Cash-in-the-market pricing or cash hoarding: how banks choose liquidity

Transcription

1 Cash-in-the-market ricing or cash hoarding: how banks choose liquidity Jung-Hyun Ahn Vincent Bignon Régis Breton Antoine Martin February 207 Abstract We develo a model in which financial intermediaries hold near-cash assets to rotect themselves from shocks. Deending on arameter values, banks may choose to hold too much or too little cash on aggregate comared to the socially otimal amount. The model, therefore, rovides a unified framework for thinking, on the one hand, about olicy measures that can reduce hoarding of cash by banks and, on the other hand, about liquidity requirements of the tye imosed by the new Basel III regulation. Key to our results is a market in which banks can obtain cash by selling or reo-ing their marketable securities. The quantity of cash obtained on this market is determined endogenously by the market value of the marketable securities and is subject to cashin-the-market ricing. When uncertainty about the value of marketable securities is low, banks hold too little cash and liquidity regulation can achieve a better allocation. When uncertainly about the value of marketable securities is sufficiently high, banks hoard cash leading to a market freeze. Keywords: money market, liquidity regulation, non conventional monetary olicy, cashin-the-market-ricing JEL: E58, G28, G2 We are grateful to Cyril Monnet, Bruno Parigi, Eva Schliehake, Alexandros Vardoulakis and Guillaume Vuillemey for helful comments. We also thank seminar articiants of the Oxford Financial Intermediation Theory (OxFIT) Conference, of the TSE-BdF conference on models of the money market, of the SAET conference in Cambridge, and at the universities of Bern, Paris-Dauhine, Padova and St. Gallen. Antoine Martin acknowledges the suort of the Fondation Banque de France in the incetion of this roject. The views exressed herein are those of the authors and do not necessarily reresent the views of the Banque de France, the Eurosystem, the Federal Reserve Bank of New York, or the Federal Reserve System. NEOMA Business School, jung-hyun.ahn@neoma-bs.fr Banque de France, vincent.bignon@banque-france.fr Banque de France, regis.breton@banque-france.fr Federal Reserve Bank of New York and University of Bern, antoine.martin@ny.frb.org

2 . Introduction This aer studies whether the amount of liquidity (near-cash assets) chosen by financial intermediaries to rotect themselves from shocks is otimal from the ersective of social welfare. Liquidity has been an imortant olicy issue since the financial crisis. The new Basel III regulation introduced liquidity requirements, in the form of the liquidity coverage ratio (LCR) and the net stable funding ratio (NSFR). This tye of regulation stems from the concern that financial intermediaries may not hold enough liquid assets. On the other hand, concerns have been exressed that some financial intermediaries may be hoarding liquid assets such as cash, leading to market freezes, articularly in Euroe during the Subrime and Euro crisis (ECB, 204). In light of these concerns, it seems imortant to understand under which circumstances the rivate decisions of decentralized financial intermediaries may lead to too much or too little liquidity in the financial system. model rovides a unified framework to think about these issues and the olicy interventions that could lead to better outcomes. We develo a simle model of banks that are subject to a liquidity shock, in the sirit of Diamond and Dybvig (983) and Holmstrom and Tirole (998). Our banks are endowed with a non-tradable roject, which we can think of as a commercial loan, that may require additional cash to successfully come to maturity. Banks also need to decide how much nearcash assets and marketable securities they want to hold. We think of near-cash assets as level high quality liquid assets, such as central bank reserves and sovereign bonds. Examles of marketable securities in our model would be cororate bonds or equities. Marketable securities have a higher exected return than cash but their return is uncertain. In addition, these securities cannot be used directly to meet the liquidity shock, but they can be exchanged against cash in a market. The market may be interreted either as a secured money market in which banks lend temorarily their securities against cash (e.g. in a reo transaction) or as a market in which the securities are sold. 2 Banks holding securities that turn out to have a low return may not be able to raise sufficient cash when they are hit by a liquidity shock. We assume that the non-marketable roject cannot be ledged or sold to other banks. That roject may not be good collateral because its value could deend on the secific skills For examle, the Wall Street Journal reorted Euroe s biggest banks are continuing to stash more money at central banks, a move that reflects their lingering fears about the financial system desite signs of imrovement. (WSJ Large Euroean Banks Stash Cash Nov ). For academic work on the toic see Acharya and Merrouche (202) for the UK, Heider et al. (205) for Euro zone, de Haan and van den End (203) for the Netherlands. See also Ashcraft et al. (20) for emirical evidence of recautionary holding of reserves by U.S. banks in As is usual in the literature (for examle Allen et al. (2009)), there is no substantial difference between a secured lending market and a market for outright urchases in our model. Indeed, since the return on marketable securities is ublicly known when the market oens in our model, there is no difference between selling marketable securities and using them as collateral in secured loans. Our

