Mandatory Disclosure and Financial Contagion

Transcription

1 Madatory Disclosure ad Fiacial Cotagio Ferado Alvarez Uiversity of Chicago ad NBER f-alvarez1 at uchicago.edu Gadi Barlevy Federal Reserve Bak of Chicago gbarlevy at frbchi.org September 13, 2013 Abstract The paper aalyzes the welfare implicatios of madatory disclosure of losses at fiacial istitutios whe it is commo kowledge that some baks have icurred losses but ot which oes. We develop a model that features cotagio, meaig that baks ot hit by shocks may still suffer losses because of their exposure to baks that are. I additio, baks i our model have profitable ivestmet projects that require outside fudig, but which baks will oly udertake if they have eough equity. Ivestors thus value iformatio about which baks were hit by shocks. We fidthatwhetheextet of cotagio is large, it is possible for o iformatio to be disclosed i equilibrium but for madatory disclosure to icrease welfare by allowig ivestmetthatwould ot have occurred otherwise. Abset cotagio, however, madatory disclosure will ot raise welfare, eve if markets are otherwise froze. Our fidigs provide isight o whe cotagio is likely to be a cocer, e.g. whe baks are highly leveraged agaist other baks, ad thus o whe madatory disclosure is likely to be desirable. JEL Classificatio Numbers: Key Words: Iformatio, Networks, Cotagio, Stress Tests First draft May We thak Aa Babus, Russ Cooper, Simo Gilchrist, Matt Jackso, Peter Kodor, H. N. Nagaraja, Ezra Oberfield, Alp Simsek, Alireza Tahbaz-Salehi, Carl Taebaum ad P. O. Weill, for their commets ad suggestios. We thak the commets from semiars participats at Goethe Uiversity, at the Networks i Macroecoomics ad Fiace coferece at the B.F.I, ad at the Summer Workshop o Moey, Bakig, Paymets ad Fiace at the Federal Reserve Bak of Chicago. The views i this papers are solely those of the authors ad eed ot represet the views of the Federal Reserve Bak of Chicago or the Federal Reserve System.

2 1 Itroductio I tryig to explai how the declie i U.S. house prices evolved ito a fiacial crisis i which trade betwee fiacial itermediaries early groud to a halt, various aalysts have sigled out the prevailig ucertaity at the time regardig which etities icurred the bulk of the losses associated with the housig market. For istace, Gorto (2008) provides a early aalysis of the crisis i which he argues The ogoig Paic of 2007 is due to a loss of iformatio about the locatio ad size of risks of loss due to default o a umber of iterliked securities, special purpose vehicles, ad derivatives, all related to subprime mortgages... The itroductio of the ABX idex revealed that the values of subprime bods (of the 2006 ad 2007 vitage) were fallig rapidly i value. But, it was ot possible to kow where the risk resided ad without this iformatio market participats ratioally worried about the solvecy of their tradig couterparties. This led to a geeral freeze of itra-bak markets, write-dows, ad a spiral dowwards of the prices of structured products as baks were forced to dump assets. Market participats emphasized the same pheomeo as the crisis was ufoldig. Back i February 24, 2007, the Wall Street Joural attributed the followig to former Salomo Brothers vice chairma Lewis Raieri, the so-called godfather of mortgage fiace: The problem... is that i the past few years the busiess has chaged somuch that if the U.S. housig market takes aother lurch dowward, o oe will kow where all the bodies are buried. I do t kow how to uderstad the ripple effects through the system today, he said durig a recet semiar. I lie with this view, some have argued that a importat step i the evetual stabilizatio of fiacial markets was the Fed s implemetatio of bak stress tests. These tests required baks to report to Fed examiers how their respective portfolios would fare uder various stress scearios ad thus the losses baks were vulerable to. I cotrast to the traditioal cofidetiality accorded to bak examiatios, the results of these stress tests were publicly released. Berake (2013) summarizes the view that the public disclosure of the stress-test results played a importat role i stabilizig fiacial markets: I retrospect, the [Supervisory Capital Assessmet Program] stads out for me as oe of the critical turig poits i the fiacial crisis. It provided axious ivestors with somethig they craved: credible iformatio about prospective losses at baks. Supervisors public disclosure of the stress test results helped restore cofidece i the bakig system ad eabled its successful recapitalizatio. 1

3 I this paper, we examie whether ucertaity as to which baks icurred losses that is, ucertaity as to where the bad apples are located ca lead to market freezes that make it desirable for policymakers to itervee ad force baks to disclose their fiacial positio. The feature that turs out to be critical for such itervetio to be beeficial i our model is cotagio, by which we mea a situatio i which shocks that hit some baks lead to losses at other baks that are ot themselves hit by these shocks. A example of cotagio i the cotext of the fiacial crisis is if the losses of baks directly exposed to the subprime market led to losses at baks that held few subprime mortgages i their portfolios. I what follows, we focus o a model of balace sheet cotagio i which baks that are hit by shocks ed up defaultig o their obligatios to other baks, so that baks ot hit by shocks ca still have their equity wiped out. We modify this model i two ways. First, we allow baks to raise additioal fuds from outside ivestors i order to fiace profitable ivestmet projects. However, we itroduce a agecy problem so that ivestors oly wat to ivest i baks with sufficiet equity. Whe ivestors are ucertai about which baks icurred losses, they may refuse to ivest i baks altogether. Cotagio exacerbates this problem, sice ivestors worry ot oly that the baks they ivest i were hit by shocks that wiped out their equity, but that these baks may be idirectly exposed to such shocks because they have fiacial obligatios from baks that were directly hit. The greater the potetial for cotagio, the more likely are market freezes to occur. Secod, we allow baks to disclose whether they were hit by shocks. To determie whether madatory disclosure is desirable, we eed to kow why baks do t simply hire a exteral auditor to coduct their ow stress test, or else release the iformatio they provide to examiers o their ow. We show that whe the extet of cotagio is small, madatory disclosure caot be welfare improvig whe baks choose ot to disclose i equilibrium, eve whe o-disclosure results i a market freeze where o bak ca raise outside fuds. But whe cotagio is large ad the cost of disclosure is low, madatory disclosure ca be welfare improvig eve though baks choose ot to disclose their fiacial situatio. Ituitively, cotagio implies that iformatio o the fiacial health of oe bak is relevat for assessig the fiacial health of other baks. Sice baks fail to iteralize these iformatioal spillovers, too little iformatio will be revealed, creatig a role for madatory disclosure as a welfare improvig itervetio. Abset these spillovers, baks iteralize the beefits of disclosure, ad so if they choose ot to disclose it must be because the cost of stress-tests exceed the beefits. I that case, forcig them to disclose will ot be desirable. Sice our model is somewhat ivolved, a overview may be helpful. At the heart of our model is a set of baks arraged i a etwork that reflects the fiacial obligatios across baks. Some of these baks are hit with shocks that prevet them from payig their 2

