Money Market Funds Intermediation and Bank Instability

Transcription

1 Fedeal Reseve Bank of New Yok Staff Repots Money Maket Funds Intemediation and Bank Instability Maco Cipiani Antoine Matin Buno M. Paigi Staff Repot No. 599 Febuay 013 Revised May 013 This pape pesents peliminay findings and is being distibuted to economists and othe inteested eades solely to stimulate discussion and elicit comments. The views expessed in this pape ae those of the authos and ae not necessaily eflective of views at the Fedeal Reseve Bank of New Yok o the Fedeal Reseve System. Any eos o omissions ae the esponsibility of the authos.

2 Money Maket Funds Intemediation and Bank Instability Maco Cipiani, Antoine Matin, and Buno M. Paigi Fedeal Reseve Bank of New Yok Staff Repots, no. 599 Febuay 013; evised May 013 JEL classification: G01, G1, G3 Abstact In ecent yeas, U.S. banks have inceasingly elied on deposits fom financial intemediaies, especially money maket funds (MMFs), which collect funds fom lage institutional investos and lend them to banks. Intemediation though MMFs allows investos to limit thei exposue to a given bank. Howeve, since MMFs ae themselves subject to uns fom thei own investos, a banking system intemediated though MMFs is moe unstable than one in which investos inteact diectly with banks. The mechanism though which instability aises in an MMFintemediated financial system is the elease of pivate infomation on bank assets, which is aggegated by MMFs and can lead them to withdaw en masse fom a bank. Key wods: Money maket funds, bank uns Cipiani, Matin: Fedeal Reseve Bank of New Yok ( maco.cipiani@ny.fb.og, antoine.matin@ny.fb.og). Paigi: Univesity of Padova (bunomaia.paigi@unipd.it). The authos gatefully acknowledge hospitality fom the Euopean Univesity Institute, whee pat of this pape was witten, and also thank Jamie McAndews, Todd Keiste, Enico Peotti, and semina audiences at the Nethelands Cental Bank and the Bank of Fance, the Maco, Money, and Intenational Economy Confeence at CESIfo Munich, the ECB/Bundesbank Joint Lunch Semina, the Univesity of Ben, and the Fankfut School of Finance fo useful comments. The views expessed in this pape ae those of the authos and do not necessaily eflect the position of the Fedeal Reseve Bank of New Yok o the Fedeal Reseve System.

3 1 Intoduction In ecent yeas, lage global banks have inceasingly elied on deposits fom nancial intemediaies, especially money maket funds (MMFs). MMFs mainly collect funds fom institutional and wholesale investos and lend them to banks. Bank deposits of institutional and wholesale investos ae not fully coveed by deposit insuance. As a esult, they need to limit thei exposue to a single banking institution and divesify thei potfolio of deposits. Intemediation by institutions such as MMFs allows lage investos to eap gains fom divesi cation, while saving on bank-monitoing costs. In the U.S., MMFs have become a vey popula nancial instument, compising oughly 0 pecent of all mutual fund assets at the end of 01. Thei assets unde management gew fom appoximately $ tillion in 005 to $3 tillion at the end of 008, although it contacted duing the nancial cisis (to $.6 tillion at the end of 01). 1 MMFs ae key povides of shot-tem funding, especially to the nancial secto. As Table 1 shows, in 01 they wee among the lagest investos in some asset classes, nancing 43 pecent of nancial commecial pape and 9 pecent of ceti cates of deposit. Table 1: MMF Investments by Asset Classes Non nancial Financial Assetbacked of Deposit Agee- Ceti cates Repuchase CP CP commecial pape ments (ABCP) 43% 43% 38% 9% 33% 75bn 07bn 117bn 434bn 591bn As a pecentage of outstanding assets. June 01. Souce: McCabe et al. (01). In the U.S., MMFs o e demandable deposits (shaes) edeemable at pa, that is, with xed net asset value (NAV). When the NAV (i.e., the value of the asset pe shae) falls below $0.995 ("beaks the buck"), the MMF is foced by SEC egulation to e-pice all its shaes. Hence, even small losses can stat a un since investos have an incentive to edeem thei shaes befoe the MMF beaks the buck. In Septembe 008, the Reseve Pimay Fund boke the buck due to its exposue to Lehman pape, causing a stampede of withdawals acoss the secto. To stem the panic, the Fedeal Reseve povided a lage amount of liquidity though emegency facilities and the Teasuy Depatment guaanteed MMF liabilities. Figue 1 shows the shap decease of asset unde management by MMFs in the U.S. afte Septembe 008; impotantly the decease is almost entiely due to the behavio of institutional investos. A simila phenomenon was obseved in August 01 when institutional investos withdew fom U.S. MMFs due to thei concens about potential losses fom exposue to Euopean banks. 1 Fo a desciption of the MMF industy, see McCabe et al. (01). 1

4 Figue 1: Assets unde management in pime U.S. MMFs. A banking system intemediated though MMFs can be moe unstable than one in which lage investos inteact diectly with banks. Since MMFs ae themselves subject to un-like edemptions fom thei own investos, they may eact to them by unning the banks in which they have deposited, hence amplifying the impact of the initial edemptions. The instability of a nancial system in which banks nance themselves though intemediaies such as MMFs is one of the diving foces behind the ecent efom e ot of the MMF industy by the SEC and the Financial Stability Ovesight Council (FSOC). In this pape, we study an economy à la Diamond and Dybvig (1983) (DD heeafte) with two banks, whose long-tem investments have stochastic and (pefectly) negatively coelated etuns. Depositing in the two banks allows agents to educe thei isk though divesi cation. We conside two maket stuctues: diect nance, whee investos deposit diectly into the banks, and MMF intemediation, whee the elationship between investos and banks is intemediated though MMFs. 3 In the model, bank bankuptcy may aise when a faction of investos unexpectedly withdaw thei funds. The withdawal occus eithe because investos have eceived a liquidity shock, o because they have eceived a pefectly infomative (negative) signal on the etun of the investment of one of the two banks. Unde diect nance, unexpected withdawals cause bank bankuptcy only if the amount withdawn is lage enough to foce the bank into liquidation. In contast, with MMF intemediation, when a faction of investos unexpectedly edeem fom the MMF, thei actions See, fo instance, Dudley (01) and Geithne (01). 3 MMFs lend to banks mostly though unsecued commecial pape and othe shot-tem investments (see Table 1). Nevetheless, ou model captues the essential economic featue of shot-tem debt ollove though MMFs decision to eithe keep o withdaw the money fom the banks.