3 of the banker so that its value is very low when the bank fails, which is recisely when the collateral is needed (Diamond and Rajan, 200). 3 In contrast, marketable securities are good collateral because their value does not deend on the survival of the bank that ledged them. In choosing the otimal amount of cash, a lanner weighs the benefit of an additional unit of cash, which can be used to continue non-marketable rojects, against the cost, which is the foregone return on tradable assets. At the otimum, the lanner chooses to hold exactly enough cash, on aggregate, to continue all non-marketable rojects, because we assume that the return on those rojects is sufficiently high. We show that banks may choose to hold too much or too little cash on aggregate, in equilibrium, deending on arameter values, comared to the socially otimal amount. When uncertainly about the value of the marketable asset is low, banks hold too little cash because they do not internalize the effect their cash-holding decision on the equilibrium rice of securities. Low cash holding imlies that the value of securities is low because of cashin-the-market ricing. This, in turns, reduces the extent to which banks hit by a liquidity shock can continue their non-marketable roject. If the uncertainty about the value of the marketable securities is sufficiently high, banks choose to hoard cash; that is, banks hold enough cash to continue their long-term roject without needing to access the market. This leads to a market freeze. Nevertheless, we show that this allocation is constrained otimal. Our work relates to the body of research analyzing liquidity choices by financial institutions, and the link with asset rices. Following Allen and Gale (2004, 2005), a number of aers have analyzed how cash-in-the-market-ricing affects banks liquidity decisions and the ossibility of central bank or ublic intervention. In articular, our aer is related to Allen et al. (2009), Freixas et al. (20) and Gale and Yorulmazer (203). An imortant difference between these aers and ours is that there can be either too much or too little cash in equilibrium in our model. Acharya et al. (200) shows how the exectation of fire sales can lead to liquidity choices that are excessive from a social viewoint. The strategic motive for cash hoarding is also analyzed in Acharya et al. (202) and Gale and Yorulmazer (203). Other aers show that liquidity hoarding may be a resonse to an exogenous increase in counterarty risk (Heider et al., 205). The interlay between endogenous liquidity choices and rivate information is analyzed in Bolton et al. (20) and Malherbe (204). Bolton et al. (20) focus on a tradeoff between outside and inside liquidity and show that the timing of trade can lead to an equilibrium with too little outside liquidity and too little cash-in-the-market-ricing. Malherbe (204) shows how in the absence of cash-in-the- 3 While we do not exlicitly model these frictions, a number of informational friction could hel exlain why the non-marketable roject cannot be sold or its return ledged. See for examle Gale and Yorulmazer (203). 2

4 market-ricing initial liquidity choices imose a negative externality on future liquidity by exacerbating adverse selection roblems. We consider a setu where information on traded assets is symmetric, and relate socially excessive liquidity to the disersion of traded assets. The remainder of the aer roceeds as follows: In section 2 we describe the model. Section 3 characterizes the allocations chosen by the lanner, under different constraints. We describe banks behavior in section 4. The definition and characterization of equilibria is done in section 5. Section 6 considers some olicy imlications of the model. Section 7 concludes. 2. Environment The economy is oulated by a continuum of identical agents called banks. There are 3 dates indexed by t {0,, 2}. At t = 0, each bank is endowed with a non-marketable investment roject, which we can think of as an industrial loan or a commercial real-estate deal, for examle, as well as one unit of cash. 4 Banks can kee the cash or invest in marketable securities, such as bonds, equities, or asset-backed securities. 5 At t =, a financial market oens in which each bank can rebalance its ortfolio of cash and securities. At t = 2, the investment roject and the securities mature, and consumtion takes lace. 2.. Cash and securities Cash has a net return of 0 and can be costlessly stored from one eriod to the next. Tradable securities generate a stochastic return at date 2. The actual return deends on the tye θ of the security, which is ublicly revealed at date. That return is H with robability of µ and L otherwise. Tradable securities generates a return R θ at date 2 with R H > R L. The exected gross return, denoted R, is assumed to be greater than. R E (R θ ) = µr H + ( µ) R L > (A) Let s [0, ] denote the share of the cash in the bank s ortfolio at t = Project Each bank holds a roject that yields y at t = 2. At t =, with robability λ, the roject is hit by a liquidity shock that requires the injection of an extra amount of cash. 6 To simlify notation, we normalize that amount to. In Aendix A, we show that our results 4 We think of cash as including near-cash assets, such as level HQLA. 5 Emirically we think of this asset as any tye of security with an International Security Identification Number (ISIN) or, in the U.S. context, a CUSIP, allowing its trade on an organized market. We do not modeled the rocess leading to the creation of this tye of security. 6 We assume a law of large number holds so that the share of rojects affected by the liquidity shock is also λ. 3

5 t = 0 t = t = 2 Banks invest in assets s in cash s in securities Liquidity shock realized λ need to inject Tye of security {H, L}, its value revealed µ µ R H R L (< R H ) Market for cash/securites trades rices: R H, R L Refinanced rojects continued Projects mature (y) Payoff from assets realized Figure : Timing generalize to the case where the amount of cash to be injected is less than. If no cash is injected, the roject is liquidated at t = and yields nothing. The roject is divisible: If a bank does not have enough cash at t = to continue the roject in its entirety, it can inject i < of cash and the roject returns i y at date t = 2. We assume that : y > R (A2) This condition imlies that, from the ersective of date 0, the return y on the extra amount of cash that needs to be injected to continue to roject in case of a liquidity shock is greater than R, the exected return on securities Timing t = 0: Each bank chooses s, the share of cash it wants to hold, with s reresenting the share of securities the bank holds. t = : Each bank learns the value of the securities it holds and whether it suffer liquidity shock. The market for securities oens. Bank with a liquidity shock decide whether to inject additional cash to continue their roject. t = 2: The return on long term roject and the securities are realized. Banks rofit are realized. 3. The lanner s roblem: Otimal allocation between cash and securities In this section we determine the date-0 otimal allocation from the oint of view of a social lanner. Since all agents are risk neutral, we define welfare as total exected outut. The social lanner aims to maximize social welfare by choosing the cash level s that 4

