Liquidity Coinsurance and Bank Capital

Liquidity Coinsurance and Bank Capital Fabio Castiglionesi y Fabio Feriozzi z Gyöngyi Lóránth x Loriana Pelizzon { March 2011 Abstract This paper analyzes whether banks use their capital to deal with liquidity uncertainty that cannot be coinsured in the interbank market. To the extent that the remuneration of capital is exible it can be adjusted to face such liquidity uncertainty. Hence, bank capital can have a risk-sharing role as it can be used to transfer bank liquidity risk to well diversi ed investors. Bank capital and the interbank market therefore can act as substitutes, implying a negative relationship between these variables. We show that banks tend to postpone the remuneration of capital when they are hit by liquidity shocks that cannot be coinsured in the interbank market. This mechanism generates a negative relationship between a bank s activity in the interbank market and its short-term variation in bank capital. We test these predictions in a large sample of US banks and nd support for the risk-sharing role of bank capital. Similar results also hold in a sample of European and Japanese banks taken from Bankscope. JEL Classi cation: G21. Keywords: Bank Capital, Interbank Markets, Liquidity Coinsurance. We thank Christa Bouwman, Fabio Braggion, Hans Degryse, Robert Hauswald, Enisse Kharroubi, Jose Jorge, Vasso Ioannidou and seminar participants at Deutsche Bundesbank Conference Liquidity and Liquidity Risk, ELSE-UCL Workshop in Financial Economics: Markets and Institutions, Frias-CEPR Conference Information, Liquidity and Trust in Incomplete Financial Markets, Fourth Swiss Winter Conference on Financial Intermediation (Lenzerheide) for helpful comments. We also thank Mario Bellia and Marcella Lucchetta for excellent research assistance. The usual disclaimer applies. y CentER, EBC, Department of Finance, Tilburg University. E-mail: fabio.castiglionesi@uvt.nl. z CentER, Department of Finance, Tilburg University. E-mail: f.feriozzi@uvt.nl. x University of Vienna and CEPR. E-mail: gyoengyi.loranth@univie.ac.at. { Univeristy Ca Foscari of Venice. E-mail: pelizzon@unive.it. 1

1 Introduction One of the fundamental roles of the banking system is to transform liquid liabilities into illiquid assets. A necessary consequence of this function is that banks face a substantial amount of liquidity risk. 1 Banks can coinsure one another against the idiosyncratic liquidity shocks by lending to each other in the interbank market. However, the interbank market is unlikely to eliminate completely this uncertainty. This occurs mainly for two reasons. First, part of this liquidity risk is likely to be systematic and, by de nition, not possible to coinsure. Second, interbank markets are typically over the counter markets and based on a limited number of pre-established connections. So an idiosyncratic shock may be impossible to coinsure in the absence of such pre-established connections. 2 When a bank is unable to obtain resources in the interbank market, the occurrence of a liquidity shock might become problematic, and ultimately force the bank to liquidate some of its assets at a re sale price. To avoid this outcome, banks can use their capital to absorb the liquidity shock by postponing the payouts to their equityholders. The possibility for banks to transfer part of their liquidity uncertainty to their investors creates a risk-sharing role for bank capital. This form of liquidity risk sharing is indeed desirable if investors are well diversi ed, and it might represent an important function of bank capital. 3 Following Gale [17], we analyze a theoretical model of bank capital as risk-sharing device and derive empirical predictions that are tested in a sample of US, European and Japanese banks. The ndings are by and large consistent with the risk-sharing role of bank capital. We model an economy with two banks that collect deposits from risk-averse depositors and capital from risk-neutral investors. 4 Banks have access to two investment opportunities: A short-term liquid asset (a storage technology) and a long-term illiquid asset. The 1 Indeed banks are subject to liquidity risk on both side of their balance sheet. On the liability side, banks face uncertain withdrawals from depositors and, more generally, the risk of rolling over short-term debt. On the asset side, banks liquidity risk can be represented by delays in the repayment of granted loans. 2 Another reason why interbank markets might o er limited coinsurance opportunities is the presence of moral hazard or adverse selection problems (see Bhattacharya and Gale [5]) 3 Alternatively, bank capital can represent a bu er to protect against solvency shocks a ecting the asset side of banks balance sheet. Or it can mitigate the incentives to take excessive risk by banks shareholders (see, among the others, Brusco and Castiglionesi [7], and Morrison and White [23]). 4 We assume away any contractual imperfection and allow banks to o er fully contingent contracts to both depositors and investors. Among other things, this assumption prevents any role for bank capital as a bu er to prevent insolvency and it clari es its role as risk-sharing device for liquidity risk. 2

