Journal of Financial Stability

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1 Journal of Fnancal Stablty 8 (2012) Contents lsts avalable at ScVerse ScenceDrect Journal of Fnancal Stablty journal homepage: Default cascades: When does rs dversfcaton ncrease stablty? Stefano Battston a,, Domenco Dell Gatt b, Mauro Gallegat c, Bruce Greenwald d, Joseph E. Stgltz d a ETH Zurch, Swtzerland b Catholc Unv. of Mlan, Italy c Polt. Unv. of Marche, Ancona, Italy d Columba Unversty, New Yor, USA a r t c l e n f o Artcle hstory: Receved 18 November 2010 Receved n revsed form 18 January 2012 Accepted 27 January 2012 Avalable onlne 8 March 2012 JEL classfcaton: D85 G01 G21 a b s t r a c t We explore the dynamcs of default cascades n a networ of credt nterln-ages n whch each agent s at the same tme a borrower and a lender. When some counterpartes of an agent default, the loss she experences amounts to her total exposure to those counterpartes. A possble conjecture n ths context s that ndvdual rs dversfcaton across more numerous counterpartes should mae also systemc defaults less lely. We show that ths vew s not always true. In partcular, the dversfcaton of credt rs across many borrowers has ambguous effects on systemc rs n the presence of mechansms of loss amplfcatons such as n the presence of potental runs among the short-term lenders of the agents n the networ Elsever B.V. All rghts reserved. Keywords: Systemc rs Networ models Contagon Fnancal crss 1. Introducton One of the most mportant ssues that the Global Fnancal Crss (GFC) has brought to the fore concerns the effects on systemc rs of the ncreasng nterdependence both among the man actors of fnancal marets and among fnancal marets across countres. In partcular, the noton of too-bg-too-fal becomes more subtle, whle the regulatory mechansms based only on a ban s own rs may fal to mtgate aggregate rs-shftng ncentves, and can, n fact, accentuate systemc rs (Acharya, 2009). Increasng nterdependence of global fnancal marets manly acheved by means of lberalzaton of captal flows may be supposed to lead to greater worldwde fnancal stablty, as rss are spread around the world. Increasng nterdependence of economc agents, on the other hand, allows for a better dversfcaton of ndvdual rs, as rss are spread around the set of connected partners: The larger the number of borrowers a lender s connected to n a networ of borrowng/lendng relatonshp, the smaller the fracton of an dosyncratc shoc (whch leads to the default of a borrower) the lender has to bear. Ths, other thngs beng equal and assumng Correspondng author. E-mal address: sbattston@ethz.ch (S. Battston). dosyncratc shocs are not correlated,.e. they are not sprngng from the same source. It s reasonable to conjecture, therefore, that ndvdual rs dversfcaton leads to a lower systemc rs. There s at least one good and obvous reason to thn that ths s ndeed the case. Consder a networ of borrowng/lendng relatonshps. Suppose agent lends 1 unt to each node n a neghborhood consstng of borrowers. When a borrower defaults (hence the dosyncratc shoc to ), the loss the lender experences (due to the non performng loan) amounts to her relatve exposure to the borrower. The relatve loss amounts to 1/. By ncreasng the number of counterpartes so that ch tends asymptotcally to nfnty, the mpact of a negatve shoc (the relatve exposure to each borrower) tends to zero. Snce the lender ht by an dosyncratc shoc does not feel the pnch and does not react to t, there wll not be further repercussons of the shoc tself. In ths case, we can rule out domno effects and default cascades. Hence enhanced rs dversfcaton through ncreasng networ densty reduces systemc rs. The GFC has cast doubt on these conclusons. The breadown of a relatvely small segment of the US fnancal system has not only spread to the other segments an obvous consequence of nterdependence but has also pushed the system on the verge of a fnancal meltdown at the tme of the Lehman Brothers banruptcy. Moreover, ths event has trggered a fnancal crss worldwde due to captal maret ntegraton. One legtmate /$ see front matter 2012 Elsever B.V. All rghts reserved. do: /j.jfs

2 S. Battston et al. / Journal of Fnancal Stablty 8 (2012) conjecture therefore, s that ncreasng nterdependence of agents and ntegraton of fnancal marets n prncple may not reduce but ncrease the rs of a systemc collapse. Emprcal research amed at estmatng systemc rs before the GFC found very lttle evdence of global vulnerablty (Bartram et al., 2007), confrmng the vew that rs dversfcaton had been pushed so far as to reduce systemc rs to a neglgble level. A remarable body of emprcal lterature on stress-testng n fnancal systems also confrmed the vew, statng that the default of an ndvdual nsttuton was typcally not able to trgger a domno effect (see Elsnger et al., 2006; Boss et al., 2004; Furfne, 2003). The emprcal evdence accumulated durng the GFC, however, has rased legtmate doubts on the adequacy of the procedure adopted to carry out these stress-tests (Haldane, 2009; Amn et al., 2010). The unravelng of the GFC has overwhelmngly shown that systemc rs s not neglgble and domno effects are lely despte the recent mpressve ncrease of rs dversfcaton (Brunnermeer, 2008). Accordng to the theoretcal lterature, Allen and Gale (2000) s the most mportant contrbuton to the analyss of fnancal contagon through credt nterlnages among bans. They show that, gven full dversfcaton of rs at the level of the ndvdual ban, the spread of an unexpected lqudty shoc and ts systemc effects depend crucally on the pattern of nterconnectedness among bans. When the networ s complete.e. densty s at ts maxmum and the amount of nterban deposts held by each ban s evenly spread over all other bans, the mpact of the shoc s easly mtgated. When the networ s connected but ncomplete, wth bans only havng few counterpartes, the system s more fragle. When the ncomplete networ assumes the typcal structure of a wheel or a cycle, the shoc may lead to a systemc collapse. In ths case, n fact, the shoc s topplng one ban after the other along the networ cycle. In the end, therefore, gven full dversfcaton of (ndvdual) rs, a complete networ s more reslent than an ncomplete one. A recent, post-crss strand of lterature has tred to dentfy the condtons upon whch an ncrease of networ densty.e. a scenaro n whch the topology of the networ tends toward completeness s not benefcal,.e. does not reduce systemc rs (see Battston et al., 2009; Stgltz, 2010; Castglones and Navarro, 2010; Allen et al., 2010; Wagner, 2010). In the present paper we contrbute to ths new lne of research by explorng the mechansms that, followng the default of an agent, may lead to an ncrease of systemc rs when connectvty ncreases. Our approach s related to the framewor put forward by (Esenberg and Noe, 2001) n order to analyze the effects of an agent s default on the cash flows of the counterpartes. Such framewor has been further studed also n Ga and Kapada (2010a) and n Cont et al. (2010), where the default of a ban decreases the value of the assets of each counterparty n the nterban maret. In ths approach, the representaton of the agents n the credt networ s stylzed and based on accountng denttes. Behavoural assumptons are ept to a mnmum: agents nether choose ther captal structure (and thus ther level of fnancal robustness), nor the partners to be connected to. Moreover, agents do not nteract strategcally. Ths statc balance sheet approach smlar n sprt to the procedure adopted to carry out stress tests on bans may loo somehow mechancal (Chrstan and Upper, 2011) but allows to characterze analytcally the emergence of systemc rs as functon of essentally two determnants: ) the fracton of defaultng counterpartes of each agent and ) the ntal fnancal robustness of each agent (Ga and Kapada, 2010a). In our paper, we model a networ of borrowng/lendng relatonshps among fnancal nsttutons ( bans ). These nsttutons are also actve on fnancal marets,.e. they trade fnancal oblgatons wth agents outsde the networ tself. For nstance, they can collect deposts from households or get short-term loans from outsde nvestors. Each agent s represented by a stylzed balance sheet. Balance sheets are nterrelated, as