NBER WORKING PAPER SERIES LIQUIDITY REQUIREMENTS, LIQUIDITY CHOICE AND FINANCIAL STABILITY. Douglas W. Diamond Anil K Kashyap

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1 NBER WORKING PAPER SERIES LIQUIDITY REQUIREMENTS, LIQUIDITY CHOICE AND FINANCIAL STABILITY Dougla W. Diamond Anil K Kahyap Working Paper NATIONAL BUREAU OF ECONOMIC RESEARCH 050 Maachuett Avenue Cambridge, MA 0238 March 206 We thank Franklin Allen, Gary Gorton, Guido Lorenzoni, Annette Viing-Jorgenon, Nancy Stokey, Nao Sudo, John Taylor, Harald Uhlig and eminar participant at the Aian Development Bank Intitute, Eat Aian Economic Seminar, Imperial College, Melbourne Intitute Macroeconomic Policy Meeting, National Bureau of Economic Reearch Monetary Economic meeting, the Bank of England, European Central Bank, Centre for Economic Policy Reearch and Centre for Macroeconomic Conference on Credit Dynamic and the Macroeconomy, Rikbank, and the Univerity of Chicago for helpful comment and Adam Jorring for expert reearch aitance. We thank the Initiative on Global Market at Chicago Booth, the Fama Miller Center at Chicago Booth and the National Science Foundation for grant adminitered through the NBER for reearch upport. Diamond ha an ongoing viiting cholar relationhip with the Federal Reerve Bank of Richmond and Chicago. Kahyap ha an ongoing viiting cholar relationhip with the Federal Reerve Bank of Chicago and i an advior to the Rikbank, ee hi web page for a complete lit of hi non-teaching compenated activitie. The view expreed herein are thoe of the author and do not necearily reflect the view of the National Bureau of Economic Reearch. NBER working paper are circulated for dicuion and comment purpoe. They have not been peer-reviewed or been ubject to the review by the NBER Board of Director that accompanie official NBER publication. 206 by Dougla W. Diamond and Anil K Kahyap. All right reerved. Short ection of text, not to exceed two paragraph, may be quoted without explicit permiion provided that full credit, including notice, i given to the ource.

2 Liquidity Requirement, Liquidity Choice and Financial Stability Dougla W. Diamond and Anil K Kahyap NBER Working Paper No March 206 JEL No. E44,G0,G8,G2 ABSTRACT We tudy a modification of the Diamond and Dybvig (983) model in which the bank may hold a liquid aet, ome depoitor ee unpot that could lead them to run, and all depoitor have incomplete information about the bank ability to urvive a run. The incomplete information mean that the bank i not automatically incentivized to alway hold enough liquid aet to urvive run. Regulation imilar to the liquidity coverage ratio and the net table funding ratio (that are oon be implemented) can change the bank incentive o that run are le likely. Optimal regulation would not mimic thee rule. Dougla W. Diamond Booth School of Buine Univerity of Chicago 5807 S Woodlawn Avenue Chicago, IL and NBER dougla.diamond@chicagobooth.edu Anil K Kahyap Booth School of Buine Univerity of Chicago 5807 S. Woodlawn Avenue Chicago, IL and NBER anil.kahyap@chicagobooth.edu

3 Introduction In September 2009 the leader of 20 major economie created the Financial Stability Board (FSB) whoe purpoe i to coordinate at the international level the work of national financial authoritie and international tandard etting bodie (SSB) in order to develop and promote the implementation of effective regulatory, uperviory and other financial ector policie. Since that time the financial ytem ha undergone a regulatory overhaul. The term macroprudential regulation ha become ynonymou with much of thi effort. A we explain in the next ection, what that mean in practice remain omewhat eluive. But, there are two tangible change that are on track to occur over the remainder of thi decade. One widely tudied et of reform pertain to the rule regarding capital requirement for bank. Le wellundertood i that, through their cooperation via the Bael Committee on Bank Superviion, the major economie have alo agreed alo to implement by 209 new rule governing bank debt tructure and requirement to hold certain type of liquid aet. To date there i a remarkable aymmetry in the economic analyi of the capital and liquidity regulation. The pioneering work of Modigliani and Miller (958) provide a olid theoretical framework for analyzing capital regulation. Any tudent taking a firt coure in corporate finance will encounter thi theory and there i a maive empirical literature that explore the theory prediction. Capital regulation for bank at the international level go back to 988 and there many empirical examination of the impact of thee regulation. In contrat, there i no benchmark theory regarding liquidity proviion by intermediarie. Indeed, financial economit even have competing concept that they have in mind when dicuing liquidity, o that there i no generally accepted empirical meaure of liquidity economit tudy. Allen (204), in hi urvey of the nacent literature on liquidity regulation, conclude by writing much more reearch i required in thi area. With capital regulation there i a huge literature but little agreement on the optimal level of requirement. With liquidity regulation, we do not even know what to argue about. Nonethele, the global regulatory community ha agreed on certain liquidity requirement (Bael Committee on Bank Superviion (203a, 204)). Two new concept, the liquidity coverage ratio and the net table funding ratio, have been propoed and bank by 209 will be compelled to meet requirement for thee ratio. Thu, it eem fair to ay we are in a ituation where practice i ahead of both theory and meaurement. In thi paper we urvey the exiting work on liquidity regulation and develop a framework for dicuing the regulation. The theory that we propoe ugget, in certain parameterization, regulation bearing ome reemblance to the liquidity coverage ratio and net table funding ratio can emerge a one which will improve outcome relative to an unregulated benchmark. However, the regulation that arie in our model would naturally differ acro bank, depending

