Liquidity Sharing and Financial Contagion

Transcription

1 Liquidity Sharing and Financial Contagion John Nash September 8, 05 Abstract I study the extent to which non-binding credit lines similar to the ones used commonly in the fed funds market can accentuate or exacerbate bank distress within the financial system. I develop a model of liquidity sharing in the banking sector in which banks endogenously opt for implicit or non-binding credit lines. These credit lines align bank screening incentives with the social optimum but also allow potential lenders to strategically respond to counter-party distress and can increase expected contagion. I show that when information is perfect, credit lines can reduce contagion because lenders are able to disconnect from distressed borrowers. However, when information is imperfect, credit lines can exacerbate contagion. I show that there exists an externality in which potential lenders do not take into account the impact of their lending decisions, not just on the borrower, but on the financial system as a whole. Finally, I show that credit lines are least effective when information is poor and potential solvency shocks are large, arguably when their importance to the economy is highest. IwouldespeciallyliketothankmyadvisorsDouglasDiamond,ZhiguoHe,AmitSeru,StavrosPanageasand Lin William Cong for their invaluable time and input. I would also like to thank Yunzhi Hu, Chenfei Lu and Gregor Matvos as well as seminar participants at Chicago Booth for their thoughtful comments and feedback. University of Chicago Booth School of Business, jnash@chicagobooth.edu

2 Introduction The fallout from the financial crisis of 008 has seen an increased level of scrutiny on the architecture of the financial system. During normal times, the financial system is thought to function like a well oiled machine, redistributing funds to sectors with relatively profitable investment opportunities and hedging risks through implicit and explicit contracts that vary in size, maturity, and execution. As a result, a web of interdependencies exists among the members of the financial system on both the asset and liability sides of balance sheets. However, recent events have suggested that the existence of all these interdependencies can exacerbate economic downturns following sequences of sufficiently bad economic shocks. For example, a financial institution with a liquidity shortage - perhaps because funds were withdrawn by a previous creditor - may look to raise liquidity from other large financial institutions with liquidity surpluses through the use of implicit or non-guaranteed credit lines. If the financial institution is unable to raise sufficient funds a downward spiral can follow, that if unchecked, will end in default. Given the interdependencies within the financial system, one default can lead to several more in the absence of an intervention. In 998 following a series of highly unprofitable trades, the hedge fund giant LTCM was unable to secure additional funding from other large financial institutions, which led to a rapid decline in the value of its equity and ultimately its demise. The subsequent fallout was largely contained because of a bailout organized by the Federal Reserve Bank of New York. I develop a model of liquidity sharing within the financial system in which inter-linkages between banks arise when banks form credit lines to help pay for liquidity shocks. Banks form credit lines to reduce the amount of liquidity they must hold, and endogenously opt for nonbinding credit lines as they align monitoring incentives with the social optimum, which in turn reduces the probability of bank distress in the economy. While healthy banks are able to use their credit lines to borrow without problem, distressed banks are exposed to potential liquidity shortages. This occurs because non-binding credit lines allow lenders to strategically respond to borrower distress, and thus give lenders the option to refuse distressed borrowers liquidity. Liquidity shortages increase the probability of default for both the distressed borrower, and any lender who provides liquidity. Under perfect information, non-binding credit lines can reduce contagion because informed lenders are able to disconnect from distressed borrowers. However, when information is imperfect, non-binding credit lines can exacerbate contagion. This occurs because a subset of informed lenders may still refuse the distressed borrower liquidity, which in turn can create a liquidity shortage for the borrower, increasing the borrowers probability of default. However, many uninformed lenders still provide liquidity, which means they remain connected to the distressed borrower and thus are more likely to contagiously default. Consequently, while it may be privately optimal for informed lenders to refuse the distressed

3 bank liquidity, it may be socially costly as it increases the probability of a default by both the distressed bank, as well any uninformed banks that continue to lend. Thus informed potential lenders impose an externality on both the distressed bank and its uninformed lenders. Finally, I show the effectiveness of non-guaranteed credit lines is closely linked to the quality of information and the amount of potential distress in the economy. In good times - when information is good and potential distress is low - non-guaranteed credit lines work particularly well as healthy banks that require liquidity are able to effectively separate themselves from their distressed counterparts. In contrast, when information is poor, the expected amount of contagion conditional on distress increases greatly. Furthermore, I show that under imperfect information, the size of distress can lead to too much or too little liquidity provision. My paper is closely related to multiple areas of economic literature. First, the model has close ties to the growing literature on network failures, contagion and systemic risk in financial markets. One seminal piece of work is that of Allen and Gale (000) who showed how regional shocks can propogate through an interbank network to produce aggregate fragility. Their work has since been expanded upon extensively in several papers including but not limited to Freixas, Parigi and Rochet (000), Cifuentes, Ferucci and Shin (005), Allen, Babus and Carletti (00), Zawadowski (0), Caballero and Simsek (0), Acemoglu, Ozdaglar and Tahbaz-Salehi (03) and Elliot, Golub and Jackson (04). For example, Acemoglu, Ozdaglar and Tahbaz-Salehi show when financial shocks are sufficiently small a complete network is the most stable financial network, but when the size of these shocks cross a critical threshold the complete network becomes the least stable. I add to this literature by examining how the ability for lenders to respond strategically to borrower distress affects the propogation of shocks in the financial network. Second, my paper is related to the literature on credit lines. From a theoretical perspective, Homlstrom and Tirole (00) have shown how firms and banks can use guaranteed credit lines to effectively manage their liquidity needs and assist with their investment decisions. Empirically, the work of Sufi (009) has established the prevalence of non-guaranteed credit lines as a source of liquidity in the economy. I contribute to this literature by examining banks preference for non-guaranteed credit lines, and the consequences of the use of non-guaranteed credit lines within the financial system. Finally, my paper is also naturally related to the large literature on the role of banks as intermediaries and the optimality of debt. An incomplete list includes Diamond (984), Rochet and Tirole (996), Kiyotaki and Moore (997), Bolton and Scharfstein (996), Diamond and Rajan (005), Hart and Moore (994), Flannery (994), Acharya and Yorulmazer (008) and Calomiris and Khan (99). I contribute to this literature by modeling the interbank liquidity provision role of banks in the presence of moral hazard, and the corresponding implications for liquidity sharing, the distribution of bank defaults and efficiency. Section outlines the model and describes the optimization problem in full generality. Section 3 presents a simple economy with perfect information in order to explain the intuition behind 3

