Diversi cation and Systemic Bank Runs

Transcription

1 Diveri cation and Sytemic Bank Run Xuewen Liu y Hong Kong Univerity of Science and Technology Thi draft: June 208 Abtract Diveri cation through pooling and tranching ecuritie wa uppoed to mitigate creditor run in nancial intitution by reducing their credit rik, yet many nancial intitution holding diveri ed portfolio experienced creditor run in the recent nancial crii of We preent a theoretical model to explain thi puzzle. In our model, becaue nancial intitution all hold imilar (diveri ed) portfolio, their behavior in the aet market i clutered: they either ell their aet at the ame time or collectively do not ell. Such clutering behavior reduce market liquidity after an advere hock and increae the probability of a panic run by creditor. We how that diveri cation, while making the nancial ytem more robut againt mall hock, increae the poibility of a ytemic crii in the cae of a larger hock; diveri cation, inducing tronger trategic complementaritie acro intitution, make a elf-ful lling ytemic crii (multiple equilibria) more likely. Becaue individual intitution either over-diverify or under-diverity in the competitive equilibrium compared with the ocial optimum, there i room for regulation. JEL clai cation: G0; G2; D82; D53 Keyword: Diveri cation, market liquidity, panic run, ytemic crie, coordination rik I am grateful to Deniz Okat for detailed feedback and uggetion. I thank Heng Chen, Itay Goldtein, Zheng Song, Tianxi Wang, Liyan Yang, and eminar participant at City Univerity of Hong Kong, Chinee Univerity of Hong Kong, Univerity of Hong Kong, HKUST, and Econometric Society China Meeting 208 for helpful comment. y Department of inance, Hong Kong Univerity of Science and Technology (HKUST), Clear Water Bay, Hong Kong. Tel: (+852) xuewenliu@ut.hk.

2 Introduction Shadow bank were at the center of the recent nancial crii of On the eve of the crii, hadow bank a well a many other nancial intitution all held diveri ed portfolio. Thee intitution were believed to be afe thank to the diveri cation of their portfolio through pooling and tranching ecuritie. Depite their high credit rating, however, thee intitution experienced run by their creditor during the crii. Remarkably, the ytemic bank panic coincided with a udden dry-up of aet market liquidity. ire-ale dicount and repo rate exhibited triking pike (Gorton and Metrick (20), Covitz et al. (203)). Commentator have attributed the dicontinuou drop in market liquidity to multiplicity jump from one equilibrium to another under elf-ful lling belief rather than to udden large hock in fundamental (ee, e.g., Brunnermeier (2009) and Brunnermeier and Pederen (2009)). Thi paper propoe a theory to explain why diveri cation at nancial intitution can be an important caue of ytemic bank panic and why diveri cation can contribute to a elf-ful lling ytemic crii. Diveri cation reult in a higher degree of imilarity in aet portfolio holding acro nancial intitution (becaue they all hold a imilar diveri ed portfolio) and thu induce their clutering behavior in the aet market. Thee intitution either ( re) ell their aet at the ame time or collectively do not ell. Such clutering behavior reduce market liquidity when an advere aggregate hock i realized. Becaue market liquidity and hence the aet liquidation value determine the extent to which an intitution can withtand it creditor withdrawal, lower market liquidity mean a higher coordinating rik among it creditor and thu a higher probability of a creditor run. At the ytem level, there i further feedback: a higher number of creditor run in the ytem, meaning more re ale, in turn decreae market liquidity. A a reult, diveri cation lead to a viciou piral of lower market liquidity and more creditor run in the ytem when an advere aggregate hock i realized. To illutrate the idea, conider the following imple example. A New York bank and a Lo Angele bank make mortgage loan in their own citie. The realized loan value will depend on the aggregate (nation-wide) economic hock a well a the local hock. Suppoe there are three poible tate of the aggregate hock: Normal, Bad and Very Bad, the economic fundamental under which are aumed to be 8, 4 and 6, repectively. (idioyncratic) hock ha two equal-probability tate, f On top of the aggregate hock, the local 5; +5g, and the local hock i perfectly negatively correlated between the two citie. Aume that each bank hold one unit of loan, and if a bank realized loan value i below the threhold 0, the bank will fail. We conider two cae: without and with diveri cation. In the rt cae, each bank hold local loan only. Then, in the Modern-day bank run occurred in the repo market, money market mutual fund (MMM), and the ABCP market. See evidence in Gorton and Metrick (20), Copeland et al. (204), Krihnamurthy et al. (204), Duygan- Bump et al. (203), Kacperczyk and Schnabl (200), and Covitz et al. (203), among other.