5 epesent a (noisy) signal on the state of the wold fo the MMF. If this signal is stong enough, the MMF will un the bank, withdawing all its funds and causing bankuptcy even if the faction of the unexpected edemptions was small enough that bankuptcy would not have occued unde diect nance. The instability of MMF intemediation stems fom the fact that the negative infomation content of an unexpected edemption fom an intemediay such as an MMF ampli es the e ect of the edemption itself. Because of this, an economy intemediated by MMFs is moe unstable than a diect- nance stuctue. The ampli cation mechanism stems fom the fact that MMFs themselves ae subject to un-like edemptions because they o e investos demandable liabilities in ode to satisfy thei liquidity needs. When an MMF expeiences lage unexpected edemptions, it uns the bank to potect all its investos, and not just those initiating the edemptions. Because of the bank s xed pomise, the MMF, eceiving negative infomation on the bank s assets, obtains a highe payo fo its investos if it uns than if it does not. Since the withdawals of funds fom the investos may be due to liquidity as opposed to infomative easons, bank bankuptcy may cause ine cient liquidation and a eduction in welfae. The ampli cation esult mentioned above may be moe geneal than the case of MMF to which we apply it in this pape: a delegated monito leans fom the behavio of the agents it epesents and by using this infomation may amplify the e ect of thei actions. This ampli cation may be ine cient if the delegated monito takes value-destucting actions to incease the welfae of the agents it epesents (fo instance, in ou model, the MMF foces the liquidation of banks assets to incease the consumption of its shaeholdes, although this destoys value in the banks). Note that in ou economy, the un on the banking system does not occu because of agents failue to coodinate as in DD. Hee, instead, banks fagility stems fom stochastic uctuations of the economy, which can be due to a liquidity shock as in Allen and Gale (000); to the elease of infomation on banks asset etuns as in Jacklin and Bhattachaya (1988); o to a combination of both as in Chai and Jagannathan (1988) and in ou model. This is consistent with the featues of the MMF industy. Indeed, the MMF un that stated when the Pimay Reseve Fund, which had invested in Lehman pape, boke the buck in Septembe 008, ts moe the de nition of a un caused by the sudden elease of infomation in the maket than an instance of self-ful lling panic unelated to economic fundamentals. Similaly, the "quiet un" by U.S. MMFs on Euopean banks in the summe of 01 was due to concen by U.S. investos on the assets held by Euopean institutions (see Chenenko and Sundeam, 013). Section descibes ou model and chaacteizes the economic function of MMF intemediation. Section 3 studies the e ect of an unexpected withdawal of funds fom the nancial system. Section 4 shows that an MMF-intemediated nancial system is moe fagile than one with diect nance. Section 5 concludes. The poofs ae in the appendix. 3

6 Figue : The economy with diect nance. The Model.1 Technology and Pefeences We descibe ou economy st with diect nance and then with MMF intemediation. Thee ae two egions, A and B. In each egion, thee is a continuum of (wholesale o institutional) 4 investos of mass M, which can be intepeted as uninsued wholesale investos, 5 fo a total population M. Each investo is endowed with one unit of unique good. In each of the two egions thee is one bank, Bank A and Bank B. The stuctue of the economy is depicted in Figue. Thee ae thee dates, 0, 1, and, and a unique good that can be consumed, stoed, o invested. Eveyone in the economy can use stoage, which etuns one unit of the good at date t+1, fo each unit invested at date t, t = 0; 1. In contast, the investment technology is available only to banks. We conside an economy whee the etuns of the investments of two banks can be eithe high o low and ae pefectly negatively coelated. We assume this in ode to maximize the gains fom divesi cation and simplify the model. The etuns of the investment technology of the two banks pe unit invested ae as follows: Retun Retun Bank A Bank B Pobability 1/ R H Pobability 1/ R H with R H > 1 >. Since fo each bank the pobability of the investment technology yielding a 4 We focus on the behavio of institutional investos (who lack insuance both when they invest in MMF and, lagely, when they deposit in banks) since, as shown in Figue 1, they wee the main dives of MMF fagility duing the ecent cisis. 5 Agents supplying funds to banks ae nomally efeed to as depositos, who deposit o withdaw thei funds. In contast, agents supplying funds to MMFs ae nomally efeed to as investos, who puchase o edeem shaes of the MMF. In ode not to saddle the eade, fom now on we will use the tem "investos," who may "supply" o "withdaw" thei funds fom the MMF o fom the bank in efeence to both diect nance and the MMFintemediated economy. 4

7 high o a low etun is 1, the net pesent value of a unit of investment is the same fo the two banks. As a esult, it is optimal to supply an equal amount of funds to both banks at date 0. Investment can also be liquidated at date 1, in which case it etuns 0 pe unit invested. 6 In each of the two egions, investos ae subject to pefeence shocks: with pobability investos must consume at date 1 ( impatient investos), and with pobability 1 they must consume at date ( patient investos). The ealization of the shock to thei pefeences at date 1 is pivate infomation. Fo simplicity s sake, we assume that investos have logaithmic utility function, so that thei expected utility is log (c 1 ) + (1 ) log (c ) ; whee c 1 and c denote date-1 and date- consumption, espectively. Fom the law of lage numbes, a faction of agents consume at date 1 and a faction (1 ) at date :. The Optimal Contact with Diect Finance As is standad in this liteatue, banks ae subject to a zeo-po t condition and, unde diect nance, choose the contact to maximize the expected utility of thei investos. To simplify notation, we expess all quantity vaiables pe unit supplied to the banking system. In paticula, we denote by i the total investment pe unit by the two banks. Moeove, we assume that R H + > 1; (1) that is, the expected net pesent value of each bank s investment is positive. This condition, as we show in the appendix, guaantees that the optimal level of investment i is positive, since the isk of banks long-tem technologies can be completely divesi ed away. The optimal contact and the optimal investment level ae: c 1 = 1; c H = R H ; c L = ; and i = 1 ; whee c H and c L epesent date- consumption if the bank has a high and a low etun, espectively. 7 The optimal contact implies that the banks stoe enough funds to satisfy withdawals fom impatient investos only and invest all the emaining funds in the longtem technology. Note that since banks have pefect negatively coelated etuns, unde the optimal contact, investos deposit an equal amount in each bank; the banks, in tun, invest a faction 1 in its long-un technology, thus allowing patient investos a deteministic etun c H + c L = RH + : The divesi cation oppotunities that aise fom investing in both banks may tun into a souce of fagility if a bank is not viable on its own given the contacts that it o es. Fomally, this occus if patient investos do not want to withdaw funds fom one bank only and wait in the othe vesus 6 The assumption is natual if we intepet the banks as investing in nancial assets at di eent matuities such as a loan that it ties to sell in an unmodeled maket; because of asymmetic infomation, maket paticipants may not want to puchase the loans at a pice >. 7 See the appendix fo the deivation. 5