6 individual banks hold at t = 0 and by redistributing cash among banks at t =, so that the banks subject to a liquidity shock have the cash they need to continue their roject. The total outut of economy is given by W (s) = max, [ λ + λ(s + c)]y + [( λ) s λc] + ( s) R () Subject to the feasibility constraint ( λ)s λc, where c is the amount of cash transferred to a bank with a liquidity shock. We define C λc as the aggregate amount of cash transferred. The first term in () is the return from rojects that mature. A roortion λ of banks are not hit by the liquidity shock and their roject will mature yielding y. In addition, a roortion λ of banks are affected by the liquidity shock. Each of these banks holds an amount s of cash and obtains c from the social lanner. In aggregate, a roortion λ + λ(s + c) of roject, each yielding y, will be continued. The second term reresents leftover cash, which cannot be negative, and the third term catures the return of tradable securities in the economy. Since the exected return of the roject, y, is greater than the exected return on securities, R, which is itself greater than the return on cash,, insection of the objective function reveals that any solution must have λ+λ(s+c) = and ( λ)s = λc. The first equality state that there should be enough cash in the economy to continue all rojects. The second equality states that there should be no leftover cash. Together, these two conditions imly s F B = λ. Next we consider the roblem of the social lanner who maximizes welfare in the economy under the constraint that a bank cannot receive more cash than the value of the securities it holds. For examle, banks may be able to hide the cash they hold and the lanner can only transfer cash if banks are comensated for the cash they give u. Studying the allocation of the constrained lanner is useful to distinguish the inefficiency related to the redistribution of cash in the interbank market from the inefficiency related to the choice of cash held by banks at date 0. The constrained-otimal, or second-best, level of cash is denoted s SB. A share µ of cash-oor banks hold H-security, which have a value greater than by assumtion, so they can receive the amount s of cash against (some of) their securities. However, a share µ of bank with liquidity shock hold L-security, which can have value lower than. Let c θ denote the transfer to banks holding θ-security, θ {L, H}. The constraint that the social lanner faces for banks holding L-security is therefore c L ( s)r L. 5

7 Note that the lanner would never transfer more than s, so the above constraint binds only if R L <. The aggregate amount of redistributed cash is given by { ( λ) s if R L, C = min {( λ) s, λ ( s) (µ + ( µ) R L )} if R L <. W To find the marginal contribution of cash to outut,, we can lug (2) into () and differentiate with resect to s. Let ŝ denote the level of s such that there is no cash left over after transfers if R L <. It is the value of s that solves λ ( s) (µ + ( µ) R L ) = ( λ) s, and is given by ŝ λ (µ + ( µ) R L ) λ (µ + ( µ) R L ) + λ < sf B. (3) It is easy to verify that W > 0 if s < ŝ. Indeed, when s < ŝ an additional unit of cash can be used to continue a larger share of the roject without resulting in any leftover cash after transfers. When s ŝ, we have W = ( λ ( µ)) + λ ( µ) (y R L (y )) R The sign of this exression deends on R L, but not on s. We can derive ˆR is the value of R L such that W = 0. ˆR R ( µ) λ (y ). (4) If R L > ˆR, then W/ < 0, so the lanner wants to minimize the amount of cash held, conditional on continuing as much of the non-marketable rojects as ossible. In contrast, if R L < ˆR, then W/ > 0, so the lanner wants to maximize the amount of cash. These results are summarized in the following roosition : Proosition.. When R L, the additional constraint faced by the lanner does not bind so that s SB = s F B = λ. 2. When ˆR < R L <, the constrained otimal level of cash s SB is ŝ (< s F B ). 3. When R L = ˆR, the constrained otimal level of cash s SB can be any value between [ŝ, ]. 4. When R L < ˆR, the constrained otimal level of cash s SB is (> s F B ). The constrained otimal level of cash deends on R L, the low return for the securities. When R L is higher than, the lanner can achieve the first-best because the constraint does not bind. When R L is smaller than, the lanner can no longer transfer the otimal amount of cash. The lanner chooses to reduce s for two reasons: First, a reduction of s (2) 6

8 is necessary to make sure that there is no unused cash after transfers. Second, reducing s allows banks to hold more securities, which can be used to acquire more cash when the return on the securities is low. For these reasons, s SB decreases as R L decreases until R L becomes smaller than ˆR at which oint it jums to. The intuition for this result is that when R L is so low, the interbank market is very ineffective at redistributing cash among banks. Hence, the lanner refers that banks hold sufficient cash to continue the roject without needing a transfer that is constrained by the value of the securities. 4. Equilibrium In this section, we derive the equilibrium allocation. At date t = 0, banks decide how much cash and securities to hold, taking into account the exected market rice of securities in terms of cash at t =. Working backward, we first derive the market rice of securities at t =, for a given choice of s, and then we consider the otimal choice of s at t = Market for securities at t = At date, a market oens in which banks can trade cash for securities. Banks that receive a liquidity shock and need to acquire additional cash to continue their roject will seek to sell some or all of their securities holding in this market. Banks that did not receive a liquidity shock, as well as banks that are unwilling to continue their roject, may want to urchase securities with their cash. Since there exists two tyes of securities H, L, we first characterize a no arbitrage condition for the rice of these securities. Because there is no informational asymmetry, the no-arbitrage condition requires that the return from urchasing either security be same. Let denote the market return on securities. This notation is convenient because the rice of security θ can be written as R θ. The no-arbitrage condition can thus be exressed as R H R H = R L R L =. (5) At t =, banks hit by the liquidity shock decide the share of their nonmarketable roject, i [0, ], to continue, with full continuation of the roject when i =. decision deends on the comarison between the return of injecting cash i in the roject, i y, and its oortunity cost, which is the return on urchasing securities, i. The bank will choose to continue its roject if This y. (6) 4...Trading strategies and suly and demand of cash Banks may be willing to suly cash to the market, and urchase securities, for two reasons. First, banks that are not subject to the liquidity shock will buy securities if their 7