4 some of the key forces of the model. Section 4 describes the equilibrium and provides the results for the general model. Section 5 discusses some potential extensions, and Section 6 concludes. Model. Setting The model has three periods, t = {0,, } and N risk-neutral banks indexed by i = {,,...,N}. Each bank maximizes their expected profits and has a charter value V i = V which is lost if it goes bankrupt. The financial system will consist of banks, their investments, and their bilateral exposures, which represent lending and borrowing relationships among banks. Each bank is funded from outside the banking sector by a unit mass of depositors D =. I do not model depositors, but D provides a default boundary for banks in the model. Bank funding is subject to liquidity shocks - the early withdrawal of funds by a fraction of depositors. At t =each bank has probability p l of receiving a liquidity shock, but only a maximum of one bank receives the liquidity shock. If a bank is hit with a liquidity shock, it is required to raise units of funds at t =. If a bank fails to raise sufficient funds, bankruptcy ensues, resulting in the liquidation of all bank assets in order to pay creditors and the loss of the bank s charter value V. On the other hand, if the bank is able to pay at t =, it will have remaining deposits of D at t =.. Timing To fix ideas, the timing in the model works as follows. At t =0, banks first raise funds from their depositors who are exogenous and outside any interbank network. Each bank i then has up to three decisions to make. First, bank i must decide how to allocate its funds between two potential investment opportunities which I will call liquid and illiquid assets. Liquid assets can be thought of as securities which can be easily traded for cash, such as government bonds, while illiquid assets represent more complex and long term investments with uncertain payoffs such as commercial and industrial loans. Second, each bank i will have the opportunity to form credit lines with other banks. The credit lines, which I will describe in more detail shortly, will take the form of promises to lend or borrow liquid assets to another bank following liquidity shocks. The credit lines will be particularly useful for a given bank as it will potentially allow for additional investment in the more profitable illiquid assets, an increased ability to weather liquidity shocks, and depending on their form, may help discipline banks to choose the socially optimal level of effort. One can Modeling depositors and requiring that they break even adds significant complexity to the model. In particular, solving for payments made to junior claimants (interbank payments) becomes significantly more complex. 4

5 think of these credit lines as implicit or explicit credit lines that banks draw down on when they have a shortage of funds. One prominent example is the fed funds market. Finally, if a bank invests in illiquid assets it must also decide how much effort e to exert at t =0. Effort can be high (e =)orlow(e =0) and can be thought of as a bank s decision to screen a project. At t =, a liquidity shock hits (at most) one random bank, and potentially, a solvency shock hits the same bank. Solvency shocks are closely tied to illiquid asset returns and will be described momentarily. However, throughout I will refer to banks that have received solvency shocks as distressed and banks that have not as healthy. If bank i receives the liquidity shock, it proceeds to raise funds using a mixture of liquid assets, borrowing, and proceeds from the liquidation of illiquid assets. Potential lenders - those who have credit lines established at t =0 with bank i - will lend to bank i, unless the credit line formed at t =0gave them the option to renege on their promise, in which case they may choose to do so. If bank i fails to raise,it defaults, liquidates all remaining assets to pay creditors, and loses its charter value V. Otherwise bank i survives until t =. At t =, returns for the illiquid asset are realized, banks return money to their depositors, pay off any interbank debt they owe, and return remaining funds to equity. Any bank who is unable to meet their liabilities, defaults, and loses its charter value V..3 Investment Opportunities As described in section., banks have two investment opportunities, liquid and illiquid assets, and must choose how to allocate their funds between them. Let i denote the fraction of liquid assets bank i chooses in its portfolio, and thus i is the fraction of illiquid assets bank i holds in its portfolio. Liquid assets are the numeraire - they have a cost of at t =0 (and t =)andpayoff at t =(and t =) - and the primary reason to hold liquid assets is insurance against liquidity shocks. Illiquid assets have a cost of at t =0and provide an uncertain return of R at t =. I denote the pdf of illiquid asset returns at t =0to be g( R) and assume that illiquid assets are more profitable investments than liquid assets (E( R) > ). In addition, each bank can partially liquidate its project prematurely at t =, but can only recover a fraction L of its full value. Unless otherwise stated I assume illiquid assets are perfectly illiquid, i.e. L =0. Consequently, if bank i cannot raise in liquidity from a combination of borrowing and its own liquid assets, it immediately defaults. I assume that L does not depend on the realization of any shocks at t =. Let l i (s i ) be the indicator variable that is equal to one if bank i receives a liquidity (solvency) shock and zero otherwise. At t =, banki s illiquid assets receive a solvency shock with probability p s (e i,l i ). I denote g ( R i )=g( R i s i = ) as the pdf of bank i s illiquid asset returns 5