the asset of one agent (lendng ban) s a lablty for another agent (borrowng ban). The ntertwned dynamcs of the ndvdual equty ratos are the drvng force of the change n the credt networ. In partcular we wll focus on changes produced by borrowers defaults, whch weaen the fnancal robustness of lenders and may therefore nduce further defaults. In ths context, therefore, from the ntal default of one or few agents may endogenously follow the default of some other agents n a full-fledged default cascade. In a nutshell, we carry out the followng exercse. We assume an ntal allocaton of assets and labltes across agents and an ntal set of defaults. We then derve a law of moton for the fnancal robustness as measured by the equty rato of the agents concerned by the default of one or more counterpartes. Fnally, we nvestgate how the sze of the default cascade s affected by the ntal dstrbuton of robustness and by the level of rs dversfcaton n the networ. The core feature of our model of the credt networ s the fact that balance sheets are nterrelated, and therefore the dynamcs of the ndvdual equty ratos are ntertwned. Ths fact s the source of the externaltes whch play a crucal role n the model. We ntroduce a dstncton between two types of externaltes whch correspond to dfferent propertes n relatons to systemc rs. Wth the frst type, the default of an agent (borrower) has an obvous and mmedate effect on the fnancal robustness of ts counterpartes (lenders) n a credt networ. The non-performng loan, n fact, translates nto a reducton of the lender s equty. However, there are no further effects of the default on the counterpartes. Whenever the maret value of total assets n the counterparty s portfolo becomes smaller than that of labltes, the counterparty n turn defaults. If some other counterpartes n turn, default on ther counterpartes a cascade of defaults may ensue. In ths baselne scenaro, whch we label as external effect of the frst type, we fnd dfferent regmes, n whch ncreasng connectvty may have a benefcal role or a detrmental one (or no role at all). When fnancal robustness s not very dfferent across agents (the degree of heterogenety and therefore the varance of equty ratos s relatvely small), ncreasng connectvty maes the system more reslent to systemc defaults. More precsely, wth ncreasng connectvty the system remans stable even at lower values of average robustness. On the other hand, ncreasng networ densty, may stmulate systemc defaults when: the ntal robustness s heterogeneous across agents (hgh varance), but the average robustness s low and there s an ntal large enough shoc. The reason why, from a systemc pont of vew, n such a stuaton t s better to concentrate rs nstead of dversfyng t s that spreadng the losses mae more agents default (snce are already fragle). We also model an external effect of the second type whch, n contrast to the frst type, nvolves an amplfcaton of losses along the chan of lendng relatons. The ambguous role of dversfcaton on systemc rs s n ths case much more pronounced. We suspect ths second mechansm to appear n several stuatons, but n ths paper we focus on one specfc case. Namely, we show how the mechansm arses f, n addton to the ngredents of the baselne model, we assume that agents borrow also short-term and are exposed to the potental run of the short-term lenders. When the agent s ht by the default of one or more of her counterpartes (for brevty, the ntal default), her short-term credtors cannot rule out that other counterpartes may default, because they do not now wth certanty the stuaton of the counterpartes. Ths means that the chances that the agent defaults have ncreased, although techncally she s stll solvent. As a result, short-term credtors have to decde whether to roll-over debt to the agent or not, tang nto

3 140 S. Battston et al. / Journal of Fnancal Stablty 8 (2012) account that the other credtors do the same reasonng. In other words, the second type of externaltes may arse when consderatons of llqudty enter nto the pcture, n the presence of mperfect nformaton. For sae of smplcty, we do not model the coordnaton game of the short-term credtors and we follow nstead a reduced form approach. We assume that credtors, due to mperfect nformaton, decde to run on an agent when () she has a low level of robustness and () the number of her defaultng counterpartes exceeds a certan threshold. In the face of a run, when the agent has to pay bac short-term debt, the frst lne of defense conssts of her lqud assets. If these are nsuffcent, however, the agent may decde to sell under dstress some of her long-term assets, n order to pay bac the remanng debt. Ths fre-sellng scenaro has been modeled n several prevous wors (e.g., Brunnermeer and Pederson, 2009). Our contrbuton here conssts n combnng the fre-sellng wth the drect loss from the default. Let us emphasze that, fre-sellng mples a further loss for the agent, n addton to the ntal loss due to the default of the counterparty. Ths addtonal repercusson can lead the agent to nsolvency, mang llqudty and nsolvency two ntertwned problems. In the case of ths externalty of second type, we fnd that: () for relatvely hgh levels of the cross-sectonal average robustness, ncreasng connectvty s always benefcal; () for low levels, ncreasng connectvty does not have any effect on systemc rs, whle () for ntermedate levels of the average fnancal robustness, ncreasng connectvty has frst a benefcal and then a detrmental effect. In ths case, the reason why dversfcaton can be detrmental s the followng. Because of the nformaton ncompleteness on the sde of the credtors, the probablty that a run occurs depends on the absolute number of defaultng counterpartes n the credt portfolo of an agent. When there are already a few defaults n the system, then a hgher level of dversfcaton mples that more of these defaultng agents can belong to the credt portfolo of the agent. If, n addton, the agent s fragle, there are more chances that the run s trggered. In summary, n our model dversfcaton s nether always good nor bad. It can have ambguous effects, and n presence of loss amplfcaton t typcally does. The precse outcome depends crucally on the allocaton of assets and labltes across agents and the structure of ther mutual exposures. In comparson wth the results on dversfcaton found n (Battston et al., 2009), t should be notced that whle, there, a detrmental effect of dversfcaton results from the dynamcs of the networ-based fnancal accelerator outsde the default cascade, here we propose an entrely alternatve mechansm whch occurs wthn the cascade. The paper s organzed as follows: Secton 2 ntroduces the model. We defne frst the structure of the agents balance sheet. Then, we descrbe the chan of events trggered by the propagaton of losses caused by counterpartes defaults. We derve the dynamcs under the two types of external effect n Sectons 2.1 and 2.2. In Secton 3, we report and dscuss the results. Secton 3.3 concludes. 2. The model We consder a set of n fnancal nsttutons ( bans ) connected n a networ of borrowng/lendng relatonshps (credt networ for short) wth each other. As a frst approxmaton, one can thn of the credt networ as the networ of nterban loans. These nsttutons are also actve on fnancal marets,.e. they trade fnancal oblgatons wth agents outsde the networ tself. For nstance, they can collect deposts, whch are labltes of the bans and assets of agents outsde the credt networ,.e. households. Table 1 Balance-sheet composton. The components of both assets and labltes are classfed n terms of maturty and n terms of whether they generate a fnancal exposure to some other agents wthn the fnancal networ. hh = households, b = bans. Maturty Nature Assets Labltes Short-term Networ A SN Short-term No Net. A SC Long-term Networ A LN Long-term No Net. A LM (credt to bans) L SN (debt to bans) (cash) D (hh deposts) (OTC credt) L LN (OTC loans, b-held bonds) (mortgages) L LH (hh-held bonds) Each nsttuton s represented by a balance sheet. In a sense, we wll provde a model of balance sheet dynamcs n the followng. A smlar approach to modelng fnancal nsttutons can be found n Esenberg and Noe (2001) and Shn (2008). We classfy balance sheet tems along two dmensons. The frst