3 Normal tate, the realized loan value of the two bank are 23 = and 3 = 8 5, which implie both bank will urvive. In contrat, in the Bad tate, the realized loan value of the two bank are 9 = and 9 = 4 5, which implie one bank will urvive and the other will fail. In the Very Bad tate, again one bank will urvive and the other will fail. In the econd cae, for implicity, aume that each bank diveri e by holding a half unit of local loan and a half unit of loan from the other city. 2 Hence, the local hock i perfectly diveri ed away. It i clear that in the Normal tate, the loan value of both bank i 8, which implie both bank will urvive. In the Bad tate, again both bank will urvive. In contrat, in the Very Bad tate, the loan value of both bank i 6 and thu both will fall. The above example illutrate that diveri cation, while having no e ect on the outcome in the Normal tate, reult in fewer of the bank failing in the Bad tate but more of the bank failing in the Very Bad tate. Our model will further endogenize the failure threhold of bank by endogenizing creditor run and market liquidity and demontrate how diveri cation impact on their interplay. In our model, there i a continuum of nancial intitution ( bank ). At the initial date, bank decide which aet to include in their portfolio. Their deciion determine the degree of heterogeneity acro bank: the greater the diveri cation, the lower the heterogeneity. In the extreme cae, for example, if all bank perfectly diverify (hold the market portfolio ), there would be no heterogeneity at all acro bank. At the interim date, creditor of a bank learn and receive noiy ignal about the fundamental value of it portfolio aet (i.e., the payo at the nal date) and decide whether or not to run. In equilibrium, bank in the higher range of fundamental value do not u er a creditor run and urvive, while bank in the lower range of fundamental value are run by their creditor and fail. The failing bank liquidate their aet in the market, where the liquidation price of a bank depend on it fundamental a well a on the market liquidity in the ytem which in turn endogenouly depend on aggregate re ale. Importantly, the diveri cation of bank a ect market liquidity at the interim date. When bank hold imilar portfolio, their elling deciion are clutered; o in a relatively bad tate (a large hock), there i an increaed number of bank under re ale and market liquidity i thu reduced for every bank, while in a relatively good tate (a mall hock) there i a decreaed number of bank under re ale and market liquidity i thu lifted for every bank. Thi in turn trigger a feedback loop in the ytem. That i, in a relatively bad tate diveri cation reult in a downward piral of a lower market liquidity and more creditor run, while an upward piral occur in a relatively good tate. Our model derive three implication. irt, diveri cation, while making the nancial ytem more robut againt mall hock, increae the poibility of a ytemic crii in the cae of a larger hock. Sytemic crie in term of ytemic bank run become le likely for mall hock but more likely for a larger hock. Second, diveri cation increae the likelihood of a elf-ful lling 2 The portfolio choice will be endogenou in the full model. 2

4 ytemic crii (multiple equilibria). When bank diverify, their portfolio look alike. Lack of heterogeneity in their portfolio induce tronger trategic complementaritie among creditor of di erent bank and make multiple equilibria more likely. Third, we how that individual nancial intitution either over-diverify or under-diverify in the competitive equilibrium compared with the ocial optimum. When a bank increae it diveri cation, it impoe externality on other bank a it action ha an impact on market liquidity. In a relatively good tate, the externality i poitive becaue increaing diveri cation of a particular bank reduce the likelihood of it u ering a creditor run and thu reduce the preure on market liquidity. However, when a relatively bad tate occur, the force become oppoite and the externality i negative. If market liquidity i more important in the bad time than in the good time, the poitive externality in the good time can be outweighed by the negative externality in the bad time, in which cae individual bank over-diverify in the competitive equilibrium. Our tudy i related to a few theoretical paper tudying diveri cation at nancial intitution and ytemic crie. Brunnermeier (2009) provide an excellent urvey on nancial crie and ytemic rik. 3 Wagner (200) (a well a Wagner (20)) build a imple and clean model howing that individual intitution over-diverify from the ocial perpective becaue diveri cation make bank more imilar. In hi model, when two bank fail jointly, there are additional cot over and above the cot of individual failure for each bank. Therefore, when an individual bank diveri e more, thi increae the probability of the joint failure and thu impoe negative externality on the other bank. Ibragimov, Ja ee and Walden (20) develop a model alo howing that diveri cation that i optimal for individual intermediarie may be uboptimal for the ociety. The negative externality in their model originate from the mechanim that when a hock dirupt all intitution imultaneouly, it take time for the nancial ytem and the economy to recover. Their focu i on the ditributional propertie of rik and the number of rik clae in the economy. Allen, Babu and Carletti (202) develop a model howing that aet commonality interacted with hortterm debt of bank can generate exceive ytemic rik. Their mechanim concern information contagion and the e ect of debt maturity. Thee extant tudie neither conider coordination rik nor model the creditor run of nancial intitution. Compared with thee earlier contribution, modeling the relation between diveri cation at nancial intitution and ytemic bank run i our focu. There i an endogenou threhold of intitution failure in our model. We how that diveri cation can have either a tabilizing or a detabilizing e ect depending on the ize of the aggregate hock. In particular, we how that diveri cation can make the ytem more fragile by increaing the poibility of multiple equilibria. In addition, di erent from thee prior tudie, our model alo how that diveri cation can introduce either a negative or a poitive externality, depending on the aggregate tate of the economy, o there can be either too much or too little 3 See alo Brunnermeier and Oehmke (203) and reixa and Rochet (2008). 3

5 diveri cation in the competitive equilibrium. Our paper i related to the literature that ue global game method to model creditor run. Morri and Shin (2004a), Rochet and Vive (2004), and Goldtein and Pauzner (2005) are the pioneering work of thi literature. 4 Morri and Shin (2009) build an analytical framework to decompoe creditor rik in a nancial intitution into inolvency rik and illiquidity rik. The recent paper of Liu (206) and Eienbach (207) tudy the interplay between aet price and creditor run. Zhou (206) how that a nancial network where bank are with more diveri ed pattern of interbank liabilitie will trigger more panic, becaue the ditribution of the total interbank repayment will become more centered, providing extra incentive for creditor to run. Our current paper follow the approach of Liu (207) in endogenizing market liquidity and howing it interaction with creditor run. Our paper add to the global game literature by howing that diveri cation can retore equilibrium multiplicity of global game: even when the preciion of creditor private ignal approache in nity, multiple equilibria can till emerge. Thi i becaue diveri cation induce tronger trategic complementaritie among the creditor of di erent bank. The paper i organized a follow. Section 2 lay out the model etting. Section 3 and 4 preent the model equilibria. Section 5 extend the model. Section 6 conclude. 2 Model Setting There are three date of the model: t = 0, and 2. All agent are rik-neutral. We dicu bank, aet, the aet market, and creditor run, in order. 2. Bank and Aet Conider an economy that conit of a continuum of tructured invetment vehicle (SIV or imply bank) with unit ma, indexed by j, and a continuum of aet type with unit ma, indexed by i. The aet may correpond to aet-backed ecuritie (ABS) in reality. The unit cot of inveting in any aet i at t = 0 i. One unit of aet i ha payo x i = i + e i at t = 2. The term e i ha the ex ante ditribution e i N(0; 2 e) and reolve it uncertainty at t = 2. We will conider two cae of e i : being perfectly correlated or imperfectly correlated acro di erent aet i. At thi tage, we aume the former cae and denote e e i for all i. The term i, 4 Some other related paper in thi literature include Dagupta (2004), Plantin (2009), Liu and Mello (20), He and Xiong (202), Xavier (204), Bouvard et al. (205), Ahnert (206), Allen et al. (207), Leonello (207), and Li and Ma (207). See Angeleto and Lian (207) for a recent extenive urvey on global game and their application. 4