8 withdawing fom both banks. That is, if: log(0:5c 1 + 0:5c 1 ) > 0:5 log(0:5c 1 + 0:5c H ) + 0:5 log(0:5c 1 + 0:5c L ): () Given the optimal contact, condition () becomes: log(1) = 0 > 0:5 log(0:5 + 0:5R H ) + 0:5 log(0:5 + 0:5 ); which is satis ed as long as R H < 3 RL 1 + R : (3) L Intuitively, condition (3) equies that R H cannot be geate than (o equal to) 3 RL 1 + R because, L othewise, each bank would be so po table that the contact it o es can stand on its own. Note that condition (1) fo an inteio solution fo i and the condition (3) that contacts ae not viable sepaately establish a ange fo R H : which has a solution fo any. 8 < R H < 3 RL 1 + ;.3 MMF Intemediation The stuctue of the economy with MMF intemediation is simila to the one unde diect nance except that, in each egion A and B, thee is one MMF MMF A and MMF B which channels the funds of its egion to the two banks. Each MMF maximizes the expected utility of its investos by investing in banks deposits (ecall that only banks can invest in the long-tem technology) and/o into the stoage technology. The stuctue of the economy with MMF intemediation is depicted in Figue 3: The isk-divesi cation poblem does not change when we intoduce MMFs in the economy. As a esult, unde the optimal contact, the nal consumption fo impatient and patient investos is the same as with diect nance. It is easy to show that this can be accomplished as long as the contacts that the two banks o e to the MMFs ae the same as those o eed to the wholesale investos with diect nance. Analogously, the contacts that the MMFs o e to thei investos simply aggegate the payouts fom the two banks: the contact pe unit invested that each MMF o es is c MMF 1 = 1 and c MMF = RH + : That is, the MMFs o e thei investos claims edeemable at pa at date 1. Finally, MMFs must shae all the funds they collect fom thei investos equally between the 8 The equality: = 3 RL 1 + ; has two equal oots = 1; and it is always satis ed fo > 0. 6

9 Figue 3: The economy with MMF intemediation. two banks. In ode to undestand the ole of MMF intemediation in the economy, let us conside the case in which banks must be monitoed o sceened othewise, thei etun is zeo at date. The need to monito the banks could aise fom the fact that the opacity of bank loans allows them to undeepot the etun to the long-tem technology o o es scope fo moal hazad to bank manages. It is well known that since monitoing has a xed-cost dimension, the duplication of monitoing costs that diect nance entails may be educed when funds ae intemediated though a delegated monito (Diamond 1984). Fo instance, let us assume that the cost to monito each bank by a deposito o by an MMF is ". Unlike banks, MMFs do not need to be monitoed because they invest in deposit contacts o eed by banks, which ae less opaque than loans. Of couse this is an exteme assumption: it can be thought of as the esult of the nomalization of the highe cost to monitoing banks vesus MMFs, which is geneally the case since MMFs invest in xed-income secuities (see Table 1). With diect nance, in each egion, depositos, who have mass M; have funds M to deposit, and theefoe spend (")(M) in monitoing costs. As a esult, the amount of deposits coming fom the investos of each egion is M (")(M) = (1 ")(M). It is immediate to show that depositos will deposit (1 ")(M) in each bank to maximize gain fom divesi cation. With MMF-intemediated nance, each MMF collects M fom the depositos of its egion only. Similaly to depositos, each MMF must monito the banks to pevent them fom unde-epoting the etun to the long-tem technology and o to pevent moal hazad by banks manages. Hence each MMF spends " to monito the banks. This cost is passed onto the depositos. Theefoe, with MMF intemediation, the funds that depositos of each egion deposit in thei MMF and that the MMF invests in the banks ae M "; which is geate than (1 ")M as long as M > 1. Hence, MMF intemediation is potentially valuable in saving monitoing costs to depositos, and as a esult, in inceasing thei level of consumption, while at the same time allowing them to enjoy 7

10 the gains fom divesi cation. 9 Since monitoing costs ae highe unde diect nance than unde MMF intemediation, the aggegate amount of funds invested in the banking system will be lowe. In the est of the pape, howeve, we disegad this and cay out ou analysis pe unit deposited in each bank. None of ou esults is a ected by this, since as we will show late, the occuence of a un depends on the popotion of funds (and not by the amount of funds) unexpectedly withdawn ealy by investos. 3 An Unexpected Withdawal of Funds 3.1 Infomation Aival The fagility of an MMF-intemediated system can be captued by consideing the e ect of an unexpected withdawal of funds in the economy with diect nance and in that with MMF intemediation. Let us assume that at date 1 some patient investos unexpectedly withdaw thei funds. Similaly to Chai and Jagannathan (1988), the unexpected withdawal can happen eithe because of infomation easons o because of a shock to agents pefeences. That is, investos unexpectedly withdaw ealy eithe because they have eceived a liquidity shock, i.e., some peviously patient investos become impatient and must consume at date 1, o because they have eceived a pefectly infomative signal that the etun of the investment of the bank in thei egion is : In eithe case the withdawal stems fom a shock to the fundamentals of the economy. Remembe that wholesale investos deposits and MMF deposits into banks ae (lagely) not coveed by deposit insuance in the U.S.. Because of this, both investos and MMFs have an incentive to eact to negative news about a bank s investment. This withdawal, which is unexpected at date 0 and hence is in excess of the liquidity available at date 1, has a di eent impact on the stability of the system unde diect nance and unde MMF-intemediated nance. Unde diect nance, the faction of funds withdawn in excess of may be su ciently low so as not to push the bank into insolvency and theefoe not to alte the equilibium descibed above. Howeve, when investos unexpectedly edeem fom the MMF, thei actions epesent a noisy signal on the state of the wold fo the MMF to intepet. Fom the size of the unexpected edemptions in excess of, the MMF will update its pio belief on the etun of the long-tem investment of the bank in its own egion, and it may un that bank by pulling all its funds away, theeby pushing it into bankuptcy. Moe fomally, we assume that a positive measue of patient investos q fom egion A unexpectedly withdaw thei funds at date 1 fom Bank A o, unde MMF intemediation, fom the MMF in egion A: 10 This assumption is in the spiit of Allen and Gale (000), who conside the 9 Note also that divesi cation can be achieved though the intebank maket; it is easy to show, howeve, that MMF intemediation saves on monitoing costs also with espect to intebank nance. 10 Of couse, since eveything is symmetical, nothing would change if the unexpected withdawal of funds occued 8