9 y ( λ)s s S Figure 2: Suly of cash return is at least as high as the return on cash,. Second, banks hit by the liquidity shock suly cash and urchase securities if the market return on cash is higher than their internal return on cash, y. The aggregate suly function of cash, illustrated in Figure 2, is given by: 0 if <, [0, ( λ)s] if =, S = ( λ)s if < < y, (7) [( λ)s, s] if = y, s if > y. Next, we can derive the demand for cash. Banks that receive a liquidity shock demand cash and suly securities to the market, rovided they have an incentive to continue their roject. This condition corresonds to inequality (6). Assuming inequality (6) holds one of two things can haen: If s + ( s) R θ (equivalently R θ ), then the bank can acquire s, the amount of cash that it needs to continue its roject. Otherwise, the bank acquires as much as it can by selling all the securities at its disosal, ( s) R θ. If inequality (6) does not hold, banks will not demand any cash. The individual demand function for cash by a bank holding θ-tye security and hit by a 8

10 R H y y λ( s) R λ( s)(µ + ( µ)r L ) R H R L λ( s)(µ + ( µ)r L ) R L λ( s) D λ( s) D (a) R L < y R H (b) R L < R H < y Figure 3: Demand of cash liquidity shock, denoted d θ, is given by: min { s, ( s) R θ } < y, d θ = [0, min { s, ( s) R θ }] if = y, 0 > y. The aggregate demand function on cash, D, illustrated in Figure 3, is given by the sum of the individual demands: where A < y, D = [0, A] if = y, 0 > y. A = λ [µ min { s, ( s) R H } + ( µ) min { s, ( s) R L }]. (8) 4..2.Equilibrium rice of security at t = This section characterizes the market equilibrium rice of securities, given a cash level s, chosen at t = 0. The rice of θ-tye securities is denoted by R θ. First, we rove the following lemma: Lemma. Given the cash holding level s chosen at t = 0, the equilibrium market return on cash always satisfies y. Proof. If the market return on cash is lower than, then holding cash would have a higher return than urchasing securities, imlying that the suly of cash in the market would be 9

11 0. Therefore, < cannot be an equilibrium. On the other hand, if market return on cash is higher than y, then all cash holders would refer to buy securities rather than continue their roject, imlying that there would be no demand for cash in the market. Therefore, > y cannot be an equilibrium. Note that since y, banks hit by a liquidity shock always refer to use their cash to continue their roject rather than to acquire securities in the market. This imlies that the suly of cash in the market is given by banks that are not affected by the liquidity shock, S = ( λ)s. (9) When < < y, the demand for cash can be written as: λ ( s) if R L, D (s) = λ ( s) [µ + ( µ) R L ] if R L < R H, λ ( s) R if > R H. Intuitively, if R L, then banks affected by a liquidity shock can sell enough assets to get the cash they need to continue their roject regardless of the return on the securities they hold. So a fraction λ of banks demand an amount ( s) of cash. If R H > R L, then banks affected by a liquidity shock can get as much cash as they would like if they hold securities with return R H, but are constrained if they hold securities with return R L. Finally, if > R H, all banks affected by a liquidity shock are constrained. There exists no equilibrium either when R L > or when R H < (see aendix). To see why R L, note that if R L >, then a bank would be able to continue its non-marketable roject even when it holds securities with a low return. Since securities have a higher return than cash, banks would increase their holding of securities to earn the extra yield. However, banks do not take into account the fact that when they hold more securities the aggregate amount of cash in the market decreases, leading to cash-in-themarket ricing. In equilibrium, the benefit of a marginal unit of cash, which comes from the ability to continue the non-marketable roject when a liquidity shock hits, must be equal to the cost from the foregone income on the marketable securities. When R L R H, the demand for cash is given by D (, s) = λ ( s) [µ + ( µ) R L ]. (0) Given the exression for suly (9) and demand (0), we can rove roosition 2, which characterizes the equilibrium rices of securities, taking as given the cash level s. Proosition 2. Under R L R H, given the level of cash holding s, the equilibrium (s) is determined as follows:. If S > D ( = ), then (s) =. The rice of security is its fundamental value. 0

12 2. If S D ( = ), then (s) is determined by S = D yielding (s) = [ ] ( λ) s R L λ ( s) µ µ. The rice of security is below its fundamental value. [ ] 3. If s = λ and R L, (s) can be any value in R L,, i.e. any value between (s) determined by S = D and Individual Portfolio Choice at t = 0 Each bank decides how much cash to hold at t = 0 to maximize its exected rofit. Given θ, the marginal return on cash deends on whether or not s + ( s) R θ. Since R L R H, it deends on which tye of security the bank holds. When the bank holds H-tye, it is not liquidity constrained, so the marginal return on cash is simly R H; the gross return on one additional unit of cash minus the oortunity cost of that additional unit. If the bank holds L-tye, then we have two cases uon the rice of L-tye security R L. When R L =, the bank is not liquidity constrained, so the marginal return on cash is similar to above and can be written as R L. When R L <, the bank is liquidity constrained. In this case, the marginal return is given by [ λy + ( λ) ] [λr L y + ( λ) R L ]. The first term in brackets corresonds to the bank s marginal return on cash: if the bank is affected by a liquidity shock, which haens with robability λ, then the marginal return on cash is equal to the marginal return on the roject, y, while if the bank is not affected by the liquidity shock, then the cash can be used to urchase securities, with a marginal return of /. An additional unit of cash also carries an oortunity cost, corresonding to the return of the securities that could have been urchased instead. This is catured by the second term in brackets: if the bank is affected by a liquidity shock, it would sell the additional securities at rice R L to acquire cash, while if the bank is not affected by the liquidity shock the additional securities would return R L. Using these exressions, we can derive the exected marginal (net) return on cash conditional on the rice of securities, which is given by { π = R if R L = ( R) ( ) + ( µ) λ y ( R L ) if R L <