6 interpreted a aet quality, ha it realization value at t =, which i independently drawn from an identical ditribution acro aet. There are three poible tate of the aggregate economy at t = : Normal tate ( = N), Bad or mall-hock tate ( = B), and Very bad or large-hock tate ( = E). In tate, the ditribution of aet quality i given by i N( ; 2 ), where = N, B and E, and N > B > E. Ex ante, at t = 0, the probability that tate will occur i, where P =. Denote e 2 e realization. and. igure illutrate the timeline of the cah ow 2 igure : Timeline of cah ow realization A bank ha one unit of capital at t = 0, of which an amount come from a continuum of it creditor (debtholder) with ma, each of them contributing unit, and an amount come from it equityholder (bankowner). 5 At t =, a creditor of a bank ha the right to decide whether to roll over hi lending to hi bank. If he decide not to roll over, hi claim i the par value at t = ; if, intead, he decide to roll over, the (promied) notional claim to him i R at t = 2, where R > i the gro interet rate. 6 The term R in the depoit contract will be endogenized. A creditor reerve return (opportunity cot) of lending i R 0, where R 0. At t = 0, each bank need to make it portfolio choice. Bank j portfolio i denoted by X j = ( ) x j + Z 0 x i di for 2 [0; ], where meaure the degree of diveri cation. Eentially, a bank portfolio can be regarded a having two component: one i local aet x j and the other i aggregate aet R 0 x idi. The aggregate aet may correpond to pooled and well-diveri ed aet in reality, like CDO, CDOquared, CDO-cubed, and o on. Thu, the portfolio payo for bank j at t = 2, conditional on the realization of aet qualitie f i g at t =, follow the ditribution X j N(( ) j + ; 2 e). We denote j ( ) j +, interpreted a the portfolio quality of bank j. Diveri cation 5 We aume that each bank ha it own creditor bae (for example, thee bank are regional bank). 6 Without lo of generality, we normalize the interim notional claim to. What matter to the model i the interet rate between t = and t = 2, i.e., the R. 5

7 potentially alo reduce aet-peci c rik e i realized at t = 2, which will be examined in Section 5. Although the portfolio quality of a bank i realized at t =, creditor are not informed of it. Neverthele, a creditor of a bank receive imperfect information (a ignal) at t = about the portfolio quality of hi bank. Speci cally, the ignal (about portfolio quality j ) for creditor h of bank j at t = i h j = j + h, where > 0 i contant (denote 2 ), and the individualpeci c noie h N(0; ). h i i.i.d. acro creditor of a bank, and each i independent of j. We aume that before creditor make their rollover deciion at t =, they are informed of the tatu of the aggregate economy, = N or B or E. Since a creditor of bank j know, hi private ignal about j i equivalent to a ignal about the quality of local aet j. In addition, it i aumed that the aggregate tate are uncontractible ex ante a in the incomplete literature. 2.2 Aet Market We follow the etting of the aet market in Liu (207). If a bank u er a creditor run (to be elaborated), it aet mut be liquidated at t = in a competitive aet market, which conit of a continuum of competitive invetor with unit ma. Invetor m ha utility function U(W m ) = exp ( W m ) ; where W m i the end-of-period wealth at t = 2, and i the rik-averion (CARA) coe cient. The rik-free (gro) interet rate between t = and t = 2 i normalized to. Invetor have private information (ignal) about bank aet portfolio qualitie. Speci cally, the ignal for invetor m about aet portfolio quality j at t = i m j = j + " m j, where 0 i contant, and the individual-peci c noie " m j N(0; ). " m j i independent acro aet portfolio for a given m and independent acro invetor for a given j, and each " m j for a given j i independent of j. Suppoe that the ytem ha, in total, a ma of ' (2 [0; ]) of bank u ering creditor run. Then, there are ' unit of aet in the ytem under re ale. 7 Denote by L j the liquidation ( re-ale) price per unit of bank j aet portfolio. 2.3 Creditor Run Let u conider a typical bank j. If it ha greater than L j proportion of it creditor declining to roll over their lending at t =, it liquidation value will not be u cient to cover the creditor claim, leading to it failure (we call thi cenario a creditor run ). 7 We will focu on the limiting cae of 0. In equilibrium, then, if a bank u er a creditor run, it will liquidate it aet entirely (i.e., there i no partial liquidation). 6

