Liquidity Management in Banking: What is the Role of Leverage?

Transcription

1 Lqudty Management n Bankng: What s the Role of Leverage? Fabana Gomez y Quynh - Anh Vo z September 2016 Abstract Ths paper examnes potental mpacts of banks leverage on ther ncentves to manage ther lqudty. We analyse a model where banks control ther lqudty rsk by managng ther lqud asset postons. In the basc framework, a model wth a sngle bank, where the possblty of sellng long-term assets when n need of lqudty s not taken nto account, we nd that the bank chooses to prudently manage ts lqudty rsk only when ts leverage s low. In a model wth multple banks and a secondary market for long-term assets, we nd that a bankng system where banks are hghly leveraged can be prone to lqudty crses. Our model predcts a typcal pattern of lqudty crses that s consstent wth what was observed durng the crss. Key words: Leverage, Lqudty Rsk, Moral Harzard, Cash-In-The-Market Prcng. JEL Codes: G21, D82 1 Introducton Lqudty played an enormous role n the global nancal crss of Durng the perod, many banks experenced d cultes because they dd not manage ther lqudty n a prudent manner. In response to the aws n banks lqudty rsk management revealed by the crss, the Basel III commttee has proposed two regulatory lqudty standards to complement ts revsed captal requrement framework n promotng the stablty of the bankng sector. Whereas the objectve of the captal requrement s to mprove the solvency of the banks, the two lqudty requrements am to promote a better lqudty The vews expressed n ths paper are those of the authors, and not necessarly those of the Bank of England or ts commttees. We are grateful to Urs Brchler, Gann De Ncolo, Hans Gersbach, Mchel Habb, Bruno Parg, Sebastan Pfel for helpful comments and suggestons. We owe specal thanks to Jean-Charles Rochet for thoroughly readng our manuscrpt at several stages, as well as for hs valuable comments. A sgn cant part of ths researd was done when the second author was a lated wth the Unversty of Zurch and receved the fundng from the ERC (grant agreement RMAC), from NCCR FnRsk (project Bankng and Regulaton) and from Swss Fnance Insttute (project "Systemc Rsk and Dynamc Contract Theory"). y Unversty of Brstol, Emal: fabana.gomez@brstol.ac.uk. z Correspondng author, Bank of England, Emal: quynh-anh.vo@bankofengland.co.uk. 1

2 rsk management. One queston then arses, as ponted out by Trole (2011): whether one should append a lqudty measure to the solvency one". Put d erently, can one trust the nancal nsttutons to properly manage ther lqudty, once ther leverage s controlled by captal requrement? In the lterature, two man theores predomnate as to the role of captal regulaton. The rst one posts that a bank s captal forms a knd of cushon aganst losses. The second one puts forward the dea that the captal regulaton can be seen as an ncentve devce to curb the excessve rsk-takng behavor of banks. Hence, so far, the bankng lterature focuses on the e ects of the banks leverage on the banks choce of credt rsk. In ths paper, we develop a model to examne potental mpacts of banks leverage on ther management of lqudty rsk. We analyse a framework where banks control ther lqudty rsk by managng ther lqud asset postons. The context we have n mnd s one of banks that are nanced by debt and equty. Due to nvestors demand of lqud nvestment, banks can ssue only shortterm debt and thus expose themselves to a lqudty rsk that stems from the maturty msmatch between asset payo s and desred redemptons. To be nsured aganst ths rsk, banks need to carry some lqud assets on ther balance sheet. We study the bank s optmal lqudty holdngs and how they are a ected by the bank s leverage. We assume spec cally that banks have choce between two types of assets. The rst one s a knd of lqud reserves that have a net return of zero. The second asset s a constant return to scale nvestment project (long-term asset) that produces a random cash ow only after two perods. Although the latter s more pro table than the former, ts capacty to generate lqudty n some future states of nature may be restrcted. Inspred of the recent crss, we model the lqudty shock by the arrval, at an ntermedate date, of some new nformaton about the qualty of the project. When good news are revealed, the lqudty rased by pledgng the project s cash ows s su cent to cover the banks re nance demand. However, f bad news are dsclosed, the project has lmted pledgeablty, whch may lead to the banks closure f they do not hold lqud reserves ex-ante. In practce when n need of lqudty, banks typcally have three optons: ether they use ther ex-ante lqudty holdngs or they borrow aganst the future cash ows generated by ther long-term assets or they sell these assets n the secondary market. We rst consder, n the basc model, the case of an ndvdual bank that could rase lqudty usng just the two rst optons. Our focus s then on the bank s precautonary motve for lqudty holdngs,.e. holdng lqudty to be nsured aganst lqudty rsk. Our man ndng s that the bank hold adequate lqudty to protect tself aganst ths rsk f and only f ts leverage rato s low. The ntuton les n the fact that when leverage s hgh, the bank s exposure to lqudty rsk s large. Buyng nsurance s then too costly, whch nduces the bank to forgo the nsurance opton and gamble. In our smple setup, there exsts a threshold of leverage below whch the bank wll choose to manage ts lqudty rsk prudently, whch mples that a properly desgned captal requrement s su cent to nduce a better lqudty rsk management. We are not clamng that ths s a general result. All we are clamng s that a restrcton on banks leverage can have postve mpact on ther ncentves to manage ther lqudty. Hence, lqudty requrement and captal requrement need to be jontly 2