6 if bank i is distressed, and g 0 ( R i )=g( R i s i = 0) as the pdf of bank i s illiquid asset returns if bank i is healthy. For simplicity, I assume that g 0 (R ) = g (R). That is, if bank i is distressed, its illiquid assets give a return that is lower by a fixed amount in all future states of the world. Thus we can view g( R i ) as a mixture of the distributions g ( R i ) and g 0 ( R i ) such that: 8 g( R < g 0 ( R i ) with probability p s (e, l) i )= : g ( R i ) with probability p s (e, l) Iassumeg 0 ( R i ) Unif[R, ] which implies that g ( R i ) Unif[R, ]. I also assume that R =which implies that lenders only default when providing liquidity to distressed banks. Bank i s effort choice is unobservable and uncontractable, but it impacts both bank i s private benefits and and bank i s probability of a solvency shock. I assume that for a given choice of effort, bank i receives some private benefits that cannot be captured by its creditors. Specifically, let e i = {0, } be bank i 0 s choice of effort, and define P e = {P 0,P } to be the private benefits received by bank i given e. IassumethatP 0 >P, and without loss of generality, I set P 0 = P and P =0. The probability bank i experiences a solvency shock p s,i (e i,l i ), depends on both whether bank i receives a liquidity shock and bank i s prior choice of effort. I define p s,i (e i,l i ) in the following way: 8 0 if l >< i =0 p s (e i,l i )= p s if l i =, e i = >: p 0 s if l i =, e i =0 In words, if bank i did not receive a liquidity shock at t =, then the probability that bank i is distressed at t =is zero. 3 However, if bank i received a liquidity shock at t =,the probability bank i is distressed depends on its prior choice of effort at t =0. Define p e s = {p 0 s,p s} as the conditional probability bank i is distressed given bank i has received a liquidity shock and made prior effort choice e = {0, }. A high effort choice (e =)att =, which could correspond to screening or screening by bank i, results in a lower probability of a solvency shock, i.e. 0 <p s <p 0 s. Thus banks face a tradeoff in their effort decision. If bank i chooses low (high) effort, it exposes itself to a higher (lower) probability of distress but is able (unable) to appropriate I do not model the underlying reason for the occurance of solvency shocks as I am mostly interested in what happens following the solvency shock. There are many potential candidates for a microfoundation present in the global economy. For example, a negative shock to house prices for a bank that is heavily invested in real estate would cause a significant reduction in the value of the bank s portfolio of assets. 3 This is again for convenience as I am primarily interested in the response of lending banks to solvency shocks that hit borrowing banks. A potentially interesting extension of the model is to allow solvency shocks to hit banks who did not receive a liquidity shock (potential lenders) and see how this effects their decision to lend. 6

7 additional private benefits per unit of investment in the illiquid asset. As is standard in the literature, I will assume that low effort (not screening) is socially suboptimal. This means that the private benefits of choosing low effort are less than the increase in the sum of expected losses from solvency shocks and expected default costs. Mathematically, this is equivalent to P<p l p 0 s p s R + ( ) V. 4 R.4 Financial Network I model the financial network as a series of credit lines between banks, initially formed at t =0, and later potentially activated at t =. The only types of credit lines I allow for are standard uncollateralized debt contracts. This is obviously a simplification of the many interconnections that exist between banks in the world. For example, in addition to borrowing and lending, banks are exposed both directly and indirectly to each other through derivatives contracts and similar investments. Specifically, I model the financial system as a directed graph G =(N, B) where N = {,,...,N} are the nodes and B = {b ij } are the edges. Each node represents a bank, while each edge represents a credit line between bank i to bank j. At t =0theedges represent the established lending relationsips, while at t =the edges will represent the credit lines that were activated. For example, if bank i forms a credit line with bank j at t =0we write this as b ij, whereas if bank i activates its credit line from bank j at t =we write b ij = B ij. If the credit line remains inactive at t =B ij =0. I denote the nature of credit lines by the choice variable z = {0, }, with z =0representing a binding credit line. For simplcity, I assume that the decision to make credit lines binding or non-binding is one decision made by the borrower about all its potential lending contracts, which implies every bank either has only binding credit lines or only non-binding credit lines. I assume that lenders face a participation constraint which requires lenders to break even in expecation - including incremental default costs that result from lending - where the expectation is conditional on the lender s information set. When lending takes place at t =, lenders do not observe the behavior of other lenders and thus instead form expectations over other bank behavior. Forming a credit line potentially provides banks with information about their counterparties that is both private and unverifiable. I assume that if bank i has a credit line with bank j, bank j becomes informed about whether bank i is healthy or distressed with probability. Formally, if we let ij be the indicator variable that is equal to if bank j receives a signal about the condition of bank i, then if b ij > 0, it follows that Pr( ij = ) =. Ifbankj receives no signal, =0,andbankj remains uninformed as to whether bank i is healthy or distressed. I allow the credit lines to take two forms - binding and non-binding. A binding credit line 4 This is taken under autarky with i =.ForaderivationseeAppendix. 7