8 A creditor payo crucially depend on the action of other creditor of the ame bank. Let denote the proportion of creditor of a bank that chooe not to roll over (i.e., chooe to call) at t =. Then, the payo for a particular creditor i given in Table. Total calling proportion 2 [0; L j ) (bank urvive) Total calling proportion 2 [ L j ; ] (bank fail ) Hold 2 min 4R; L j Xj 3 5 L j Call L j Table : Creditor-run payo tructure If 2 [ L j ; ], a creditor run occur and the bank fail at t =. In thi cae, all creditor hare the liquidation value L j at t =, but thoe who have not called will pay an extra fee (e.g., legal cot) to get their money back. Thi etup of a rt-mover advantage of withdrawing (calling) follow the etup in Eienbach (207). A Eienbach argue, the rt-mover advantage, not depending on the equential ervice contraint inherent in depoit contract, i more repreentative of market-baed funding without a equential ervice contraint a in Cole and Kehoe (2000). If 2 [0; L j creditor. Thu, at t = 2, ), the bank need to liquidate unit of it aet to raie cah to pay it calling L j L j unit of aet remain. Since the number of taying creditor at t = 2 i, thee creditor total notional claim i R. Hence, a taying creditor will 2 3 have payo min R; the par value at t =. X L j j = min 4R; L j Xj 5 at t = 2. A creditor who call obtain or a cleaner and impler analyi, we follow Morri and Shin (2009) to implify the payo tructure of the creditor-run game. Morri and Shin (2009) aume that if there i not a run, new creditor will eventually be found and the balance heet revert to it initial tate after the failed run. Baically, they are auming that the partial liquidation of aet ha no long-run e ect (in the language of Vive (204)). Concretely, if a bank ha le than L j proportion of it creditor withdrawing, partial liquidation will occur but the bank can till urvive to t = 2, in which cae Morri and Shin (2009) aume that the bank balance heet revert to it initial tate. Eentially, after an unucceful run, the aet ide of the balance heet of the bank i retored to X j and the liability ide revert to the total notional debt value R claimed by creditor. 8 In hort, the 8 or example, a long a the bank i till alive (after an unucceful run), it can buy back it (partially) liquidated aet with re nancing from it original withdrawing creditor or new creditor. Alternatively, a long a the total amount of early withdrawal at t = i le than L j, the bank i able to raie thi amount of cah temporarily (for intance, from ome outide deep-pocketed invetor) by uing it aet a collateral; new creditor will then eventually be found to replace the withdrawing creditor and the re nancing amount from the new creditor i ued 7

9 aumption of Morri and Shin (2009) give the impli ed payo tructure in Table 2. Total calling proportion 2 [0; L j ) (bank urvive) Total calling proportion 2 [ L j ; ] (bank fail ) h Hold min R; X j Call i L j L j Table 2: Simpli ed creditor-run payo tructure Note that Table 2 i identical to Table if we et = 0 for the payo of holding in the cae of 2 [0; L j )). That i, a long a 2 [0; L j ), the bank continue a if it had not experienced any withdrawal in Table 2. Liu (207) how that the impli cation of Morri and Shin (2009) doe not change model reult qualitatively, only quantitatively. 2.4 Timeline At t = 0, the liability ide of the balance heet of a bank i given by (, ). On the aet ide, the degree of diveri cation,, i decided by a bank. Moreover, a bank chooe the (notional) interet rate R in the depoit contract ubject to creditor participation. At date t =, for a given, creditor move rt by making their rollover deciion, and bank move later by conducting aet ale in the aet market baed on the total withdrawal requeted by their creditor. We olve the equilibrium by backward induction: from t = to t = 0. 3 Equilibrium at t = We are intereted in the equilibrium where every creditor ue a threhold (monotone) trategy. The trategy i given by h j 7 ( Call h j < Hold h j, where h j i the ignal of creditor h in bank j and i the rollover threhold. Becaue bank are identical ex ante, we naturally conider the ymmetric equilibrium in which creditor of all bank ue a common trategy, i.e., the threhold i not bank-peci c. We will how that an upper dominance region exit for the bank-run game in our model. An upper dominance region exit becaue in our model the interim liquidation value of a bank i fundamental-dependent a in Rochet and Vive (2004). 9 to repay the temporary borrowing. 9 Goldtein and Pauzner (2005), following the original model of Diamond and Dybvig (983), aume that the 8

10 3. Equilibrium De nition Becaue creditor are alo informed at t = of the tate of the aggregate economy, = N or B or E, before they make their rollover deciion, the equilibrium can be tate-dependent. ormally, we de ne the equilibrium at t = a follow. n o De nition The equilibrium at t = i characterized by triplet ; L j ; ' for tate, where i the rollover threhold for creditor, L j i the liquidation price of bank j aet, and ' i the total ma of bank under re ale, uch that ) given creditor rational expectation of ', they et their rollover threhold a ; 2) given the rollover threhold, the ma of failed bank in the ytem i ' ; and 3) given the total re ale ', the equilibrium price of bank j aet in the aet market i L j. We derive the equilibrium in three tep. Aet market equilibrium To reduce notational clutter, we drop the tate index upercript for now. The portfolio choice of an invetor in the aet market at t = i given by max fqj m g E exp ( W m ) j m j ; fl j g () Z.t. W m = qj m (X j L j )dj, where qj m i the quantity of demand for bank aet j for invetor m. or implicity, we follow the trading game (mechanim) in Vive (204b) and Benhabib, Liu and Wang (206) to focu on the fully-revealing equilibrium of the aet market, i.e., the equilibrium in which nancial price fully reveal the fundamental of the trading aet. In our context, it i the equilibrium in which f j g i fully revealed to the invetor through the nancial price. 0 The rt-order condition of () implie that q m j exit and demand d j ), and that Z q j dj = j = q j for any m (i.e., a repreentative invetor Given that the total ma of bank u ering creditor run i ' in tate, the market clearing condition dictate Lemma follow. Z 2 e q j dj = '. interim liquidation value of a bank i fundamental-independent, o an upper dominance region doe not exit in their model. 0 Alternatively, intead of f jg being fully-revealed through the nancial price, we can aume that the preciion of invetor private ignal approache the limit 0 a in Morri and Shin (2004b), jut a 0 for the preciion of creditor private ignal. Note that thi part of the model i not a focu of our paper, and we can adopt either alternative. 9 L j.