3 desgned n an optmal way to avod overregulaton. Next, we extend our basc settng nto a mult-banks contexts that allow to take nto consderaton the possblty of asset sales. Precsely, we analyze a three-bank settng n whch banks that are n need of lqudty can sell ther long-term asset n the secondary market. We assume that because of asset spec cty, the only potental purchasers of one bank s asset are other banks. Hence, the market prce depends on the overall amount of lqudty avalable n the system for asset purchases. Allowng for asset sales has two nterestng mplcatons. Frst, gven that the market prce of long-term asset depends on the aggregate lqudty of the bankng system, the dstrbuton of leverage n the system should be matter for banks lqudty pro le. Moreover, t s also nterestng to see how the mpact of banks leverage on the banks choce of lqudty holdngs has consequences on deleveragng and re sales n the bankng sector durng lqudty crses. The second mplcaton les n the fact that there exsts an addtonal reason for banks to hold lqudty besde the precautonary motve. The dea s that banks that survve the lqudty shock have opportunty to buy assets put for sale by banks that have lqudty demand. If such assets are sold at prce below ther fundamental value, banks that do have enough lqudty stand to make wndfall pro ts from purchasng assets. By characterzng the ratonal expectaton equlbra, we derve a set of results that shed lght on these ssues. We nd that fundng lqudty and market lqudty of long-term asset are postvely related. We also nd that a bankng system that conssts of hghly leveraged banks can be prone to lqudty crses. Our model predcts a typcal pattern of the crses: Hgh leverage results n low ex-ante lqudty holdngs of banks. Then, when a lqudty shock s realzed, many banks have trouble n honorng ther debt oblgatons and thus have to sell ther nvestment at re-sale prces, whch causes the falure of banks that are n need of lqudty. Ths pattern s consstent wth what was observed durng the crss. The organzaton of the paper s as follows. After dscussng the related lterature n the next secton, we descrbe, n Secton 3, the basc model. In Secton 4 we analyse the banks optmal lqudty holdngs and the e ect of bank s leverage on ths choce. Secton 5 consder multple bank settng and the consequences of permttng asset sales. Fnally we conclude n Secton 6. 2 Related Lterature To the best of our knowledge, the present paper s the rst one that addresses the mpact of banks leverage on the banks ncentves to manage ther lqudty rsk. Stll the nsghts on whch our model bulds are related to varous lteratures. The dea that the lablty structure of a bank may have e ect on ts asset composton s lnked to the large lterature that evaluates the foundaton for the mposton of captal regulaton. See, among others, Rochet (1992), Besanko and Kanatas (1996), Blum (1999), Repullo (2004) 1. Ths lterature studes how the ncentves of banks to take excessve rsk 1 For an excellent revew of ths lterature, see Frexas and Rochet (2008), VanHoose (2007). 3

4 can be curbed by requrng banks to mantan an adequate captal rato. Whle the focus of ths lterature s the mpact on the banks ncentves to take credt rsk, our paper ams to examne the e ect on ther ncentves to manage ther lqudty rsk. In our paper, the reason for banks to hold lqudty s based on two assumptons: () ex-ante uncertanty about the lqudty needs; () lmted pledgeablty due to asymmetrc nformaton. Those two assumptons are smlar to those used by Hölmstrom and Trole (1998) to analyse the lqudty demand of corporate sector and the role of government n supplyng lqudty. The man d erence les n the fact that n Hölmstrom and Trole (1998), lqudty shocks arse as producton shocks to the rms technologes. The sze of the shocks s exogenous and especally ndependent of the rms balance sheet characterstc. We rather derve lqudty needs as beng determned n equlbrum by asset-lablty msmatch. Such d erence explans why n Hölmstrom and Trole (1998), the rms lqudty demand does not depend on ther lablty structure whereas n our framework t does. We beleve that lqudty shocks arsng from technology shocks as n Hölmstrom and Trole (1998) are sutable for non- nancal enterprse whle our formulaton s more reasonable n the context of nancal nsttutons. The present paper s also related to several contrbutons that use the "cash-n-themarket-prcng" mechansm proposed by Allen and Gale (1994, 2004, 2005) to understand the nancal fraglty. Bolton et al. (2011) construct a framework to analyse the optmal composton of nsde lqudty (.e. the cash reserves held by nancal ntermedares themselves to meet ther lqudty demand) and outsde lqudty (.e. the lqudty holdngs of other nvestors wth a longer horzon). They examne the asset allocaton between cash and long-term nvestment of two types of agents, short and long-run nvestors. Short-run nvestors (SRs) can be ht by a lqudty shock that takes the form of a late maturty of ther nvestment. The key novelty of ther analyss s the focus on the tmng of lqudty trades. They assume that SRs had the choce of ether mmedately respondng to the lqudty shock by sellng ther assets at re-sale prces, or takng a chance that the shock mght be short-lved at the rsk of havng to rase lqudty at a later date under much worse condton. They then analyse how the expectaton about the tmng of lqudty trades a ects the nvestment decson of SRs. Our paper nstead ponts to the e ect of the banks leverage on ther choce of nvestment. A more closely related to our work s the paper of Acharya and Vswanathan (2011b) that bulds a model to understand the de-leveragng of the nancal sector durng crses. They examne how adverse shocks that materalze n good economc tmes, represented by hgh expectatons about economc fundamentals, lead to greater de-leveragng and asset prce deteroraton. In ther framework, banks ssue short-term debts to nance ther nvestment n long-term rsky assets. In response to a lqudty shock, banks rase new debt and, f necessary, sell ther assets. Good expectatons about the qualty of those assets enable low-captal banks to be funded ex-ante and the resultng dstrbuton of leverage n the economy can potentally lead to more serous re-sale problems when adverse shocks arse n good tmes. Our multple banks setup wth asset tradng s n fact nspred of Acharya and Vswanathan (2011b) s settng. The man d erence s that we allow banks to hold lqudty to self-nsure aganst lqudty shock, whch enables us to shed lght on 4

5 how banks ncentves to manage lqudty rsk are a ected by ther lablty structure. Several papers study the banks choce of nvestment between lqud and llqud assets. They d er n the determnants they focus on. Acharya et al. (2011a) examne the e ect of polcy nterventons to resolve bank falure on ex-ante bank lqudty. Malherbe (2014) provdes a model n whch the fear of future market llqudty due to adverse selecton may trgger hoardng behavor today. Heder et al. (2015) analyse banks lqudty holdngs to shed lght on how banks prvate nformaton about the rsk of ther assets a ects the tradng and prcng of lqudty n the nterbank market. Acharya et al. (2015) studes how the lqudty choces of rms are shaped by the rsk-sharng opportuntes n the economy. Our paper consders the e ect of banks lablty structure on ther lqudty choces. Fnally, some papers address the optmal desgn of bank lqudty requrements. Calomrs et al. (2014) develop a theory of lqudty requrements whch focuses on the role of cash n ncentvzng banks to properly manage ther default rsk. They argue that because cash s rskless asset and cash holdngs are observable, banks can commt to exert e ort on rsk management by holdng a su cent amount of cash. Hence, n Calomrs et al. (2014), a lqudty requrement that takes the form of a narrow cash reserve requrement can be used to provde banks ncentves to reduce credt rsk. The present paper consders the tradtonal role of cash holdngs n lmtng the lqudty rsk and examne whether the banks ncentves to manage ths rsk are a ected by ther captalsaton. Walther (2015) constructs a model to analyse how nancal regulaton n the form of restrcton on maturty msmatch can be used to avod socally wasteful re sales. It s found that there exsts stuatons where re sales arse n the decentralzed compettve equlbrum. In that case, mposng a lnear constrant on banks s su cent to restore e cency. Such constrant can be mplemented by regulatory tools such as Net Stable Fundng Rato or Lqudty Coverage Rato. In Walther (2005), banks are ex-ante dentcal and banks short-term debt takes the form of collateralzed debt wth exogenous har-cut. In contrast, n our model banks are ex-ante heterogenous and the nterest rate on the shortterm debt s derved n equlbrum dependng on the banks choces of asset composton. Walther (2005) does not examne how the banks decsons on maturty msmatch are n uenced by ther captal rato as we do n the present paper. Fnally, Santos and Suarez (2016) present a model where the ratonale for bank lqudty standards s an mprovement of the e cency of the decson of the lender of last resort. They consder a dynamc model n whch recevng lqudty support from the lender of last resort may help banks to cope wth nvestor runs. In ther settng, holdng lqudty s costly because t forces banks to forgo valuable nvestment opportuntes, but t can be e cent. The reason s that, when a run happens, lqudty holdng ncreases the tme avalable before the lender of last resort must decde on supportng the bank, whch facltates the arrval of nformaton on the bank s nancal condton and mproves the e cency of the decson taken by the lender of last resort. 5