4 obligatios to other baks i full. Cosequetly, eve baks ot hit by shocks are vulerable to losses. All baks, icludig those hit by a shock, have access to profitable projects that require them to raise outside fuds. However, because of a agecy problem that is preset at each bak, outside ivestors will oly wat to ivest i baks that have eough equity. Baks that wat to raise fuds from outsiders ca disclose at some cost whether they were hit by a shock. This disclosure must be made before a bak kows which otherbakswere hit with shocks, ad thus before it kows its ow equity value. Outside ivestors see all the iformatio that is disclosed ad the decide which baks to ivest i ad uder what terms. If eough baks choose ot to disclose their state, ivestors will be ucertai as to which baks were hit by shocks. Fially, baks lear their equity ad decide what to do with ay fuds they raised. I particular, baks that lear their equity has bee wiped out will take actios that yield them private beefits at the expese of their ivestors. This framework allows us ot oly to draw the coectio betwee cotagio ad the desirability of madatory disclosure, but also to show which features of the uderlyig fiacial etwork are more likely to give rise to cotagio ad market freezes, e.g. the degree of leverage baks have agaist other baks i the etwork, the magitude of losses, ad the umber of baks hit by shocks, both relative to the umber of baks ad i absolute level. I additio, our approach leads us to derive expressios for cotagio probabilities for a particular etwork whe there are multiple bad baks, a result that may be of iterest for researchers workig o cotagio idepedetly of our results regardig disclosure. The paper is structured as follows. Sectio 2 reviews the related literature. Sectio 3 develops the model of cotagio we use i our aalysis. I Sectio 4 we modify our model so that baks ca raise additioal fuds, ad we itroduce a agecy problem that makes ivestors leery of ivestig i baks with little equity. I Sectio 5, we itroduce a disclosure decisio. We the examie whether o-disclosure ca be a equilibrium outcome, ad if so whether madatory disclosure ca be welfare improvig relative to that equilibrium. Sectio 6 cosiders more geeral etwork structures. Sectio 7 cocludes. 2 Literature Review Our paper is related to several differet literatures, specifically work o i) fiacial cotagio ad etworks, ii) disclosure, iii) market freezes, ad iv) stress tests. Turig first to the literature o cotagio, various chaels for cotagio have bee described i the literature. For a survey, see Alle ad Babus (2009). We focus o models of cotagio based o balace sheet effects i which a bak hit by a shock is uable to pay its obligatios, makig it difficult for other baks to meet their obligatios. Examples of 3

5 papers that explore this chael iclude Kiyotaki ad Moore (1997), Alle ad Gale (2000), Eiseberg ad Noe (2001), Gai ad Kapadia (2010), Caballero ad Simsek (2012), Acemoglu, Ozdaglar, ad Tahbaz-Salehi (2013), ad Elliott, Golub, ad Jackso (2013). These papers are largely cocered with how the patter of obligatios across baks affects the extet of cotagio, ad whether certai etwork structures ca reduce the extet of cotagio. Our focus is quite differet: Rather tha explorig which policies might mitigate the extet of cotagio, we examie whether policies ca be used to mitigate the fallout due to cotagio oce it occurs, e.g. restartig trade i markets that would otherwise remai froze. Sice our model posits that baks coected via a etwork ca commuicate iformatio about themselves, we should poit out that there is some work o commuicatio ad etworks, e.g. DeMarzo, Vayaos, ad Zwiebel (2003), Calvó-Armegol ad de Martí (2007), ad Galeotti, Ghiglio, ad Squitai (2013). However, these papers study eviromets i which agets commuicate to others o the etwork. By cotrast, we study a eviromet where agets commuicate iformatio about the etwork, specifically the locatio of bad odes i the etwork, to outsiders. The other major literature our work relates to cocers research o disclosure. Two good surveys of this literature iclude Verrecchia (2001) adbeyer et al. (2010). A key result i this literature, first established by Milgrom (1981) adgrossma (1981), is a uravellig priciple which holds that all private iformatio will be disclosed because agets with favorable iformatio will wat to avoid beig pooled together with iferior types ad receive worse terms of trade. Beyer et al. (2010) summarize the various coditios subsequet research has established that are ecessary for this uravellig result to hold: (1) disclosure must be costless; (2) outsiders kow the firm has private iformatio; (3) all outsiders iterpret disclosure i the same way, i.e. outsiders have o private iformatio (4) iformatio ca be credibly disclosed, i.e. the iformatio disclosed is verifiable; ad (5) agets caot commit to a disclosure policy ex-ate before observig the relevat iformatio. Violatig ay oe of these coditios ca result i equilibria where ot all relevat iformatio is coveyed. We show that o-disclosure ca be a equilibrium outcome i our model eve whe all of these coditios are satisfied. We thus highlight a distict reaso for the failure of the uravellig priciple that is due to iformatioal spillovers: I order to kow whether a bak i our model is safe to ivest i, outside ivestors eed to kow ot just the bak s ow balace sheet, but also the balace sheets of other baks. Ours is certaily ot the first paper to explore disclosure i the presece of iformatioal spillovers. A particularly importat predecessor is Admati ad Pfleiderer (2000). Like i our model, their setup allows for iformatioal spillovers ad gives rise to o-disclosure equilibria. However, these equilibria rely crucially o disclosure beig costly; whe the cost 4