11 Lemma The liquidation price of bank j aet in tate i given by L j = j ' k, where k 2 e and ' k meaure market liquidity in tate. Proof. See Appendix. The reult of Lemma i in the pirit of Groman and Miller (988). When the rik-avere market maker ector i forced to aborb more riky aet, the price of each riky aet i a ected and reduced becaue of the limited rik-aborbing capacity of the market maker ector. A in Brunnermeier and Pederen (2009), market liquidity i meaured a the degree to which the market price of an aet i depreed below it fundamental value. Market liquidity in our model i thu meaured by the term ' k. Creditor-run equilibrium for an individual bank Conidering that L j i fundamental ( j )-dependent by Lemma, when j i u ciently high, bank j will urvive even if everyone of it creditor withdraw. That i, an upper dominance region exit. Therefore, we only need to focu on threhold equilibria (ee Morri and Shin (2003) and Vive (204a)). h i Denote by D( j ; R) E min R; X j j j the expected payo of the debt at t = 2 conditional on the realization of j at t =, where X j N( j ; e). j > 0. We conider the limiting cae of ignal preciion: 0. The threhold equilibrium of the creditor run game i given by where L j = (D( ) ) L j = L j, (2) ' k. The proof i provided in Appendix. The intuition i the following. To the marginal creditor who receive ignal h j =, he perceive that (i.e., the proportion of hi peer creditor chooing to call) i uniformly ditributed within [0; ]. Hence, in hi eye, the probability of bank urvival i L j and that of bank failure i L j. In the cae of bank urvival, holding ha an advantage over calling with the additional payo being D( ), which i the LHS of (2). On the contrary, in the cae of bank failure, calling ha an advantage over holding with the additional payo being, which i the LHS of (2). Becaue the marginal creditor i indi erent with calling and holding, the equality of (2) follow. Moreover, under the limit 0, the marginal creditor aement of L j i L j = ' k. We can rewrite (2) a ' k = D( ) + : (3) Let denote the threhold uch that if and only if a bank realized portfolio quality j < will the bank fail at t = in tate. Under the limit 0, it i eay to how that = : 0

12 The limit 0 alo implie that in equilibrium all creditor of a bank are in the ame poiton ex pot, i.e., either all of them decide to roll over or none of them doe o. Thi in turn implie that in equilibrium a bank either completely liquidate it aet or doe not liquidate any fraction of it, i.e., there i no partial liquidation. Bank failure in the ytem Given, the portfolio quality ditribution acro bank in the ytem at t = (conditional on the realization of tate ) i j N( ; 2 2 ). Bank with realized portfolio quality j urvive at t = while other fail. Hence, the total ma of failing bank in the ytem i given by ' =, (4) where () tand for the c.d.f. of the tandard normal and () denote it p.d.f. Lemma 2 The equilibrium at t = i given by the ytem of equation (3) and (4) for = N, B and E under the limiting cae of 0. Two-way feedback exit between market liquidity (' k) and the creditor-run threhold Proof. See Appendix. > 0 in > 0 in (4). The two-way feedback between market liquidity and creditor run ha been tudied in Liu (207). Here we are intereted in the e ect of diveri cation on the feedback. Speci cally, we have the following propertie ( > 0 < 0 if > if < : (5) 3.2 Characterization of the Equilibrium Combining (3) and (4) yield one equation: D( ) k + = : (6) The equilibrium at t = i fully characterized by equation (6). When the liquidation value L j i exogenouly given, the creditor-run game, characterized by equation (2), ha a unique equilibrium. Similarly, when the market liquidity ' k i exogenouly given, the creditor-run game, characterized by equation (3), alo ha a unique equilibrium. Under endogenou market liquidity, the creditor-run game may have multiple equilibria (Liu (207)). Mathematically, the rt term on the LHS of (6)

13 may be decreaing in while the econd term i increaing in, o the function on the LHS with repect to may be non-monotonic and thu multiple olution to equation (6) are poible. In particular, a ect the likelihood that multiple equilibria exit. Write the LHS of (6) a function V ( ; ; ), where V ( ; ; ) = D( ) + k. igure 2 plot the function under a et of parameter value, where = :5, = 4, k = 2, = 0:6, R = :2, e = 0:, and = 0:. igure 2: unction V ( ; ; ) Lemma 3 follow. Lemma 3 Conider the limiting cae of 0. or a given, when i high enough, the equilibrium at t = i alway unique. or a given, when i high enough, multiple equilibria may exit. Proof. See Appendix. Even when the preciion of creditor private ignal approache in nity ( 0), multiple equilibria can till emerge. Equilibrium multiplicity arie becaue diveri cation make bank more imilar and thu introduce tronger trategic complementaritie among creditor of di erent bank. When market liquidity i endogenou, a creditor of a bank e ectively face a coordination problem not only with creditor of the ame bank but alo with creditor of ome other imilar bank. When increae, there are more of uch imilar bank and thu the coordination become more di cult, making multiple equilibria more likely. 2

14 3.3 Equilibrium Outcome or a given, we conduct analyi on two range of. We tart the analyi from = 0 (i.e., no diveri cation). To make the reearch quetion intereting, we focu on a u ciently high uch that when = 0, equation (6) admit a unique olution to for = N, B and E (ee Lemma 3). We have two reult, ummarized in Propoition and 2 below. Propoition When i mall enough, ceteri paribu, the equilibrium at t = i unique. In tate = E, greater diveri cation can lead to the feedback loop of a lower market liquidity and more creditor run k) > 0, E k) > 0, k) > 0). In tate = B, the E loop can be in the oppoite direction k) < 0, B k) > 0, k) > B Proof. See Appendix. The condition for realizing the equilibrium outcome in Propoition i that B i u ciently high and E i u ciently low uch that when = 0, the unique olution to equation (6) for = B and E ati e B < B and E > E. We alo chooe a u ciently high N uch a N < N 3 for the unique olution to equation (6) when = 0. 2 Propoition tate that diveri cation reult in more creditor run under a large hock and fewer creditor run under a mall hock. The mechanim i the following. irt, diveri cation reult in a higher degree of imilarity among bank and thereby their clutering action in the aet liquidation/ re ale market. Thi i characterized by j N( ; 2 2 ), where an increae in lead to a reduction in variance. Hence, in the large-hock tate, there i an increaed number of bank under re ale while in the mall-hock tate there i a decreaed number of bank under re ale. Therefore, diveri cation reult in market liquidity becoming higher in the mall-hock tate but lower in the large-hock tate, > 0 < 0 by (5). Second, there i a further feedback loop between market liquidity and creditor run a hown in Lemma 2. igure 2 illutrate the e ect in Propoition. Thi mean that in equilibrium there are le than 50% of bank failing for tate = B while there are more than 50% of bank failing for tate = E. 2 We are mainly intereted in tate = N and E. A for the tate = N, although qualitatively the poitive feedback loop a in tate = B alo applie, the magnitude of the feedback i mall becaue we have choen a u ciently high N uch a N N > 3 in equilibrium for = 0. That i, even under no diveri cation ( = 0), N N almot no bank will u er a creditor run (i.e., i cloe to 0). Hence, diveri cation ha almot no e ect. 3