4 on certain bank characteritic, o they do not mimic exactly the one that are on track to be implemented. The critical ingredient in our model are the following. Firt, we conider bank which are patially eparated and hence do not compete aggreively for depoit. Treating the bank a monopolit implifie the analyi by allowing u to ide-tep ome complication that arie from having to model the depoit market equilibrium. The model can alo be interpreted a a decription of the aggregate banking ytem, which for many financial tability and regulatory dicuion i the object of primary concern and under thi interpretation ignoring the depoit competition i perhap more natural. Second, we aume that intermediarie provide liquidity inurance for cutomer who have uncertain withdrawal need (or conumption deire). We build on the Diamond and Dybvig (983), henceforth DD, model of banking in which bank provide thi inurance by relying on the law of large number to eliminate idioyncratic cutomer liquidity need. For thoe familiar with DD, we make two modification. The firt i allowing the bank to invet in a liquid aet that ha a rate of return exceeding the return from liquidating illiquid aet and thu i the efficient way to arrange to pay cutomer that need liquidity. Thi introduce a tradeoff between lending and holding liquidity a in Bhattacharya and Gale (987) and everal paper of Allen and Gale (997 and other). The other modification to DD i the form of run rik that the bank face. Bank are aumed to have a good aement of the aggregate need of their cutomer for fundamental reaon. But, they alo know that ome cutomer will receive a ignal about the bank which could lead to a run. The unpot that we conider are a metaphor for people being concerned with the health of the bank, but not having a fully formed et of belief about the bank olvency tatu. In making their deciion we aume that cutomer are unable to fully evaluate the ability of the bank to honor depoit. Given the complexity of modern bank it eem realitic to preume that mot cutomer cannot preciely determine their bank maturity mimatch and hence it vulnerability to a run. The imperfect information create a challenge for the bank becaue their cutomer will not necearily know if the bank i prudently holding liquidity or not, which reduce the incentive to hold liquidity. In the event that a run doe occur, we depart from DD and Enni and Keiter (2006) to allow for the poibility that not all cutomer eek to withdraw their fund. We believe it i ueful to analyze partial run for two eparate reaon. One i that in practice there do eem to be ome ticky depoit that do not flee even in time of coniderable banking tre. In addition, even before trouble occur it i uually clear which type of depoit are prone to running. So thi allow u to talk about policie for different type of withdrawal rik. Within thi environment we can ae the vulnerability of the financial ytem to run under different regulatory arrangement. In the baeline cae, we aume that bank imply maximize 2

5 their profit and ee which type of equilibria arie. A uual in DD tyle model, the outcome depend critically on how depoitor form belief. It i poible, under certain parameter configuration, that the pure elf-interet motive of the bank will ufficient to inure that the ytem will be run proof even if depoitor had no detailed information about a bank liquidity holding. In thee ituation, added liquidity could not influence whether a given depoitor would chooe to join a run if one wa feared. We decribe everal reaon why depoitor may not be able to ue ome type of dicloure of a bank liquidity holding to determine if the holding i ufficient to allow it to urvive a run. To fix idea, one can conider whether a bank would chooe to hold thi ufficient amount of liquidity even if it choice between liquid aet and illiquid loan wa completely unobervable. In circumtance where depoitor cannot be ure about how change in liquidity holding impact the robutne of bank to run, the bank will typically face a tenion in deciding how much to fortify themelve againt the rik of a run. They can alway chooe to be ufficiently conervative to be able to withtand a wort cae of fundamental withdrawal a well a a panic. But in order to do that, they will engage in very little lending, and the forgone profit from deterring the run will be high. The additional liquidity to urvive a run will turn out to be exceive whenever a run i avoided. Hence, it i poible they will make more profit from added lending which would leave them unable to alway be able to utain a run. We next allow regulatory intervention that place retriction on preent and poibly on future bank portfolio choice. In the baeline et up, the bank have perfectly aligned incentive to prepare to ervice fundamental aggregate withdrawal need. So the regulatory challenge i to determine whether a requirement that ditort their private incentive toward being more robut to a run will improve outcome. We allow for regulation that i inpired by the two impending Bael rule. One variant require an initial liquidity poition that mut be etablihed before depoitor make their intention clear. Thi can function like the net table funding ratio that i propoed a part of the Bael reform. A econd option i a mandate to alway hold additional liquid aet beyond thoe needed for the fundamental withdrawal. Thi impoe both preent and future minimum holding of liquid aet. Thi regulation look like a traditional reerve requirement for the bank, but can alo be interpreted a a kind of liquidity coverage ratio that i part of the Bael reform. One point of contention regarding the liquidity coverage ratio that ha emerged i whether required liquidity can be deployed in the cae of a crii. Goodhart (2008) framed the iue nicely with a now famou analogy of the weary traveller who arrive at the railway tation late at night, and, to hi delight, ee a taxi there who could take him to hi ditant detination. He hail the taxi, but the taxi driver replie that he cannot take him, ince local bylaw require that there mut alway be one taxi tanding ready at the tation. 3