6 of disclosure is zero i their model, iformatio will be disclosed. The reaso our framework allows for o-disclosure eve whe disclosure is costless is because it allows for iformatioal complemetarities that are ot preset i their model. I particular, disclosure by a bak i our model is ot eough to establish whether that bak has positive equity, sice this requires iformatio about other baks i the etwork. This feature, which has o aalog i their model, is why o-disclosure equilibria ca arise i our framework despite satisfyig all of the coditios listed above. However, Admati ad Pfleiderer (2000) are similar to us i showig that iformatioal spillovers ca make madatory disclosure welfare-improvig. 1 Aother differece betwee our model ad theirs is that i their model agets commit to disclosig iformatio before they lear it, while i our model baks kow their losses before they choose whether to disclose it. I additio, our setup allow us to study the role of cotagio for disclosure, somethig that caot be deduced from their setup. Our paper is also related to the literature o market freezes. As i our model, this literature has emphasized the importace of iformatioal frictios. Some of these papers emphasize the role of private iformatio, where agets are reluctat to trade with others for fear of beig exploited by others who are more iformed tha them. Examples iclude Rocheteau (2011), Guerrieri, Shimer, ad Wright (2010), Guerrieri ad Shimer (2012), Camargo ad Lester (2011), ad Kurlat (2013). Other papers have focused o ucertaity cocerig each aget s ow eed for liquidity ad the liquidity eeds of others which discourages trade. Examples iclude Caballero ad Krishamurthy (2008) ad Gale ad Yorulmazer (2013). Oe differece betwee our framework ad these papers cocers the source of iformatioal frictios. Sice i our framework the ucertaity cocers iformatio that ca i priciple be verified such as the bak s balace sheet, it aturally focuses attetio o the possibility that the iformatio agets are ucertai about will be revealed. By cotrast, previous papers have focused o private iformatio o idividual assets that may be more difficult to verify, or iformatio that o agets are privy to ad thus caot be disclosed. Fially, there is a emergig literature o stress tests. O the empirical frot, Peristia, Morga, ad Savio (2010), Bischof ad Daske (2012), Ellahie (2012), ad Greelaw et al. (2012) have looked at how the release of stress-test results i the US adeuropeaffectedbak stock prices. These results are complemetary to our aalysis, which is more cocered with ormative questios regardig the desirability of releasig stress-test results. There are also several recet theoretical papers o stress tests, e.g. Goldstei ad Sapra (2013), Goldstei ad Leiter (2013), Shapiro ad Skeie (2012), Spargoli (2012), ad Bouvard, Chaigeau, ad de Motta (2013). I these papers, baks are ot allowed to disclose iformatio o their ow. 1 Foster (1980) adeasterbrook ad Fischel (1984) also argue that spillovers may justify madatory disclosure, although these papers do ot develop formal models to study this. 5

7 Thus, these papers sidestep the mai questio we are after, amely whether it is possible for baks to choose ot to disclose eve whe forcig all baks to disclose is desirable. 3 A Model of Cotagio We begi with a bare-boes versio of our model where baks make o decisios. This allows us to highlight how cotagio works i our model ad to motivate our measure of cotagio. Our approach to modellig cotagio follows Alle ad Gale (2000), Eiseberg ad Noe (2001), Gai ad Kapadia (2010), Caballero ad Simsek (2012), ad Acemoglu, Ozdaglar, ad Tahbaz-Salehi (2013) i focusig o the role of balace sheet effects. Formally, there are baks idexed by j {0,..., 1}. Each bak i is edowed with a set of fiacial obligatios Λ ij 0toeachbakj i. Followig Eiseberg ad Noe (2001), we take these obligatios as give without modellig where they come from. For much of our aalysis, we follow Caballero ad Simsek (2012) i restrictig attetio to the special case i which Λ ij = { λ if j =(i +1) (mod) 0 else This case is kow as a rig etwork or circular etwork, sice these obligatios ca be depicted graphically as if the baks are located alog a circle as show i Figure 1, with each bak owig λ uits of resources to the bak that sits clockwise from it. I Sectio 6, we show that our aalysis ca be exteded to a larger class of etworks. However, sice the circular etwork is expositioally coveiet, we prefer to focus o this etwork iitially. I additio to the obligatios Λ ij, each bak is edowed with some assets that ca be liquidated if eeded. We do ot explicitly model the value of these assets, ad simply set their value fixed at some value π>0. A fixed positive umber of baks b are hit with egative et worth shocks, where 1 b 1. We refer to these as bad baks. We thus geeralize Caballero ad Simsek (2012), who oly cosider the case of b = 1. Each bad bak icurs a loss φ, whereφ represets a claim o the bak by a outside sector, i.e. by a etity that is ot ay of the remaiig baks i the etwork. The obligatio φ is seior to the obligatios to other baks i the etwork. That is, all of a bak s available resources must first be used to pay its seior claimat, ad oly the ca bak j make paymets to bak j + 1 from ay remaiig fuds. For example, φ could represet a margi call agaist the bak followig a drop i the value of some asset the bak used as collateral. We shall refer to all remaiig baks as good. Let S j = 1 if j is a bad bak ad 0 otherwise. The vector S =(S 0,..., S 1 )deotesthe state of the bakig etwork. By costructio, 1 j=0 S j = b. Shocks are equally like to hit 6 (1)

8 ay bak, i.e. each of the ( b) possible locatios of the bad baks withi the etwork are equally likely. I particular, Pr (S j =1)= b for ay bak j. We ow aalyze the fiacial positio of baks give our seiority rules. Baks ca be either isolvet meaig they are uable to fully repay their obligatio λ to aother bak or solvet ad able to fully repay, although they may have to liquidate some of their edowmet to do so. The mai feature we wish to highlight is that eve good baks may be forced to liquidate their assets or wid up isolvet because of their exposure to bad baks. Let x j deote the paymet bak j makes to bak j +1, ad y j deote the paymet bak j makes to the outside sector. Bak j has x j 1 + π resources it ca draw o to meet its obligatios. Give our restrictios o the seiority, it must first pay the outside sector. Let Φ j φs j deote the obligatio to the outside sector. The the paymet y j must satisfy y j = mi {x j 1 + π, Φ j } (2) Bak j ca the use ay remaiig resources to pay bak j + 1, to which it owes λ. Hece, the paymet bak j makes to bak j + 1 is give by x j = mi {x j 1 + π y j,λ} (3) Substitutig i for y j yields a system of equatios ivolvig oly the paymets betwee baks, {x j } 1 j=0, that characterizes these paymets: x j =max{0, mi (x j 1 + π Φ j,λ)}, j =0,..., 1 (4) (4) ivolves equatios ad ukows. Give a solutio {x j } 1 j=0, we ca defie the implied equity of bak j as the value of ay residual resources after a bak settles all paymets, i.e. e j =max{0,π Φ j + x j 1 x j } (5) Although e j is redudat give the paymets x j, equity will tur out to be importat later o whe we expad the model. While x j ad e j both deped o the state of the etwork S, i.e. x j = x j (S) ade j = e j (S), we shall omit the explicit referece to S whe this depedece does ot play a essetial role. Our first result is to establish that (4) has a geerically uique solutio { x j Propositio 1: } 1 j=0.2 For a give S, thesystem(4) has a uique solutio { x j } 1 j=0 if φ b π. 2 Our result is a special case of Theorem 2 i Eiseberg ad Noe (2001) ad Propositio 1 i Acemoglu, Ozdaglar, ad Tahbaz-Salehi (2013). The latter establishes uiqueess for a geeric etwork Λ ij but does ot provide exact coditios for o-uiqueess as we do for the particular etwork we aalyze. 7