dmenson s the maturty (short-term or lqud vs. long-term or llqud assets/labltes A/L herefter). The second dmenson s the relaton (or lac thereof) of the a/l to the credt networ. Some of the a/l, n fact, represent a credt nterlnage as they create a fnancal exposure to some other agents n the networ. Some other a/l, on the other hand, do not represent a credt nterlnage because they are ssued and purchased n fnancal marets outsde the credt networ. On the assets sde of the balance sheet of ban, short-term assets A S are lqud,.e. they can be promptly sold on the maret, whle A L are long term llqud assets,.e. assets whch can be lqudated only at the cost of a non neglgble loss of maret value. A N, = S, L are assets that represent labltes of some other agents n the networ (for nstance nterban loans). There are also two types of assets that are traded on fnancal marets,.e. they do not have another fnancal nsttuton as a counterparty: A SC can be assmlated to cash avalable to fnancal nsttutons (e.g. ban reserves); A LM are long term assets such as mortgages or long term bonds. Smlarly, on the labltes sde, L S represents short-term debt whle L L s long term debt. L N, = S, L are agent s labltes that represent assets for some other agents n the networ: L SN can be thought of as loans obtaned on the nterban maret, whle L LN are loans negotated over-the-counter wth other agents. The agent s labltes towards households are represented by deposts, D, whch are short-term n the sense that they are subject to potental wthdrawal at any tme. Bonds ssued by the ban and held by households L LH are long term because they mply an oblgaton of rembursement only at maturty on a tme scale longer than the short term. Table 1 summarzes the composton of the balance sheet. The dfference between total assets A and total labltes L s net worth or the equty base. In the followng we wll focus on the fnancal rato: = ASC + A LM + A SN + A LN A LN L SN D L LN L LH = A L A LN, (1) whch s the rato of equty over the long-term networ-related component of assets. Ths equty rato s an ndcator of fnancal robustness. The exercse we carry out n the paper conssts frst n dervng a law of moton for the fnancal rato, n presence of defaults of counterpartes. We assume to start from a gven ntal allocaton of assets and labltes across agents and thus wth a gven dstrbuton of fnancal robustness across agents. We also assume some ntal defaults. These ntal defaults may or not trgger other defaults. In any case, we compute recursvely the effect of these defaults on the balance sheets of the counterpartes and the counterpartes of the counterpartes, downstream along all paths n the

4 S. Battston et al. / Journal of Fnancal Stablty 8 (2012) from the lqudaton of the assets of the defaulted counterpartes, so that a 1. 1 We wll dscuss the case a < 1 at the end of Secton 3.2. Moreover, n the scenaro we are loong at, wthn the duraton of a default cascade, agent s assets at tme t are affected by the banruptcy of agent j ndependently of the perod n whch her banruptcy has occurred. It s then convenent to rewrte the equaton above as follows: A LN (t) = A LN (0) aa LN j j (t), (3) Fg. 1. Representaton of the credt relatons between agent and ts oblgors n the smplest case (left). Another scenaro: agent has oblgatons to both long-term credtors ( 1 4) and short-term credtors (h 1 h 3), whle t has long-term clams on some oblgors (j 1 J 4)) (rght). networ. At the end of ths cascadng process there s a number of default accumulated. We wll study how the cascade dynamcs s affected by the dstrbuton of ntal level of robustness across agents and on the networ structure. At the end of one cascadng process all agents are replaced by new agents. Ths s a sort of stress test for the whole fnancal networ, n whch the shoc s represented by the ntal default, and the response s gven by the fnal number of defaults. In terms of nformatonal set, agents are assumed to now whch agents have defaulted, but they do not now the exposures of ther counterpartes to other agents. Only n the extreme case of fully connected networ, connectons are trvally nown to everyone. Moreover, agents do not now wth certanty the level of robustness of ther counterpartes. Therefore, they cannot antcpate whether there wll be a large default cascade or not. Fg. 1 shows a vsual llustraton of the stuatons leadng to two dfferent types of external effects. In the frst stuaton (left), the default of some counterpartes affect agent, but there s no further repercussons. In the second stuaton (rght), the presence of short-term lenders creates the potental for further repercussons on herself, as t wll be clear n the followng secton External effect of the frst type Suppose that counterparty j of agent defaults on her long-term oblgatons at tme t. Ths affects A LN (t),.e. the networ-related long-term component of the balance sheet n the same perod, whch wll be reduced by a fracton a of the nomnal value of the oblgaton of agent j to, A LN (t) : j A LN (t) = A LN (t 1) aa LN j (I) (t), (2) j where (I) (t) ndcates f agent j defaults at tme t. The parameter a j measures the fracton of funds that agent s assumed to loose, n the short-term, when the counterparty j defaults. For sae of smplcty, here, ths fracton s assumed to be the same across agents, but t could be made heterogeneous. Wth a = 0, bans do not loose anythng, whch mples that default do not have any externalty on the counterpartes. It s not unfrequent that defaults propagate on the tme scale of days or wees, as t has been the case durng the varous epsodes observed durng the fall of Such short tme horzon maes reasonable the followng assumptons. Frst, agents are not able to modfy ther exposures to other agents n reacton to the defaults occurrng n the fnancal networ. Secondly, agents are not able to recover, wthn the duraton of the cascade, most of the proceeds where A LN (0) represents the ntal value of networ-related longterm assets (n the followng the ntal condton wll be denoted as the begnnng of a cascade of defaults), and j (t) ndcates f agent j has defaulted at tme t or n any perod before t. Consderng the nexus of credt nterlnages, the total loss to agent due to non-performng loans ( bad debt ) can be ndcated by j ALN j j (t). Agent becomes nsolvent when net worth becomes negatve: A SC + A LM + A SN + A LN a A LN j j (t) L SN D L LN L LH < 0, j (4) where to smplfy the notaton, we dropped the ndcaton of the tme 0. Let us denote the relatve exposure of to j n terms of long term labltes as follows: A LN j W j = A LN. (5) From ths defnton follows that the matrx W of relatve exposures s non-negatve and row-stochastc. Dvdng both sdes of Eq. (4) by A LN and recallng the defnton of the equty rato above we conclude that solvency requres the followng quantty to be postve (t) = (0) a W j j (t), (6) j (0) represents the ntal fnancal robustness of. The term j W j j (t) accounts for all relatve losses (due to the default of some of her counterpartes) experenced by agent snce the begnnng of the cascade. 2 Notce that ths measure of fnancal robustness s an nverse measure of aggregate asset rs. Thus, the same argumentaton of the paper can be recast n terms of asset rs. Let us now assume that each agent wth counterpartes, has roughly comparable exposure to them. Then, t s W j = 1/ (unform rs sharng). Let us ndcate wth f the number of defaultng partners of agent. Hence j W j j (t) = f / s the fracton of defaultng counterpartes, whch measures the relatve mpact of defaults on fnancal robustness. In the case of unform rs sharng, therefore, the law of moton of the equty rato above can be wrtten as follows: (t) = (0) a f. (7) Eq. (7) mples that the fracton of defaultng counterpartes s the man determnants of the default cascade (as also found n (Ga and Kapada, 2010a)). However, t may not be obvous at a frst thought that the same level of relatve loss f /, can occur wth 1 For nstance one can assume that defaulted bans assets are dstrbuted to depostors frst and to other credtors next, as n (Ior et al., 2006). 2 Notce that the dynamcs of nsolvency s completely ndependent of ntroducng and normalzng as we have done. The procedure smply allows to rewrte the dynamcs n a convenent way.