15 igure 3: Ampli cation loop triggered by diveri cation in tate = B and E When increae further, equilibrium multiplicity become poible. Propoition 2 follow. Propoition 2 When i high enough, ceteri paribu, the equilibrium at t = can be uch that there i a unique equilibrium for tate = B and there are multiple (typically three) equilibria for tate = E. Proof. See Appendix. Propoition 2 highlight that when diveri cation i high enough, a u cient deterioration in fundamental ( ) of the aggregate tate can trigger a elf-ful lling ytemic crii (multiple equilibria). That i, under a high degree of diveri cation, the economy can experience a dicrete jump when the fundamental ( ) woren to a certain point: there i a regime change from equilibrium uniquene to equilibrium multiplicity and the bad equilibrium among the multiple equilibria can be realized. The jump correpond to the evidence dicued in the introduction that a dicontinuou drop in market liquidity i accompanied by ytemic creditor run on bank. igure 4 illutrate the e ect in Propoition 2. Rewrite the LHS of (6) a V ( h ; ; ) = n h io h k D( h ) +. Hence, the olution with repect to h to V ( h ; ; ) = give the bet repone function h = r ( ; ). igure 4 plot function h = r ( ; ). Under = B (a mall hock), the unique equilibrium i repreented by point A; under = E (a large hock), the equilibrium can be either B or B The curve for tate = E ha three interection with the 45 0 line. The middle interection correpond to an untable equilibrium. The other two correpond to table equilibria. 4

16 igure 4: Equilibrium multiplicity in tate = E 4 Equilibrium at t = 0 We move on to tudy the bank deciion at t = 0. We analyze the competitive equilibrium and the contrained econd-bet equilibrium, in order. 4. Competitive Equilibrium In the competitive equilibrium, every bank take the market liquidity (' k) a given (their rational expectation) in tate = N, B and E, while the market liquidity (' k) i the equilibrium outcome of the diveri cation () deciion of other bank. irt, let u conider the diveri cation deciion of an individual bank j. It objective i to maximize it expected equity value (or, equivalently, bank value) ubject to the participation condition (IR) of it creditor. In the limiting cae of 0, all creditor of a bank are in the ame poition ex pot, i.e., either all of them run on the bank or none of them doe o. We nd the participation condition of a creditor a follow: 2 R 0 = X 6 4 Z j ' k j d j = {z } bank fail at t= E min R; X j j j j d 7 j = 5 : {z } Z + bank urvive to t=2 (7) 3 5

17 The rt and econd term within the outermot bracket in (7) correpond to the two cae of j < and j >, repectively. In the cae of j <, the bank fail at t = and creditor divide the liquidation value L j = j ' k; that i, each creditor obtain L. In the cae of h i j >, the bank urvive to t = 2 and the debt value for each creditor i min R; X j. The h h i i expected debt value conditional on a realized j i E min R; X j j j with X j N( j ; 2 e). Ex ante, at t = 0, the bank know that j will be drawn from the ditribution j N( ; 2 2 ), j o the term d repreent the (unconditional) probability denity of j. Similarly, the expected equity value of the bank i given by X where the term max (X j 2 Z j R; 0) j j ] d 6 E [max (X j 4 j = {z } bank urvive to t= ; (8) R; 0) repreent the equity value at t = 2. Note that the bank equityholder obtain nothing if the bank fail at t = (i.e., when j < ). Bank j optimal diveri cation choice, j, i therefore given by the following program: " X Z j = arg max E [max (X j j =.t. (3) (creditor run) (7) (IR of creditor) ) j # R; 0) j j ] d (bank j peci c ( ; R) for a given ' k). (9) The objective function in (9) i to maximize the bank equity value. A for the contraint, creditor of bank j take ' a given, and chooe bank-peci c ( ; R) in repone to each that bank j (it owner) chooe. In other word, the contraint give the mapping (; ' ) ( ; R). We can rewrite the objective function in (9) and tranform Program (9) to an equivalent optimization problem: " X Z j = arg max.t. j = (3) (creditor run) (7) (IR of creditor) ( j ' j Z j # k) d + E (X j j j ) d j = ) (bank j peci c ( ; R) for a given ' k). The objective function in (0) i to maximize the total value of bank j (i.e., it debt value plu equity value). Thi i equivalent to maximizing the bank equity value, becaue creditor of a bank, in total, claim a contant reidual value, R 0. In fact, by adding the RHS of (7) multiplied (0) 6

18 by to (8), we have the bank value, which i exactly the term in the objective function of (0). Second, uppoe all other bank ` 6= j chooe a. The market equilibrium then determine the triplet ( ; ' ; R); in particular, market liquidity ' k i determined. That i, (3) (4) (7) 9 >= >; = () i identical to (3)-(4) and (7) with being replaced by. inally, by ymmetric equilibrium acro bank, we have (other bank determine ' k given their ). () j =. (2) Lemma 4 The competitive equilibrium at t = 0 that determine the optimal diveri cation i given by (0)-(2). X The rt-order condition of (0) implie 2 R j = ( j ' k) R j = j j + j 0 j 3 + j 0 j 3 A5 d j + ' k A5 d j d d d d where d d i the rt-order derivative of (), and () i the olution to the ytem of equation (3) and (7) in the contraint of Program (9) for a given '. In the rt-order condition of (3), individual bank j take ' a given (i.e., not a function of it choen ). (3) 3 = 0, Contrained Second-bet Equilibrium In the contrained econd-bet equilibrium, the ocial planner take the market liquidity ' k a endogenou and he recognize that diveri cation,, endogenouly impact on the market liquidity ' k. Her optimal diveri cation choice, SB, i given by X SB = arg max t. (3), (4), and (7). Z ( j ' j k) d j = {z } failing bank in the ytem at t= j E (X j j j ) d j = 7 5 {z } urviving bank at t=2 Z + 3 (4) 7