11 ealization of an additional state of natue that was assigned a pobability zeo at date As a esult of the shock q, the total amount of withdawals at date 1 fom egion A s investos is + : Note that, since we assumed that condition (3) holds, that is, that Bank B s contact is not viable on its own, agents eceiving negative infomation about Bank A will also withdaw fom Bank B in the diect- nance case. 1 In a DD economy, a un occus because of agents failue to coodinate. Hee, instead, similaly to Jacklin and Bhattachaya (1988), Chai and Jagannathan (1988), and Allen and Gale (000), the fagility of the nancial system stems fom stochastic uctuations of the economy. In othe wods, we focus on "essential" bank uns, which cannot be uled out by assuming that agents ae able to coodinate. In the emainde of the pape, we follow the appoach used by Allen and Gale (000): we assume that agents ae able to coodinate on the non-un supeio equilibium when it exists; and we study whethe this equilibium is moe esilient to shocks unde diect nance o unde MMF-intemediation. 3. Infomation Aggegation and Updating As we mention befoe, the unexpected withdawal of funds can stem fom a liquidity shock o fom the elease of negative infomation on Bank A s assets. Similaly to Chai and Jagannathan (1988), we assume that the pobability that the unexpected withdawal is infomative is inceasing in q: P (fshock due to infomational easong) = f(q); f 0 (q) > 0 that is, the highe the faction of withdawals in excess of, the moe likely it is that it happens fo infomation easons. This is consistent with the idea that when thee is a elease of negative infomation to the maket, we obseve lage withdawal of funds. Note that in ode to keep the algeba simple, fom now on we will conside the case f(q) = q that is, the pobabilities that the unexpected withdawal is infomative o that it is due to a pefeence shock ae q and (1 inceasing function f(q). q); espectively. 13 All the esults we pesent, howeve, hold fo any Unde MMF intemediation, MMF A sees the unexpected withdawal of funds by its investos in egion B: 11 Moe ecently, Gennaioli et. al (01) agued that investos may not take into consideation cetain highly impobable isks, such as the pobability that the shae pice of a money maket mutual fund may fall below 1. 1 Note that if patient investos ae negatively infomed they will always nd it convenient to withdaw because c 1 > c L (as we explain below, even if Bank A goes into bankuptcy, infomed patient investos will always be able to ecoup c 1 ): 13 In an addendum available on equest fom the authos, we show that f(q) = q can be deived fom a simple infomational stuctue. 9

12 and intepets this as (impefect) bad news on the etun of the assets of Bank A. In paticula, afte obseving the unexpected withdawal ; the MMF A updates the joint pobabilities that Bank A and Bank B have a high o a low etun in the following manne: 14 P (RA H ; RB H jq) = 0(q) + 0(1 q) = 0; P (RA H ; RBjq) L = 0(q) + 0:5(1 q) = 0:5(1 q); P (RA; L RBjq) L = 0:5(q) + 0(1 q) = 0:5q; P (RA; L RB H jq) = 0:5(q) + 0:5(1 q) = 0:5: Note that when the shock is infomative, the elease of infomation is not about the state of natue; athe, it is about the etun of one of the two banks (Bank A). 15 In othe wods, the pobability of the etun of Bank B being high o low is not a ected by the aival of negative infomation on Bank A. In fact, afte obseving (1 bad emains )q, the conditional pobability of Bank B being good o P ( Bjq) = P (R H B jq) = 0:5: Nevetheless, since the contact that Bank B o es is not viable on its own, because of condition (3), the destuction of divesi cation oppotunity stemming fom the elease of infomation on Bank A may also send Bank B into bankuptcy, an issue that we will analyze below. The fact that unexpected withdawal is an impefect signal on the bank long-tem investment geneates confounding as in Chai and Jagannathan (1988). If withdawals wee always due to the elease of infomation, any ealization of q, howeve small, would geneate the collapse of Bank A and also the collapse of Bank B because of assumption (3). Finally, we assume that the MMF does not lean the ealization of the banks asset etun duing its egula activity of monitoing. As will be appaent fom the est of the pape, should we emove this assumption, the fagility of an MMF-intemediated system would be even geate. 3.3 Bankuptcy The excess withdawal of funds mattes because it may cause bank bankuptcy. To study its impact on the banking system, we need to make some assumptions on how the banks assets ae split in case of bankuptcy. In paticula we assume that: - Banks abide by the sequential sevice constaint when facing withdawals at date 1, both unde diect nance and in the MMF-intemediated economy. - Patient investos withdawing thei funds ealy do so at the beginning of the queue. This captues the notion that since they ae potentially infomed about bank asset etuns, they ae 14 In ode to popely de ne the pobabilities conditional to the unexpected excess withdawal q, one can think of the unexpected excess withdawal as being an event with a low enough pobability that the optimal contacts (unde both stuctues) ae abitaily close to those descibed in the pevious section. 15 In othe wods, the zeo-pobability event can be thought of as a change in the etuns to the long-un technologies in the two states of natue, which become ( A ; RL B ) and (RL A ; RH B ): 10