9 I the kife-edge case where total losses across bad baks, bφ, are equal to the aggregate value of the asset edowmets of baks, π, there ca be multiple solutios if λ is sufficietly large. However, these solutios are equivalet to oe aother i the sese that across all such solutios, the outside sector is paid i full, so y j =Φ j for all j, ad the equity values {e j } 1 j=0 of all baks are the same, so e j = 0 for all j. The oly differece across solutios are the otioal amouts baks default o to other baks. 3 I what follows, we will iitially restrict attetio to the case of φ< π, so total losses b icurred by bad baks bφ caot be so large that they exceed the total resources of the bakig system, π. Although Acemoglu, Ozdaglar, ad Tahbaz-Salehi (2013) show that explicitly allowig for large losses ca yield importat isights o the ature of cotagio, allowig for large shocks yields few isights for our purposes. I particular, whe φ> b π, there are two possible outcomes depedig o the value of λ. Wheλ is small, the distributio of equity values {e j } 1 j=0 is idepedet of φ, ad so the implicatios of this case ca be uderstood eve if we restrict φ< π. Whe λ is large, oe of the baks have equity b whe φ> π. Sice we are iterested i decisios whe baks are usure about their equity, b the case where baks kow their equity to be zero is of little iterest. At the same time, we do t wat the loss per bak φ to be too small, sice as the ext propositio shows, φ π implies bad baks are solvet ad so there is o cotagio. Propositio 2: Ifφ π, thex j = λ for all j ad e j = π for ay j for which S j =0. The above isights suggest the followig restrictio o φ: Assumptio A1: Lossesatbadbaksφ satisfy π<φ< b π. Whe φ>π, bad baks will be isolvet: Eve if these baks receive the full amout λ from the bak that is idebted to them, they will have less tha λ resources to pay their obligatios. The equity of each bad bak must therefore be 0 uder Assumptio A1. To uderstad the ature of cotagio i this ecoomy, it will help to begi with the case of oe bad bak, i.e. b = 1, as i Caballero ad Simsek (2012). Without loss of geerality, let bak j = 0 be the bad bak. Give that bak 0 receives x 1 from bak 1, the total amout of resources bak 0 ca give to bak 1 is mi {x 1 + π φ, 0}. We show i Propositio 3 below that uder Assumptio A1, there is at least oe bak that is solvet ad ca pay its obligatio λ i full. From this, it follows that bak 1 must be solvet, sice if ay bak j {1,..., 2} were solvet, it would pay bak j + 1 i full, who i tur will pay bak j + 2 i full, ad so o, util we reach bak 1. Hece, x 1 = λ. Give that x 1 = λ, derivig the equity of the remaiig baks is straightforward. Sice 3 Eiseberg ad Noe (2001) also show i their Theorem 1 that {e j } 1 j=0 is uique eve if {x j} 1 j=0 is ot. 8

10 φ>π, the bad bak will fall short o its obligatio to bak 1 by the amout 0 = mi {φ π, λ}. Sice bak 1 is edowed with π>0 resources, it ca use them to make up some of the shortfall it iherits whe it pays bak 2. If the shortfall 0 >π, bak 1 will also be isolvet, although its shortfall will be π less tha shortfall it receives. The first bak that iherits a shortfall that is less tha or equal to π will be solvet, with a equity positio that is at least 0 but strictly less tha π. Hece, we ca classify baks ito three groups: (1) Isolvet baks with zero equity, which icludes both the bad bak ad possibly several good baks; (2) Solvet baks whose equity is 0 e j <π.wheb = 1, there will be exactly oe such bak; ad (3) Solvet baks that are sufficietly far from the bad bak ad have equity equal to π. Sice equity will figure promietly i our aalysis below, it will be coveiet to work with the case where e j ca take o oly two values, 0 or π. For b = 1, this requires that 0 = mi {φ π, λ} be a iteger multiple of π. For geeral values of b, we will eed to impose that both φ ad λ are iteger multiples of π. Formally, we have Assumptio A2: φ ad λ are both iteger multiples of π. For b = 1, Assumptio A2 implies that the oe solvet bak with equity less tha π has exactly zero equity. The umber of good baks with zero equity whe b = 1 is thus k = { 0 φ π = mi π 1, λ } π Caballero ad Simsek (2012) refertok as the size of the domio effect of a bad bak o good baks. With more tha oe bad bak, k still captures the potetial for cotagio of ay give bad bak, i that at most bk good baks ca have their equity wiped out. But for reasos that will become apparet whe we tur to the case where b>1, the actual umber of good baks whose equity falls because of their exposure to bad baks ca fall below this amout. As such, we will eed to itroduce a differet metric to measure cotagio. This metric will deped o k as well as o other parameters. Two coditios are required for k i (6) to be large. First, a large k requires losses φ to be large. This is because whe φ is small, a bad bak will still be able to pay back a large share of its obligatio λ, ad so fewer baks will ultimately be affected by the loss. Secod, a large k requires the obligatio λ be large. Ituitively, whe λ is small, baks are ot very idebted to oe aother, ad i the limit as λ 0, there will be o cotagio to good baks regardless of how large losses φ at bad baks are. As λ rises, what matters is ot so much that the bad bak s obligatio to oe of the other good baks grows, but that (6) 9