6 One way to interpret the Goodhart conundrum i to recognize that, broadly peaking, there are two way to think about the purpoe behind liquidity regulation. One motivation can be to make ure that bank can better withtand a urge in withdrawal hould one occur. From thi perpective mandating that the lat cab cannot depart the tation eem foolih. Another poible motivation i to deign regulation aimed at reducing the likelihood of a withdrawal urge in the firt place. Our model help highlight the potential incentive propertie of regulation and can potentially explain why mandating the preence of ome unued liquidity could be beneficial. In tudying how private and ocial incentive for liquidity choice diverge, our main concluion from analyzing the two Bael-tyle regulation i that they may improve outcome relative to the one that arie from pure elf-interet, but each bring potential inefficiencie. Hence, we briefly alo decribe the olution of the mechanim deign problem for a ocial planner who ha le information about withdrawal rik than the bank doe and eek to optimally regulate bank to avoid run. That olution provide a natural benchmark againt which to judge the Bael-tyle regulation. The remainder of the paper i divided into five part. Section two contain our elective overview of previou work. We organize thi into three ub-ection. We begin with an overview of the emerging policy propoal and reearch regarding macroprudential regulation. We then hone in on the enormou and rapidly growing literature on capital regulation. We provide our perpective on how to group thee paper and highlight everal recent excellent urvey on the pure effect of capital regulation. We cloe with a review of the mot relevant paper for our quetion that motivate u about liquidity regulation. Section three introduce the benchmark model. We explain how it work under complete information. We alo derive a generic propoition that hold with incomplete information that decribe when the bank preferred liquidity choice will be ufficient to deter a run. Generically, however, privately choen level of liquidity need not be ufficient to deter run. So thi open the door for regulation that might do o. In ection four we analyze the two type of liquidity regulation that are akin to the one contemplated under the Bael proce. We firt demontrate that a particular type of regulation that require the bank to hold liquid aet equal to a fixed percentage of depoit at all time can potentially deter run. Thi work becaue the liquidity mandate, combined the bank elfinteret to prepare to ervice predictable depoit outflow, lead the bank to hold more overall liquidity than it would otherwie. Becaue depoitor undertand thi, it remove the incentive to run in ome cae. We alo conider alternative aumption about depoitor knowledge and the information available to regulator and ae the vulnerability of the bank to run in thee cenario. In ection 5, we decribe a couple of extenion of the baeline model. The firt ketche a mechanim deign problem where the regulator doe not have all of the bank information and 4

7 eek to implement run-free banking. We fully characterize the olution to thi problem in Diamond and Kahyap (206), here we decribe the main finding from thi exercie. It turn out that a regulator with ufficient tool can induce the bank to hold the proper amount of liquidity depite the private information advantage poeed by the bank. We alo briefly dicu capital regulation. We explain why a a tool for managing liquidity problem, capital requirement can be relatively inefficient compared to the other regulation that we have reviewed. Obviouly in a richer model where both credit rik and liquidity rik are preent, capital and liquidity regulation can erve different purpoe. We decribe ome of thee difference. Section ix preent our concluion. Beide ummarizing our finding, we alo poe a few open quetion that are natural next tep to conider in addreing the iue analyzed in thi paper. 2. Literature Review Reearch on financial regulation ha exploded ince the global financial crii, and the number of regulatory intervention and tool ha alo expanded maively. To review all of thi work would require a book. To keep our review manageable, we limit our dicuion to focuing on the theoretical underpinning and rationale behind thee change. 2. Macroprudential regulation Clement (200) provide the intereting hitory of the origin and evolution in the meaning of the phrae macroprudential. Hi bet etimate i that the term appeared firt in 979 in the document of the committee that wa the fore-runner to the Bael Committee on Bank Superviion. The firt public document uing the term which he can identify wa a report by the committee now known a the Committee on the Global Financial Sytem. It defined macrorpudential policy a promoting the afety and oundne of the broad financial ytem and payment mechanim. The phrae took on added prominence when it wa the focu of a September 2000 peech by Andrew Crockett (who wa then the General Manager of the Bank for International Settlement (Crockett (2000)). He defined the objective of macroprudential policy to be limiting the cot to the economy from financial ditre, including thoe that arie from any moral hazard induced by the policie purued. Crockett rational for calling for macroprudential policie wa hi belief that optimal choice for a ingle intitution could create problem for the financial ytem For a divere et of perpective on the changing pot-crii regulatory landcape ee Čihák, Demirgüç-Kunt, Martínez-Pería and Moheni-Cheraghlou (203), Financial Stability Board (205), Claeen and Kodre (204), Bael Committee on Bank Superviion (203), and Fiher (205). 5