13 able to jump ahead of the line. 16 The assumption e ects the fact that institutional investos ae pone to un in a cisis (See Figue 1). - Analogously and fo the same easons, if one MMF makes unexpected withdawals, it is st in the queue with espect to the othe MMF. 17 Bankuptcy in eithe maket stuctues is esult of an essential bank un, that is, a un due to eithe the elease of pivate infomation o to a shock to investos pefeences. It is not the esult of a sunspot unelated to economic fundamentals (e.g., a wave of pessimism), but stems fom the unexpected withdawal of funds (1 )q by patient investos and, in the MMF-intemediated economy, fom the infomation that such a withdawal conveys to the MMF. We now contast the e ect of the unexpected withdawals of funds (1 ) q; unde diect nance and in a MMF-intemediated economy, and we study how it may cause bank uns in the two economies Bankuptcy with Diect Finance In the case of diect nance, the unexpected withdawal of funds fom Bank A will push it into bankuptcy if the popotion q of patient investos who withdaw thei funds at date 1 is such that: ( + (1 ) q)c 1 > 1 i + i: (4) That is, bankuptcy will occu when the bank s date-1 liabilities, pe unit deposited in the bank, (LHS of 4) exceed its date-1 assets (RHS). 18 Given the optimal contact descibed above, condition (4) becomes: + (1 ) q > + (1 Hence, the bank goes bankupt if and only if q > : Note that since q (0; 1) ; a necessay condition fo bankuptcy to occu unde diect nance is ): < 1 : (5) Fom now on, we concentate on ealizations of q such that the bank does not go bankupt with diect nance, and we show that the same ealizations of q may instead tigge bankuptcy unde MMF intemediation. That is, we assume: q : (6) 16 Note that liquidity withdawes will also ty to jump ahead of the queue as they ae awae that the bank/mmf may not be able to seve latecomes if thee ae excess withdawals at date These assumptions allow us to chaacteize the equilibium in the economy in the simplest possible way. As will be clea, howeve, the fagility of an MMF-intemediated economy does not stem fom the paticula bankuptcy assumption that we adopted, but fom the ability of MMFs to aggegate infomation among thei investos. 18 (1 ) (1 ) Obseve that only q patient investos withdaw fom Bank A since the othe patient investos have deposited in Bank B. 11

14 Let us make two obsevations. Fist, since the popotion of patient investos who unexpectedly withdaw at date 1 ae ealy in the queue, they ae able to withdaw c 1 = 1 as long as + (1 ); (7) whee the LHS of (7) is the amount of funds withdawn at date 1 by the patient investos, and the RHS ae the bank s assets at date Obviously, as long as the level of withdawals is such that the bank is not pushed into bankuptcy, that is, as long as q, the bank is always able to pay c 1 = 1 to the patient investos withdawing thei funds ealy. As a esult, as mentioned above, a patient investo knowing that the etun of Bank A s assets is low (because he eceived the infomation shock) nds it optimal to withdaw since c 1 > c L. Second, if bankuptcy occus, impatient investos (fom both Banks A and B) do not necessaily get c 1 since thee ae not enough esouces in the bank, even afte liquidating all the long-tem assets. Moeove, even if bankuptcy does not occu, patient investos do not eceive the optimal contact at date since some (o all) of the long tem investment has been liquidated. As a nal emak, obseve that since, then Bank A will neve liquidate all its long-tem assets when it obseves an excess withdawal of funds; it only liquidates whateve is needed to epay the popotion of patient investos who withdaw thei funds ealy Bankuptcy with MMF Intemediation Even if both the MMFs and the banks issue the same claims demandable at pa at date 1, upon obseving unexpected edemptions MMF A behaves di eently fom the bank in the diect- nance case. In fact, the MMF can withdaw its funds fom Bank A at the contact c 1 = 1, while in diect nance, when Bank A liquidates ealy to meet the unexpected withdawal, it does so at < 1. Theefoe, if afte obseving the unexpected edemptions, MMF A believes that Bank A 0 s etun is low with high enough pobability, it withdaws all its funds, and not only what is needed to meet the unexpected edemptions. This ampli cation mechanism makes the MMF-intemediated stuctue moe unstable than diect nance. Of couse, the fact that MMF A withdaws all its funds fom Bank A does not necessaily imply that Bank A is bankupt, which only happens when: ( + which, given the optimal contact, becomes: (1 ) )c 1 > 1 i + i; 19 The condition (7) ; when computed fo the highest possible level of withdawal, i.e., fo q = 1; becomes: which is always tue fo > 1 3 : (1 ) < ( + (1 )); 1

15 (1 ) + > + (1 ); o < 1 : Note that < 1 is the same as condition (5), which makes bankuptcy possible in diect nance fo a high enough ealization of q. 3.4 The MMF Reaction to an Unexpected Withdawals of Funds We now investigate the continuation equilibium of MMF A afte obseving an unexpected withdawal of funds: MMF A will be able to o e c 1 to all its investos withdawing thei funds at date 1 as long as: ( + ) c 1 ( + (1 )); (8) whee the LHS of (8) is the oveall withdawal of funds fom MMF A, and the RHS is the value of the combined assets of both banks A and B at date 1. MMF A is able to o e c 1 because it has demandable claims on both banks and it fee ides on the claims of the othe MMF. Since c 1 = 1 unde the optimal contact, (8) becomes + (1 ): (9) Note that since we ae only consideing ealizations of q such that the banking system does not go bankupt unde diect nance 0 (i.e., q ), condition (9) becomes: (1 ) + (1 ); which, as in the case of the analogous condition with diect nance, is always satis ed. As a esult, in MMF A both impatient and patient investos edeeming ealy eceive c 1. This allows us to study the eaction of the MMF upon obseving the unexpected edemptions disegading the welfae of the investos edeeming ealy. In paticula, MMF A must choose the popotion by which it meets the unexpected edemptions (1 )q by withdawing funds fom Bank A and fom Bank B. These popotions, which we denote by and (1 0 In contast, fo q = 1, the condition would be: ); ae the esults of the MMF e-optimization upon obseving the unex- 1 < + 1 > + Note that, in this case, if the MMF withdaws all its funds ealy at the ate c 1 fom Bank A and B, the banks will neve go bankupt. This is because all thei combined assets ( + (1 )) ae equal to o geate than the MMF maximum withdawal, which is equal to + (1 ): 13