11 more of the resources of the bakig system ed up i the hads of the bad bak, where they are diverted to seior claimats. This starves the bakig system of equity, leavig fewer resources for baks located dowstream from the bad bak. The higher is λ, the fewer resources that remai for baks, ad hece the larger the umber of good baks that fall victim to cotagio. Ideed, we show i Propositio 4 below that for sufficietly large λ, the outside sector will icur o losses, ad all losses will be bore by baks withi the etwork. 4 Armed with this ituitio, we ca ow move to the geeral case of a arbitrary umber of baks, i.e. 1 b 1. We begi with a prelimiary result that uder Assumptio A1, at least oe bak will be solvet ad ca pay its obligatio i full. Propositio 3: If φ< b π, there exists at least oe solvet bak j for which x j = λ, ad amog solvet baks there exists at least oe bak j with positive equity, i.e. e j > 0. As i the case with b = 1, there will be three types of baks whe b>1: (1) Isolvet baks with zero equity; (2) Solvet baks whose equity is 0 e j <π; ad (3) Solvet baks that are sufficietly far away from a bad bak whose equity e j = π. Sice we kow there is at least oe solvet bak j, we ca start with this bak ad move to bak j +1. Ifbak j + 1 is good, it too will be solvet ad its equity will be e j+1 = π. We ca cotiue this way util we evetually reach a bad bak. Without loss of geerality, we refer to this bad bak as bak 0. By the same argumet as i the case where b = 1, Assumptio A2 implies that baks 1,..., k will have zero equity, where k is give by (6): Eve if all of these baks are good, each will iherit a shortfall of at least π ad will have to sell off its π assets. If ay of these baks are bad themselves, the shortfall subsequet baks will iherit will be eve larger, ad so equity at the first k baks will be zero. If bad baks are sufficietly spread out across the etwork, i.e. if there are at least k baks betwee ay two bad baks, the umber of good baks with zero equity would equal bk, this i additio to the b bad baks whose equity is also wiped out. But more geerally, a bak that is exposed to a bad bak may be bad itself. I this case, the umberofgoodbaks with zero equity may fall below bk, depedig o the size of the obligatios across baks λ. We ow show that whe λ is large, the locatio of the bad baks withi the etwork will ot matter, ad exactly bk good baks will have zero equity regardless of whether bad baks are spaced out or ot. But whe λ is small, the umber of good baks with zero equity ca be smaller tha bk ad will deped o how close bad baks are located to oe aother. We begi by showig that for sufficietly large λ, all baks will be able to make some 4 Per Elliott, Golub, ad Jackso (2013), icreasig λ i our setup implies more itegratio but ot more diversificatio. However, ulike i their model where greater itegratio meas firms swap their ow equity for that of other firms, here greater itegratio implies greater exposure to shocks at other baks while leavig baks equally vulerable to their ow shocks. Hece, the effect of higher λ is (weakly) mootoe. 10

12 paymet to the bak they are obligated to regardless of where the bad baks are located, i.e. regardless of the state of the bakig etwork S =(S 0,..., S 1 ). Propositio 4: Uder Assumptio A1, x j (S) > 0 for all j ad all S iff λ>b(φ π). Whe λ b (φ π), there exist realizatios of S for which x j (S) = 0 for at least oe j. If each bak j ca pay some positive amout to bak j +1, the each bak j must pay the outside sector i full give whose claims are seior to all other claims. Hece, Propositio 4 implies that for large λ, seior claimats will be fully paid i all states. This i tur implies that the the total amout of resources left withi the bakig etwork is the same regardless of where bad baks are located. Sice Assumptio A2 implies baks ca have equity equal to either 0 or π ad total equity is the same for all S, it follows that the umber of baks with zero equity is the same for all S wheever λ>b(φ π). Formally: Propositio 5: Uder Assumptios A1 ad A2, if λ>b(φ π), the umber of good baks with zero equity is equal to bk regardless of the state of the bakig etwork S. Next, cosider the case where λ is small. With fewer resources flowig to bad baks, some bad baks may default o seior claimats. But the more baks default o their obligatios to the outside sector, the larger the umber of baks who ca maitai their equity. For sufficietly low values of λ, specifically whe λ<φ π, we ca explicitly characterize the distributio of the umber of baks with zero equity. At such low values of λ, badbaks would default o seior claimats, ad thus default i full o their obligatio λ to the ext bak. This implies that regardless of where the bad baks are located, the cotagio from each bad bak is limited to wipig out the equity of the k = λ baks that come after it. π Deote the total umber of baks with zero equity, icludig the b bad baks, by ζ. The umber of baks with zero equity ζ is ow a radom variable, with a support that rages from b + k, whe all bad baks are located ext to each other, to bk + b whe there are at least k good baks betwee ay two bad baks. By cotrast, for λ>b(φ π), the umber of baks with zero equity ζ has a degeerate distributio with all of its mass at bk + b. To obtai a exact distributio for ζ for λ<φ π, we exploit the fact that for λ<φ π, our model correspods to a discrete versio of a well-studied geometric problem i applied probability kow as the circle-coverig problem first itroduced by Steves (1939). I this problem, a give umber of poits are draw at radom locatios alog a circle of legth 1, ad the arcs of a give legth less tha 1 are draw startig from each of these poits ad proceedig clockwise. The oly radomess is the locatio of the arcs. The circle-coverig problem ivolves determiig the probability that the circle is covered by the arcs ad the distributio of the legth of the regio that is ot covered. I our settig, the umber of bad baks is aalogous to the umber of poits draw at radom, while the potetial for 11