8 a a whole. He wa explicitly focued on the ditinction between the uperviory challenge for monitoring an individual intitution and thoe for protecting the aggregate financial ytem. Crockett did not offer precie microeconomic foundation for why the private action of individual actor would not be aligned with ocial welfare, but he did give a few example where he aw the potential for divergence. One poibility he cited i that one bank eeking to limit it credit expoure could chooe to cut lending to it client, but if all bank did thi a credit crunch could enue that would trigger a receion. A econd example wa the poibility of what we would now dub to be a fire-ale where all agent imultaneouly cut back on aet expoure due to falling price and in the coure of doing o exacerbate the price decline. A third problem arie if many lender horten the maturity of their funding to a particular borrower, then the rik of a run can increae o that they are all more vulnerable. Our view i that Crockett potlight on the divergence between the narrow private interet of individual intitution (or upervior monitoring a ingle intitution) and the interet of overall ociety i exactly the right focu for conidering macroprudential policie. Indeed, thi literature would be well-erved to move in the direction where all macropurudential paper tart by clarifying why (and when) ocial and private interet diverge. The challenge for both for reearcher and policymaker i the difficulty in formalizing and prioritizing the exact reaon for the divergence. To clearly ee the problem, compare three prominent perpective on macroprudential regulation that have followed Crockett. Firt, variou BIS document (e.g. Clement (200)) now interpret Crockett a having identified two type of problem that are to be addreed. One relate to the build-up of rik over time, that are often now referred to a the pro-cyclicality of the financial ytem or the time dimenion of the macroprudential policy problem. The other relate to the ditribution of rik within the financial ytem, the o-called cro-ectional dimenion of the problem. Many official ector document adopt the convention of eparating time-erie and cro-ectional macrorprudential problem. A Clement (200) note, while the BIS work in thi area ha been relatively precie in the way thee iue are dicued, the uage of the term in the public phere ha on occaion been looe. It i not uncommon for it to be employed almot interchangeably with policie deigned to addre ytemic rik or concern that lie at the interection between the macroeconomy and financial tability, regardle of the pecific tool ued. In contrat, Hanon, Kahyap and Stein (20) tart with a particular view of how modern financial crie unfold, and why both an unregulated financial ytem, a well a one baed on capital rule that only apply to traditional bank, i likely to be fragile. Their perpective, appealing to the model in Stein (202), preume that bank will find it cheaper to fund themelve with hort term debt than equity, o that bank have limited incentive to build trong equity buffer in normal time. If, in a crii, uch bank uffer ubtantial loe, then the market value of debt claim can fall below the face value, which will deter them from raiing 6

9 new equity (Myer (977)). Conequently, in thi cae the bank are likely to comply with capital regulation by hrinking their aet bae. Hence, Hanon et al argue that the goal of macroprudential regulation hould be to control the ocial cot aociated with exceive balance-heet hrinkage on the part of multiple financial intitution hit with a common hock. A recent urvey by the Norge bank taff, Borchgrevink, Ellingrud and Hanen (204), argue that in fact there are ix market failure that can give rie to macroprudential concern. Thee are pecuniary externalitie, interconnectedne externalitie, trategic complementaritie, aggregate demand externalitie, market for lemon and deviation from full rationality. Not urpriingly they conclude Becaue of the diverity of thee categorie, policy leon diverge. There i yet no workhore"model for policy analyi. Though they do argue that capital and liquidity regulation hould tuned to aggregate condition, not jut thoe of individual bank, and that borrower hould be ubjected to time-varying policie that aim to force them to internalize the cot of exceive borrowing. 2 We hare the Borchgrevink et al (204) concluion that the macroprudential literature at thi point remain in ufficient flux that it i too oon to reach firm concluion about where it will lead. Hence, for the remainder of our analyi we focu on capital and liquidity regulation where the range of iue to be conidered can be narrowed and where pecific global policie are being implemented. 2.2 Capital regulation For an overview of the literature on capital regulation, it i ueful to ort paper along two dimenion. The firt regard what i aumed regarding the Modigliani-Miller (958) (henceforth MM) capital tructure propoition. A in all model of corporate finance, abent failure of one of the MM propoition any choice regarding capital tructure will be inconequential. There have been four primary MM violation that have drawn attention in the literature. One concern that exitence of depoit inurance. If certain part of a bank capital tructure i protected from loe by the government, that can create rik-hifting incentive for equity holder. In many model, bank manager working on behalf of the equity owner face an incentive to gamble after advere hock that goe unchecked becaue depoitor are immune from loe that they would uffer if the gamble fail. 2 Other have alo choen to organize their analye around ditinction between the kind of tool that can be deployed. For example, Aikman, Haldane, and Kapadia (203) claify tool into three group: thoe that operate on financial intitution balance heet; thoe that affect the term and condition on financial tranaction; and thoe that influence market tructure. While Cerutti, Claeen, and Laeven (205) preent empirical analye comparing 2 type of different regulation. 7

10 A econd ditortion i concern over guarantee to protect equity holder of bank from loe. Uually thi i couched a a problem of having ome bank that are aumed to be too big or too-interconnected to fail. But, in the recent global financial crii, there were alo cae in ome countrie where equity owner of maller, non-ytemic bank were inulated from loe due to political connection. A third violation regard the MM aumption of complete financial market. With incomplete market, an intitution that create new ecuritie could be valuable. In the banking context, depoit are a leading example of pecial ecurity that bank might create. Finally, there are many model where either aymmetric information or moral hazard problem are conidered. Some of the prominent example include the poibility that borrower know more about their invetment opportunitie than lender, or that borrower can hift the rikine of their invetment after receiving funding. So unlike much of the reearch on non-financial corporation, the trade-off theory of capital tructure, whereby firm prefer debt for it tax advantage and balance thoe benefit againt cot of financial ditre, ha not figured prominently in the banking reearch on capital regulation. Rather, regulation i uually jutified on the ground of addreing one of thee other four problem. The type of regulation that can be welfare improving will differ depending on which of thee other friction i aumed to be preent. The econd important dimenion one which the literature can be organized concern the economic ervice that bank are aumed to provide. 3 Broadly, there are three type of ervice that have been modeled. The firt preume that certain financial intitution can expand the amount of credit that borrower can obtain (ay, relative to direct lending by individual aver). The micro-founded theorie typically aume that borrower can potentially default on loan and o any lender ha to be diligent in monitoring borrower (Diamond (984)). By concentrating the lending with pecialied agent, thee monitoring cot can be conerved and the amount of credit extended can be expanded. A econd widely poited role for intermediarie i helping people and buinee hare rik (Benton and Smith (976), Allen and Gale (997)). There are many way to formalie how thi take place, but perhap the implet i to recognie that becaue bank offer both depoit and equity to aver, they can create two different type of claim that would be backed by bank aet. Thee two choice allow aver to hedge ome rik aociated with lending and thi hedging improve the conumption opportunitie for aver. More broadly, thee theorie uppoe that bank help pool and tranche rik. 4 3 The next few paragraph are taken from Kahyap, Tomoco and Vardoulaki (204) 4 For intance, if there are tranaction cot aociated with buying ecuritie, a bank that make no loan but hold traded ecuritie could till be valuable. 8