16 pected withdawal of funds. Since the MMF knows it is able to pay its investos edeeming ealy the amount c 1 = 1; the popotions and (1 ) ae deived only by looking at the welfae of the emaining patient investos (i.e., (1 )(1 q)). Note that although the excess withdawal of funds occus in egion A (and, with pobability q, it e ects bad infomation on Bank A s longtem investment), in geneal the MMF will decide to meet the unexpected edemptions by pulling funds fom both banks. The eason is that although the expected etun on Bank A assets has deceased (wheeas that on Bank B assets has not), it may not be optimal to meet the unexpected edemptions exclusively fom Bank A, as the two banks povide a hedge one against the othe. The optimal is given by 0 B = ( RH 1)(1 q) + ( R H +1) ( q)( RH 1) 1 q 1 C ; 1A : (10) In the inteest of space, we do not epot the deivation of the optimal level of in the text of the pape, but descibe it in the appendix. Note that, as shown in the appendix, if q = 0; then = 1 ; that is, in the limit, when the unexpected withdawal of funds does not contain any infomation on Bank A, it is met by withdawing funds equally fom both banks. Note also that if q = 1; then = 1; that is, if the unexpected withdawal is so high that the MMF knows that the etun on Bank A assets is low, it is met by withdawing fom Bank A only. Finally, ecall that the total amount withdawn at date 1 by MMF A is +(1 )q. Since both banks have invested in the long-tem asset a faction i = 1 of each unit deposited, the oveall liquidation of the long-tem assets pe unit deposited in the banking system is : Given the optimal faction withdawn by MMF A fom Bank A; the withdawal pe unit of deposit fom Bank A is ( + ) and that fom Bank B is (1 )( + ): Moeove, the amount of Bank A 0 s assets liquidated at date 1 is ; and the amount of Bank B 0 s assets liquidated at date 1 is (1 ) : Because of the liquidation of both banks assets to meet the unexpected withdawal of funds, the payo s that the banks can a od to o e at date will change; the payo s o eed by Bank A ae: and those o eed by Bank B ae: bc H ;A = max(r H (1 ); 0); (11) bc L ;A = max( (1 ); 0); (1) bc H ;B = max(r H (1 (1 ) ); 0); (13) bc L ;B = max( (1 (1 ) ); 0): (14) 14

17 4 The Fagility of MMF Intemediation Having chaacteized the continuation equilibium conditional on the unexpected withdawals of funds at date 1, we now show that thee ae levels of withdawals and edemptions such that thee is no bank bankuptcy with diect nance, but bankuptcy occus with MMF intemediation. We ae looking fo a condition on q such that MMF A, afte having eceived the unexpected edemptions, pefes to withdaw all its holdings fom Bank A and tigge its liquidation, 1 opposed to liquidating only the minimum fom both banks to satisfy the unexpected edemptions and keep the est in the banks. Recall that MMF A maximizes the expected utility of the investos fom egion A only. The expected utility of the MMF A investos if the MMF decides to withdaw only (1 to foce Bank A 0 s liquidation, is: as )q; and not EU Non-Liquidation = ( + ) u (c 1 ) + (1 ( + )) (15) {z } 0 " # 0:5(1 q)u( bch ;A + bcl ;B ) + 0:5u( bch ;B + bcl ;A ) + 0:5qu( bcl ;B + bcl ;A ) whee bc H ;i; bc L ;i i = A; B ae the payouts of Banks A and B at date afte MMF A withdaws its funds, fom equations (11), (1), (13), and (14) : Note that since the MMF knows it can pay its investos edeeming at date 1 the amount c 1 = 1; the st tem dops out fom (15). In contast, the expected utility of MMF A investos if the MMF decides to foce Bank A into liquidation is: whee EU Liquidation = ( + ) u (bc 1 ) + (16) (1 ( + )) 0:5u( ch + bc 1 ) + 0:5u( cl + bc 1 ) ; + (1 ) bc 1 min( ; 1) = min( [ + (1 )] ; 1); (17) (1 ) + is how much the MMF obtains if it foces Bank A to liquidate all its assets at date 1. that (17) is the level of consumption that MMF A is able to pay to those withdawing at date 1: + (1 ) ae Bank A s assets at date 1 if it liquidates all its long-tem investment; in the denominato = is the measue of impatient investos that MMF A satis es with its deposits in Bank A, and (1 ; Note )= is the measue of patient investos that MMF A satis es with its deposits in Bank A. Note that if the MMF A foces Bank A into bankuptcy, it will not necessaily be able to pay all its investos c 1 = 1 and, theefoe, geneally, bc 1 < c 1 : 1 Since we assumed that < 1, Bank A goes bankupt if MMF A withdaws all its assets (see Section 3..). MMF B neve withdaws in excess of since it has no infomation on the etun of the long-tem asset. 15