6 Fgure 1: The rsky nvestment opportunty 3 The basc model In ths secton, we descrbe the problem of an ndvdual bank that seeks to manage ts lqudty rsk. We consder an economy that lasts for three dates t = 0; 1; 2: There s a bank wth balance sheet of sze normalzed to 1. We assume that the bank s funded at date 0 by equty (of amount E) and short-term debt (of amount 1 E). The face value of short-term debt repad at date 1 s denoted by D. The bank has access to two nvestment opportuntes. The rst one s a storage technology, referred to as cash, that has a net return of zero. The second nvestment opportunty s a constant return to scale project, referred to as long-term asset, that requres a start-up nvestment at date t = 0 and generates an uncertan cash ow at date t = 2. Fgure 1 summarzes the payo structure of the project. Precsely, f the bank nvests 1 at date 0 n ths project, t receves ~y > 0 at date 2 wth probablty, and zero wth the complement probablty. ~y s not known at date 0 but wll be revealed at date 1. At date 0, we only know that there are two possble states at date 1: hgh or low state. In the hgh state, whch happens wth probablty, ~y s equal to y H whereas n the low state, ~y takes a lower value y L. We assume that the realzaton of ~y s observable but not ver able. Therefore, the short-term debt repayment cannot be contngent on such nformaton. Assumpton 1 The nvestment project has postve NPV: E (~y) = y H + (1 ) y L > 1 Observe that on average, nvestng n the project s more pro table than holdng cash. However, gven the mstmach of the tmng between the bank s lqudty needs and the project s cash ow, the bank may optmally choose to nvest a postve amount n the storage technology. At date 1, the bank has two sources of lqudty to repay ts short-term debt. The rst one s the amount of cash t holds from date 0. The second one s the new borrowng t 6

7 can make by pledgng the date 2 - cash ow generated by the project. The extent to whch the bank can pledge ts future cash ow may be constraned by a moral hazard problem. Spec cally, we assume that between date 1 and date 2, after rasng new funds and before the project s cash ow s realzed, the bank can swtch nvestment to another asset wth probablty of success 1 and success cash ow y 1. Assumpton 2 The moral hazard problem matters only n the low state: 1 < ; y H > y 1 > y L and 1 2 y L > 1 y 1 Hence, the new asset has a lower success probablty but ts success return s hgher than that of the project n the low state. Assumng that 1 y 1 < 1 2 y L ensures that nvestng n the 1 -asset s a negatve net present value nvestment for the bank. The man mplcaton of Assumpton 2 s that whle n the hgh state the bank can pledge the full value of ts long-term asset to outsde nvestors, the bank s borrowng capacty n the low state s strctly lower than the expected present value of ts future cash ows. We could alternatvely nterpret the asset swtch as the fact that the bank refrans from montorng. Precsely, f a hgh value of ~y s realzed at date 1, the qualty of the nvestment project happens to be very good and no ntermedate montorng needs to be done. However, when a low value s realzed, the project s return depends on the bank s montorng actvtes. If the bank refrans from them, t can save on the montorng cost and thus, receve more n case of success. Nevertheless, the success probablty wll be reduced. The realzaton of the low state can be seen as the materalzaton of a lqudty shock that put constrants on the amount of lqudty the bank can rase and makes the rollover of ts short-term debt problematc. If the bank cannot rase enough lqudty, t wll be closed and the bank s nvestment project s lqudated. We assume that the lqudaton value s equal to `, whch s ndependent of the state. ` can be nterpreted as the mnmum possble value of the asset (for nancal assets) or as the resale prce (for physcal assets) Assumpton 3 The value of the nvesment project to the bank s nancers s less than the value to the bank: ` < y L Assumpton 3 s just ed n stuatons where the management of the bank s assets requres sophstcated expertse that the bank s nancers do not have. Assumpton 4 y H + (1 ) ` > 1 Assumpton 4 means that the expected payo of the project, even f t s lqudated early, s postve 2. Ths assumpton mples that at date 0, t s stll worth for the bank to nvest n the project even f the bank may be closed f the lqudty shock s realzed. The tmng of the model, whch s summarzed n Fgure 2, s as follows: 2 Note that Assumpton 1 s automatcally sats ed f both Assumptons 3 and 4 are true. 7

