lik. I sectio 5.1, I discuss the effects of imposig a small cost for likig, which would imply that if two baks are idifferet betwee formig ad ot formig the lik, the they do ot form the 1

22 lik, whereas if two baks are idifferet betwee keepig ad severig a lik, the they severe the lik. Properties 1 to 8 characterize the beefits ad the costs of formig or severig liks for ay give pair of baks i ad j, spaig all combiatios of the umber of liks that they have i a etwork g,ηi. For istace, cosider the case of bak i, with η i = η + 1, ad j, with η j = η, that do ot have a lik. Property 3 characterizes the icetives for bak i to form the lik, while Property 1 characterizes the icetives for bak j to form the lik. These properties are useful i derivig the followig result. Propositio 3 Let g,ηi be a pairwise stable etwork. The there are at least max{ η, } baks that have η coectios with baks i the same regio. The mai ituitio for this result is as follows. The threshold umber of coectios η determies what exterality each bak has o the etire bakig system. Whe a bak that is affected by the early-withdrawal shock has less tha η coectios with baks i the same regio, all baks default by cotagio as Propositio 1 shows. However, if a bak that is affected by the early-withdrawal shock has at least η coectios with baks i the same regio, the oly the cosumptio of the late cosumers of her eighbors is egatively affected. These two types of exteralities drive baks icetives to form or severe liks. I particular, baks weigh the beefit of formig liks that allow them to avoid defaultig by cotagio, agaist the implicit cost that late cosumers icur. Properties 3 4 imply that baks are willig to icur the implicit cost for the late cosumers, if they ca avoid default. However, they are better off if they free-ride o others liks. The trade-off that a lik ivolves is best illustrated by the followig case. Cosider two baks, i with η i = η 1, ad j with η j η, that do ot have a lik. It follows from Property 3 that bak j beefits from formig with i, as she exchages a situatio whe the failure of i iduces its ow failure, for a situatio whe the failure of i results i a lower utility for her late cosumers. However, bak i does ot iteralize the effects that its ow failure o other baks. That is, the utility gaied whe bak i is affected by the early-withdrawal shock is ot suffi ciet to compesate for the utility lost whe bak j is affected by the liquidity shock, if r is ot too large. I cosequece, the lik betwee i ad j is ot formed, as the formatio of a lik requires the coset of both baks.

23 However, the result described i Propositio 3 does ot deped o r beig small eough. This is because the trade-off described above geeralizes to most pairs of baks. I particular, Properties 3 4 ad 7 8 imply that oly baks that have less tha η 1 liks uequivocally beefit from formig or maitaiig liks with each other. Thus, a bak that has less tha η 1 liks seeks to form liks with other baks, util she has at least η liks ad o other baks accepts a lik with her, or util the oly baks with whom she does ot have a lik have at least η liks. I cosequece, there is always a set of baks that have at least η liks. Propositio 3 has sigificat implicatios for the stability of the bakig system. I particular, whe η is small, i equilibrium most of the baks have suffi ciet liks to prevet a shock i oe of the istitutios spreadig through cotagio. From 13 it follows that the higher r or R is, the smaller is η. However, eve as η icreases, i a stable etwork, at least half the baks will have a suffi cietly may liks such that the losses they may geerate are small eough. The followig result characterizes the probability of systemic risk i a equilibrium iterbak etwork. Corollary 1 i Let g,ηi be a pairwise stable etwork. The the probability that all baks default is at most mi{ ϕ η, ϕ }. ii There exist pairwise stable etworks i which all baks have η liks ad the probability that all baks default is 0. This result idetifies a upper boud o the probability that all baks default by cotagio i state S. Corollary 1 is a immediate cosequece of Propositio 3, ad a proof is omitted. I a etwork g,ηi all baks default by cotagio whe a bak that has less η liks with baks i the same regio is affected by the early-withdrawal shock. The result follows sice there are at most mi{ η, }, ad each baks is affected by a early-withdrawal shock with probability 1/. Propositio 3 idetifies may iterbak etworks that ca be pairwise stable. The followig result refies the characterizatio, ad, more importatly, provides a rakig of equilibria based o their implied expected welfare. 3