18 At date 1, MMF A withdaws fom Bank A all its holdings (as opposed to only the unexpected withdawal (1 )q) if the expected utility of its investos upon total withdawal fom Bank A (EU Liquidation ) is geate than the expected utility upon keeping funds in Bank A (EU Non-Liquidation ). 3 the unexpected withdawal (1 Thus, we can establish a level of q such that MMF A, afte having obseved )q, pefes to withdaw all its holdings fom Bank A and tigge its liquidation (if < 1 ), as opposed to liquidating only the minimum fom both banks to satisfy the unexpected withdawal of funds. The following poposition compaes the stability of MMF intemediation and diect nance. Poposition 1: Fo any values of R H and satisfying condition (1) and fo 0:5, thee 1 is an inteval of ealizations of q fo which bankuptcy occus with MMF intemediation, but not with diect nance. The poof is in the appendix. Poposition 1 establishes that a MMF-intemediated system is moe fagile than diect nance. That is, thee is a set of ealizations of q such that a MMF-intemediated system collapses wheeas an economy unde diect nance would not. This happens because MMFs have demandable claims on the banks and give thei investos demandable liabilities in ode to satisfy thei liquidity needs, which, in tun, makes MMFs liabilities subject to un-like edemptions. The unexpected ealy edemptions may contain negative infomation on Bank A s assets, which may make it optimal fo the MMF to un the bank. Note that the MMF decides to un the bank in ode to potect all its investos, and not just those unexpectedly withdawing ealy. Indeed, given the bank s xed pomise at date 1, the MMF obtains a highe payo fo its investos if it uns than if it does not. The eason is that MMF A s demandable claims on both banks allow it to fee ide on the claims of the othe MMF in both banks to satisfy the excess edemptions fom depositos in egion A. Futhemoe since the unexpected ealy edemptions may be due to liquidity as opposed to infomative easons, bank bankuptcy unde MMF intemediation may cause ine cient liquidation of the long-tem investment. Obseve that the highe instability of a MMF-intemediated economy is not due to the lack of insuance fo MMF investments. Indeed, we ae compaing MMF intemediation with a diect nance stuctue in which all depositos ae wholesale investos and, theefoe, uninsued. It is immediate to show that the highe fagility of MMF-intemediation would suvive in an economy in which banks eceived some of thei deposits fom etail insued investos. Note also that, in this economy, MMF intemediation geneates nancial fagility even though MMFs maximize the welfae of thei investos. The instability does not aise fom any fiction (such as agency poblems), but simply fom the ability of MMFs to aggegate pivate infomation and use it to the bene t of its investos. 3 Fo simplicity s sake, we assume in the poof that MMF A pefes to liquidate when the inequality holds weakly. 16

19 As mentioned above, bankuptcy in an MMF-intemediated economy occus because thee is a theshold of q, such that any ealization of q geate than that leads MMF A to withdaw its funds fom Bank A and, as a esult, Bank A collapses. In the following poposition we povide an uppe bound fo such a theshold. Poposition : Fo any values of R H and satisfying condition (1) and fo 0:5 1, let us de ne by ^q the theshold such that any ealization of q geate than ^q leads to bankuptcy unde MMF intemediation. We can show that ^q ~q (RH+1)(RL+1) log (R H + ) : log R H +1 The poof is in the appendix, whee we also show > H > 0; that is, ~q inceases with both and R H. When is highe, the negative infomation conveyed by the excess withdawal is less impotant; theefoe, a highe level of withdawal is needed fo the MMF to cause the bank s bankuptcy. Similaly, when R H inceases, the highe etun in the high state of the wold inceases the expected utility fom not withdawing fom the bank; as a esult, a highe excess withdawal is needed fo the MMF to cause bankuptcy. Note that the contact o eed by Bank B may not be viable on its own, given the conditional infomation that the MMF has on Bank B etuns (see Section 3:1). In paticula, it is easy to show that if assumption (3) holds, bankuptcy of Bank A tigges bankuptcy of Bank B. The bankuptcy of Bank B stems fom the loss of divesi cation oppotunity once Bank A goes bankupt. In othe wods, the pesence of two banks o es wholesale investos hedging oppotunities; howeve, when a bank is liquidated, this hedging oppotunity vanishes, which may cause the othe bank to be liquidated too. This channel of banking contagion due to loss of divesi cation has been studied by Cipiani, Matin and Paigi (013). 4.1 Fagility: an Example In this Section, we povide a numeical example of an economy in which an unexpected edemption (1 )q causes bankuptcy unde MMF intemediation, but not unde diect nance. The bankuptcy of one Bank unde MMF intemediation causes the othe bank to go bankupt too (contagion). Conside an economy whee = 0:5 and R H =. Assume that the liquidation value = 0:49 <. Theefoe, since < 1=; by condition (5) bankuptcy is possible unde diect nance. Also assume that the faction of impatient equals 0:8. With a logaithmic utility function, the 17

20 optimal contact o eed by the banks is c 1 = 1 c H = c L = 0:5: Moeove, since R H < 3 RL = :; by condition (3) the optimal contacts o eed by Banks A (1+ ) and B ae not viable sepaately. is (1 Conside a level of excess withdawal q = 0:35, which implies that the unexpected withdawal )q = 0:(0:35) = 0:07. Since, q < = 0:498, by condition (5) such level of unexpected withdawal does not cause bankuptcy in an economy with diect nance. What happens instead with MMF intemediation? Upon obseving the unexpected edemption, the MMF A will update upwads the pobability that etun of Bank A long-tem investment yields : Given this infomation, if MMF A decides to withdaw only what is needed to meet the unexpected withdawal of funds fom its investos (1 nothing fom Bank B (i.e.; the optimal equals 1): )q, it would withdaw only fom Bank A and Because of this, upon obseving the withdawal, Bank A would only be able to o e bc H = R H (1 ) = 1:44 and bc L = (1 ) = 0:18; wheeas Bank B would not have to modify its payouts. As a esult, fom (15) the expected utility of MMF A investos would be 0:10: What would happen if MMF A decides to pull all its funds fom Bank A? Since is elatively high (0:8), MMF A would be able to pay bc 1 = c 1 = 1 to all investos withdawing ealy. As a esult, fom (16) the expected utility of MMF A investos would be 0:004; highe than if MMF A decides not to pull out all its funds fom Bank A: Finally it is easy to veify that given the decision to pull out fom Bank A, MMF A would also nd it convenient to withdaw its funds fom Bank B: That is, we found a level of unexpected withdawals such that with diect nance thee is no bankuptcy, wheeas with MMF intemediation both Bank A and Bank B go bankupt. 5 Conclusion In this pape we show that MMF intemediation allows uninsued investos to limit thei exposue to a single banking institution and eap the gains fom divesi cation. Howeve, a banking system intemediated though MMFs is moe unstable than one in which investos inteact diectly with banks because MMFs have demandable claims on the banks and ae themselves subject to uns fom thei own investos. The mechanism though which instability aises is the elease of pivate infomation on bank assets, which is aggegated by MMFs and lead them to withdaw en masse fom a bank. Ou esults povide a theoetical undepinning fo the idea that an MMF-intemediated - nancial system can be paticulaly fagile. This fagility has been the main dive of the ecent 18