8 Fgure 2: The tmelne At date 0, the bank chooses the composton of ts assets. Denote by c the amount of cash t holds. Thus, 1 c wll be nvested n the project. At date 1; rst, the value of ~y s observed. Then, the bank tres to rase funds to pay back ts short-term debt. If the bank cannot rase su cent lqudty, t s lqudated. At date 3 2, between date 1 and date 2, the bank may swtch ts nvestment to the 1 -asset. At date 2, the project s cash ow s realzed and payments are settled. Before proceedng wth the analyss of the bank s optmal cash holdngs, some addtonal remarks are n order. Frst, we take as gven the bank s maturty msmatch: bank s credtors want short-term debts, whle borrowers need long-term credt. We assume that the bank cannot ssue long-term debt. Second, does our bank s short-term debt correspond to wholesale or retal debt? One of the man d erences between the two types s the fact that retal deposts are nsured but wholesale deposts are not. Ths d erence mples that the repayment promsed to retal depostors does not depend on the bank s choce of assets whle the repayment to wholesale credtors does. In ths model, motvated by the nancal crss , we refer to the wholesale debts such as the ones held by Money Market Funds. The debt repayment s thus endogenously determned n our framework by the break-even condton of the bank s debtholders. Concernng our formulaton of the lqudty shock, note that n the present framework, ths shock does not come from the uncertanty about the amount of short-term debt that need to be repad, as commonly assumed n the model wth retal deposts. Indeed, n our setup, the bank knows exactly as of date 0 how much debt t has to repay. What t does not know s ts fundng capacty at the tme the repayment s made. If everythng goes well,.e. there are good news about the qualty of the bank s asset, the bank s borrowng capacty s not constraned and thus, the need to re nance the short-term debt does not create any problem to the bank. However, when bad news about the bank s nvestment 8

9 are revealed, ts capacty to rase funds s restrcted. As a consequence, the bank may fal to roll-over ts debt. The above-descrbed scenaro s analogous to what happened n the crss. Pror to the crss, banks nanced a growng porton of ther subprme mortgage loans wth short-term debts such as repos or asset-backed commercal papers (ABCP). Everythng worked very well untl an ncrease n subprme mortgage defaults was rst noted n February 2007, whch was followed by a deteroraton of the banks short-term fundng market. Many banks then experenced d cultes n rollng-over ther short-term debts. 4 Analyss We now analyse the bank s optmal nvestment decson. Our man objectve s to study how much cash the bank wll hold on ts balance sheet and how ths decson s a ected by the bank s leverage. We wll proceed n two steps. Frst, we determne the bank s borrowng capacty at date 1. Then, we examne ts optmal cash holdngs at date Borrowng Capacty At date 1, the bank has to repay ts short-term debt D. It has c unts of cash, whch mples that ts lqudty needs are D c. The bank can rase ths amount by ssung new debt repad at date 2. We now determne how much the bank can borrow at date 1 by pledgng the future cash ow generated by ts long-term asset. If the hgh state s realzed at date 1, the moral hazard problem does not matter, the bank can pledge the full value of ts asset to nvestors,.e. they can borrow up to y H, and there s no problem n rollng over ts short-term debt. If the low state s realzed, the ncentve compatblty condton whch ensures that the bank does not swtch to the rsker asset s as follows: (y L f) 1 (y ) where f s the face value of the new debt ssued aganst one unt of long-term asset. After smpl caton, ths yelds: f y L 1 y 1 1 = f f represents the maxmum cash ow that can be pledged to outsde nvestors (.e. f s the maxmum pledgeable ncome). The bank s maxmum borrowng capacty (per unt of long-term asset) n the low state s thus f. Notce that f < y L. We make an addtonal assumpton as follows: Assumpton 5 ` < f Assumpton 5 ensures that new borrowng s a better way to rase lqudty for the bank than partal lqudaton of ts long-term asset. The followng lemma summarzes the bank s stuaton at date 1: 9

10 Lemma 1 At date 1: () If D c 1 c f, the bank can always roll over ts debt. () If D c 1 c > f, the bank s lqudated when beng ht by a lqudty shock (.e. when the low state s realzed). We refer to the rst stuaton as the one where the bank s lqud. The second stuaton s referred to as the case where the bank s llqud. 4.2 Optmal Cash Holdng Polcy In the next step, we study the bank s decson regardng the amount of cash held. Gven two possble stuatons of the bank at date 1, we wll rst determne how much cash the bank holds n each stuaton. Then, we characterze the optmal cash polcy of the bank. If the bank chooses c so that t wll be lqud at date 1, the amount of cash held by the bank s determned by the followng program 3 : l = Max 0c [(1 c) y H f H ] + (1 ) [(1 c) y L f L ]g where f s, s = H; L s the face value of the new debt ssued at date 1 n the state s: f s = D c for all s subject to the break-even condton of short-term nvestors: and the lqudty condton: D + (1 ) D = 1 E (1) D c 1 c f (2) Pluggng (1) nto (2) and nto the objectve functon, we can rewrte the above program as follows: l = Max 0cy H + (1 ) y L 1 + E c (y H + (1 ) y L 1)g (3) subject to (1 E f ) c ( ) (4) Ths program makes clear the trade-o drvng the bank s cash holdng decson. The cost of holdng cash s the foregone return of the long-term asset, whch explans why the term "c (y H + (1 ) y L 1)" s deducted from the bank s expected pro t. The bene t of holdng cash s to provde nsurance aganst the lqudty shock at date 1, whch s re ected n Constrant (4). Note that ths constrant matters only f f < 1. One unt of cash at date 0 generates one unt of lqudty at date 1 whereas the amount of lqudty rased 3 The superscrpt "l" refers to lqudty. 10

11 aganst one unt of long-term asset s f. Clearly, holdng cash makes sense only when f < 1. We make the followng assumpton to ensure the role of cash n our model: Assumpton 6 f < 1 At the optmum, the bank holds an amount of cash that s just su cent to overcome the lqudty shock,.e. c l = max 1 E f ; 0. Note that when E s hgh enough (.e. E ), the lqudty shock s low, the bank s lqud even though t holds zero cash. Hence, the bank s expected pro t when choosng to be lqud at date 1 s: 1 E f l = y H + (1 ) y L 1 + E max ; 0 (y H + (1 ) y L 1) We now turn to the amount of cash the bank holds f t chooses to be llqud at date 1. The bank problem n ths case s wrtten as follows 4 : ll D c = Max (1 c) y H 0c1 subject to the break-even condton of short term nvestors: and the llqudty condton: D + (1 ) (c + (1 c) `) = 1 E (5) D c 1 c > f Pluggng (5) nto the objectve functon, we get: ll = Max 0cy H + (1 ) ` 1 + E c (y H + (1 ) ` 1)g subject to (1 E f ) > c ( ) Hence, c ll = 0 at the optmum. Snce the only bene t of cash s to provde nsurance aganst the lqudty shock, t s ntutve that f the bank decdes to be llqud at date 1, t wll hold zero cash. The bank s expected pro t when choosng to be llqud at date 1 s then: ll = y H + (1 ) ` 1 + E Fnally, to determne the optmal cash polcy of the bank, we have to compare l and ll. We see that the condton: s equvalent to l ll 1 E f (1 ) y L (1 ) ` max ; 0 (y H + (1 ) y L 1) (6) 4 The superscrpt "ll" refers to llqudty. 11