11 A third cla of model, which complement the econd, uppoe that the financial ytem create liquid claim that facilitate tranaction. There are variou motivation behind how thi can be modelled. In DD tyle model, an intermediary can cro-inure conumer need for liquidity by exploiting the law of large number among cutomer. But doing o expoe bank to the poibility of a run, which can be diatrou for the bank and it borrower and depoitor. Calomiri and Kahn (99) and Diamond and Rajan (200) explain that the very detructive nature of a run i perhap helpful in diciplining the bank to work hard to honour it claim. So the fragility of run i potentially important in allowing both high amount of lending and large amount of liquidity creation. Gorton and Winton (2003) give a much more complete review of thee three clae of theorie and one clear concluion that emerge i that depending on which of thee three ervice i preumed to be operative, and which of the MM failure are preent, one can reach very different concluion about the efficacy of capital regulation in improving welfare. For intance, in model where liquidity creation i not one of the ervice provided by bank, the cot of mandating higher amount of equity financing are often modet. Likewie, the benefit of protecting taxpayer from having to bail out bank or depoitor by forcing more equity iuance are potentially ubtantial. Rather than reviewing the reult from many paper on capital regulation we refer intereted reader to everal recent urvey including Brooke et al (205), Martynova (205), Rochet (204) (and the reference therein). Both Brooke et al. and Rochet attempt to compare the macroeconomic cot and benefit of higher level of required capital and ue a variety of calculation to ae them. In both cae, the benefit are preumed to be a reduction in likelihood and potential everity of financial crie (and the aociated reduction in output). While the cot of higher capital requirement are the poible potential reduction in lending and loe of output. One humbling obervation from both of thee paper i that depite drawing on many different type of evidence, empirically etimating the net effect i difficult and there i ubtantial uncertainty about the overall net effect. One other important obervation i that mot of the paper in thee review are not very informative regarding liquidity regulation, or the potential interaction of liquidity and capital regulation becaue in the environment being analyzed there i no value to liquidity creation (and hence no cot to limiting it). Indeed, Bouwman (205), in a review article, emphaize the dearth of reearch on potential interaction between capital and liquidity regulation and argue that it i critically important to develop a good undertanding of how capital and liquidity requirement interact. 2.3 Liquidity Regulation A mentioned in the introduction, there are far fewer paper that eek to invetigate the purpoe and effect of liquidity regulation. Allen (204) offer a urvey of thi nacent literature and we 9

12 hare the entiment of the concluding paragraph of hi urvey. He write, much more reearch i required in thi area. With capital regulation there i a huge literature but little agreement on the optimal level of requirement. With liquidity regulation, we do not even know what to argue about. It i poible to again ue a imilar kind of two-way to claification regarding capital regulation to decribe much of the thinking on liquidity. Trivially, if the economic ervice offered by a bank do not include the proviion of liquidity, then regulation that focue on liquidity will not be particularly intereting to conider. It i poible that in uch environment regulating liquidity could make ene to achieve other aim, uch a upplementing or ubtituting for capital requirement. However, if maturity tranformation i not one of the output of the financial ytem, aement of the efficacy of liquidity regulation in uch model will be incomplete. Put bluntly, if there are no cot to limiting liquidity proviion per e, then obviouly the cot of regulation that have thi effect cannot be fully aeed. It i worth noting that will mot of the literature on liquidity and liquidity regulation label the intitution that that undertake thi activity a bank. However, a became evident in the global financial crii thi activity i hardly limited to bank. Figure, reproduced from Bao, David and Hong (205) how the total amount of runnable funding inide the U.S. financial ytem over the pat 30 year. We draw three concluion from their etimate that are worth bearing in mind throughout the ret of the dicuion. Firt, there ha been a izable increae in the amount maturity tranformation over the lat 20 year. From 995 until 205, the cale of uch activity roe by 50% a meaured relative to Gro Dometic Product. Second, a far back a 985 a much of thi activity ha occurred outide the banking ytem a inide it. Third, the decline immediately after the GFC wa izable. The drop in repurchae agreement and money market fund were epecially pronounced, but even a a percent of GDP, the level in 205 i very imilar to the level in 2005 (jut before the frenzied period ahead of the GFC). Hence, maturity tranformation i till happening on a ubtantial cale even after the GFC and all of the variou regulatory reform that have been introduced. Given thi evidence, we focu only on paper where one of the ervice of the financial ytem i to provide liquidity. Among thee it i helpful to eparate them into paper that model liquidity proviion in the ame way or imilarly to DD, and thoe that introduce other mechanim. 0