19 In Program (4), the objective function i to maximize the aggregate value of all bank in the economy, including the failing bank at t = (the rt term) and the urviving bank at t = 2 (the econd term). The appearance of the objective function in (4) i exactly the ame a that of (0) (i.e., the two objective function take the ame form), which make the comparion between the competitive equilibrium and the contrained econd-bet equilibrium meaningful. The contraint of Program (4) give the mapping ( ; ' ; R). Lemma 5 The contrained econd-bet equilibrium at t = 0 that determine the optimal diveri - cation SB i given by (4). X The rt-order condition of (4) implie R j = ( j ' k) j A5 2 + j 0 j d j + ' k 2 0 ( ) 3 + R j = j j A5 2 + j 0 j d j Z 6 + ( k) d' 4 j = d d j {z } internalizing the impact on market liquidity d d d d where d d and d' d are repectively the rt-order derivative of () and ' (), the olution to the ytem of equation (3), (4) and (7) in the contraint of Program (4) for a given. (5) 3 = 0, 7 5 Let u compare the two rt-order condition. In (5), the ocial planner internalize the price impact when deciding her, o there i an additional term in (5) relative to (3). Now we can compare and SB. Propoition 3 follow. Propoition 3 The competitive equilibrium can be higher or lower than the contrained optimum SB. That i, an individual bank either over-diveri e or under-diveri e. Proof. See Appendix. We dicu the intuition behind Propoition 3. 4 Baed on the dicuion in Propoition, it i eay to ee that diveri cation ha almot no e ect on a bank expected value for tate = N. So we only need to conider the two other aggregate tate ( = B and E). When an individual bank increae it diveri cation, it impoe an externality on other bank a it action ha an impact on market liquidity. When tate = B occur, the externality i poitive becaue greater diveri cation of thi particular individual bank reduce it likelihood of u ering a creditor run and 4 In both the competitive equilibrium and the econd-bet contrained equilibrium, we aume that agent will coordinate on the e cient equilibrium in the cae that there are multiple equilibria ex pot at t =. 8

20 hence it probability of undergoing re ale, which in turn reduce the preure on market liquidity and thereby decreae the probability of creditor run for all other bank. However, when tate = E occur, the force become oppoite and the externality i negative. 5 Eentially, greater diveri cation of an individual bank increae market liquidity in the good time but reduce market liquidity in the bad time, while market liquidity a ect the probability of creditor run for all other bank. Therefore, ex ante at t = 0, an individual bank either over-diveri e or under-diveri e in the competitive equilibrium compared with the contrained optimum. 5 Model Extenion - Imperfect Correlation of e i In the main model, for implicity, we have aumed that e i i perfectly correlated acro aet i. In thi ubection, we how that a long a e i cannot be completely diveri ed away, our model reult hold true. It i important to emphaize that in reality the number of aet (and bank) i not in nite and, therefore, e i cannot poibly be completely diveri ed away even when e i i independent acro aet. Now we conider the cae where e i i correlated acro i to ome degree. Speci cally, we aume that e i ha two component: e i = ^e + ~e i, where ^e N 0; 2^e, ~ei N 0; 2 ~e which i independent acro i, 2^e = 2 e, and 2 ~e = ( ) 2 e. That i, the rt component, ^e, i undiveri able while the econd component i diveri able, and meaure the proportion of the former. We can interpret ^e a the ytemic rik of the economy, which i undiveri able. Note that the benchmark cae in which e i i perfectly correlated acro i i a pecial cae of the current etup by etting = (i.e., both part of e i are undiveri able). Thi alternative aumption ha two impact on the main model. irt, invetor in the aet market who hold the diveri ed portfolio can diverify away a part of the rik aociated e i. That i, in Lemma, we have L j = j ' k, where k 2 e intead of k 2 e. Second, bank aet portfolio alo have a lower variance of payo. It i eay to how that the payo ditribution of bank j portfolio, conditional on it portfolio quality j, become X j N j ; 2^e ~e. (6) Clearly, greater diveri cation (a higher ) till decreae the conditional variance of X j (where 5 Wagner (200) model aume that there are only two aet and the upport of an aet payo i peci ed a [0; ]; a bank fail when it realized aet value i le than the (depoit) liability d. In addition, hi model implicitly aume parameter condition 2d. Under thee condition, the externality in hi model i alway negative. Technically, our model generalize hi model by conidering a continuum of bank and a continuum of aet. Conequently, the externality in our model can be poitive or negative, depending on the realization of the aggregate tate. 9

21 V ar(x j j j ) 2 [ 2 e; 2 e]). With thi new etup, equation (3) i replaced by where ^D( h j ; ; ) E min ' k = ^D( ) +, (7) i j j with the ditribution of X j being given by (6). Equation R; X j (4) doe not change. The ytem of equation (7) and (4) give the creditor-run equilibrium. Lemma 6 Suppoe e i i imperfectly correlated acro aet. The equilibrium at t = i given by the ytem of equation (7) and (4) under the limiting cae of 0. Under the u cient condition that i high enough, ceteri paribu, the concluion in Propoition and 2 do not change. Proof. See Appendix. 6 Concluion In the recent nancial crii of , the banking ytem experienced ytemic run by creditor. Thi paper explain why diveri cation at nancial intitution can be an important caue of ytemic bank run. We how that diveri cation, while making the nancial ytem more robut againt mall hock, increae the poibility of ytemic creditor run when a larger hock hit. In particular, diveri cation increae the likelihood of a elf-ful lling ytemic crii (multiple equilibria). The underlying mechanim i that diveri cation make nancial intitution more imilar in term of their aet portfolio holding and thu induce their clutering behavior in the aet liquidation market. Such clutering behavior reduce market liquidity after an advere aggregate hock and contribute to a viciou piral of a lower market liquidity and more creditor run in the ytem. Our model ugget that ex pot intervention meaure to mitigate ytemic creditor run include the injection of liquidity into the nancial ytem, which i crucial to breaking the viciou cycle of feedback. 6 Ex-ante policy meaure can include regulation on nancial intitution aet portfolio holding to prevent their over-diveri cation which can impoe a negative externality on the entire ytem. 6 See alo Liu (206). 20