12 Fgure 3: The bank s optmal cash holdng polcy Note that the LHS of Inequalty (6) s the expected loss n value due to early lqudaton of the long-term asset whle the RHS represents the cost of buyng nsurance aganst lqudty rsk (.e. holdng cash) for the bank. Clearly, the bank chooses to be nsured only f the nsurance cost s lower than the loss n the value. Inequalty (6) results n a condton on the bank s leverage as follows: E ( ) y H + (1 ) ` 1 y H + (1 ) y L 1 = E (7) The followng proposton summarzes the characterzaton of the bank s optmal cash holdng polcy: Proposton 1 When the bank s undercaptalzed (.e. E < E ), t chooses to be llqud and holds zero cash. The bank chooses to be lqud only when t s well captalzed (.e. E E ). In that case, the bank holds an amount of cash equal to max 1 E f ; 0 and c the lqudty coverage rato (.e. D ) s ncreasng wth the leverage. We graphcally represent n Fgure 3 the cash holdng polcy characterzed n Proposton 1. We rst observe that the bank holds su cent cash to be nsured aganst the lqudty shock f and only f the bank s captal rato s hgh enough. Ths result s due to the fact that when the bank s captal rato decreases, the bank holds more debts, whch exposes t to a hgher lqudty shock. Ths hgher exposure n turn leads to a hgher cost of nsurance. We see clearly n Inequalty (7) that the nsurance cost s decreasng wth the bank s captal rato E. When ths rato s too low, buyng nsurance aganst the lqudty shock becomes too costly, whch nduces the bank to gamble. The second concluson obtaned n Proposton 1 concerns the ncreasng relatonshp between the lqudty coverage rato and the leverage of the bank when t s well captalzed. The ntuton behnd t s straghtforward. Once the bank chooses to be lqud, the amount of cash t holds s ncreasng wth ts exposure to lqudty rsk. 12

13 Proposton 1 brngs out the postve mpact that a restrcton on the bank s leverage can have on ts ncentves to manage ts lqudty. In the present model, captal requrements can perfectly do the job of mprovng the management of lqudty rsk by banks. As noted n the ntroducton, although we do not clam that ths s a general result, ths nsght s tself nterestng n the sense that t shows that any proposal concernng a lqudty requrement needs to be jontly consdered wth the captal regulaton n order to avod overregulaton. Another nterestng nsght derved from Proposton 1 pertans to the mpact of a decrease n the lkelhood of the lqudty shock on the captal rato threshold: Corollary 1 The captal rato threshold E s decreasng wth the probablty (1 the lqudty shock happens. ) that Corrolary 1 states that the captal rato threshold ncreases when the lkelhood of the shock decreases. Put d erently, the captal rato threshold s hgher for the lqudty rsk that has smaller probablty of occurrence. Corollary 1 thus mples that t s much more d cult to nduce banks to properly manage the tal rsk. 5 Multple Banks Settng wth Asset Sales In the prevous settng, we assume away the possblty that a secondary market for long-term assets s opened at date 1, whch allows banks to sell them when n need of lqudty. In ths secton, we examne the consequences of permttng the sales of longterm assets. 5.1 Envronment We consder a model wth three banks, called A; B and C. Bank ( = A; B; C) has an amount of equty equal to E. Each banks has access to the same nvestment technologes and s subject to the same moral hazard problem as descrbed n the prevous secton. We assume that the lqudty shock represents a common exposure of three banks: at date 1, the realzaton of ~y s the same to all of them. The d erence wth the prevous setup s that to pay back ther short-term debt, banks now have three optons nstead of two: () The amount of cash held from date 0 () The new debt ssued aganst the date 2 - cash ow generated by the project () The proceeds from sellng ther long-term assets. Wth regard to the secondary market where banks can sell ther asset, we assume, n accordance wth Assumpton 3 about the spec cty of the banks asset, that potental purchasers of a bank s long-term asset are other banks. Hence, n the present paper, we dstngush between asset sale and asset lqudaton. Asset sale corresponds to the transfer of the asset from one specalst to the other who has the same ablty to redeploy t. As to asset lqudaton, t s equvalent to the transfer of the asset to a non-specalst who can extract a much lower surplus from the assets than specalst. 13

14 In order to characterze the equlbra of the present economy, we proceed as follows: we rst examne the market for asset sales and analyze how the market prce s determned. Then we study the banks ncentves for lqudty holdngs. Fnally, we nvestgate the exstence and the man features of d erent ratonal expectaton equlbra. 5.2 Borrowng Capacty As n the basc model, f the hgh state s realzed, all banks can roll over ther debt. If the low state s realzed, the maxmum borrowng capacty (per unt of long-term asset) for each bank s f. 5.3 Market for Asset Sales We now analyze the secondary market of long-term assets. For ths purpose, we ntroduce some addtonal notatons as follows: denotes the bank s lqudty demand (per unt of long-term asset) at date 1: c = D for = A; B; C 1 c p s the unt prce of the long-term assets. ( = A; B; C) ndcates the fracton of long-term assets sold by bank to cover ts lqudty need. ( = A; B; C) s the volume of long-term assets acqured by bank. A. Demand and Supply of Long-Term Assets We start wth the determnaton of the ndvdual suppy and demand functons. Snce the maxmum fundng capacty (per unt of long-term asset) for each bank s f, banks who have to sell ther long-term assets are the ones wth exceedng f. In contrast, banks that have lower than or equal to f are n excess of lqudty and thus, can buy assets. The fracton of asset sold by each bank ( = A; B; C) wth greater than f s determned as follows: (1 c ) p + (1 c ) (1 )f D c (8) In Inequalty (8), the LHS s the total lqudty bank could rase. It s the sum of respectvely the proceeds from sellng a fracton of long-term asset and the lqudty obtaned by ssung new debt aganst the remanng fracton 1. After smpl caton, we get: = mn 1; f p f (9) Observe that fundng lqudty expands wth asset sales f and only f the unt prce p s greater than f. We assume for now p f and wll show later that t s ndeed the 14