5 142 S. Battston et al. / Journal of Fnancal Stablty 8 (2012) very dfferng probablty, dependng (1) on the values of f, and the probablty p of ndvdual defaults, and (2) on the presence (or lac thereof) of correlatons. Let us llustrate ths statement wth some examples. If defaults are uncorrelated, the probablty P{ f, } of occurrence of f defaults among partners follows a bnomal dstrbuton, P{ f, } = p f (1 p) f. Ths probablty f decreases sharply wth f, for f > p, for gven and p. In partcular, for a gven value of p and f /, the larger s, the less probable t s to observe f smultaneous defaults. For nstance, consder p = 0.1 and f / = 0.2. The probablty to have 2 defaults out of 10 counterpartes (P{2, 10} 0.2) s much larger than the probablty to have 20 defaults out of 100 (P{20, 100} 0.001). Thus, when defaults are not correlated, the event consstng of several smultaneous defaults among the counterpartes of an agent wth a large portfolo s qute rare. If nstead defaults are correlated, the probablty of smultaneous defaults s not bnomal anymore. In contrast to the uncorrelated case, t may be more lely to observe many smultaneous defaults than only few ones. Unfortunately, there s no smple way to descrbe mathematcally such probablty, as t depends on the structure of the correlaton (Frey and McNel, 2003). Notce that, even f the average ndvdual default probablty s nown (or can be estmated from the frequency of defaults n the whole populaton), the probablty of jont defaults remans unnown f the correlaton s unnown. But the correlaton of defaults n a networ depends tself on the default probablty and the networ structure. For our purposes, t suffces the assumpton that ether defaults are uncorrelated or they are completely correlated, as t wll be clear n Secton 3. In the dynamcs of presented above, the shoc caused by the default of one agent s unformly spread among the credtors. Ths s the external effect of type 1. The external effect of defaults s, n a sense, conservatve because t s dvded across the agents who bear some fnancal exposure to the defaultng partner but t does not get amplfed durng ts propagaton. In a more realstc settng, n whch the lender can recover at least part of the nterban clam on defaulted agents, the external effect of type 1 would be mtgated. In the present framewor, for sae of smplcty and based on the assumpton (see earler) that the legal settlement of the banruptcy taes much longer than the unravelng of the cascade, we rule out repossesson of the assets of the defaulted agents on the part of the lender External effect of the second type In ths secton, we focus on another type of external effects n whch the losses due a counterparty default can cause addtonal losses to the agent. Losses are n a sense amplfed along each connecton. In ths paper, we nvestgate how ths type of effect can arse n a specfc scenaro 3 n whch a lqudty run and the consequent fre-sellng on the asset sde generates a further repercusson to the agent, n addton to the frst drect loss due the counterparty default. In a sense, the external effect of the second type combnes the externalty of the frst type wth the fre-sellng scenaro already well nvestgated n the lterature (see e.g., Brunnermeer and Pederson, 2009). In the followng, we frst dscuss how llqudty orgnatng from the default of one or more counterpartes of an agent can lead to her nsolvency, due to the possblty of a run of ts short-term lenders. Notce that the run s not modelled explctly. We assume that the run occurs or not dependng on the level of robustness of the agent and on the number of her counterpartes n default. Based on such assumpton, we derve a law of moton for robustness that ncorporates the external effect of second type. An agent facng the request to repay a part of her labltes - denote such amount as L SN becomes llqud n case her lqud assets are not suffcent to cover for the payment: A SN + A SC L SN < 0. (8) An agent can well be llqud even f she remans solvent.e. even f net worth remans postve (see Eq. (4)). In terms of nformatonal set, we have n mnd the stuaton n whch short-term credtors do not now wth certanty the level of robustness of the counterpartes of the agent. In the presence of ths nformaton mperfecton, the default of one or more of the counterpartes of, can trgger the llqudty of, even f she can absorb the shoc due to the defaults and reman solvent. Ths happens n two stages. Frstly, the default of some counterpartes s a shoc that reduces the asset sde of and thus, n presence of uncertanty on the future, ncreases the chances of default of. More specfcally, recall that, as dscussed earler, the probablty of occurrence of the relatve loss due to defaults, f /, depends n non-trval way on the values of f,, the probablty p of ndvdual defaults, and on the presence of correlaton. Thus, when short-term credtors of agent observe a number of smultaneous defaults among her counterpartes, they face uncertanty along several dmensons: Has the probablty of default ncreased? Were there nterdependences among the counterpartes who defaulted? Does ths mean that the remanng counterpartes are also lely to fal shortly after? Fnally, s there somethng systematcally wrong n the way the agent has chosen her counterpartes? In other words, mperfect nformaton on the fnancal health of the counterpartes of agent and the structure of the correlatons among ther possble defaults, mples that short-term lenders cannot exclude that the probablty of default of has ncreased, even f at the moment she s stll solvent. It follows and ths the second step that t s ratonal for the short-term lenders of to consder the opton of refusng to roll-over debt to. Ths s even more so, gven that, at the same tme, all other short-term lenders may also end up decdng not to roll-over,.e., that there could be a run on agent. To contnue wth the setch of the stuaton we magne here, let us assume that, as a result of the defaults among s counterpartes, some of s lenders refuse to roll-over ther short-term loans and agent wll have to pay bac short-term debt. She wll frst try to satsfy ths need usng her lqud assets Af. If these are not suffcent,.e. L SN > A S, then may decde to sell some of her long-term assets. For the sae of smplcty, n the followng we assume that the frst lne of assets sold under dstress conssts of securtzed mortgages. If the maret of such type of securtes s lqud the agent s able to re-balance her lqudty poston wthout any loss. However, f the maret s not very lqud agent may be forced to a fre-sellng,.e. to sell below maret prce. Then, the nomnal value of assets to be sold exceeds, n absolute value, the value of labltes to be repad: A LM = q( L SN A S ), (9) where q = p (maret) /p (fre) 1 s the rato of the maret prce over the sellng prce. 4 Whether and when credtors wll eventually run 3 Other scenaros, e.g., nvolvng credt dervatves would be plausble canddates, but they are not nvestgated here. 4 Incdentally, notce that n Ga and Kapada (2010b), as a result of counterpartes defaults and short-term lender refuse to roll-over, agents decde to hoard lqudty