22 Appendix A Proof Proof of Lemma : Conidering that the price L j fully reveal the fundamental j (ee the trading mechanim in Vive (204) and Benhabib, Liu and Wang (206)), an invetor doe not rely on hi private information in trading. Hence, all invetor are baically the ame (a a repreentative invetor). Thu, the objective function of () can be tranformed into maximizing = Z Z q j ( j q j ( j L j )dj L j )dj Z V ar e 2 Z 2 2 e q j dj q j dj 2 : The rt-order condition with repect to q j implie ( j L j )dj 2 e Z q j dj dj = 0; that i, R q j dj = j 2 e L j j ' k, where k 2 e.. Becaue R q j dj = ' by the market clearing condition, we have L j = Proof of Lemma 2: The ditribution of X j under a higher j ha rt order tochatic dominance over that under a lower j. Becaue function min R; X j h i i non-decreaing in X j and h i trictly increaing for ome range of X j, D( j ; R) E min R; X j j j i increaing in j. To reduce notational clutter, we drop the tate index upercript for now. Given that all other creditor of a bank ue threhold, the bank when realizing aet quality a j ha a j proportion of it creditor withdrawing. Moreover, the bank with realized aet quality j will have it aet liquidation value a L j = j of the bank failure, denoted by, i given by 'k 'k. Hence, by the nature of credit run, the threhold = : (A.) That i, the bank fail if and only if j < and individual creditor rationally anticipate thi. Given the bank failure threhold, an individual creditor threhold i given by the following 2

23 indi erence condition: Z + j = (D( j ) 0 ) j A = q + Z j = 0 j A, q + (A.2) where the prior of j i j N( ; 2 2 ) and. 2 2 Combining (A.) and (A.2) mean that in (A.2) i given by the implicit function (A.) or = + 'k. Write the LHS of (A.2) a Y L ( ; ). We tranform Y ( ; ) by changing variable to z = j q + Y L ( ; ) = and obtain Z z=z 0 where z 0 ati e the joint equation r D z (z) dz, By (A.3), we have = 'k r, z 0 + +, z 0 r + +, z 0 = + ( ) q + = + 'k j z = + + q + + ( ) = q + = j z=z 0 : = 'k 'k q + + z 0 + (A.3) 'k A : So it follow that lim z 0 = 0 'k : 22

24 Thu, under the limit 0 for a given, we have = and Z lim Y L ( ; ) = (D( ) ) 0 'k = (D( ) ) 'k : (z) dz Similarly, writing the RHS of (A.2) a Y R ( ; ), we have lim Y R ( ; ) = 0 Therefore, (2) i proved. 'k. We alo need to prove that a creditor roll over when hi ignal i higher than and otherwie withdraw. An individual creditor take a given. Writing the LHS minu the RHS of (A.2) a ~Y ( ) and changing variable to z = j 8 >< ~Y ( ) = >: Z + z= r + Z r + z= We ~ Y ( ) =@ > 0. It i traightforward to q + h q D + z + (z) dz, we obtain + + > 0 in (4). ^Y ( ; ') ( 'k) (D( ) + ) = 0, + > 0 in (3), write (3) a i (z) dz 9 >= : >; and we > 0 < 0, > 0. Proof of Lemma 3: It i eay to how Write the LHS of (6) a function V ( ; ; ), where V ( ; ; ) = D( ) + ( ; ; = k D( ) D ( ) k. (A.4) or a given, when ( ;; > 0 hold for any ; when i high ( ;; > 0 alo hold for any, o V ( ; ; ) = ha a unique olution with repect to. 23

25 or a given, when i cloe enough ( ;; < 0 around =, conidering that lim k j = =. Alo, under being cloe enough ( ;; > 0 when i u ciently higher or lower than. That i, when i cloe enough to, V ( ; ; ) i non-monotonic in : it i increaing in initially, and then decreaing in around =, and then increaing in again, a the non-monotonic curve in igure 2 depict. Thu, equation V ( ; ; ) = typically admit three olution. Proof of Propoition : By Lemma 3, under = 0, V ( ; ; ) = ha a unique olution with repect to a long a i high enough. Hence, given uch a, when i u ciently cloe to 0, V ( ; ; ) = till admit a unique olution with repect to. We alo ( ; ; = D( ) + 2 k. ( ; ; ( > 0 < 0 if < if > : Applying the implicit ( ;; ( ;; )=@. Hence, the following comparative tatic i < 0 > 0 if < if > : Note that we focu on the equilibrium uch that when = 0, B < B and E > E. Hence, when increae, B i decreaing and move further below B while E i increaing and move further above E. Proof of Propoition 2: Note that V ( ; ; ) i increaing in. Hence, when i increaing, the non-monotonic curve in igure 2 hift upward. Therefore, by Lemma 3, when i high enough, it i a poible that for a lower equation V ( ; ; ) = admit multiple olution with repect to, and for a higher equation V ( ; ; ) = admit a unique olution with repect to. Proof of Propoition 3: The two rt-order condition, (5) and (3), are di erent. So the two program have di erent optimum. can be higher or lower than SB. or ome et of parameter value, i higher than SB. 24