two banks have di erent depositor bases and face uncertain liquidity needs. Most of the time their liquidity shocks are idiosyncratic, but they also face some probability of receiving a symmetric liquidity shock. The two banks participate in an interbank market which allows them to coinsure against idiosyncratic liquidity shocks. However, the interbank market is of little help in case of a symmetric shock. We refer to liquidity risk that cannot be coinsured in the interbank market as undiversi able (liquidity) risk. The presence of undiversi able liquidity uncertainty creates a scope for the use of bank capital as a risksharing device. That is, some of the undiversi able risk can be transferred to risk-neutral investors. We assume that collecting resources from risk-neutral investors is costly, so that banks would have no capital were the liquidity uncertainty purely idiosyncratic. Clearly, the optimal level of capital crucially depends on the level of liquidity uncertainty that cannot be coinsured in the interbank market. We show by means of examples that this relationship might not be monotonic. In fact, while we would expect the optimal level of bank capital to decrease when the undiversi able uncertainty reduces, this only happens for some parameters con gurations. This is due to the fact that a reduction in undiversi able uncertainty also has an e ect on a bank s portfolio choices. In particular, a lower level of undiversi able uncertainty induces banks to reduce the investment in the liquid asset and, as in Castiglionesi et al. [8], this can produce higher consumption volatility for depositors. In this case, the optimal level of bank capital can increase because it helps moderate this volatility by transferring it to risk-neutral investors. An interesting insight that can be derived from this analysis is that the amount of liquidity uncertainty that a bank cannot insure in the interbank market can be an important determinant of bank capital. To the extent that such risk is a persistent bank characteristic, it might be responsible for at least some of the large explanatory power that bank xed e ects have in regressions explaining banks capital structure (Gropp and Heider [19]). Unfortunately it is di cult to empirically measure the bank-level undiversi able liquidity risk. Therefore, to obtain testable empirical predictions, we make use of the following general insight of the model. Bank capital should not be remunerated in states of the world where the marginal utility of consumption for depositors is high. In particular, when an undiversi able liquidity shock hits and liquidity needs are high in both banks, depositors per-capita consumption tends to be low and its marginal utility high. Hence, it is optimal to postpone bank capital remuneration when interbank markets activity is low. 3

The decision about when to remunerate bank capital clearly a ects the short-term value of bank capital and its short-term variation. Namely, when bank capital is remunerated, its short-term value tends to drop, both for an accounting reason and because it results in a lower remuneration in the future. The postponement of bank capital remuneration instead means higher future payouts to investors, and the short-term value of bank capital should increase as a consequence. Because the remuneration of bank capital occurs (is postponed) when the activity in the interbank market is high (low), the model predicts that a bank s activity in the interbank market has a negative correlation with (i) the short-term value of bank capital and (ii) the short-term change in bank capital. In the empirical part of the paper we test predictions (i) and (ii) and we nd support for both of them. The main results are obtained in a sample of 2,954 commercial banks in the US. We use their Call Reports to build a quarterly panel dataset spanning from the rst quarter of 2005 till the third quarter of 2010. In particular, for these banks we obtain information on their balance sheet items as well as on their activity in three di erent interbank markets: (a) Unsecured interbank lending and borrowing, (b) Repos and Reverse Repos with maturity longer than one day, and (c) Lending and borrowing on the overnight Federal Funds market. We consider the sum of the absolute value of the net positions in (a) as our main measure of interbank market activity. However, our results also hold when we add the absolute values of the net position in (b) and (c) to (a). As for capital, we adopt a broad de nition including book values of equity and reserves, as well as subordinated debt and hybrid capital. In this way we intend to include any source of funding with a long maturity and no collateral, whose remuneration is exible enough to be potentially used to absorb non diversi able liquidity shocks. To test prediction (i) we use a regression panel approach that allows us to estimate the conditional correlation between a bank s interbank market activity and its capital, including standard controls as well as bank xed e ects. We nd evidence of a negative and signi cant relationship, with both our measures of interbank activity. To test prediction (ii) we instead regress the quarterly variation of bank capital on our measures of interbank activity. We use standard controls also in these regressions as well as instrumental variables to address potential endogeneity issues. Again, we nd evidence of a negative and signi cant relationship which is robust to the choice of the measure of a bank s interbank market activity. Overall we consider this evidence as strongly supportive of the risk-sharing role of bank capital. We also perform a similar analysis in a sample of 877 large European 4