13 cotagio k, expressed relative to the total umber of baks i the etwork, correspods to the legth of each arc. The regio of the circle covered by arcs is aalogous to the fractio of baks with zero equity. The discrete versio of this circle-coverig problem has bee aalyzed i Holst (1985), Ivcheko (1994), ad Barlevy ad Nagaraja (2013). As Holst (1985) otes, the discrete versio ca be aalyzed usig Bose-Eistei statistics. This isight ca be used to obtai a exact expressio for the distributio of ζ. However, for our purposes oly the expected value of E [ζ] matters, which ca be obtaied usig results i Ivcheko (1994) ad Barlevy ad Nagaraja (2013). This expectatio is summarized i the ext lemma. Lemma 1: Uder Assumptios A1 ad A2, the expected umber of good ad bad baks with zero equity, ζ, is give by E [ζ] = ( b)! ( k 1)! ( 1)! ( b k 1)! where k is defied by (6) ad is equal to λ π give λ<φ π. Fially, for itermediate values of λ betwee φ π ad b (φ π), the umber of baks with zero equity ζ will agai be radom, with support ragig betwee b + λ >b+ k ad π = bk + b, wherek is defied i (6). For these itermediate values of λ, the distributio of b φ π baks with zero equity is aalogous to a circle coverig problem i which the legth of the arcs is ot fixed but rather depeds o the locatio of the poits draw at radom. As far as we kow, this variatio of the circle-coverig problem case has yet to be studied. However, i Propositio 6 below we establish some comparative static results for E [ζ] for this case. To recap, whe b>1, how may good baks will ed up with zero equity ca be radom. To summarize the extet of cotagio i this case, cosider what happes if we chose a good bak at radom. The extet to which good baks are exposed to losses at bad baks will be reflected i the distributio of the equity of this good bak, i.e. how likely it will be to have to liquidate its edowmet ad ed up with a equity below π. The smaller the probability that the equity value is equal to π, the more good baks that ted to have equity below π, ad thus the greater the extet of cotagio. Formally, defie p g as the probability that a good bak retais all of its equity, i.e., p g =Pr(e j = π S j =0) (7) We will use p g as our measure of cotagio: A value of p g close to 1 implies a good bak is highly likely to avoid liquidatig its resources, so losses at bad baks have a small effect o good baks, while a value of p g close to 0 meas a good bak will be very likely to be wiped out because of direct or idirect exposure to bad baks. As we discuss below, for more 12

14 geeral etworks e j will take o more tha just two values, ad we will eed to track the distributio of e j. For ow, usig the defiitio of k i (6), we ca compute p g as follows: p g = = bk+b z=b+k bk+b z=b+k Pr (e j = π S j =0,ζ = z)pr(ζ = z) z E [ζ] Pr (ζ = z) = b b. Ituitively, the expected umber of baks with positive equity is E [ζ]. Sice oly good baks ca have positive equity, ad there are always b good baks, the fractio of good baks with equity equal to π is just the ratio of the two. The ext propositio summarizes how p g varies i our model depedig o the uderlyig parameters: Propositio 6. Uder Assumptios A1 ad A2, λ/π ( b i ) i=1 if 0 <λ<φ π i p g = Ψ ( b,, φ, ) λ if φ π λ b (φ π) π π ( 1 b φ 1) if b (φ π) <λ b π (8) where the fuctio Ψ is weakly decreasig i φ/π ad i λ/π. Propositio 6 reveals that p g depeds o the magitude of the losses at bad baks φ, the depth of fiacial ties λ, the umber of bad baks b, ad the total umber of baks. Oe feature we poit out ow ad revisit below is that the effect of bak losses φ o p g depeds o λ. For small λ, specifically for λ<φ π, chages i φ have o effect o p g. This is because icreasig φ oly affects seior claimats but ot other baks i the etwork. For large λ, icreasig φ lowers p g. That is, whe baks are more strogly itegrated, a shock that results i bigger losses at bad baks will wipe out the equity of a larger umber of good baks. Essetially, high values of λ allow losses at bad baks to affect more good baks. For much of our aalysis we ca treat p g as fixed, although we will occasioally retur to the comparative statics of what drives p g. For b =1,p g reduces to k 1 ad reflects both the probability a good bak has zero equity ad the fractio of good baks with zero equity. For b>1, the fractio of good baks with zero equity may be a radom variable, so p g reflects the probability a good bak has zero equity ad the average fractio of good baks with zero equity. Remark 1: For some applicatios, it would be more coveiet to have the fractio of baks with zero equity also determiistic. Oe way to achieve this for geeral b is to icrease the umber of baks ad exploit the law of large umbers. I particular, suppose we hold 13

15 the potetial for cotagio k i (6) fixed ad keep the fractio of bad baks b costat at some value θ, but let.letζ deote the (radom) umber of baks with zero equity whe there are baks i the etwork. Whe λ<φ π, we ca appeal to Theorem 4.2 i Holst (1985) to establish that ζ coverges to a costat as. Likewise, the fractio of good baks with zero equity, ζ, also coverges to a costat. This costat will equal p b g, which recall is just the expected fractio of good baks with zero equity. Takig the limit of (8) as for the case where λ<φ π reveals that p g coverges to a simple expressio: lim p g =(1 θ) k (9) Ituitively, a good bak will oly have positive equity if each of the k baks located clockwise from him are good. As, the probability that ay oe bak is bad coverges to θ idepedetly of what happes to ay fiite collectio of baks aroud it. Hece, the probability that all of the relevat k eighbor baks are good is (1 θ) k. While the locatio of baks with zero equity remais radom whe the size of the etwork becomes large, the fractio of good baks with positive equity ζ will exhibit o radomess i the limit. b For ay give θ, the limitig value of p g ca rage betwee 0 ad 1 as k varies from 0 to arbitrarily large iteger values. Note that sice k = mi { λ, φ 1}, values of k that exceed π π 1 1 will violate the secod iequality i Assumptio A1, which requires that φ be less θ π tha = 1. However, this restrictio ca essetially be dispesed with for large values of, b θ sice the probability that equity is wiped out at all baks becomes exceedigly small eve without this assumptio. The limitig case as is thus useful ot oly for elimiatig ucertaity regardig the extet of cotagio, but also for demostratig that the cotagio measure p g i a circular etwork ca assume the full rage of possible values, from early o cotagio (p g 1) to early full cotagio (p g 0). Fially, i some of our subsequet aalysis we will eed the ucoditioal probability that a give bak chose at radom has positive equity. Deote this probability by p 0. Sice there are exactly b bad baks ad b good baks, ad sice all bad baks have zero equity uder Assumptio A1, p 0 ca be expressed directly i terms of p g : p 0 = b p g + b ( 0= 1 b ) p g (10) 4 Outside Ivestors ad Bak Equity We ow build o the model of cotagio from the previous sectio by allowig baks to raise exteral fuds i order to fiace productive opportuities. Although all baks ca use the 14