6 S. Battston et al. / Journal of Fnancal Stablty 8 (2012) e. refuse to roll over to s an ssue that has been extensvely nvestgated n the vast lterature on ban runs. Most wors fnd that nvestors run when the shoc httng the ban s relatvely large and the ban s already relatvely fragle (Rochet and Vves, 2004). In lne wth ths lterature and motvated by the prevous dscusson, we assume (wthout modelng the coordnaton game among credtors) that there s a run of all the credtors f the number of defaults s larger than a certan threshold that ncreases wth the robustness of the agent, f > (0), (10) where s a scale factor, wth 0 1. In other words, there s a run when several counterpartes default and the agent has already low robustness. Notce that when the run occurs, agent has to repay an amount correspondng to the total aggregate short-term loan she receved,.e. L SN > L SN L SN, snce all the credtors run. Ths s done by reducng AL M accordng to Eq. (9). As a result, n case of a run after the defaults, the equty of decreases as follows: A (t) L (t) = A L a A LN j j (t) A S A LM + L SN j = A L a f A LN j0 A S q(l SN A S ) + L SN = A L a f A LN j0 (q 1)(L SN A S ). After normalzng by A (LN), we fnally obtan the followng dynamcs: (0) a f b (t) f (0) < f (0) a, (11) f otherwse where the parameter b = (q 1) LSN A S A LN, (12) measures the mpact on agent of the cost of the run (q 1) L SN A S, relatve to the long-term networ related assets. Notce that, n addton to the shoc due to s counterparty defaults, n case of run, agent, faces now a further decrease of equty. Ths amplfcaton of the ntal shoc characterzes what we call external effect of the second type. The dea that a lqudty run can trgger a fre-sellng s well-nown. Our contrbuton here s to cast ths wthn the framewor of the default cascade and show that ts effect s to amplfy the ntal losses. Notce also, that f we rule out runs for all agents, ether because they do not mae use of short-term credt, or because the threshold s never reached ( = 0), ths s equvalent to set b = 0, so that we recover the dynamcs wth external effects of the frst type (Eq. (6)). For the sae of smplcty, n the followng we focus on the case n whch the parameter b s homogenous across agents, b = b for all. 3. Default cascades The ntal default of one or more agents n the credt networ trggers the default of other agents,.e. a cascade or avalanche of by wthdrawng short-term lendng from other agents n the networ. Here we do not focus on the lqudty evaporaton ssue, but on the propagaton of nsolvency. defaults. The development of a cascade n the model s generated applyng recursvely the dynamcs of Eq. (11). The cascadng process s determnstc and termnates after a fnte number of steps. 5 In the fnal state of the cascade, a certan fracton s of the agents has defaulted. Ths s the sze of the cascade, whch, n our framewor, captures the magntude of the systemc rs to whch the credt networ s exposed. In the followng, we nvestgate how to determne the sze of the cascade and the role played by dversfcaton for fnancal stablty. The fracton of defaults at the end of the cascade (.e., the sze of the cascade) can be computed as the stable fx pont of a recursve equaton for the cumulatve fracton of falures. We wll derve ths equaton frst n the presence of externaltes of the frst type 6 and then n the general case of external effects of the second type External effects of the frst type We frst focus on the dynamcs descrbed n Secton 2.1 by Eq. (6). It s nstructve to llustrate a smplfed verson of the computaton method. Suppose the networ s a regular graph,.e. each agent has the same number of connectons. The fracton s(t + 1) of agents who wll default n the future tme step of the dynamcs represented by Eq. (6) s smply the fracton of agents who happen to go below the default threshold n the current tme step. In a large system, such fracton approxmates the probablty that the robustness of a randomly chosen agent s below the threshold. Moreover, n a mean-feld approxmaton, we can replace the expected fracton of defaults among the counterpartes of each agent wth the current fracton of defaults n the populaton. We can thus wrte { } s(t + 1) = P f < 0 P{ < s(t)} (13) Assumng that agents who default are not replaced or refnanced durng the cascade, then the cascade sze s at least the ntal fracton of defaults s(1). We, thus, obtan the followng equlbrum condton: s = F(s) = max{s(1), (s)}, (14) x where (x) = p()d s the cumulatve dstrbuton of robustness up to the value x. Ths yelds a recursve equaton n s whose soluton s the sze of the cascade. One can mprove the computaton usng a better estmaton of expected number of defaults among the counterpartes of a gven agent. However, the procedure s conceptually the same. The outcome of the cascade depends of course on the ntal dstrbuton of robustness across agents. If agents have low robustness, the cascade tends to be larger because the default of some agents s more lely to cause the default of other agents. In the followng, we wll assume that the ntal dstrbuton of robustness when the cascade starts can be approxmated by a Gaussan wth mean m and standard devaton. The values of mean and standard devaton have an mpact on the shape of the functon n Eq. (14): a decrease of robustness shfts the functon F(s) to the rght, whle a decrease of maes the slope steeper. In Fg. 2 (left and rght), the functon F(s) s plotted for some values of m and and two examples of fxed ponts are llustrated. Notce that f the ntal robustness s dstrbuted across agents accordng to a Gaussan dstrbuton, there s always a postve probablty that some agents have a value of robustness below 0, 5 Ths number s smaller than the number of agents snce agents are not re-started f faled durng the cascade (Klenberg, 2007). 6 Notce that ths method dffers from the graph generatng functon approach used n (Ga and Kapada, 2010a), whch bulds on (Callaway et al., 2000).