13 When R L < (, the R) exected marginal return on cash is the sum of two comonents. The first term,, reresents the ecuniary or fundamental net return on cash, which can be negative. The second term reresents the additional value of cash that comes from its use in continuing the long-term roject when a liquidity shock hits. The relative size of these two terms will determine whether there is cash hoarding or cash-in-the-market ricing in equilibrium. cash. The equilibrium cash holding level s given must maximize the exected return on From this together with the roosition 2, we can characterize market equilibria, reresented by the air of the rice and the level s of individual cash holding. Proosition 3. = R. (otimal). When R L = R H = R, there exist an equilibrium with s = λ and 2. When ˆR < R L < R, there exist an equilibrium with s (< s SB ) < λ and <. (cash-in-the market ricing) s and are determined by s = λ (µ + ( µ) R L ) λ (µ + ( µ) R L ) + ( λ) = R ( µ) λ () (y ) ( R L ) (2) 3. When R L < ˆR, there exist an equilibrium with s = and =. (cash hoarding, with the rice being the fundamental value) The roof is rovided in the aendix. Figure 4 illustrates the equilibrium rice and cash level uon the value of security with low quality R L. 5. Discussion 5.. Comarison with otimal allocation An equilibrium with cash hoarding can only exist when R L is lower than ˆR. 7 When it exists, this equilibrium corresonds to a situation where the return on cash that comes from its use to continue the non-marketable roject exceeds, in equilibrium, the return from holding securities. In such a situation, the interbank market functions very oorly as a way to redistribute cash between banks. As a consequence, the interbank market freezes; all banks hold as much cash as they need and no trade takes lace. Since there is cash hoarding in this equilibrium, the rice of securities always corresonds to their fundamental value. The level of cash is constrained otimal. 7 Such an equilibrium may not exist, deending on the value of other arameters and if ˆR is lower than 0. 2

14 R 0 ˆR R R L s (a) Equilibrium λ (S F B ) s SB s 0 ˆR R R L (b) Equilibrium cash holding level s Figure 4: Equilibrium rice and cash holding level 3

15 When ˆR < R L < R, the equilibrium level of cash holding by banks is lower than the otimal amount, s < λ, and even lower than the constrained otimal amount, s < s SB. In this case, and s are determined by () and (2). Because banks are holding so little cash, equilibrium is always lower than. In other words, there is cash-in-the market ricing. As already discussed, in this equilibrium banks do not internalize the effect their cash-holding decision has on the equilibrium rice of securities. Banks hold too little cash, which imlies that the value of securities is low because of cash-in-the-market ricing. This, in turns, reduces the extent to which banks hit by a liquidity shock can continue their roject. The threshold ˆR reresents the value of R L below which individual banks shift their behavior from holding too little cash to holding too much cash, comared to the socially otimal level. Equation (4), shows that this threshold increases as the roortion of low quality securities, µ, and the roortion of rojects hit by liquidity shock, λ, increases. If both of these values are high, then the value of R L below which banks hold too much cash becomes relatively high. Indeed, if µ tends to zero and λ tends to, then ˆR tends to. In words, if the share of securities with a low return and the share of non-marketable rojects that are hit by a liquidity shock are high the interbank market may not be very effective at redistributing cash among banks. The market is likely to freeze as banks attemt to self-insure Comarative statics 5.2..Disersion of return on securities Using (4), we can figure out the grahical reresentation of roosition 3 in the ( R L, R ) sace, which is illustrated in Figure 5. For a given level of R, a change in R L corresonds to a change in the disersion of the return on securities. When there is no disersion, R L = R, the first best can be achieved in equilibrium. As disersion increases (R L decreases), we have first cash-in-the-market rincing. Then, as disersion increases further there can be cash hoarding in equilibrium. The corresonding behavior of cash holdings is shown in figure 4 (b). Interestingly, we have a non trivial/non monotonous imact of the level of disersion, with a significant change in the cash holding as disersion increases. Note that the first best level of cash, s F B = λ, is unaffected by the exected return on securities R, which is assumed to be greater than the return on cash. The intuition is that if nothing revents the efficient redistribution of cash at date, then it is efficient to hold just enough cash to continue all long-term rojects, since they have a high return, but no more since the return on cash is lower than the return on securities. Proosition 4. There exist non-monotonous relationshi between cash holding level of banks 4

16 R R L = ˆR underinsurance (cash-in-the market) otimal (R L = R) 0 overinsurance (fundamental value) R L Figure 5: Tye of equilibria and disersion of return on securities. Cash holding level of banks decreases in the regime with low and moderate disersion (R L < R) and jums to at R L = R, then it remains unchanged as disersion increases in the regime with high disersion (R L > R) Exected return on securities ( R) Now, we analyze the imact that changes in exected return on securities ( R) have on cash holding behavior of bank. In order to control the imact driven by changes in disersion of return on securities and to focus on ure imact of change in R, we conduct a comarative analysis given that disersion remains unchanged. Let σ denote coefficient of disersion defined as standard deviation of returns on securities divided by their exeected return. σ is therefore given by σ 2 µ ( R H R ) 2 + ( µ) ( RL R ) 2 R 2 Using R µr H + ( µ) R L, σ can be simlified as σ ( 2 ( µ) R ) L < 2 ( µ) (3) R 8 We show more clear non-monotonous relationshi in a general set-u with x < resented in the aendix A. In this extension, the relationshi between cash holding level and disersion of return on securities are negative when diersion is low and moderate (R L < R) whereas this relationshi becomes ositive when disersion is high (R L > R). 5