and Japanese commercial banks, obtaining yearly data from 2005 to 2009 from Bankscope. Also in this case results support the risk-sharing role of bank capital. These ndings would be di cult to rationalize if bank capital mainly had an incentive function. Indeed, if the main role of bank capital were to provide incentive to avoid excess risk taking then more capital would translate into lower bank insolvency risk. At the same time more capital results in easier access to the interbank market and larger interbank market activity. This in turn would imply a positive relationship between the level of bank capital and interbank activity. Even if our paper does not directly address normative issues, our results have important implications for regulation. Theoretically, we highlight the degree of undiversi able liquidity risk that each bank faces as an important determinant of bank capital. Moreover, we provide evidence that is consistent with this insight. The current debate on bank capital regulation emphasizes its role as incentive function (see Squam Lake Report, "Stanford" view ect.). We clearly do not want to dismiss this important role, but our results point to the fact that the role of bank capital as risk sharing device has been overlooked. We suggest, that when determining capital requirements, regulators may focus on identifying measures that indicate how much undiversi able risk a bank is subject to. Otherwise, any intervention to regulate bank capital will likely a ect the functioning of the markets in which banks coinsure their liquidity risk in a non-trivial way. Our paper is related to both theoretical and empirical works in banking. On the theory side, the paper closest to ours is Gale [17]. He also considers the risk-sharing role of bank capital but, contrary to us, his analysis focuses on regulatory aspects. For this purpose, Gale [17] considers spot markets as a way to co-insure against the liquidity shocks. By modeling the interbank market as a device to decentralize the social planner solution, our framework is similar to Allen and Gale [3]. However, following Castiglionesi et al. [8], we assume that aggregate uncertainty is perfectly anticipated by economic agents. More importantly, we analyze the relationship between the liquidity insurance provided by the interbank market and by bank capital, which is outside the scope of the two previous contributions. 5 5 There is also an extensive theoretical literature on capital regulation based on the incentive function of bank capital. The results are not conclusive since while bank capital requirements usually decrease risk taking the reverse is also possible (see, Kim and Santomero [22], Furlong and Keeley [16], Gennotte and Pyle [18], Rochet [24], Besanko and Kanatas [6] and Hellman et al. [21]). Among the recent contributions, Diamond and Rajan [11] rationalize bank capital as the trade o between liquidity creation, costs of bank 5

On the empirical side, our paper relates to two strands of the literature: The rst one on bank capital and the second on interbank markets. Flannery and Rangan [12] and Gropp and Heider [19] look at the determinants of banks capital holding. Flannery and Rangan [12] argue that the main cause of capital build-up of large US banks in the 1990s was an increased market discipline due to legislative and regulatory changes, resulting in the withdrawal of implicit government guarantees. Gropp and Heider [19] study the determinants of banks capital structure and address the questions of whether these determinants di er from those of non- nancial rms. While they do not nd evidence on the di erences, they argue that the most important determinants of banks capital structure are time-invariant bank xed e ects. Moreover, deposit insurance and capital regulation do not seem to have a signi cant impact on banks capital structure. Regarding the interbank market, Fur ne ([13], [14] and [15]) analyzes banks screening and monitoring activity in the Federal Funds market, and the behavior of this market during Russia sovereign default. Cocco et al. [9] look at the importance of relationships as an important determinant of banks ability to access the Portuguese interbank market. In a recent contribution, Afonso et al. [1] examine the impact of the nancial crisis of 2008, speci cally the bankruptcy of Lehman Brothers, on the functioning of the Federal Funds market. They argue that while banks became more restrictive in which counterparties they lent to, the nancial crisis did not lead to a complete collapse of the Fed Funds market. The novelty of our approach is to look at the co-determination of banks capital holding and their interbank market participation. To the best of our knowledge this relationship has not been explicitly addressed both in the theoretical and in the empirical banking literature. The remainder of the paper is organized as follows. Section 2 presents the model. Section 3 analyzes the optimal risk-sharing allocation chosen by a social planner. Section 4 shows how the e cient allocation can be decentralized by the presence of interbank markets. Section 5 characterizes the e cient allocation and it analyzes how the participation in the interbank market a ects bank capital and short-term changes in bank capital. Section 6 presents the data we used to test the model s predictions and the results of our regressions. Sections 7 concludes. The Appendix contains the proofs. distress and the ability to force borrower repayments. Allen, Carletti and Marquez [2] analyze the role of market discipline as a rationale to hold bank capital. 6