7 144 S. Battston et al. / Journal of Fnancal Stablty 8 (2012) Fg. 2. Plot of R.H.S. of the smplfed fxed pont Eq. (13) for the cascade sze. Examples of curves for m = 0.4, 0.6 and = 0.2 (left). The curve for m = 0.6 has two stable ponts, close, respectvely, to 0 and 1 and one unstable pont. In the case of m = 0.4 there s only one stable pont concdng wth the ntal fracton of defaults, s(1) Despte the large fracton of ntal defaults the cascade stops mmedately. Examples of curves for m = 0.4, = 0.05 and = 0.4 (rght). In the case of = 0.05, from an ntal value s(1) 0.1, the fracton of defaults ends up close to a complete default, s 0.9. because the support of the Gaussan stretches to the whole real lne. 7 In general, there s an expected number of ntal defaults that depends smply on the mean and standard devaton of the dstrbuton. We denote these ntal defaults as endogenous. However, n lne wth the sprt of a stress testng exercse, we are nterested n nvestgatng the response of the fnancal networ, gven a certan dstrbuton of robustness, to shocs of varyng ntensty. Therefore, n addton to the endogenous defaults, we assume that, at tme 0, a fracton y 0 of randomly chosen agents default. By varyng y 0, we vary the ntensty of the shoc httng the fnancal networ at the begnnng of the cascadng process. We denote these defaults as exogenous. Notce that n ths modellng framewor one can also nvestgate the effect of systematc shocs httng all agents n the same way. Indeed, one can study, as we do n the followng, the outcome of the cascade for varyng levels of the average robustness across agents. Shftng the mean of the dstrbuton down, say from m 1 to m 1 s equvalent to assumng that a systematc shoc has ht all agents before the dynamcs starts. The followng proposton characterzes the sze of the cascade. Proposton 1. Consder the process of Eq. (6). Assume the networ of frms s a regular random graph wth degree. Assume also the ntal probablty dstrbuton of robustness s Gaussan 8 wth mean m and varance 2 = 2 /: p((t = 0)) Gauss(m, ). Let (x) = x p()d denote the cumulatve probablty dstrbuton of up the value x, and s 0 = (0) denote the fracton of frms whose robustness s below zero at t = 0. Moreover, assume that at tme t = 0 there s a fracton y 0 of exogenous defaults. Then: 1. The fracton s of falures at the end of the cascade process s the soluton of s = max{s 0 + y 0, F(s, m, )} (15) 7 It would be nterestng, to now the emprcal dstrbuton of the varant of equty rato, that we have ntroduce n ths paper. However, dfferently from the case of the usual equty rato, there are no readymade statstcs and one would need to loo at the balance sheets of the nsttutons n order to estmate the amount of assets that have a networ valence. 8 A smlar computaton could be carred out for other probablty dstrbutons of robustness p(). F = af s f (1 s) f f f =1 2. A stable fxed pont always exsts. (16) Notce that Eq. (15) provdes an analytcal expresson for the cascade sze. 9 The result holds also for random graphs n the lmt of small degree varance. It s possble to extend the result to the case of heterogenous networs wth a gven degree dstrbuton p(). In the expresson of F, one needs to also sum over varyng level of the connectvty degree, and to weght each term by the correspondng value of probablty p(). However, n so dong we would ntroduce addtonal dmensons to the parameter space. Ths case wll be nvestgated n future wor. In ths paper we want to focus on the effect of the average level of the dversfcaton and how t affects the cascade sze s, dependng on the values of the parameters, m and the exogenous shoc y 0. We fnd, that typcally, the cascade sze s ether s 0 + y 0 tself (.e., no new defaults are nduced by the ntal ones), or a full cascade of the whole fnancal system s trggered. In partcular, there are dfferent regmes, n whch ndvdual rs dversfcaton may have ether a benefcal (stablzng) macroeconomc role, or a detrmental one, or no role at all. Further wor should am at dervng some general results on the exact transton boundares between these regmes n the space of the parameters, m and y 0. However, even f these general results are not yet avalable, the proposton above allows to prove (smply by exstence) a few nterestng results and to draw some relevant mplcatons for fnancal stablty. Fndngs are llustrated n scenaros, related to specfc regons of the parameter space. To help the reader gettng a general pcture, the detaled descrpton of each fndng s preceded by a concse but smplfed statement. All results concern the default cascade sze dependng on the followng parameters: the standard devaton and mean m of the robustness dstrbuton, the sze of the exogenous shoc y 0 and the connectvty degree of the agents n the networ. In order to better understand the mplcatons of the fndngs, let us recall the meanng of the parameters and ther range of varaton. 9 As for all trascendental equatons,.e., nvolvng non-polynomal functons, the soluton has to be computed numercally. But ths can be done wth arbtrary precson. Ths s not the same as fndng the cascade sze wth smulatons.