13 Figure : Bao, David and Han (205) Etimate of Runnable Funding in the U.S. Among the DD tyle model, we focu on three that are cloely related to our analyi. Enni and Keiter (2006) have a DD tyle model (related to Cooper and Ro [998]) which determine how much liquidity bank need to hold to deter run. They compute the amount of exce liquidity the bank mut hold to buffer it againt a run by all depoitor, and alo determine the optimal amount to promie depoitor. In their model with full information, when depoitor deire afe bank, there will be private incentive to hold enough exce liquidity to deter a unpot-baed run. They do not tudy regulation becaue there i no need for any under their aumption, but we will ee that ome of the ame force that are preent in their model arie in our. Vive (204) analyze a quetion imilar to that in Enni and Keiter (2006): what are the efficient combination of equity capital and liquidity holding to make a bank afe when it ubject to run baed on private information about it olvency? He tudie a global game where a bank can be inolvent or illiquid. The need for regulation i not conidered explicitly, but he doe examine what capital and liquidity level would make the bank afer. He find that capital and liquidity are differentially ucceful in attending to inolvency and illiquidity. In particular, if depoitor are very conervative (and which make them more inclined to run in the

14 model), increaed liquidity holding which reduce profit by inveting more in liquid aet can enhance tability. Farhi, Goloov, and Tyvinki (2009) invetigate a DD model where conumer need bank to invet and where the conumer can trade bank depoit. Abent a minimum liquidity regulation, it i profitable to free ride on the liquidity held by other bank, becaue bank offer rate which ubidize thoe who need to withdraw their depoit early (which i the pirit of Jacklin(987)). A floor on liquidity holding remove the incentive for thi free riding. Among the non-dd model, one that i related i Calomiri, Heider, and Hoerova (204). They have a ix period model where bank can potentially engage in rik-hifting o that when bank uffer loan loe they may not be able to honor their depoit contract. Cah i obervable and mandating that bank mut have minimum level of cah reerve can limit the rik-hifting. Santo and Suraez (205) examine another role for liquidity when run occur lowly: it allow time to decide if the bank aet are ufficient to imply olvency abent a run. Thi channel i forecloed in our et-up with aet which are free of rik. More generally, our approach i cloely related to the mechanim deign approach to regulation of monopolit in Baron and Myeron (982). They alo were intereted in invetigating how regulation could be tructured to induce the party being regulated to efficiently ue information that i private. 3. Baeline Model We begin by decribing a baeline et up in which the timing and preference are a in DD. We then modify certain informational aumption to bound the poible outcome. Throughout we maintain that there are three date, T= 0,, and 2. The interet rate that bank mut offer are taken a given, motivated by a monopoly bank which mut meet the outide option of depoitor to attract depoit. Equivalently, the ingle bank can be thought of a repreenting the overall banking ytem. For a unit invetment at date 0, the bank offer a demand depoit which pay either r at date or r 2 at date 2. Thi effectively offer a gro rate of return r 2 /r between date and 2 which i equal to the exogenou outide option (uch a government bond) for depoitor between thee date. Eentially, the bank offer one period depoit which equal the interet rate on the outide option. We will aume that depoitor are ufficiently rik avere that they would like the banking ytem to upply one period depoit that are rikle. Hence, when we conider intervention they will be deigned to deliver a thi a the only poible equilibrium. 2

15 The reidual claim after depoit are paid i limited liability equity retained by the banker. All equity payment are made at date 2. 5 The bank can invet in two aet with contant return to cale. One i a liquid aet (which we will interchangeably refer to a the afe aet) that return R >0 per unit inveted in the previou period. The other i an illiquid aet for which a unit invetment at date 0 return at date 2 an amount that exceed the return from rolling over liquid aet (R 2 > R * R ). The illiquid aet (which we will interchangeably refer to a loan) can be liquidated for θr 2 date, where θr 2 < R and θ 0. Thee retriction imply that when the bank know it mut make a payment at date, it i alway more efficient to do that by inveting in the afe aet rather than planning to liquidate the loan. We alo aume that banking i profitable even if the bank invet excluively in the liquid aet, 2 o that r R and r2 R. Thi i a ufficient condition to guarantee that requiring exce liquidity will not make the bank inolvent (though it till will reduce the efficiency of invetment). In addition, we aume that bank profit from inveting in illiquid aet when depoitor hold their depoit for two period (borrowing hort-term repeatedly to fund long-term illiquid invetment) i greater than from inveting in liquid aet when depoitor hold their r2 r depoit for only one period (or < ). Thi implie that a bank i mot profitable when in R2 R can finance loan returning R 2 with depoit for two period at cot r 2 (a compared with financing liquid aet for one period). Thi econd aumption i ued only to obtain ome reult on optimal liquidity holding. There are many poible reaon to preume that the illiquid aet can be liquidated for only θr 2. For intance, in DD liquidation can be thought of a a non-tradable production technology. Alternatively it could reflect the bank lending kill, implying that it would be worth le to a buyer than to the bank becaue (compared to the bank) the buyer would be able to collect le from a borrower, a in Diamond-Rajan (200). Nothing in our analyi hinge on why thi dicount exit, though we do init that it i operative for everyone in the economy including a potential lender of lat reort. Alo, our aumption that θ i a contant implie that we are not modeling a ituation where the ale price depend only on the amount of remaining liquidity held by potential buyer (a in Bhattacharya-Gale (987), Allen-Gale (997) and Diamond (997)). For fundamental reaon, a fraction t of depoitor want to withdraw at date and -t want to withdraw at date 2 in tate. The realization of t are bounded below by t 0 and above by t The banker will know the realization of t when the aet compoition choice i made. Thi aumption i meant to capture the fact that bank have uperior information about their 5 We could introduce another incentive problem for the banker to motive a minimum value of equity at all date and tate, but for now the bank will operate efficiently a long a equity remain poitive in equilibrium. 3