16 fuds they raise profitably regardless of their equity positio, we itroduce a moral hazard problem that implies oly baks with eough equity will use the fuds as iteded. Specifically, we allow baks to divert the fuds they raise to achieve private gais, a temptatio that is mitigated by the equity a bak would give up i that case. More geerally, there are various actios baks ca udertake whe their equity is low that would be agaist the iterests of outside ivestors, e.g. ivestig i risky projects or gamblig for resurrectio. I this sectio, we focus o the full-iformatio bechmark i which baks ad outside ivestors kow which baks are bad ad thus the equity of each bak. I this case, allowig baks to raise fuds has o impact o cotagio. I particular, sice outside ivestors are oly willig to fiace baks with eough equity, baks that would have had zero equity i the origial model will ot be able to raise ew fuds. Lettig baks raise fuds merely accetuates the iequality betwee baks with zero ad positive equity. While this leads to o ew isights regardig cotagio, it does itroduce a reaso for why bak equity ca matter for the allocatio of resources: Bak equity facilitates gais from trade that would ot occur i its absece. Whe we allow baks to withhold iformatio about whether they were hit by shocks or ot, as we do i the ext sectio, policy ca potetially affect what agets believe about the equity at ay give bak ad thus whether trade takes place. Formally, suppose that outside ivestors which ca be the same origial outsiders that have seior claims agaist baks or a ew group of outsiders ca choose whether to ivest with ay of the baks i the etwork. Baks have profitable projects they ca udertake, but fudig these projects requires outside fiacig. For simplicity, we assume that each bak has a fiite umber of profitable projects it ca udertake. We set the capacity of the bak to 1 uit of resources. O their ow, outside ivestors ca ear a gross retur of r per uit of resources. Baks ca ear a gross retur of R o the projects they udertake, where R>r. Thus, there is scope for gais from trade. We restrict baks ad outside ivestors to trasact through debt cotract that are juior to all of the bak s other obligatios. Allowig for equity cotracts would ot resolve the moral hazard problem we itroduce below, ad so we ivoke this assumptio for coveiece oly. Let rj deote the equilibrium gross iterest rate bak j offers outside ivestors for ay fuds they ivest i the bak. We assume that the outside sector is large eough that rj is set competitively, i.e. the expected gross returs from ivestig i abakequalr. Hece, r j r, ad the most a bak ca ear from raisig fuds is R r. After baks raise fuds from outsiders, they ca choose to either ivest the fuds they raised ad ear a retur R, or divert the fuds to a project that accrues a purely private beefit v per uit of resources. These private beefits caot be seized by outsiders. Outside ivestors caot moitor baks ad prevet them from divertig fuds. However, if the bak 15

17 fails to pay the required obligatio rj,theycagoafterayassetsthebakows. We wat v to be large eough to esure that baks with zero equity would choose to divert so the moral hazard problem is bidig but ot so large that eve a bak that keeps its π worth of assets will be tempted to divert fuds. To satisfy the first coditio, we eed v>r r, i.e. the private beefit v exceeds the most a bak ca ear from udertakig the project. To esure that a good bak will ot be tempted, we eed to make sure that the payoff after udertakig the project, π + R rj, exceeds the payoff from divertig fuds, v +max { π rj, 0}, i.e. the bak would ear v i private beefits but would have to liquidate at least some of its assets to meet the promised obligatio of rj. Comparig the two expressios implies we eed v<r max { rj π, 0 }. Sice a bak that ca be etrusted ot to divert fuds eed ot offer more tha r to outsiders, the coditio that esures baks with assets worth π ca credibly promise to ivest the fuds they raise is if v<r max {r π, 0}. The coditios o v we eed ca be summarized as follows: Assumptio A3: The private beefits v to a bak from divertig 1 uit of resources it raises from outsiders are either too high or too low, specifically R r <v<r max {r π, 0} (11) Note that the secod iequality i (11) implies v<r, so diversio is socially wasteful. I the full iformatio bechmark, baks kow the state S, i.e. they kow the locatio of the bad baks. I Sectio 3, we showed that whe baks had o optio to raise ad ivest fuds, there were ζ baks with zero equity ad ζ with equity π. Weowshowthatwhe baks ca raise fuds, the ζ baks that origially had o equity will ot be able to raise fuds ad will thus remai with zero equity, while the remaiig ζ baks would be able to raise fuds ad raise their equity to π + R r. Allowig baks to raise fuds uder full iformatio would ot chage the patter of cotagio i our origial model. To derive this result, defie a ew variable I j [0, 1] as the amout outsiders ivest i bak j. Sice Assumptio A3 ivolves strict iequalities, baks will either divert the fuds they raise or ivest. Let D j = 1 if bak j decides to divert the fuds ad 0 otherwise. Recall that y j deotes the obligatio of bak j to its most seior creditors ad x j its paymet to bak j +1. Letw j deote its paymet to outsiders who ivest i bak j. Thewehave y j = mi {x j 1 + π + R (1 D j ) I j, Φ j } x j = mi {x j 1 + π + R (1 D j ) I j y j,λ} w j = mi { x j 1 + π + R (1 D j ) I j y j x j,rj I } j 16

18 Fially, the equity at each bak j is give by e j =max{0,x j 1 + π + R (1 D j ) I j y j x j w j } Let {ŷ j, x j } j=1 deote the paymets to seior creditors ad to baks, respectively, if outside ivestors could ot fud ay bak, i.e. if I j = 0 for all j. Likewise, defie {ê j } j=1 as the equity positios give {ŷ j, x j } j=1, i.e. ê j =max{0,π Φ j + x j 1 x j } Note that ê j correspods to the equity positios we solved for i the previous sectio. Our claim is that uder full iformatio, e j =0wheeverê j =0,ade j > 0wheeverê j > 0. Propositio 7: Give Assumptio A1-A3, with full iformatio, e j =0foraybakj for which ê j =0,ade j > 0 if ê j > 0. Moreover, I j = 0 if ad oly if ê j =0. Propositio 7 shows that eve though allowig bakrupt baks to raise fuds potetially provides these baks with a way to make up their shortfalls, uder full iformatio such baks would ot be able to raise fuds. Thus, with full iformatio, cotagio from bad baks to good baks persists as before. The role of allowig firms to raise fuds will tur more iterestig whe we allow for icomplete iformatio, i.e. whe outsiders are usure which baks are bad. This is the case we tur to i the ext sectio, where we allow firms to choose whether to disclose their fiacial positio. However, eve uder full iformatio, allowig baks to raise fuds itroduces oe ovelty. I particular, we ca ow assig a social cost to cotagio, eve though policymakers ca do othig to prevet it i our model: Whe bak balace sheets are liked, shocks drai more equity away from the bakig system ad redirect it to seior creditors, reducig the scope for trade. Abset this reductio i trade, cotagio merely redistributes resources betwee bakers ad seior claimats. 5 Disclosure We ow itroduce the last compoet ito our model allowig baks to decide whether to disclose their fiacial positio before raisig fuds. If eough baks decide ot to disclose, outsiders must decide whether to ivest i baks ot kowig exactly where all of the bad baks are located. This allows us to explore the mai questios we are after: Uder what coditios will market participats be usure about which baks icurred losses, ad i those cases would it be advisable to compel baks to reveal their fiacial positio? This sectio is orgaized as follows. After we describe how we model disclosure, we 17