17 Using roosition 3, we obtain the following roosition for R (, y), which is relevant range of R: Proosition 5.. When σ = 0, in other words R = R H = R L, equilibrium level of cash holding is s = s F B = λ indeendent of level of R. 2. When σ is too high such that σ ˆσ, equilibrium level of cash holding is s = (cash hoarding) indeendent of level of R. 3. When 0 < σ < ˆσ, equilibrium level of cash holding is given by the following: (a) If R > R 0, where s λ ( µ + ( µ) k R) = λ ( µ + ( µ) k R ) + ( λ) ( = R ( µ) λ y ) ( k R) k (4) (5) σ 2 ( µ) (6) (b) If R < R0, equilibrium level of cash holding and rice is resectivelys = and =. ˆσ and R 0 are given by ˆσ = R 0 = 2 ( µ) (y ) (( µ) λ + ) y ( µ) λ 2 ( µ) [( µ) λ (y ) + ] 2 ( µ) [( µ) λ (y ) + ] σ ( µ) λ (y ) Proosition 6 follows. Proosition 6. Equilibrium level s of cash holding is (weakly) decreasing to R. More recisely, as R decreases, s increases till R0 and then jums to and remains unchanged. Proof is relegated to the aendix. Figure 6 illustrates roosition 6. It is noteworthy that unlike relationshi between cash holding level of banks and disersion of return on securities, there exist a monotonous relationshi between cash holding level and exected return on securities Probability of liquidity shock Now, we analyze how the change in robability of liquidity shock affects the individual bank s decision on liquidity holding. Accordingly, we investigate on the effect of change in robability λ of liquidity shock on the equilibrium level of cash holding (s ). Proosition 7 follows. 6

18 s λ y R0 R y R0 R (a) Equilibrium (b) Equilibrium s Figure 6: Return on securities vs. and s s * s * (λ ) λ λ 0 s * (λ 0 ) R L Figure 7: The effect of increase in robability of liquidity shock 7

19 Proosition 7. Given R L > ˆR, an increase in λ, from λ 0 to λ > λ 0 results in. an increase the amount of cash s held by banks; 2. an increase in the threshold below which the equilibrium shift from cash-in-the-market ricing to cash hoarding. Figure 7 illustrates the above roosition. 6. Intervention olicy 6.. cash-in-the-market ricing and olicy intervention 6...Liquidity regulation When the economy faces cash-in-the-market ricing, a liquidity requirement, such as the LCR, can imrove welfare. For the sake of illustration, consider the case with ˆR < R L < R H, where the market equilibrium exhibits cash-in-the-market ricing (s < λ). In equilibrium, welfare is given by: W (s) = ( λ + λs + C m ) y + [( λ) s C m ] + ( s) R, where C m is the aggregate amount of cash that banks faced with a liquidity shock can obtain in the market. It is easy to verify that C m = ( λ) s. Plugging this exression into W (s) yields total welfare at the market equilibrium cash level s W (s ) = ( λ + s ) y + ( s ) R The marginal contribution of cash to welfare, W, is y R, which is always ositive. Therefore, requiring banks to hold more cash than s imroves welfare. Not surrisingly, the solution for the welfare maximizing level of cash that banks should be required to hold is same as the level of cash chosen by the constrained social lanner. This is summarized in the next roosition. Proosition 8.. When the economy dislays cash-in-the-market ricing, liquidity requirements, such as the LCR can imrove welfare. 2. In such a situation, liquidity requirement can restore first-best when R L < R H and can restore contained otimal level of welfare (second-best) when ˆR < R L <. By forcing banks to hold more liquidity, a requirement like the LCR increases the value of securities and ermits more long-term rojects to be continued, resulting in higher welfare. An interesting imlication of this result is that the aroriate level of the LCR should deend in art on the disersion of the return of marketable securities. More recisely, the 8

20 constrained otimal level of cash becomes lower as the disersion of the return on securities is greater. Since the disersion of returns is likely to be greater during eriods of crisis, our model suggests that the liquidity requirement should be lower in times of stress than in good time Announcement of asset urchase by the central bank Another olicy intervention that can imrove welfare when the economy faces cash-inthe-market ricing is for the central bank to announce at t = 0 that it will urchase the securities at their fundamental value at t =, which imlies that =. To analyze the effect of asset urchase by the central bank, we slightly modify our basic model. We suose that at t = 0 the central bank levies a tax τ < on banks in the form of cash. Hence, the amount of cash held by banks is s [0, τ] and the amount of securities is s τ. At t =, banks can obtain the cash by trading their securities either with other cash-rich banks at market rice or with the central bank at the rice =. The central bank can urchase securities until it runs out of cash. The central bank chooses τ to maximize social welfare. The following roosition describes an equilibrium: Proosition 9.. When the economy dislays cash-in-the-market ricing, the central bank can imrove welfare by levying a tax τ > 0 at t = 0 and using the roceeds to urchase securities at their fundamental value ( = ) at t =. (a) When R L < R, such intervention with τ = λ can restore the first-best allocation. (b) When ˆR < R L <, such intervention with τ = s SB can restore the constrained otimal allocation. 2. In both cases, banks choose to hold no cash s = 0 in equilibrium. The market for securities is inactive and all trading occurs with the central bank. Proof. Here, we will demonstrate that when ˆR < R L <, intervention with τ = s SB can restore the constrained otimal allocation. The other case can be roved in a similar way. When ˆR < R L <, we know that the constrained otimal allocation is attained when the total cash in the economy is s SB and if all of the cash redistributed to banks that need it. Thus, it is sufficient to demonstrate that banks have no incentive to deviate from s = 0 and that the cash held by the central bank will be transferred to the banks with liquidity shock. First, note that this olicy is feasible, since τ = s SB and the central bank sets CB =. If the market equilibrium rice is <, then all banks that need cash sell their securities to the central bank. Since s SB is the maximum amount of cash that is demanded when the securities are traded at their fundamental value, all demand for cash can be satisfied by 9