21 egulatoy e ots of the industy by the SEC and FSOC. Ove the ecent decades, banks have elied moe and moe on nancial intemediaies, such as money makets funds, to nance thei investment. Ou esults, suggest that this tend, while poviding investos with valuable divesi cation oppotunities, may incease the instability of the banking system. 6 Bibliogaphy Allen, F. and D. Gale (000), Financial Contagion, Jounal of Political Economy, 108, 1, Chai, V.V. and R. Jagannathan (1988), Banking Panics, Infomation, and Rational Expectations Equilibium, The Jounal of Finance 43, 3, Chenenko, S. and A. Sundeam (013), Fictions in Shadow Banking: Evidence fom the Lending Behavio of Money Maket Funds, Havad Business School, mimeo, Febuay. Cipiani, M., A. Matin, and B.M. Paigi (013), "Divesi cation and Bank Contagion," mimeo. Diamond, D.V. (1984), Financial Intemediation and Delegated Monitoing, Review of Economics Studies, Diamond, D.V. and P. Dybvig (1983), Bank Runs, Deposit Insuance, and Liquidity, Jounal of Political Economy, 91, 3, Dudley, W. (01), "Fo Stability s Sake, Refom Money Funds," Bloombeg.og, Aug 14. Geithne, T. (01), Lette to the Membes of the Financial Stability Ovesight Council, Sept Gennaioli, N., A. Shleife, and R.W. Vishny (01), Neglected Risks, Financial Innovation, and Financial Fagility, Jounal of Financial Economics, 104, Jacklin, C. and S. Bhattachaya (1998), "Distinguishing Panics and Infomation-based Bank Runs: Welfae and Policy Implications," Jounal of Political Economics, 96, 3, McCabe, P.E., M. Cipiani, M. Holsche, and A. Matin (01), "The Minimum Balance at Risk: A Poposal to Mitigate The Systemic Risks Posed by Money Makets Funds," FRBNY Sta Repot No. 564, July. 7 Appendix 7.1 The Optimal Contact with Diect Finance We deive optimal contact as the solution to the planne poblem in an economy with diect nance. Note that although each bank A and B o es potentially di eent contacts c A 1 ; c A;H ; c A;L ; c B 1 ; c B;H ; c B;L it is tivial to show that, unde the optimal contact, bank contacts would be identical and investos would invest an equal amount in each bank. Theefoe, to simplify notation, we denote the optimal contact by c 1 ; c H ; c L : Denote with s stoage to date pe unit of deposit. The optimal contact is the solution to 19

22 the following optimization poblem: Max u(c 1 ) + (1 )[u( ch + c L )]; w..t. c 1 ; c H ; c L ; i; s s.t. date 1 : c 1 = 1 i s; date : (1 )c H = ir H + s; date : (1 )c L = i + s; i + s 1; i 0; s 0; whee, ecall, the second utility tem comes fom the fact that by investing 1 in each bank, and since banks have pefect negative coelation, patient investos obtain a deteministic etun at date. Substituting the equality constaint The FONCs ae: Max u( 1 i s ) + (1 )[u( i(rh + ) + s )]; (1 ) w..t. i; s s:t i + s 1; i 0; s 0: u0(c 1 ) + u 0 ( ch + c L ) (RH + ) u0(c 1 ) + u 0 ( ch + c L which, with the natual log utility function, becomes = 0 ) = 0: whee ; > 0 i = 0; s = 0; 1 c c H + c L 1 c 1 + R H + + = = 0: c H + c L i = 0; s = 0; whee ; > 0: 0

23 Thee ae thee cases: Case 1), with s = 0; i > 0: Then the multiplie = 0 and the st constaint, R H + = 0: c 1 c H + c L The solution to the optimization poblem is inteio and i = 1 : The second constaint, = 0; c 1 c H + c L whee > 0: Fo the constaint to be satis ed, it must be the case that 1 + < 0; c 1 c H + c L that is, (R H + ) < 0; o R H + > ; which is the condition (1) fo an inteio solution. Case ), with s > 0; i = 0: Then, the multiplie = 0; and the second constaint, 1 + = 0: c 1 c H + c L Fo the constaint to be satis ed, it must be the case that (1 ) + = 0, that is, 1 s s s = (1 ): The st constaint, (R H + ) + = 0; c 1 c H + c L which since 0 implies (1 ) (R H + ) + 6 0; 1 s s that is, R H + 6 ; in which case the banks net pesent value is smalle than zeo, and the optimal contact implies zeo investment in the long technology. Case 3), with s > 0; i > 0: Then, both multiplies ; = 0; and the constaints become: = 0; c 1 c H + c L and R H + = 0; c 1 c H + c L which can neve be the case unless R H + = : As mentioned in the text, we assumed that condition (1) holds, that is, R H + >, which implies i > 0, s = 0. 1

24 7. The Optimal Withdawal by the MMF Recall that MMF A chooses how much to withdaw fom Banks A and B assuming that it can still obtain c 1 fo all its investos edeeming ealy (that is, bc 1 = c 1 ). This allows us to disegad the welfae of the investos edeeming ealy fom the MMF A. As a esult, the optimal withdawal of MMF A fom the two banks is the esult of the following maximization poblem: Max w::t: e 0:5(1 q)u( ch;a + c L;B ) + 0:5u( ch;b + c L;A ) (18) +0:5qu( cl;a + c L;B ) s.t. c H;A = max(r H (1 e ); 0); c L;A = max( (1 e ); 0); c H;B = max(r H (1 (1 ) e ); 0); c L;B = max( (1 (1 ) e ); 0); whee ; e and (1 ) e epesent the faction of withdawal that the MMF A will do in Bank A and B espectively, and c i;j epesents date- consumption if the etuns ae low o high, i = L; H, by bank j = A; B. Let us analyze the thee tems in (18) we need to maximize sepaately: Tem 1: c H;A + c L;B which is deceasing in e : Tem : which is inceasing in e. (RH R (1 e L (RH R + 1) L = (RH (1 e )) + ( (1 (1 e ) )) ) + (1 (1 ) e )) = ( ( e RH 1) + 1); c H;B + c L;A = (RH (1 (1 ) e )) + ( (1 e )) ; = RL (RH R (1 (1 e ) ) + (1 e )); L = RL (RH R + 1) ((1 e R H )( 1) + 1); L =