8 from bank j to bank i is a promise from bank j to always lend an amount B ij if bank i requests liquidity at t =. Interest rates for binding credit lines are set at t =0subject to lenders participation constraints at t =0. In contrast, a non-binding credit line from bank j to bank i is also a promise from bank j to lend an amount B ij if bank i requests liquidity at t =, except that bank j is able to renege on the promise. Therefore interest rates for non-binding credit lines are fixed at t =and are conditional on the lenders information set at t =due to the lender s ability to renege. It is possible that no interest rate exists that satisfies lenders participation constraints, in which case no lending will take place. Notice that lenders with non-binding credit lines have two potential actions at t =- lend or renege - and can choose their action based on the information set they face at t =. I denote the action lend as a and renege as a. I define a strategy by lenders with non-binding credit lines to be a mapping from the lenders information set to an action taken by the lender. Denote a strategy played by lender as a pair (a i,a j ) where a i is the action taken when =0(no signal) and a j is the action taken when =. Since lenders receive a binary signal and face a binary choice, there are four pure strategies a lender could potentially play; always lend (a,a ), lend only if no signal of distress (a,a ), renege only if no signal of distress (a,a ), and always renege (a,a ). Finally, since the signal lenders receive are both private and unverifiable, I assume that when activating their credit lines, borrowers make a take it or leave it offer to their potential lenders. By making this assumption I avoid any hold up problems. I do not allow the borrower to mix over which lenders it requests liquidity from, but I do allow borrowers to play mixed strategies over the promised payment it requests on its debt when using non-binding credit lines..5 Bank Optimization Problem Define the function i ( i,e i,l i,s i, i,g,n, B, {R i } in ) to be bank i s profits at t =. Define the functions A i ( i,e i,l i,s i, i,g,n, B, {R i } in ) and L i ( i,e i,l i,s i, i,g,n, B, {R i } in ) as bank i 0 s assets and liabilities at t =. Finally define the function i ( i,e i,l i,s i,g,n, B) to be the state contingent fraction of bank i s illiquid assets that remain following any liquidation at t =,the function D i ( i,e i,l i,s i, i,g,n, B, {R i } in ) to be the payment made by bank i to its depositors at t =, and the function F i, i ( i,e i,l i,s i,g,n, B, {R i } in ) to be the payment on interbank debt from bank i to bank i. To save on notation, I will hereon take the brackets to be implicit and shorten the respective functions to i, A i, L i, i, D i and F i, i.notethatasoft =0, each of these functions is a random variable, and as of t =, all with the exception of i remain random variables. First, consider the function i. The assumption that L =0implies that if bank i receives a liquidity shock, and cannot raise sufficient funds from a combination of its own liquid assets and the interbank market, it will have to liquidate its entire portfolio of illiquid assets. In contrast, 8

9 if bank i can raise sufficient funds, no liquidation will be required. Thus we have: i = i +B i () Given all decisions made prior to t =, and any realization of {R i }, we can write bank i s assets at t =as the sum of (i) liquid asset holdings, (ii) illiquid asset holdings, (iii) interbank P debt receipts of j F i,j, and (iv) in the event that a liquidity shock hit bank i at t =,we must substract off payments made to the depositors that withdrew early: A i = i +( i ) i R i + B ij + X i F i,i l i () Let F i,j be the maximum repayment by bank i to bank j at t =. Then in a similar fashion, liabilities can be written as the sum of payments owed to depositors D l i and other banks P F i i, i : L i = D l i + X F i, i (3) i Before I specify interbank payments F i, i (and F i,i ), consider the function D i which defines the payment made to depositors at t =. Depositors are senior to interbank debt at t =and hence are paid first. Thus bank i pays its depositors D i at t =such that: D i =min(a i,d l i ) (4) Bank i s interbank debt is junior to its depositors, and hence its counterparties are entitled to the remaining assets of bank i up to a fixed amount specified in the interbank debt contract. Iassumethatallbanks i = {,,...,j} that bank i borrows from are of equal seniority. If we define F i = P Fi, i,banki pays its counterparties a total of: 5 F i = max min A i D i, F i, 0 Any credit line must satisfy the lenders participation constraint. This requires lenders to be compensated for the principal they lend and the expected default costs incurred from lending. Define the function ˆ d,j ( i,e i,l i,s i, i,g,n, B, {R i } in ) to be the change in probability of default for bank j resulting from lending to bank i (as opposed to not lending), and denote this as ˆ d,j for short. The lenders participation constraint requires: E( F i,j ) B i,j +ˆ d,j V (5) 5 I make the simplifying assumption that when bank i is only able to partially repay the principal it borrows from its counterparties, then all repayments to counterparties are proportionate to each counterparties share of the total principal borrowed. For a technical discussion of this assumption see the appendix. 9