15 case. The extent of asset sales s decreasng wth the asset s prce. A bank wll have to sell all of ts exstng nvestment when the prce p falls below ts lqudty demand. Wth regard to the asset demand of bank who has excess lqudty, note that no bank would acqure assets at a prce hgher than ther expected payo. Hence, f p > y L ; should be equal to zero. If f < p < y L, s determned as follows: (1 c + ) f (D c ) p (10) The LHS of Inequalty (10) s the total lqudty avalable to bank for asset purchase. It conssts of ts spare debt capacty from exstng assets, (1 c ) f (D c ), plus the lqudty that can be rased aganst assets to be acqured, f. After some arrangements, we have: = (1 c ) f p f Notce that f p = f, the lqudty rased aganst assets to be acqured s su cent to pay for the assets, whch mples that the demand for the assets s n ntely hgh. To summarze, the long-term asset s demand functon of bank who has lower than or equal to f s as follows: 8 >< ( ; p) = >: 0 f p > y L (1 c ) f p f f f < p < y L any value between 0 and (1 c ) f p f f p = y L 1 f p = f (11) B. Unt Prce of Long-Term Assets Now, we turn to nvestgate the unt prce of long-term assets. Because of lmted market partcpaton, the prce s determned by the amount of lqudty avalable n the market for asset purchases, whch n turn depends on the banks cash holdng decson at date 0. In ths secton, we characterze the equlbrum prce for all possble dstrbutons of lqudty n the bankng system at date 1. In the followng, ; j and k can be ether A; B or C. () j k f : All banks are lqud and can roll over ther debt. Therefore, there s no asset tradng at date 1. () f < j k : Bank s lqud whle banks j and k have to sell some of ther long-term assets f the low state s realzed. From the ndvdual demand and supply functons characterzed n respectvely (11) and (9), we can compute the total supply S(p) as well as the total demand D(p) as follows: S(p) = (1 c j ) mn 1; j f p f + (1 c k ) mn 1; k f p f and D(p) = (1 c ) f p f for p < y L 15

16 The equlbrum prce sats es the market-clearng condton: ED(p) D(p) S(p) = 0 If excess demand ED(p) s postve for all p < y L ; then n the equlbrum p = y L. There exsts three subcases: (.1) f p j k : We have: mn 1; j f p f = 1 and mn 1; k f p f = 1 Ths case corresponds to the stuaton where both banks j and k cannot rase enough lqudty even when they sell all of ther long-term assets. They are thus closed at date 1. Ther assets are sold by ther debtholders to bank. The excess demand ED(p) s then computed as follows: ED(p) = (1 c ) f p f (1 c j + 1 c k ) Hence, the condton that excess demand be zero leads to the followng relatonshp: p = f + (1 c ) (f ) 2 c j c k (12) whch mples that the equlbrum prce s gven by: p = mn f + (1 c ) (f ) ; y L 2 c j c k (.2) f < j < p k : In ths case, bank j can overcome the lqudty shock (13) after sellng a fracton j, whch s strctly less than 1, of ts long-term assets. However, bank k has to sell all of ts long-term assets and thus, wll be closed at date ollowng the occurrence of the lqudty shock. Zero excess demand s equvalent to: ED(p) = (1 c ) (f ) (1 c j ) j f! p f p f + (1 c k ) = 0 Therefore, the equlbrum prce s represented as follows: p = mn f + (f ) (1 c ) j f! (1 c j ) ; y L 1 c k (14) (.3) f < j k < p: Both banks j and k survve the lqudty shock and contnue to operate at date 1. The excess demand functon s gven by: ED(p) = (1 c ) (f ) (1 c j ) j f + (1 ck ) ( k f ) p f p f 16

17 We see that n ths case, the equlbrum prce s determned by a postve excess demand condton as follows: (1 c ) (f ) p f (1 c j) j f + (1 ck ) ( k f ) p f (15) whch means that n the equlbrum the prce p equals to y L. () j f < k : Two banks and j are lqud whereas bank k s llqud and must sell ts long-term assets at date 1 f the low state s realzed. Smlarly to Case (), the total supply and total demand are computed as follows: S(p) = (1 c k ) mn 1; k f p f There are two subcases: D(p) = (1 c ) f p f + (1 c j) f j p f for p < y L (.1) f < p k : Bank k has to sell all of ts long-term assets and then, s closed at date 1. The equlbrum prce s gven by: p = mn f + (f ) (1 c ) + f! j (1 cj ) ; y L 1 c k (.2) f < k < p: Bank k survves the lqudty shock. As n Case (.3), the (16) prce s at t frctonless value of y L and the postve excess demand condton s as follows: (1 c ) (f ) + (1 c j ) f j p f (1 c k) ( k f ) p f (17) (v) f < j k : All banks are llqud. Therefore, at date 1, no bank s able to absorb the assets put for sales, whch mples that long-term assets are transfered to non-specalsts. In other words, at date 1, all banks are closed and ther nvestment s lqudated at the value `. From these representatons of the equlbrum prce, several observatons can be made. Frst, we see that the prce of long-term assets never falls below f (as remarked n (9)). Ths s because the buyer banks can always rase f of lqudty aganst each addtonal unt of asset they purchase. Typcally, the equlbrum prce s determned by the sum of f and an other term that captures the e ect of spare lqudty n the system. Let us look at, for nstance, Equaton (12) n Case (.1). The spare lqudty n the system s represented by (1 c ) (f ), the excess lqudty held by bank. Whether or not the prce devates from the asset s value depends on the magntude of ths spare lqudty. If t s hgh enough, the RHS of Equaton (12) s then greater than y L, whch mples that excess demand s postve for all p < y L. Hence, the prce equals to the asset s expected payo,.e. p = y L (see (13)). In the other case, the prce s strctly less than 17