8 S. Battston et al. / Journal of Fnancal Stablty 8 (2012) The average fnancal robustness m, ranges n [0 1]. A value m = 0 means no equty. Because the dstrbuton s Gaussan and thus symmetrc, t also means that half of the agents have robustness below 0. Instead, m = 1 means that equty s as large as long-term networ-related assets (see Eq. (1)). Gven the range of m, values of the standard devaton of robustness > 0.5 can be consdered as large, whle values < 0.1 are small. Fnally, snce y 0 represents the ntal fracton of defaultng agents, values y 0 > 0.3 are qute large, whle y 0 < 0.05 represent typcal stuatons n normal course of the economy. Scenaro 1. A fragle system s prone to systemc default, even f there are no exogenous shocs. Even n the absence of exogenous defaults, y 0 = 0, for any value of the standard devaton, the cascade sze s tends to 1 for decreasng values of m. Fg. 2 shows how the fx pont soluton of Eq. (13) vares as a functon of m and. A decrease n m shfts to the rght the functon F. The slope of F decreases wth. However, t s clear that no matter how steep s the slope (.e., small ) there s always a value of m small enough so that F(s) > s for all n ]0 1[and thus the only stable pont s the one closer to s = 1. The same result apples to Eq. (15). Thus, when the average robustness across agents s low enough, the endogenous defaults trgger a systemc default even n the absence of any exogenous shoc. In ths case, dversfcaton s rrelevant, as shown n Fg. 3. Indeed the sze of the cascade remans constant wth and close to 1, when the average robustness s low. Scenaro 2. Dversfcaton prevents systemc defaults when the fnancal condton overall s not too bad. There exsts a range of the parameter values (,m) [0 0.15] [ ], y 0 < 0.1, where the cascade sze s decreases wth dversfcaton. Fg. 3 (rght) shows that, for a gven m, s eventually drops to small values as ncreases. Thus, when fnancal robustness s not very dfferent across agents (small varance of robustness) and the exogenous shoc s not large, then dversfcaton maes the system more reslent to systemc defaults. More precsely, wth a larger dversfcaton, the system remans stable even at lower values of average robustness. Ths result s n lne wth the pro-connectvty vew mentoned n the Introducton. However, results are not always n ths drecton. Scenaro 3. Dversfcaton may n some cases lead to systemc defaults. There exsts a range of values, e.g., (,m) [0.4 1] [ ], where the cascade sze s ncreases wth dversfcaton, as shown n Fg. 4 (left). Ths s a qute counterntutve result whch requres a more detaled explanaton. Frst of all, t s not always necessary a bg loss to cause the default of an agent. If her robustness s low,.e. her equty s small relatve to her assets, even a relatvely small loss may be suffcent to push the agent beyond the threshold of default. Ths depends on the structure of assets and labltes, as defned by the solvency condton n Eq. (4). What happens then, when an agent suffers a loss bgger than the crtcal one,.e. the loss whch would be exactly suffcent to mae her default? The excess loss.e. the dfference between the actual and the crtcal loss does not have any addtonal effect on the agent and her credtors. Then consder the case n whch ntal robustness s heterogeneous and many agents are fragle. Wth a low level of dversfcaton, the ntal default of an agent causes a bg loss to counterpartes that are already fragle. These counterparty fal, but many more could have faled, would the loss been shared more broadly. So the momentum of the ntal loss s dampened on the way. Instead, when the level of dversfcaton s hgher, then every default adds only a lttle loss to each counterparty. Thus, when the accumulated losses eventually exceed the solvency threshold, they do so by lttle margn, so that no loss s lost, so to say. In other words, the momentum of the ntal defaults s not dampened on the way. Ths mples that there can be stuatons n whch the ntal endogenous and exogenous defaults are capable of trggerng a systemc default when the dversfcaton s hgh, whle the cascade stops on the way when the dversfcaton s low. Ths feature obvously depends on the way the propagaton of dstress has been defned n the model, whch ncludes a sort of lmted lablty of agents. Stll, the result rases an nterestng ssue, n partcular n connecton to how the losses assocated to defaults are socalzed. Scenaro 4. Dversfcaton has no effect when the system s fragle, relatvely to the exogenous shoc. There exsts a range of parameter values,.e. (,m) [0 0.4] [0 0.2], wth y 0 > 0.1 where s does not change wth (see Fg. 4, rght). Thus, f robustness s not very heterogeneous and the average s low, then varyng the level of dversfcaton has no effect. The systemc default occurs anyway. There s no gan n dstrbutng losses more wdely because the exogenous shoc s already bg enough to noc down everybody. There s also no gan n concentratng the losses among fewer counterpartes because the most robust agents are not much more robust than the wea ones External effects of the second type In ths secton, we apply the procedure already descrbed n the prevous secton to the dynamcs defned n Eq. (11). We also mae an addtonal assumpton concernng the heterogenety of robustness across agents. One may argue that, f agents have a hgh level of dversfcaton n ther fnancal exposure, then ths affects not only the loss they face as a result of the default of a counterparty but also the varatons n the ntal robustness from an agent to another. To put t straght, magne each agent dversfes ts exposure, wth equal weghts, on the entre set of other agents. Then, the ndvdual robustness may stll vary, snce t depends on the choce of the level of short-term vs. long-term assets and labltes. However, dfferences n robustness between any two agents should decrease because all agents have almost the same portfolo of A LN assets. Notce that f the standard devaton u of the robustness decreases, then, as long as the average m s postve, the fracton of endogenous defaults,.e. the agents that are already ntally below 0, decreases. Ths can be expected to mae the system more reslent. Thus, n order to verfy f the results found n the prevous sectons stll hold when the heterogenety decreases wth the dversfcaton, we assume that the varance of the robustness scales as 2 = 2 /. Ths s a mared decrease n heterogenety varance tends to zero for large whch corresponds then to qute a conservatve scenaro. Notce that the assumpton s n lne wth (Battston et al., 2009), where the fnancal robustness s the result of an endogenous dynamc process. The sze of the cascade under the dynamcs ncludng credt runs s characterzed n the followng proposton. Proposton 2 (.). Consder the process of Eq. (11). Assume the networ of frms s a regular random graph wth degree. Assume also the ntal probablty dstrbuton of robustness s Gaussan 10 wth mean m and varance 2 = / : p((t = 0)) Gauss(m, ). Let x (x) = p()d denote the cumulatve probablty dstrbuton of up the value x, and s 0 = (0) denote the expected fracton of frms whose robustness s below zero at t = 0 (endogenous defaults). In addton, assume that at tme t = 0 there s a fracton y 0 of exogenous defaults. Then: 10 A smlar computaton could be carred out for other probablty dstrbutons of robustness p().

9 146 S. Battston et al. / Journal of Fnancal Stablty 8 (2012) Fg. 3. Cascade sze dagram. The fracton s of defaulted agents s plotted n gray scale as a functon of dversfcaton degree and average robustness m. Case: absence of exogenous shoc y 0 = 0, ntermedate varance, = 0.2 (left). Case: y 0 = 0.1, = 0.1 (rght). 1. The fracton s of falures at the end of the cascade process s the soluton of the equaton s = max{s 0 + y 0, G(s, m,, b)} (17) G = ( s f (1 s) af f (1 ( f f )) f =1 ) af + + b ( f ) 2. A stable fxed pont always exsts. (18) Agan, we study how the level of dversfcaton mpacts on the cascade sze s. In addton to the parameters, m and y 0 (respectvely, standard devaton and mean of robustness, and exogenous shoc) already at play n Secton 3.1, we have here one more parameter, namely the cost of the credt run, b. As before, we observe dfferent regmes for the behavor of the cascade sze. The meanng of the values of the parameters m, and y 0 has been explaned n the prevous secton. For the scale factor a value close to 0 means that short-term lenders are very lttle senstve to the defaults among s counterpartes. For nstance f f = 3 and gamma = 0.01, then a run occurs only of agent has an ntal robustness smaller than f = 0.03 (see Eq. (11)). Fnally, to gve a concrete dea of the meanng to the values of the parameter b, consder the followng relatvely severe scenaro: a margnal cost of fre-sellng of 50%,.e., q 1 = 0.5; a small amount of cash and a large rato of short-term networ-related lablty to long-term networ-related assets, so that, e.g., L SN A SC /A LN = 0.8. Ths yelds a value b = 0.4 for the mpact of the cost of the run (see Eq. (12)). Thus, values b < 0.1 correspond to mld cost of runs whle values b > 0.3 correspond to sgnfcant cost of runs. Scenaro 5. In the absence of runs and large exogenous shocs, dversfcaton prevents systemc defaults. There exsts a range of parameter values [0 0.3], y 0 < 0.1 where cascade sze s decreases wth dversfcaton for a gven average robustness m [0.11] (see Fg. 5, left). Thus, when the exogenous shoc s not very large, then dversfcaton maes the system more reslent to systemc defaults. More precsely, wth a larger dversfcaton, the system remans stable even at lower values of average robustness. In partcular, the result suggests that, as long as robustness s postve, there s always a level of dversfcaton (provded the number of agents s also large enough) so that systemc defaults dsappear. Agan, n ths parameter range the behavour supports the proconnectvty vew. Scenaro 6. In the presence of runs, dversfcaton has an ambguous effect. There exsts a range of parameter values, e.g., (,m) [0 0.5] [0.10.5], b > 0.2, > 0.02, where the cascade sze s frst decreases and then ncreases wth dversfcaton (see Fg. 5, Fg. 4. Cascade sze dagram (see Fg. 3). Case: y 0 = 0, = 0.8 (left). Case: y 0 = 0.2, = 0.2 (rght).