10 Since I assume borrowers make take it or leave it offers, this will bind with equality in equilibrium. Note that banks must decide at t =0whether these credit lines are binding at t =. The terms set for the promised payment will depend on the nature of the credit line, and I discuss the nature of the credit lines and their impact on bank decisions further in Sections 3 and 4. Finally, we can write i as the difference between bank i 0 s Assets, A i and Liabilities, L i : i = max (A i L i, 0) = max( i +( i ) i Ri + X j F j,i D X j F i,j, 0) (6) Moreover, let d i be the indicator function that is equal to one if bank i defaults at t =and zero otherwise. Default occurs whenever a banks liabilities are greater than its assets at t =, and in this case, a banks profits are zero due to limited liability. With this notation, bank i s optimization problem can be written as: max i ( i, i,b ij )=E( i )+( E(d i ))V +( i )Pi e i,e i,b ij,z i Here the expectation is taken over both the occurance of liquidity shocks, l i and solvency shocks, s i (at t = ) and the realization of the illiquid asset returns R i (at t = ). Then because V is a constant, we can rewrite the expecation of the bank default indicator function as a probability function for bank i s default. Define the function d,i ( i,e i,l i,s i,g,n, B, {R i } in ) as the probability that bank i defaults at t =. Again to save on notation I will often drop the brackets and simply write d,i. Thus we can rewrite the banks objective function as: max i ( i,e i,b ij )=E( i )+( d,i )V +( i )Pi e i,e i,b ij,z i This is bank i s objective function in its most general form. In section 3 I will analyze a simplified economy in which the above maximization problem will take on a simpler form. In section 4 I will return to discuss the general form of the objective function as well as given a full equilibrium analysis. 3 Economy with Perfect Information To illustrate the main mechanism for liquidity sharing and reneging in my model, it is useful to start with a simplified economy with perfect information ( =). That is, if bank i and bank j formed a credit line at t =0, the latter can perfectly observe at t =whether the former is distressed or healthy. If the credit line is non-binding, bank j will have the option of basing its lending decision off this information. I will first consider the banks static choice problem when banks are not permitted to form 0

11 credit lines at t =0. I provide conditions where banks choose = and fully insure themselves against liquidity shocks. I then turn my attention to what happens when banks are allowed to form credit lines at t =0andexamine the consequences of liquidity sharing in the presence of solvency shocks. I will consider what happens when credit lines are both binding and nonbinding. 3. Autarky (No Liquidity Sharing) Consider an autarky economy with N banks i = {,,...N} that are unable to lend to each other. As described in section.5, we can write the profits of bank i i as the difference between its assets A i and liabilities L i : i = max (A i L i, 0) = max ( i +( i ) i R D, 0) Note that there are no F i,j here for now as the interbank lending channel has been shut down. In similar fashion to section.5, we can proceed to write bank i s objective function as: max i ( i,e i )=E( i )+( i,e i i )P e i +( d,i )V The following proposition details when i = and e i =0is bank i s optimal choice under autarky. Note that i = corresponds to full insurance against liquidity shocks but no protection from solvency shocks (which would require i =). Proposition. If (p l,p s,,v,,, P ) satisfy (7) and (8) (Below), then banks opt to insure against liquidity shocks but not solvency shocks, i.e. =.If(p l,p s,,v,,, P ) satisfy (9), banks will choose e =0. (p l ) R / p l p s ( ) R p l p s V p l p s R R R Proof. See appendix. + p l ps R V P e (7) ( p l p s )( R) p l (p 0 s p s) ( R R ) + V apple P e (8) < P (9) Proposition ensures =, and implies neither liquid assets ( =0) nor illiquid assets ( =) are redundant in the autarky economy. Under Proposition the bank objective function

12 equals: i =( ) R p l p s + p lp e s R ( ) +( )P e + p l p e s V R There are four distinct parts here. The first term is the expected returns from liquid and illiquid assets in excess of what is owed to depositors. The second term is the expected losses taken by depositors - these flow to bank i. The third term is the private benefits bank i receives from choosing e =0, and the fourth term is the expected charter value. Importantly, as all banks are identical, Proposition applies to all banks, and thus when Proposition holds aggregate liquidity in the N bank economy without liquidity sharing equals N. Restricting banks such that they cannot lend to each other can be inefficient. Recall that at most one liquidity shock hits the economy, and thus by assumption banks cannot be hit simultaneously with a liquidity shock. This implies that if liquidity cannot be shared between any banks i and j, at least (N ) units of liquidity will not be used to pay for a liquidity shock at t =. However, if banks could share liquidity, each bank could potentially reduce their holdings of the liquid asset, shifting investment to the more profitable illiquid asset, and still have sufficient funds to cover potential liquidity shocks. Proposition also provides a condition under which individual banks will against screening their illiquid assets. In later sections when I introduce credit lines we will see that the type of credit line may enhance or weaken bank screening incentives. Finally, recall that the social optimality of effort required: p l (p 0 s p s) R R + V > ( )P Which does not imply e =. This happens because the deposit rate does not adjust to the choice of effort by banks, and thus even though not screening is socially suboptimal, banks may opt to do so as the losses are disproportionately borne by the depositors. Although this is present in all versions of the model, it is not new, nor surprising. One can think of the decision not to screen as a special case of risk-shifting by the bank which has been well documented in the literature. 3. Liquidity Sharing Suppose now that banks can agree at t =0to form credit lines which will allow each bank to borrow from others should it receive a liquidity shock. For the moment, I do not differentiate between binding and non-binding credit lines. Since I assume liquidity shocks only hit one bank at a time, I do not have to deal with cases where multiple banks would like to borrow. Profits