18 the asset s value, whch re ects a re-sale dscount. The same observatons apply to Cases (.2) and (.1). For Cases (.3) and (.2), we see that the market should ether be n excess demand or n excess supply. If the market s n excess supply, then the prce should equal f, whch volates the condtons characterzng these two cases. Therefore, Cases (.3) and (.2) only occur when the spare lqudty n the system s hgh enough so that the equlbrum prce s at ts frctonless value of y L. These observatons lead to the followng lemma: Lemma 2 The equlbrum prce of long-term assets s represented by ether (13) or (14) or (15) or (16) or (17), dependng on the dstrbuton of lqudty n the bankng system. It has the followng propertes: 1. It s ncreasng n the fundng lqudty of the long-term asset. 2. It s lower than the asset s value when the spare lqudty n the bankng system s low. 5.4 Banks Incentves for Lqudty Holdngs Gven the above representatons of the asset prce, we are now equpped to analyse the banks cash holdng decsons. From a date 0 perspectve, each bank must choose how much cash t holds on ts balance sheet. Its decson s a ected by ts expectaton about the prce of long-term assets at date 1, whch depends on other banks decsons. To gan nsghts on the banks ncentves to hold cash, let us rst formulate n ths secton the program that determnes a bank s cash holdng decson gven the choces of two other banks j and k. Then, we wll examne how the possblty of acqurng asset cheap a ects the banks motvaton for holdng cash. A. The Banks Optmzaton Problem We rst wrte the program that determnes the amount of cash bank wll hold f choosng to be lqud. Notce that ts decson wll depend on whether or not t has the opportunty to purchase some long-term assets at date 1. Hence, we dstngush between two stuatons: wth and wthout asset tradng at date 1: When two other banks j and k also choose to be lqud, no long-term assets are put for sale at date 1 by other banks. Bank s expected pro t f choosng to be lqud s then computed as follows 5 : l_ntr D c D c = Max (1 c ) y H + (1 ) (1 c ) y L c 2[0;1] subject to the break-even condton of short-term nvestors: D + (1 ) D = 1 E (18) 5 In what follows, the superscrpts "l_ntr", "l_tr", "ll_ntr" and "ll_tr" refers respectvely to "lqudty and no tradng", "lqudty and tradng", "llqudty and no tradng" and "llqudty and tradng". 18

19 and the lqudty constrant: f (19) After subtractng the break-even condton from the objectve functon, we can rewrte the above program as follows: l_ntr = Max c 2[0;1] subject to Program } l_ntr fy H + (1 ) y L 1 + E c (y H + (1 ) y L 1)g (20) c 1 E f When at least one of the two other banks chooses to be llqud, some assets wll be put for sale at date 1 n the secondary market, whch gves bank the occason to buy some assets f t has avalable lqudty. Gven that bank s demand for the asset s represented by de ned by (11), bank s expected pro t f choosng to be lqud s thus: l_tr D c D c + p = Max (1 c ) y H + (1 ) (1 c + c 2[0;1] ) y L (21) subject to the two same constrants (18) and (19). The rst bracket n (21) s smply what bank gets n case of success at date 2 f a hgh state s realzed at date 1. It s the d erence between the project s cash ow and the face value of the new debt ssued at date 1. The second bracket s the bank s pro t followng the realzaton of the low state. Note that at date 1, when the low state s observed, bank purchases a volume of assets put for sales by llqud banks, whch means that t holds 1 c + unts of long-term assets after the trade. To pay for the transacton, bank has to borrow an addtonal amount equal to p, whch explans why the face value of ts new debts s now D c +p. After some rearrangements, we get: Program } l_tr l_tr = Max c 2[0;1] ( y H + (1 ) y L 1 + E c (y H + (1 ) y L 1) + (1 ) (y L p) ) (22) subject to c 1 E f (23) As for the program determnng the amount of cash bank holds f choosng to be llqud, there are also two cases: If both other banks choose to be llqud, bank wll have to lqudate ts long-term nvestment when the low state s realzed at date 1. Its expected pro t s then: ll_ntr D c = Max (1 c ) y H c 2[0;1] 19

20 subject to the break-even condton of short-term nvestors: and the llqudty condton: D + (1 ) (c + (1 c ) `) = 1 E (24) c < 1 E f (25) Agan, subtractng (24) from the objectve functon, the above program becomes: subject to: ll_ntr = Max c 2[0;1] Program } ll_ntr fy H + (1 ) ` 1 + E c (y H + (1 ) ` 1)g (26) c < 1 E f If at least one of two other banks choose to be lqud, bank can sell ts long-term assets to lqud banks n the secondary market when t needs lqudty. Its expected pro t s now computed as follows: ll_tr = Max c 2[0;1] 8 < : + (1 ) max where s de ned by (9) and subject to h h (1 c ) y H D c (1 c ) (1 ) y L D c (1 c ) p ; 0 9 = ; (27) and D + (1 ) mn [D ; (c + (1 c ) p + (1 c ) (1 ) f ] = 1 E c < 1 E f The second term n (27) s the bank s expected pro t followng the realzaton of the low state. Note that f bank s llqud, at date 1 n the low state, t wll sell a fracton of ts long-term assets and ends up wth the remanng fracton 1. When ths term takes a strctly postve value, only a part of the nvestment s sold and thus, bank survves the lqudty shock. Otherwse, bank has to sell all of ts nvestment and ts expected pro t s equal, by lmted lablty, to zero. After smpl catons, we obtan: Program } ll_tr ll_tr = Max c 2[0;1] ( y H + (1 ) y L 1 + E c (y H + (1 ) y L 1) (1 ) (1 c ) (y L p) ) (28) subject to c < 1 E f B. Speculatve Motve of Holdng Cash 20

21 To shed lght on the banks motvaton for holdng cash that stems from the opportunty to acqure other banks asset, we compare two Programs } l_ntr and } l_tr. We see that the possblty of acqurng other banks assets at date 1, whch happens wth probablty (1 ), generates an addtonal pro t of (y L p) to the lqud bank. We refer to t as tradng pro t and denote t by T P. The rst remark s that T P s strctly postve f and only f p < y L. In other words, the opton to buy assets a ects the banks cash holdng ncentves only when assets are traded at re-sale prce. As a mean to analyse the e ects of one addtonal unt of cash held by bank on ts tradng pro t, we compute the rst dervatve of T P wth respect to c as follows: dt P dc = (y L p) d dc dp dc (29) As long as p < y L, we have: d = dp dc p f p f dc whch mples that (29) yelds: dt P = (y L p) dc p f (y L p) p dp f + (30) dc Observe that one addtonal unt of cash held by bank has two e ects on the tradng pro t. The postve e ect, captured by the rst term n (30), s the mpact on the bank s excess lqudty avalable to acqure the asset: one more unt of cash ncreases the excess lqudty by, whch allows bank to buy more assets for a gven prce p. However, one addtonal unt of cash held by bank wll also ncrease the asset prce, whch decreases the volume of assets bank can buy for a gven level of excess lqudty as well as the pro t per unt of asset acqured. The optmal amount of cash the lqud bank holds, when t expects to have the opportunty to purchase some assets at date 1, s then determned by the followng FOC: ( (yh + (1 ) y L 1) + (1 ) dt P dc + = 0 1 E c f = 0 and 0 (31) where s the Lagrange multpler assocated wth the lqudty constrant (23). We state n the followng proposton the amount of cash bank holds when t chooses to be lqud. For ths purpose, we de ne the followng varable : = (1 ) ( ) (y L f ) y H + (1 ) y L 1 + (1 ) ( ) (32) Proposton 2 In a model wth three banks, f a bank chooses to be lqud, ts cash holdngs are as follows: 21