24 Propositio 4 Let r < z + η z +1+ η 1. The, i ay pairwise stable etwork each bak has at most η liks. Moreover, the expected welfare i pairwise stable etworks is icreasig i the umber of baks that have η liks. While cotagio ca occur i ay pairwise stable iterbak etwork i which at least oe bak has less tha η liks, the probability that all baks default decreases with the umber of baks that have η liks. The result follows as the welfare loss iduced whe a bak that has less tha η liks is affected by the early-withdrawal shock is smaller the the welfare loss whe a bak that has η liks is affected by the early-withdrawal shock. Moreover, a iterbak etwork i which all baks have η liks ad cotagio ever arises ca be supported as a equilibrium. 7 Such iterbak etwork isures the highest welfare of all pairwise stable etworks. Up to ow, I have aalyzed equilibrium etworks o the cotiuatio path iduced by the deposit cotract 1 ad the portfolio. Clearly it is importat to isure that baks fid it optimal to offer the deposit cotract 1, ad to hold the portfolio. The followig propositio shows that this is the case, at least if the probability of state S is suffi cietly small. Propositio 5 There exists ϕ 0, 1 such that there is a equilibrium i which each bak offers the deposit cotract 1 ad holds the portfolio, for ay ϕ ϕ. Rather tha derivig the equilibrium deposit cotract ad portfolio allocatio that maximizes the expected utility of cosumers for each ϕ, Propositio 5 shows that each bak s best respose is to offer the deposit cotract 1 ad the portfolio whe all other baks offer the deposit cotract 1 ad the portfolio if ϕ is suffi cietly small. The ituitio is as follows. Because the deposit cotract 1 ad the portfolio is optimal whe ϕ = 0, a bak that ivests more i the liquid asset provides a lower utility to her cosumers i state S 1 ad S. I Lemma 1 i the appedix, I show that a bak that deviates from the cotract 1 ad the portfolio, while all other baks cotiue to choose the cotract 1 ad the portfolio, must be self-suffi ciet. That is, the bak 7 The existece of such a equilibrium etwork is coditioal o whether symmetric etworks i which each baks has a total of + η liks exist. Lovasz 1979 discusses i detail coditios for the existece of a symmetric etworks with odes, i which each ode has + η liks. 4

25 that deviates does ot hold ay iterbak deposits. The lemma follows from the feasibility coditios itroduced i Sectio.. I other words, the lemma relies o the assumptio that there are o defaults, ad that a bak delivers at date 1 ad the deposit cotract she had promised at date 0, i states S 1 ad S. Give this, I show that there is o patter of iterbak deposits such that a bak that deviates from the cotract 1 ad the portfolio exchages positive amouts with other baks i the system. Otherwise, there must be at least aother bak that defaults o the cotract 1, either i state S 1 or S. Therefore, the bak that deviates must offer a deposit cotract for early ad late withdrawals, ad hold a portfolio as whe she is i autarky. This implies that the expected utility loss i states S 1 ad S for the bak that deviates is strictly positive. Thus, eve though the bak may obtai a lower utility i state S whe she offers the cotract 1 ad holds the portfolio tha whe she deviates, the deviatio is sub-optimal if the probability of the aggregate shock is suffi cietly small. Propositio 5 formalizes the reasoig outlied i Alle ad Gale 000. Although they do ot explicitly write dow the feasibility costraits at date 1 ad, they argue that i order to avoid default, a bak has to make a large deviatio, rather tha a small oe. The distortio this causes i the other states will ot be worth it, if the probability of the aggregate shock is suffi cietly small. This result is ideed cosistet with the umerical fidigs i Dasgupta 004, who illustrates that it is optimal for baks to fully isure agaist regioal liquidity shocks, as log as the probability of cotagio is small. 5 Discussio 5.1 The model with likig costs As described i Sectio 4, there is a implicit cost associated to formig liks. I particular, a bak i must cosider the utility loss for her late cosumers that ca potetially arise whe formig a lik with a bak j that has η j η liks. This implicit cost itroduces a atural trade-off agaist the beefits that a lik ca brig. However, there is o cost associated to formig a lik with a bak that has less tha η liks. I fact, as Properties 7 8 imply, there are may cases i which baks are idifferet betwee formig or sever- 5