16 cutomer. Indeed, ome early theorie of banking uppoed that the advantage of tying lending and depoit making wa that by watching a cutomer checking account activitie a bank could gauge that cutomer creditworthine (Black(975)). Meter, Nakamura and Renault (2007) provide direct evidence upporting the aumption that bank can learn about cutomer credit need by monitoring tranaction account. Drawing on a unique data et from a Canadian bank, they demontrate the bank i able to infer change in the value of borrower collateral that i poted againt commercial loan by tracking flow into and out of the borrower tranaction account. At thi bank they document that the number of prior borrowing in exce of collateral i an important predictor of credit downgrade and loan writedown. Mot importantly, the bank ue thi information in making credit deciion. Loan review become longer and more frequent for borrower with deteriorating collateral. 6 In what follow, we make the implifying aumption t i alway known exactly by the bank, but the analyi alo goe through o long a the bank i imply better informed than the depoitor and the regulator. To undertand agent incentive, note that if the ex-pot tate i and there i not a run, a fraction f =t will withdraw r each, requiring r t in date reource, and thi will leave a fraction -t depoitor at date 2 who are collectively owed r 2 (-t ) (in date 2 reource). If we let α be the fraction of the bank portfolio that i inveted in the liquid aet and (-α ) be the portion inveted in the illiquid one, then the bank profit, and hence it value of equity in general will be (-α )R 2 + (αr-f r )R -( -f )r 2 if f r αr Value of equity = (f () r -αr) Max{0, -α - )R 2 - (-f )r 2} if fr > αr θr 2 Becaue we are auming that the bank know t, it own elf-interet will lead it to make ure to alway have enough inveted in the liquid aet to cover thee withdrawal. So abent a run, the profit are very intuitive and eay to undertand. The firt term in () when fr αr repreent the return from the illiquid invetment, the econd reflect the pread on the afe aet relative to depoit (recognizing that any leftover fund are rolled over), and the third term reflect the funding cot of the remaining two period depoit. When fr > αr, the bank need to pay out more than it liquid aet are worth at date. To honor it promie the bank mut liquidate illiquid aet worth θr 2 each, implying that each unit of withdrawn in exce of α R remove θr loan from the bank balance heet. Thee loan would each be worth R 2 at date 2. For a 2 6 Norden and Weber (200) alo find that credit line uage, credit limit violation and cah inflow into checking account are unuual in the period preceding default by mall buinee and individual in Germany. 4

17 bank in thi ituation that can honor all early and late withdrawal the reidual profit go to the banker (otherwie the bank i inolvent). Given our aumption about interet rate and liquidation dicount, if actual withdraw, f, were known, the bank would chooe to hold enough liquid aet to avoid needing to liquidate any loan. We know that at all time, even abent a run in tate, f tr. A a reult, the bank will alway have an incentive to chooe tr AIC α α A a reult we refer to R holding of the bank. AIC α a the automatically incentive compatible liquidity It i intereting to conider what happen when a run i poible. We uppoe that a fixed number Δ of the patient depoitor are highly likely to ee a unpot. All depoitor (and the bank) know Δ and upon eeing the unpot they mut decide whether they believe that the other who ee it will decide withdraw their fund early. A mentioned earlier, the unpot i intended to tand in for general fear about the olvency of the bank, o the inference problem relate to their conjecture about whether other invetor might panic. In that cae, they have to decide whether to join the run. 7 So in general f > t i poible. If the bank will be inolvent with a fraction of withdrawal of any amount le than t+δ, then we aume each depoitor who ee then unpot will withdraw and f =t +Δ. Thi will give zero to all who do not withdraw, and the goal of bank or it regulator i to prevent thi outcome from ever being a Nah equilibrium. We will refer to a bank a untable if it aet holding admit the poibility of a run. Alternatively, we refer to a bank a table if it aet holding eliminate the poibility of a run. In addition, we will aume that if the bank i exactly olvent at f =t +Δ, no depoitor who doe not need to withdraw (and only ee the unpot) will withdraw. Thi condition etablihe exactly how much liquidity i needed to deter a run (a oppoed to providing a floor which mut Stable be exceeded). We define the minimum table amount of liquidity holding, a a the minimum fraction of liquid aet in tate which eliminate the poibility of a run. Thi implie Stable that a bank witha a will be run-free. 3. Complete information. We preume that depoitor deire run-free bank depoit. A a firt benchmark, uppoe that depoitor know all of the choice and information which bank know, and thu oberve α, Δ and t. In thi cae, the need to attract depoit will force the bank to make itelf run-free. If, 7 Uhlig (200) how that partial bank run in a DD tyle model can arie if there other type of diperion in agent belief. For intance, if depoitor are highly uncertainty avere and differ in their etimate of θ that heterogeneity can lead to a partial bank run in hi etup. 5