21 the central bank and there is no trade in the market. Therefore, < cannot be an equilibrium. If the market equilibrium rice is =, then banks with liquidity shock are indifferent between selling their securities to the central bank and selling them in the market. However, when ˆR < R L and =, the marginal return on cash exected at t = 0 is always negative. As a consequence, banks will not hold any cash. Therefore, s = 0 is an equilibrium given τ = s SB and CB =. In such an equilibrium, all the cash s SB is transferred to banks with liquidity shock since it is equal to demand for cash by these banks. The central bank s intervention described in the roosition above resembles the Quantitative Easing (QE) olices that several central bank conducted during the recent crisis along some dimensions. These olicies can increase the value of securities by flooding the market with reserves. This increase brings the value of the securities back to their fundamental value. Flooding the market with reserves tyically deress activity in the interbank market and this haens in the model too. With the cash injections from the central bank, there is no need for banks to trade with each other Cash hoarding and olicy intervention. When R L < ˆR, there is cash hoarding. The decentralized individual choice of cash holding is clearly higher than the otimal amount as defined by the first-best. However, our result shows that the market equilibrium is constrained otimal. Indeed, in such situation the value of cash to continue the long-term roject is so high that it more than comensate for the fact that cash has a lower exected fundamental return than securities. As a result, the central bank cannot imrove welfare excet, unless it is willing to urchase securities at a value that is greater than their fundamental value, which would lead to losses Mutual insurance In this tye of situation, an insurance scheme among banks could restore the first best. If banks can commit to share the available cash, then they face the same roblem as an unconstrained lanner. This tye of insurance mechanisms can be costly to imlement and difficult to enforce, so that they might not be set u ex ante. This might be the case for examle, if the economy is rarely exected to be in a situation where banks would want to hold so much cash. That said, the tye of clearing house arrangements that arose during banking anics, described in Gorton (985), have characteristics that are similar to an insurance mechanism that would restore the first best in our model. In resonse to a anic, clearinghouse member banks would join together to form a new entity overseen by the Clearinghouse Committee. 20

22 As Swanson (908) ut it, describing the crisis of 860, there was the virtual fusion of the fifty banks of New York into one central bank (. 22) Central bank s lending against non-marketable roject It is also ossible for a central bank to achieve the unconstrained otimal allocation if it can lend against the return of the non-marketable long-term roject. While this result requires assuming that the central bank can choose actions that are not ermissible to market articiants, it is consistent with emirical evidence. In ractice, some central banks do lend against such assets. In Euroe, assets which do not have an International Securities Identification Number (ISIN) are tyically very illiquid and cannot be traded in a market on short notice, if at all. Such assets would corresond well to the non-marketable asset of our theory. Nevertheless, the ECB regularly rovides liquidity to banks against this tye of collateral which includes credit claims and non-marketable retail mortgage-backed debt instruments (RMBDs) as long as it comlies with some eligibility criteria. 0 The use of non-marketable assets ledged at the Eurosystem oen market oerations as ercentage of the total collateral ledged increased steadily with the Subrime and Euro crisis, juming from about 0% in 2007 to more than 25% in 202. We do not exlicitly model the frictions that revents market articiants from urchasing, or lending against, non-marketable assets. One reason might be that revealing the information necessary to evaluate the quality of such assets could rovide a cometitive advantage to the bank s cometitors. Providing this information to the central bank, in contrast, is not an issue. In order to analyze the effect of lending against roject by central bank, we make similar assumtion as in the revious subsection. The central bank levies a tax with τ at t = 0 before banks allocate their resource between cash and securities. At t =, banks can obtain the cash either by trading their securities on the market or by borrowing from the central bank ledging their roject (non-tradable security). We assume that ) there is no informational asymmetry between banks and the central bank and therefore the quality of roject is known to the central bank; and 2) that the central bank lends the amount of debt that banks want under the constraint of its resource τ with net interest rate r and with haircut 0. The central bank chooses the level of tax τ that maximizes the social welfare W. The following roosition describes an equilibrium: 9 See Gorton (200), Chater 2. 0 These criteria include, notably, that the asset must be denominated in Euro, the debtor must be located in the Euro area, and the debtor s default robability, as defined by the Basel accord, must be lower than 0.4%. Details on non-marketable acceted in guarantee to ECB loans to credit institutions can be found in Sauerzof (2007) and in Tamura and Tabakis (203). See ECB (203),. 8. 2