10 S. Battston et al. / Journal of Fnancal Stablty 8 (2012) Fg. 5. Cascade sze dagram (see Fg. 3). Case: y 0 = 0, b = 0, = 0.3 (left). Case: y 0 = 0.01, b = 0.4, = 0.1, = 0.3 (rght). Fg. 6. Cascade sze dagram (see Fg. 3). Case: y 0 = 0, b = 0.4, = 0.05, = 0.3 (left). Case: y 0 = 0.05, = 0.1, b = 0.5, = 0.3. rght; Fg. 6, left and rght). Ths fndng s n star contrast wth the one of the prevous pont and results drectly from the credt runs. When agents observe that one of ther oblgors has a large number of defaultng counterpartes, compared to her robustness, they decde to wthdraw ther short-term funds from her. The run mposes a cost that often pushes the agent nto default, although not always. Ths depends on her ntal level of robustness and also on the cost b of the credt run (see Eq. (12)). The boundary curve separatng the regme where cascades are small from where they are large has a rebound for ncreasng. Wth larger values of b such rebound grows steeper. Ths means that hgher costs of credt runs mply stronger adverse effects of dversfcaton. However, for smaller values of the rebound starts only at larger values of. Thus, a hgher threshold for the credt run to occur mples that the adverse effect of dversfcaton steps n only at larger values of. Scenaro 7. Dversfcaton has no effect when the system s fragle, relatve to the exogenous shoc. There exsts a range of parameter values,.e.,m [00.4] [00.2], wth y 0 [0.051], = 0.1 and b > 0.2, where the cascade sze s s constant wth dversfcaton. An example can be seen n Fg. 6, rght) for m < Thus, f the average robustness s low, then the level of dversfcaton has no effect because the systemc default occurs anyway. Robustness of the results n case of partal asset recovery. It s nterestng to dscuss how the results are affected n case bans are able to recover n part the funds nvested n contracts wth defaulted counterpartes. Ths corresponds to the case of the parameter a < 1. As shown n Fg. 7 (top left and rght), n absence of runs, 11 dversfcaton s, as before, always benefcal but the cascades are reduced n sze. In the presence of runs (Fg. 7, bottom left and rght), even wth values a = 0.25 or a = 0.5 the ambguous role of dversfcaton perssts. The case a = 1 (not shown) s very smlar to a = To understand the reason, notce that n Eq. (11), the parameter a affects drectly only the term n the dynamcs that accounts for the externaltes of the frst type. The fact that agents recover part of the funds mples that n the face of defaultng counterpartes ther robustness decreases less and thus the run s less probable to occur. However, n case short-term credtors do mae a run, the cost of the run s not decreased by havng a < 1, because the damage created by the run s ndependent of the losses due the defaulted counterprtes. The cost of the run s already captured by the parameter b. Therefore, the overall effect of a < 1 s to mae the system more robust agan large cascades when dversfcaton s small. However, when dversfcaton s large, then large cascades are stll trggered as n the case a = See the note regardng the expressons absence of runs or presence of runs. 12 Of course n the case a = 0 and wth no ntal shoc (y 0 = 0) (not shown), then no cascade occur.

11 148 S. Battston et al. / Journal of Fnancal Stablty 8 (2012) Fg. 7. Cascade sze dagram (see Fg. 3). Case: a = 0.25, y 0 = 0.05, b = 0, = 0.05, = 0.3 (top left). Case: a = 0.5, y 0 = 0.05, = 0.1, b = 0, = 0.3 (top rght). Case: a = 0.25, y 0 = 0.05, b = 0.4, = 0.05, = 0.3 (bottom left). Case: a = 0.5, y 0 = 0.05, = 0.1, b = 0.5, = 0.3 (bottom rght) Concludng remars In ths paper, we contrbute to the lterature on the effects of rs dversfcaton on systemc rs by developng a new model of default cascades n fnancal networs n whch two nds of external effects of the defaults may occur. An externalty of the frst type occurs when the loss ncurred by an agent facng the default of a counterparty s smply proportonal to the relatve exposure of the agent to her counterparty. An externalty of the second type, nstead, depends also on the absolute number of defaults among the counterpartes. In ths paper, we have shown how the second type of externalty may arse n presence of possble runs on the agent by her short-term lenders. We have nvestgated how the number of defaults n the system depends on the dversfcaton under varous condtons. In partcular we have tested how the mpact of dversfcaton depends on the average robustness of the agents, the degree of heterogenety of fnancal condtons across agents (cross-sectonal varance), the sze of the exogenous shocs and cost of credt runs. We have shown that credt rs dversfcaton has ambguous results, especally n the presence of credt runs. Indeed the beneft of dlutng the loss of the defaults s counterbalanced by the fact that agents are more exposed to credt runs when they have many counterpartes. Ths analyss contrbutes the followng message to the debate on polces amng at enhancng fnancal stablty: Indvdual rs dversfcaton may have ambguous effect at systemc level. In partcular, networ structure and heterogenety of levels of fnancal robustness across agents should be carefully taen nto account when tryng to devse polces that enhance the reslence of the fnancal system. Our wor can be extended n varous drectons. One lne of further research we want to nvestgate n the future s to what extent these results are affected by assumng dfferent networ structures. Another one concerns the endogenous evoluton of lns over tme. Fnally, one could try to replcate and enrch these results n the 3- dates theoretcal framewor that s more commonly followed n fnancal economcs (Allen et al., 2010). Acnowledgements Authors acnowledge the fnancal support from: the ETH Competence Center Copng wth Crses n Complex Soco-Economc Systems (CCSS) through ETH Research Grant CH ; the European Communty Seventh Framewor Programme (FP7/ ) under Soco-economc Scences and Humantes, Grant agreement no (POLHIA); the European Commsson FET Open Project FOC no ; the Swss Natonal Scence Foundaton Grant no. CR12I /1); the Insttute for New Economc Thnng, Grant no. IN INET Fnancal Fraglty and Systematc Rs. Polcy Implcatons of Heterogeneous Agents Models.