18 given depoitor knowledge of α, Δ and t, the bank would remain olvent in a run, then it never i individually rational to react to the unpot and there will be no run. Propoition how that it i poible that the bank will not need to ditort it holding of liquidity to implement run-free banking. tr Propoition : If the bank chooe α =, and if t + < 2 AIC R tr t r + (- )θr 2- r2θ R ΔrR (equivalently θ ), r - r θ R R -r t - r R -t invetor will not run and the bank i table with α ( ) ( -Δ) 2 t r =. AIC R 2 (f r -α R) Proof: If f r > a R, the bank' equity i poitive when (-α - )R 2 - (-f )r2 0. So, when θr 2 the automatically incentive compatible level of initial liquid value of equity i decreaing in f and equal zero when f AIC AIC ity α i choen ( α = ) 2 2 R 2 * if t +Δ i le than f, then the depoitor alway know the bank will be olv Nah equlibrium with a run. QED * tr R = tr R, the tr R + (- )θr - r θ. Therefore, r - r θ ent and there i no The propoition imply tate the condition when the bank i ufficiently profitable and liquid, o that by holding only enough of the liquid aet to ervice fundamental withdrawal, the bank will AIC Stable nonethele be olvent in the event of a run. Under thee condition, a a. When the condition for Propoition fail, becaue loan are quite illiquid or the bank i not very profitable, then to deter run, a bank mut hold more liquidity than i needed to meet normal withdrawal. One ueful cae to contemplate, i when loan are totally illiquid (θ=0). In thi cae, the bank mut alway hold enough liquidity to fully finance the run becaue there i no Stable r AIC tr other way to get acce to liquidity, or α = ( t + ) > α =. Therefore, the bank mut R R alway hold more liquidity than i needed for normal withdrawal in order to deter a run. More generally, whenever 6

19 tr t r + (- )θr - r θ 2 2 R Δr table R + or θ< then the bank mut increae α to α to r - r θ 2 R 2( R -r t ) - r 2 R ( -t -Δ ) t > definitely deter the run, where α table i uch that t α + = table R + (-α table r - r θ 2 θr - r θ 2 2. Thi yield table ( t + ) r + θ (( -t - ) r - R ) 2 2 α =. R -θ R 2 So when it i ufficiently illiquid, merely preparing to ervice fundamental withdrawal will not alway be enough to deter a run. ) Thi threhold tell u how much liquidity i needed when there i full information uch that all variable including t are known and all partie undertand the bank incentive. Under the tr condition or Propoition, the bank will chooe α= and no unued liquidity i held from R date to 2. Becaue depoitor might chooe to run and the incentive for thi mut be removed, thi will not be enough liquidity when the condition for Propoition do not hold. To alway deter a poible run, the bank will have to hold unued liquidity, Stable AIC Stable AIC α= α >. Thi will require that ome (α α )R Ut ( ) > 0, to be held from date to 2, after the normal withdraw are met at date. If the bank i free to ue all of thi unued liquidity if a run hould occur, then depoitor can ee that the liquidity i preent and will never chooe to run. Once the run i deterred, the liquidity will be in exce of what i needed. Thi i the implet example of the benefit of holding unued liquidity or leaving extra taxicab at the train tation. With full information available to all partie, market force will produce run-free banking. Alternatively, uppoe the depoitor do not oberve t or α, but the bank and a regulator do. Then the following arrangement i poible, but only by regulation. Propoition 2: With full information available to the bank or regulator, a bank (or a regulator) * AIC Stable eeking to deter run will chooe α =max{α,α } rt Stable Proof: The bank i automatically table when α o the regulator would alway want to R rt maximize lending and allow the bank to follow it elf-interet and elect that level ( R ) of Stable liquidity. Otherwie, the minimum amount of liquidity that i needed i α. AIC Stable More generally, for arbitrary anticipated withdrawal of t, α and α will differ and if liquidity, θ, i not too high or too low, and their relationhip will be imilar to what i hown in a 7

20 Figure 2. For very low level of anticipated withdrawal, where the condition in Propoition hold, the bank i ufficiently olvent that chooe to hold more ex-ante liquidity than i need to be table, o that run are impoible. At ome point, however, thi ceae to be true and the amount needed to jut be olvent in a run i higher than the bank would hold out of pure elfinteret. So in thi cae run deterrence would require a higher level of initial liquidity. Thi obervation will be helpful in undertanding ome of the regulatory tradeoff that we ubequently explore. α= fraction in liquid aet Figure 2: Comparion of Automatically Incentive Compatible and Stable liquidity choice and the implied amount of unued liquidity held from date to 2. Note; Parameter value are =0.3, θθ=0.5, R =.,R 2 =.33,r =r 2 =. Note that becaue ome liquidity mut be unued, it will appear that there i an unneeded amount of liquidity. With full information, thi amount will erve to deter run and will be choen at date 0 with all knowing the amount of normal withdrawal at date t. If more than a fraction t were to withdraw at date, the unued liquidity could be ued becaue all would know that a run wa occurring. The bank, or regulator acting for depoitor, could ue liquidity holding to deter run in the efficient way which maximize lending. Becaue depoitor alway deire run-free depoit, with full information, bank would be forced to hold the extra unued liquidity becaue otherwie no depoit would be attracted. To ummarize, with complete information, a bank will be forced to hold enough liquidity to deter run and it deire to maximize profit will aure that it hold no more than thi amount. 8