23 Proosition 0. Given that r = 0, we can have an equilibrium achieving the otimal allocation in which the central bank levies tax with the amount τ = λ and banks choose their cash holding level s = 0 and holds only securities λ. The market will collase and the central bank substitutes market but otimal allocation is achieved. Proof. First, we characterize the decentralized equilibrium given τ = λ. Then we demonstrate that the social welfare given by τ = λ and s is first-best. Since the central bank levies τ = λ, banks know that the central bank has enough resources to lend to all banks with a liquidity shock. Hence, banks would only hold cash to urchase securities. This imlies that the (net) exected return on additional cash for banks is always R. Since banks do not want to sell securities below their fusubsectionndamental value, the only rice at which the market could clear is =. However, at that rice R < 0, and banks do not want to hold any cash. As a result, given τ = λ, market equilibrium is s = 0. The total outut in the economy in this equilibrium is given by W = y + ( λ) R, which is the first-best level of welfare. The central bank has no incentive to deviate from τ = λ. 7. Conclusion Our aer rooses a simle theoretical framework to analyze the liquidity choice of banks. Banks require liquidity to meet unexected shocks and can obtain this liquidity from other banks by selling marketable securities in a market. Interestingly, our theory suggests that banks may hold either too much cash or too little cash, deending on arameter values. The banks choice if cash holding deviates from the social otimum when the return on marketable securities is sufficiently variable. In this case, a bank needing liquidity but holding marketable securities with a low return will not be able to obtain as much cash as it would like. When the exected return of securities is sufficiently high, then banks will always hold too little cash, comared to the social otimum. The intuition for this result is that banks do not internalize the effect of their cash holding on the market rice of securities. Because of cash-in-the-market ricing, the fact that banks hold too little cash, lowers the rice of securities. This, in turn, means that banks cannot continue the long-term roject to the extent they would like. A liquidity constraint, such as the LCR, that forces banks to hold enough liquidity can restore the constrained otimum in such situations. The return on marketable securities is sufficiently variable and the exected return is not too high, then banks will choose to hold too much cash. In this situation, the value of cash for the urose of continuing the long-term roject more than comensate for the 22

24 fact that cash has a lower fundamental return than securities. This can be thought of as a state of crisis, where the market for securities shuts down and banks hoard cash. Central bank intervention can restore the first best in this situation; in articular if the central bank can lend against the value of the non-marketable roject. Aendix A. Extension: The general case with liquidity shock x Until now, we have assumed that bank needs the amount of liquidity to continue its roject when it is affected by liquidity shock. In this aendix, we relax this assumtion by considering that the amount of liquidity needed to continue roject is x. In this general case, the assumtion (A2) is relaced by y x > R (A2 ) First, we determine the date-0 otimal allocation from the oint of view of a social lanner. The total outut of economy becomes { [ W x (s) = min, λ + λ( s + c ]} x ) y + [( λ) s λc] + ( s) R (7) The consideration similar to the section 3 yields that the otimal level of cash for economy is s F x B = λx when the social lanner has no constraint in redistribution of cash. The constrained-otimal, or second-best, level of cash is denoted by s SB x. This is given by the following roosition : Proosition.. When R L ( λ)x λx, the lanner is not constrained to redistribution of cash so that s SB x = s F x B = λx. 2. When ˆR x < R L < ( λ)x λx, the constrained otimal level of cash ssb x where and ŝ x is ŝ x (< s F B x ) λ (µx + ( µ) R L ) λ (µ + ( µ) R L ) + λ, (8) ˆR x R λ ( µ) ( y (9) x ). 3. When [ R L = ˆR x, the constrained otimal level of cash s SB x ŝ x, x R L R L ]. can be any value between 4. When R L < ˆR x, the constrained otimal level of cash s SB x is x R L R L ( > s F B x ). 23

25 The roof is rovided in the aendix E. The constrained otimal level of cash deends on R L, the low return for the securities. When R L is higher than ( λ)x λx, the lanner can achieve the first-best because the constraint does not bind. When R L is smaller than ( λ)x λx, the lanner can no longer transfer the otimal amount of cash. The lanner chooses to reduce s for two reasons: First, a reduction of s is necessary to make sure that there is no unused cash after transfers. Second, reducing s allows banks to hold more securities, which can be used to acquire more cash when the return on the securities is low. For these reasons, s SB decreases as R L decreases until R L becomes smaller than ˆR at which oint it jums to x R L R L. The intuition for this result is that R L becomes so low that the lanner refers that banks hold sufficient cash to continue the roject in full scale even when banks hold L-tye security. In other words, s SB becomes such that s + ( s) R L = x. We can easily verify that our main model is a articular case with x =. Now, we turn to the decentralized decision on cash holding by individual banks. Consider first aggregate demand and suly of cash in the market. The equilibrium rices and cash holding level satisfy the following lemma 2: Lemma 2.. Given the cash holding level s chosen at t = 0, the equilibrium market return on cash always satisfies y x. 2. Given the market rice of securities, the equilibrium cash holding level s always satisfies s + ( s ) R H x. Proof. The roof is similar to that of lemma and aendix B.2. Given that y x, banks hit by a liquidity shock always refer to use their cash to continue their roject whereas banks that are not affected by liquidity shock always suly their cash in the market. The aggregate suly of cash in the market is S = ( λ) s The demand for cash deends on whether banks hit by liquidity shock is constrained by the value of their securities or not. Under s + ( s ) R H x, banks with the H-security is never constrained. The aggregate demand deends on the value of L-security. The demand for cash can be written by: { λ (x s) if x s + ( s) R L D (s) = (20) λ [µ (x s) + ( µ) ( s) R L ] if x > s + ( s) R L Now we consider individual decision on cash holding level by banks. They decide their cash holding level at t = 0 maximizing their exected rofits. Their decision deends on the 24