19 provide coditios uder which there exists a o-disclosure equilibrium where o bak discloses its S j. We the examie whether madatory disclosure ca improve welfare relative to this equilibrium. While we give a rigorous aswer to this questio, our essetial isight is captured i Theorem 1, which shows that madatory disclosure caot improve welfare whe cotagio is small but ca improve welfare whe cotagio is large ad disclosure costs are ot too large. Fially, we examie whether other equilibria besides o-disclosure are possible. While we provide coditios uder which multiple equilibria exist, we argue that our mai result poits to a geeral tedecy for isufficiet disclosure i the presece of cotagio rather tha to the eed to help coordiate agets to a superior equilibrium. 5.1 Modellig Disclosure To model disclosure, suppose that after ature chooses the locatio of the b bad baks, each bak j observes S j,butots i for i j. At this poit, all baks simultaeously choose whether to icur a utility cost c 0 ad disclose their ow S j.thecostcis meat to capture the effort of coductig ad documetig the result of stress-test exercises. I priciple, c could reflect the cost of revealig iformatio about tradig strategies that rival baks ca exploit. But it is ot obvious whether we should treat these as costs a social plaer would face, so we prefer to iterpret c as the costs of ruig stress-tests. Outside ivestors observe these aoucemets ad the decide what terms to offer baks (if ay). After outsiders choose whether to ivest i baks, the state of the etwork S is revealed, ad baks lear their ow equity. Oly the do baks decide whether to ivest the fuds they raised or divert them. Fially, profits are realized ad obligatios are settled. Note that a bad bak with S j = 1 will ever fid it beeficial to disclose if c>0. As such, we ca describe each bak s decisio by a j {0, 1}, wherea j =1measbakj discloses it is good ad a j = 0 meas it does ot disclose ay iformatio. Outside ivestors thus observe the vector a =(a 1,..., a ) ad choose whether to provide fuds to ay of the baks. For simplicity, we force outsiders to oly offer debt cotracts, so the terms offered to baks ca be summarized as a amout of resources each bak j receives, Ij (a), ad a iterest rate rj (a) bakj must repay its ivestors. As will become clear, allowig for equity cotracts would ot be of much help give the problem is that baks already have too little equity. 5.2 Existece of a No-Disclosure Equilibrium Our first questio is uder what coditios o-disclosure ca be a equilibrium, i.e. each bak is willig to set a j = 0 if it expects a i =0fori j. This case is of iterest sice it implies outsiders must be ucertai as to the locatio of bad baks, i lie with our 18

20 discussio i the Itroductio. As our equilibrium cocept, we use the otio of sequetial equilibria itroduced by Kreps ad Wilso (1982), which requires that off-equilibrium beliefs correspod to the limit of beliefs from a sequece i which players choose all strategies with positive probability but the weight o suboptimal actios teds to zero. This rules out arguably implausible off-equilibrium path beliefs. For example, without this restrictio, off the equilibrium path outsiders could believe all baks that do t report are bad,eve though oly b baks are bad. Likewise, without this restrictio outsiders ca form ay beliefs about the eighbors of bak j if bak j deviates from equilibrium ad chooses ot to disclose, eve though bak j kows othig about other baks whe it decides o disclosure. We ow show that the existece of a o-disclosure sequetial equilibrium depeds o two parameters the cost of disclosure c ad the degree of cotagio p g. For o-disclosure to be a equilibrium, each good bak must weakly prefer ot to disclose, i.e. set a j =0, whe it aticipates other baks will ot disclose. To solve for the optimal disclosure decisio, we eed to establish which baks if ay outsiders fud whe o bak discloses ad whe a sigle good bak discloses, sice this determies the bak s payoffs. If o bak discloses, the probability that a radom bak has positive equity is p 0 = ( 1 b ) pg as defied i (10). Uder Assumptio A3, baks that lear they have zero equity would divert fuds ad leave othig for ivestors. Assumptio A2 implies remaiig baks have equity π. Whether these baks ivest or divert depeds o how much rj they promise outside ivestors i equilibrium. The ext lemma summarizes whe baks would divert fuds: Lemma 2: Assume Assumptio A3 holds. For ay bak j where pre-ivestmet equity is π, D j = 0 is optimal if ad oly if rj (a) r π + R v. I other words, if outside ivestors charge a rate above some threshold r, baks will divert fuds regardless of their equity. I priciple, outsiders might still fud baks at a rate above r, sice they ca cout o grabbig the equity of baks with positive equity. However, it turs out that the equilibrium iterest rate charged to ay bak ever exceeds r: Lemma 3: Assume Assumptios A2 ad A3 hold. I ay equilibrium, rj (a) r for ay bak j that receives fudig, i.e. for which Ij (a) =1. Uder Assumptio A3, the maximal rate r is bigger tha the outside optio of outside ivestors r. 5 We ow argue that if p 0 is small, specifically if p 0 < r /r < 1, the outsiders will ot fiace ay bak i a o-disclosure equilibrium, i.e. I j = 0 for all j. Abset ay iformatio o S, the rate outside ivestors must charge to ear as much as from their outside optio is r p 0.FromLemma3,bakscaotchargeabover i equilibrium. Hece, the oly possible o-disclosure equilibrium whe p 0 < r /r is if Ij = 0 for all j, or else outsiders 5 Cosider the two cases r >πad r π. Ifr >π, the secod iequality i (11) implies r <R+π v r. If r π, the secod iequality i (11) implies v<r,adhecer = π + R v>π r. 19