13 for bank i under this scenario are: i = max (A i L i, 0) = max ( i +( i ) i R i + F j,i D i F i,j, 0) Bank i 0 s objective functions (subject to participation constraints of its lenders) is: max i ( i,e i,b ij,z i )=E( i )+( i )P e +( d,i )V i,e i,b ij,z i Notice that the one difference from the autarky case is that in addition to choosing an effort level, e i and an amount of liquid assets to hold, i,banki must also choose what link to form with the other banks j, b ij and the type of credit line z i.alsonotethatb ji does not enter bank i s objective function as bank i breaks even on all lending to bank j. Not only does securing a lending promise from other banks - binding or non-binding - benefit bank i by reducing its expected liquidation costs, it ex-ante allows bank i to redistribute its portfolio towards the more profitable illiquid assets. For example, suppose that bank i secures a binding lending promise from bank j of B ij. Bank i could potentially reduce its holdings of liquid assets by amount B ij without increasing its expected liquidation costs, and redistribute all these funds into illiquid assets which would increase the expected returns from its assets A i by B ij E( R i ). Before analyzing the types of credit lines, I define the equilibrium in the economy with perfect information. Definition. An Equilibrium at t =0consists of vectors of investment and screening decisions by banks!,! e, a set of credit lines between banks i and j {b ij }8i, j {,...,N}, a vector of credit line decisions by banks! z, and a set of beliefs over bank lending decisions at t =such that:. Given! and beliefs, all banks maximize i -Nobanki can be strictly better off by changing ( i,e i,b ij,z i ).. Bank lending behaviors at t =are consistent with beliefs held at t =0. 3. All lenders participation constraints, borrowing constraints, and where appropriate - incentive compatability constraints - are satisfied. The following two subsections describe properties of binding and non-binding credit lines. Throughout I will assume i < hold sufficient liquid assets to pay for potential liqudity shocks., as banks have no incentive to form credit lines if they already Moreover, note that given liquid asset holdings from other banks i,banki sliquidasset P holdings must satisfy i > i, where i =min 0, j j. The lower bound exists because 3

14 a neccesary condition for any credit line is for bank i to have the ability to repay its creditors in the future. Bank i cannot repay its creditors at t =if the liquidity it borrows does not prevent bank i from liquidiating its illiquid assets at t =. Since bank i can never borrow more than the total liquidity held by other banks, P i i, it can never hold less than i and simultaneously satisfy all other bank s participation constraints. 3.3 Binding Lending Relationships First lets consider credit lines that are binding. A binding credit line between bank i and bank j is a binding promise for bank j to lend to bank i should bank i receive a liquidity shock in the future. Since I assume all credit lines take the form of standard debt contracts, the promised payment must satisfy bank j s participation constraint, and be set in advance to avoid any potential hold up problems. Note that this implies that the promised payment cannot depend on whether bank i is distressed or healthy. Thus if bank i forms a binding credit line with bank j of size B ij at t =0,bankj will be required to lend bank ib ij if bank i is hit with a liquidity shock at t =. In return bank i must make an expected payment to bank j at t =equal to the amount bank i borrowed plus bank j s expected additional default costs from lending. That is, if bank i borrows from bank j it sets a promised payment on its debt to bank j that satisfies: E( F i,j )=B i,j +ˆ d,j V Where the expectation is taken using bank j s t =0information set. The following proposition outlines the relevant properties of binding credit lines. 6 Proposition. Suppose banks i = {,,...,n} hold i liquid assets and Proposition holds. If bank i chooses to form binding credit lines with banks j {,,...n}, thenthefollowingare true:. Bank i will not screen.. If bank i holds i liquid assets, the total size of bank i s binding credit lines will equal P j B ij = B i = i. 3. If bank i prefers binding credit lines to autarky for some i [ i,,), banki will hold i = i liquid assets. Proof. See Appendix. 6 In the Appendix I also describe how bank i may go about deciding which other banks it wants to borrow from when using binding credit lines as well as conditions for some of the properties listed below. 4