2 The Model The basic model is similar to Gale [17], and provides a rationale for the use of bank capital based on risk sharing. Consider a three-date economy (t = 0; 1; 2) with a single good available at each date for both consumption and investment. There are two assets: a short-term or liquid asset that matures in one period with a return of one, and a long-term or illiquid asset that requires two periods to mature and delivers a return R > 1. The short asset represents a storage technology (one unit of the good invested at t = 0; 1 produces one unit at t + 1), while the long asset captures long-term productive opportunities (one unit invested at t = 0 produces R units at t = 2, and nothing at t = 1). Clearly, the choice of a portfolio of assets re ects a trade-o between returns and liquidity. There are two banks i = A; B in the economy, and two groups of agents. The rst group is a continuum of risk-neutral agents that we call investors. They are endowed with a large amount of the consumption good at t = 0 and nothing at t = 1; 2. Investors cannot consume a negative amount at any time, and their utility is 0 c 0 + 1 c 1 + c 2 ; where 0 > R, and 0 > 1 > 1: The second group is given by risk-averse agents that we call depositors. They are endowed with 1 unit of the consumption good at t = 0, and nothing at t = 1; 2. Following Diamond and Dybvig [10], depositors can be of two types, early consumers who only value consumption at t = 1, or late consumers who only value consumption at t = 2. The type of an agent is not known at t = 0. When consumption is valuable, the agent s utility is u(c), where u : R +! R is continuously di erentiable, strictly increasing and concave, and satis es the Inada condition lim c!0 u 0 (c) = 1. We assume that each bank has a unitary mass of depositors. The uncertainty about the preference shocks for the second group of agents is resolved in period 1 as follows. First, a liquidity shock is realized, which determines the fraction! i of early consumers in each bank i = A; B. Then, preference shocks are randomly assigned to the consumers in each bank so that! i agents become early consumers. The preference shock is privately observed by consumers, while the aggregate shocks! i are publicly observed. The bank shock! i takes the two values! H and! L, with! H >! L. We assume that with probability p > 1=2 the two banks have opposite shocks, that is, when a bank has 7

high liquidity needs (i.e.,! i =! H ), the other bank has low liquidity needs with probability p. In this case there is in principle room for trading on an interbank market to diversify away the liquidity shock. However, with probability 1 p both banks face high liquidity needs and the interbank market is of little help. The economy is therefore characterized by three possible states of the world S 2 S = fhh; LH; HLg. In state HH both banks have high liquidity needs, while in states LH and HL they are hit by di erent shocks. Table 1 summarizes the probability distribution of the liquidity shocks. Table 1: Banks liquidity shocks State S A B Probability HH! H! H (1 p) LH! L! H p=2 HL! H! L p=2 Notice that in states LH and HL, the average fraction of early consumers is constant and equal to! M =! H +! L ; 2 whereas it is clearly! H in state HH. Hence, there is some aggregate uncertainty on liquidity needs that is maximum when p = 1=2. 6 Notice that, as we assume p 1=2, any increase in p represents a reduction in aggregate uncertainty on liquidity needs. Agents cannot trade directly with one another, but the banking sector makes up for the missing markets. In particular, the activity of each bank develops as follows. At t = 0 each bank collects the initial endowment of its depositors and an amount e 0 of resources from investors. Therefore, the amount e will henceforth be referred to as bank capital. The bank invests an amount y in the short asset and an amount 1 + e y in the long asset; then, in period 1, after the aggregate shock S is publicly observed, the consumer reveals his preference shock to the bank and receives the consumption vector c S 1 ; 0 if he is an early consumer and the consumption vector 0; c S 2 if he is a late consumer. Similarly, 6 In fact, the aggregate liquidity shock can be measured by the average fraction of early consumers in the economy, and it can either be! M with probability p, or! H with probability 1 of this binary random variable is maximum when p = 1=2. p. Clearly, the variance 8