22 1. Gven that two other banks choose to be lqud: c l_ntr = 1 E f 2. Gven that at least one of two other banks chooses to be llqud: a) If both banks j and k choose to be llqud or as long as p = y L : c l_tr = 1 E f b) In the other case,.e. among two other banks, one bank chooses to be lqud, say bank j, one bank chooses to be llqud and s closed, say bank k: c l_tr = max " 1 E f ; 1 E f + where " j s the excess lqudty held by bank j,.e. " j = (1 c j ) f j p # (1 ck ) " j " j Proof. Appendx Some more ntutons underlyng Proposton 2 are worth provdng here. Frst, as we already noted, the possblty of acqurng assets provdes banks wth an addtonal reason to hold cash only when assets are traded at re-sale prce. If banks expect that p = y L, the tradng pro t s zero and banks hold cash only for nsurng themself aganst the lqudty shock, whch explans why the lqud bank just holds the mnmum amount of cash,.e. c = 1 E f, when t expects p = y L. Second, even when the tradng pro t s strctly postve, bank stll decdes to hold a mnmum amount of cash f t expects all other banks to be llqud. The reason s that when bank s the only potental buyer of assets, one addtonal unt of cash held by bank wll have a strong e ect on the prce, whch compensates or even outwegh the e ect on ts excess lqudty. Hence, ts tradng pro t s decreasng wth ts cash holdngs. Thrd, when the lqud bank expects to have compettors n the secondary market of the asset, the e ect of ts cash holdng on the prce s weaker. As a consequence, bank may hold more than the mnmum requred amount f ts compettors cash holdngs are low. 5.5 Equlbra We study now the exstence and the man features of ratonal expectaton equlbra. We focus on pure strategy equlbra, whch can be one of the followng types: () all three banks are lqud; () one bank s lqud and two banks are llqud; () two banks are lqud and one bank s llqud; (v) all three banks are llqud. We wll characterze the condtons on the banks leverage under whch each of those equlbra exsts as well as ther propertes. 22

23 Equlbrum de nton: A quadruple (c A ; c B ; c C ; p ) s a ratonal expecton equlbrum f and only f: (1) c s the optmal cash holdngs of bank ( = A; B; C) gven p (2) p s the equlbrum prce nduced by the choces (c A ; c B ; c C ) Before proceedng wth the characterzaton, we de ne the followng thresholds: E 1 = ( ) y H + (1 ) f 1 y H + (1 ) y L 1 E 2 = E 1 ( ) y H + (1 ) y L 1 E 3 = E 1 ( ) E 4 = E 1 ( ) 1 4 y L f 2 y H + (1 ) y L (y L f ) y H + (1 ) y L 1 E 5 = ( ) y H + (1 ) ` 1 2 (1 ) (y L f ) y H + (1 ) y L 1 where s de ned by Expresson (32). It s easly to check that E 1 > E 2 > E 3 > E 4 > E 5. We begn by statng the characterstcs of two extreme equlbra where all banks choose ether to be lqud or to be llqud. Proposton 3 In a model wth three banks: (a) An equlbrum where all banks choose to be lqud exsts f and only f E E or all = A; B; C In ths equlbrum, each bank holds an amount of cash equal to c = 1 E f for all (b) An equlbrum where all banks choose to be llqud exsts f and only f E < E 5 for all = A; B; C In ths equlbrum, each bank holds a zero amount of cash,.e. c banks wll be closed at date 1 f the low state s realzed. = 0 for all. All Proof. Appendx To gan some ntuton on the constructon of those equlbra, consder for nstance the equlbrum where all banks choose to be lqud. From Program } l_ntr, we see clearly 23

24 that the optmal amount of cash each bank holds f choosng to be lqud gven that two other banks also choose to be lqud s equal to 1 E f. Next, we have to make sure that no bank has ncentves to devate. Evdently no bank should devate by holdng more than 1 E f. If a bank devates by holdng less,.e. by choosng to be llqud, we could show that ts expected pro t s as follows: 6 de = y H + (1 )f 1 + E Therefore, to ensure no devaton, the followng condton must be sats ed for all : y H +(1 ) y L 1+E 1 E f (y H + (1 ) y L 1) y H +(1 ) f 1+E whch yelds E E or all Now, we turn to the equlbrum where one bank s lqud and two banks are llqud. Proposton 4 In a model wth three banks, an equlbrum where one bank chooses to be lqud and two banks choose to be llqud exsts f and only f n the bankng system, one bank has captal rato greater than E 5 whle two other banks have captal rato lower than E 4. In ths equlbrum, at date 0, the lqud bank holds an amount of cash: c = 1 E f whereas each llqud bank holds zero cash. At date 1, two llqud banks are closed f the low state s realzed. Ther long-term assets are sold to the lqud bank at a re-sale prce: Proof. Appendx p = f < y L The constructon of ths equlbrum s smlar to that of two equlbra characterzed n Proposton 3. We rst determne the amount of cash each bank holds n the equlbrum. Then we establsh the condtons under whch no bank has ncentves to devate. As shown n Part 2(a) of Proposton 2, when two other banks choose to be llqud, the bank who chooses to be lqud wll hold 1 E f as cash. Concernng the banks that choose to be llqud, ther cash holdngs are determned by Program } ll_tr. Thanks to the observaton that the lqud bank holds an amount of cash that s just su cent to overcome the lqudty shock, we know that the lqudty avalable n the market for asset purchases wll be the one rased aganst the assets to be acqured. As a result, the asset prce equals f, whch mples that llqud banks must sell ther entre nvestment. Replace p = f and = 1 n the program } ll_tr, we see that the amount of cash llqud banks carry on ther balance sheet s zero. Wth regard to the no-devaton condtons, we could prove that f the lqud bank 6 The superscrpt "de" refers to devaton. 24