15 Proposition. states that if bank i opts against screening in the absence of any credit lines, then the use of binding credit lines will not alter bank i s screening behavior. To see why recall that under autarky, bank i opted not to screen as it could profit by pushing the losses incurred from not screening onto its depositors whose payment was capped at a fixed amount D. The use of binding credit lines effectively increases bank i s total debt by a fixed amount in certain states of the world, and thus only serves to weakens bank i s screening incentives. However, bank i s potential lenders recognize both bank i s lack of screening incentives and bank i s inability to commit to screening, and thus ex-ante request a higher promised payment on their debt than what would be required if bank i chose to screen. The higher promised payment required by bank i s counterparties both reduces bank i s profits and increases bank i s expected default costs following a liquidity shock. In fact, bank i s inability to commit to screen when using binding credit lines harms bank i as the costs associated with the higher promised payment outweigh the private benefits bank i receives from not screening. The failure of binding credit lines to provide incentives for banks to screen is the main reason why banks consider the non-binding alternative that is discussed in the next section. Proposition. notes that if bank i engages in binding credit lines, it will never borrow more than what it needs to pay for the liquidity shock. Intuitively, since depositors are both senior to any interbank borrowing, and not always paid in full, additional borrowing by bank i above what is needed to pay for the liquidity shock ( i ) increases the expected payment received by depositors. Since bank i s counterparties must break even, they will demand a higher promised payment, which reduces bank i s profits and increases its default risk. Finally, Proposition.3 states that if bank i engages in binding credit lines, it will hold as few liquid assets as possible. Intuitively, when considering binding credit lines, bank i trades off the increased profits from shifting its portfolio towards illiquid assets against an increase in the expected payment to depositors and an increase in its probability of default. Since the benefits are linear in i and the costs can be shown to be convex (concave in i ), if it is profitable for bank i to use binding credit lines, it follows that bank i should hold as few liquid assets as possible. To illustrate the increased default risk faced by borrowers (relative to autarky), suppose for simplicty N =,andbankschoose i = j =. Under a binding credit line, borrowers have assets worth A i =( i ) R i and liabilities of size L i = + F i, with default occuring when A i < L i. Since conditional on bank choices at t =0and the realization l i =at t =, A i is a monotone increasing function of R i and L i is constant, we can define a default boundary R where if R i >R,bankidoes not default. A higher value for R implies a higher probability of default for the borrower. It should be immediately obvious that R = L i /( i ). Moreover, because F i >B i = due to the fact that borrowers require compensation for their default risk, 5

16 it follows that L i > i,andthatr >. However, under autarky, i =,andthusr =. Thus we have when banks engage in binding credit lines R is higher than if they remained in autarky. The borrowers increased default risk comes from the fact that borrowers must compensate lenders for (i) states where the borrower is unable to repay the lender in full and (ii) lender default risk. Importantly, because banks use standard debt contracts to borrow and lend, R is not state contingent, which means that banks face higher default risk both when they are distressed and when they are healthy. Thus under binding credit lines, potential distress is sufficient to generate default in states where default otherwise would have not occurred. This completes my discussion of binding credit lines. It should be noted that all the properties of binding credit lines discussed here carry over to the economy with imperfect information ( <) as all contract terms in a binding credit line are established at t =0and do not depend on any uncertainty resolved at t =with the exception of which bank receives the liquidity shock. The next section focuses on an alternative to binding credit lines which provide a very different set of incentives for individual banks and their counterparties. 3.4 Non-Binding Lending Relationships I now consider banks that use non-binding credit lines. A non-binding credit line between bank i and bank j is an implicit promise for bank j to lend to bank i should bank i receive a liquidity shock in the future. However, it is not a binding promise, and thus bank j could opt not to lend to bank i. Since I assume all credit lines take the form of standard debt contracts, the promised payment must satisfy bank j s participation constraint. Thus if bank i forms a non-binding credit line with bank j of size B ij at t =0,bankj will have the option to lend bank ib ij if bank i is hit with a liquidity shock at t =.Ifbankjdoes lend, bank i must make an expected payment to bank j at t =equal to the amount bank i borrowed plus bank j s expected additional default costs from lending. That is, if bank i borrows from bank j it sets a promised payment on its debt to bank j that satisfies: E( F i,j ) B i,j +ˆ d,j V Where the expectation is taken using bank j s t =information set. For the duration of this section I assume that solvecy shocks are sufficiently large such that lenders only plausible strategy is to lend (not lend) when they do not (do) receive a signal of borrower distress. That is, lenders play the strategy (a,a ) as described in Section.5. I relax this assumption in the economy with imperfect information. The following proposition outlines the relevant properties of non-binding credit lines. Proposition 3. Suppose banks i = {,,...,n} hold i liquid assets and Proposition holds. 6

17 If bank i chooses to form non-binding credit lines with banks j {,,...n}, thenthefollowing are true:. Bank i screens if R + q (V ) + V (V ).. If bank i holds i liquid assets, the total size of bank i s non-binding credit lines will satisfy P j B ij = B i i, and under perfect information, bank i is indifferent across all options that satisfy this requirement. 3. If bank i prefers non-binding credit lines to autarky for some i [ i,,), banki will hold i = i liquid assets. Proof. See Appendix. Proposition 3. is the main reason a bank might prefer non-binding credit lines to their binding counterparts. If Proposition 3. holds, we have that non-binding credit lines align bank screening incentives with the social optimum. Non-binding credit lines provide screening incentives by substantially increasing the cost of not screening. Specifically, if bank j plays strategy (a,a ), then whenever bank i is distressed, bank j will take action a (since =), and bank i is forced to default at t =. Not only is the probability of default significantly higher if bank i were to choose e =0(p 0 s vs p s), it is particularly costly as it occurs at t =which forces bank i to liquidate all illiquid assets immediately and lose all potential profits that it may have otherwise received at t =should it have survived. Proposition 3. is in stark contrast to Proposition. from the proceeding section. Whereas binding credit lines only serve to exaccerbate bank incentives to no screen, non-binding credit lines encourage banks to screen. This disciplining incentive associated with non-binding credit lines is analogous to short-term debt in a setting where banks are disciplined by the threat of creditors not rolling over bank debt. For example, prior research by Calomiris and Khan (99) and others have shown how short-term debt can discipline bank risk-taking behavior. However, unlike existing research, Section 4 will show how non-binding credit lines can impact the banking system as a whole. Just as in Section 3.3, Proposition 3. states that bank i will not form a credit line than it is less than the liquidity it requires when holding i liquid assets. Although the result is almost identical, the intuition is very different. Specifically, with non-binding credit lines, and lender strategies of (a,a ), banks who receive liquidity shocks know potential lenders will (not) provide liquidity when they are healthy (distressed). Furthermore, because information is perfect, all lenders behave synchronously. That is, liquidity shocked banks will either receive funding from all their potential lenders, or from none at all. Since there is no benefit to receiving an amount greater than i, banks ex-ante only secure promises of total size i. 7

18 However, when information is imperfect, Proposition 3. will not neccessarily hold. Specifically, banks may anticipate that uninformed lenders may provide liquidity to a distressed borrower, and thus respond ex-ante by securing additional promises to cover for the informed borrowers who do not provide liquidity. I explore this further in Section 4. To understand Proposition 3.3, note that engaging in non-binding credit lines when there is perfect information causes bank i to default immediately if it is distressed at t =. This happens because bank i requires i in liquidity but receives nothing from its counterparties as they take action a. 7 As a result, conditional on a screening decision by bank i, non-binding credit lines increase bank i s default costs by p l p s R V. Moreover, since >, an early default has the additional cost of denying bank i any potential profits at t =. However, notice that this is true for non-binding credit lines of any size >0 as the probability bank i defaults is fixed at p s under perfect information. Thus if we take the limit as! 0, we can view engaging in non-binding credit lines under perfect information as akin to signing a contract with fixed cost p l p s R R + V, and benefits that are linear in i. Thus if bank i decides to use non-binding credit lines, it will be optimal to hold as few liquid assets as possible. Moreover, relative to binding credit lines, non-binding credit lines under perfect information allow banks to reduce their risk of default when healthy. 8 Recall, that under binding credit lines, healthy borrowers have a positive probability of default as their debt contracts with lenders require the healthy borrower to compensate the lender for the losses incurred from lending to distressed borrowers. In contrast, non-binding credit lines ensure that lenders require no compensation when solvency shocks do not occur, which reduces the debt burden for healthy borrowers. 3.5 Equilibrium and Discussion As previously discussed, the choice between binding and non-binding credit lines presents several tradeoffs for banks. However, the primary trade-off is between the improved screening incentives provided by the non-binding credit lines and the reduced probability of an early default guaranteed by the binding credit lines. In either case, if banks use binding or non-binding credit lines, Propositions.3 and 3.3 imply that all banks would like to hold as few liquid assets as possible. Thus if banks use any form of credit line in a symmetric equilibrium where there is perfect information, it must be that 8i, j i = { N } and B ij = { N }. Thus compared to when banks remain in autarky and opt to fully insure, the option for banks to form credit lines allows banks to still remain fully insured against liquidity shocks, but also simultaneously hold more of the relatively profitable illiquid assets. 7 Again, in Section 4, counterparties will only take action a if their incentive compatability constraint cannot be satisfied when taking a. 8 This is not necessarily the case under imperfect information. 8

19 The next proposition outlines the equilibrium choice of banks in the economy with perfect information. Proposition 4. Suppose that Propositions -3 are satisfied. Also assume that banks plays strategy (a,a ) under a non-binding credit line. It follows that all banks hold i = N. Banks use nonbinding credit lines iff 0 d p l p s V > p l E((A 0 i L 0 i ) (A i L i ) l i = ) + ( i )P (0) and binding credit lines if the above inequality does not hold. Moreover, banks pick e =if the above condition holds, and e =0if it does not hold. Proof. See Appendix. Proposition 4 outlines the equilibrium choices of all banks. There are a few interesting takeaways here. First, when Proposition 3. holds, the nature of credit lines and the choice of screening by bank i are interchangeable decisions - picking one effectively determines the other. Second, all banks hold as few liquid assets as possible, i = N. Notice that unlike in autarky, the efficient level of liquid assets are held as P i i =. As a result, bank profits are substantially larger, as is social surplus. However, there are more defaults in the economy with liquidity sharing, which is perhaps unsurprising given that there is both substantially less liquidity, and higher leverage among banks than there was under autarky. Third, although the efficient amount of liquidity is held in equilibrium, if (0) does not hold, the distribution of liquidity is not always ex-post efficient. That is, banks who receive liquidity shocks do not always receive the liquidity they need to survive such shocks, as banks who do not receive the shocks may opt to hold onto their liquid assets. This happens because when (0) is not satisfied, the equilibrium involves non-binding credit lines, and thus potential lenders will opt to hold onto their liquidity rather than lend it to the distressed bank. If the solvency shocks are not too large, it is efficient for lender s to provide liquidity to the distressed bank. However, lenders are unwilling to provide liquidity as borrowers have insufficient future funds to meet all lender s outside options. Finally, note that contagious default only ever occurs when (0) is satisfied. This is because under non-binding credit lines with perfect information, potential lenders playing strategy (a,a ) never lend to distressed borrowers. In the economy with imperfect information, strategy (a,a ) will not be sufficient to rule out contagious default as uninformed banks may lend to distressed borrowers. In fact, with imperfect information contagious default can be higher under nonbinding credit lines than under their binding counterparts. I return to this possibility in Section 4. The next subsection presents a brief numerical example of many of the properties described in Sections