after the state S has been revealed, investors receive the consumption vector (d S 1 ; d S 2 ) 0: 7 Therefore, a risk sharing contract, also called an allocation, o ered by the bank is fully described by an array fy; e; c S t ; d S t g: S2S;t=1;2 As in Allen and Gale (2000), the existence of di erent groups of banks with di erent liquidity needs can capture di erent level of aggregation. Each bank in the model could indeed correspond to a speci c nancial institution, or to the representative bank in a speci c banking sector, a geographical region, etc. For our purposes, the economy represents a set of banks connected through an interbank market (to be explicitly introduced in section 4) together with their depositors and investors. In this sense, the parameter p represents a measure of the deepness of the interbank market, as it gives the probability of nding a bank with di erent liquidity needs to, potentially, trade with. In what follows we are interested in studying the e ects of the interbank market on the incentives to hold bank capital. Since our focus will be on an interbank market able to decentralize the rst-best allocation, we start in the next section to characterize optimal risk sharing and we will introduce the interbank market in section 4. 3 Optimal Risk Sharing In this section we abstract from the interbank market and consider the problem faced by a planner that chooses an allocation to maximize the sum of ex-ante expected utilities of depositors, maintaining investors at their reservation utility (i.e., the utility they can obtain by consuming their endowment at t = 0). The planner is unable to observe the preference shock of individual depositors but can observe banks aggregate liquidity shocks. Notice that total liquidity needs in the economy are the same in states HL and LH, and it is therefore optimal for the planner to move resources from one bank to the other to make the agents consumption plans constant in this case (i.e., c HL t t = 1; 2). = c LH t and d HL t = d LH t With a slight abuse of notation we can de ne a new state space S 0 = fh; Mg with the understanding that M = fhl; LHg and H = fhhg. An allocation can now be described 7 Agents are in a symmetric position ex-ante, and we assume that they are treated equally, that is, risk averse agents are all given the same contingent consumption plan, summarized by c S t similarly, risk neutral agents are all given the same contingent consumption plan d S t S2S;t=1;2. 9 for S2S;t=1;2 and,

by an array fy; e; fc s t; d s tg s2s 0 ;t=1;2 g, and it is said to be feasible if for each s 2 S0 and t = 1; 2, we have e 0; d s t 0; and! s c s 1 + d s 1 y; (1) (1! s )c s 2 + d s 2 (1 + e y)r + y! s c s 1 d s 1; (2) p( 1 d M 1 + d M 2 ) + (1 p)( 1 d H 1 + d H 2 ) 0 e: (3) The rst two constraints guarantee that there are enough resources at t = 1 and t = 2 respectively, to deliver the planned amount of consumption in each state s. Whenever y! s c s 1 d s 1 > 0 we say that there is positive rollover in state s, that is, some resources are stored through the liquid asset between t = 1 and t = 2. In this case the ex-post social value of liquidity is clearly the lowest possible as it exceeds the needs in the economy. The third constraint guarantees that investors get at least their reservation utility. 8 The planner s problem is therefore to choose a feasible allocation to maximize p! M u(c M 1 ) + (1! M )u(c M 2 ) + (1 p)! H u(c H 1 ) + (1! H )u(c H 2 ) : (4) Notice that in state H each banks s consumption needs must be satis ed with the resources available within the bank. In fact, in state H, both banks have a total demand for liquidity (from both consumers and investors) equal to! H c H 1 + d H 1 and from (1) we see that the available amount of the short asset within each bank is in fact enough to satisfy the internal demand (i.e., y! H c H 1 + d H 1 ). Things are di erent in state M: In this case in order to implement the rst best, the planner has to move resources between the two banks. For example, with no rollover in state M, the amount of liquid resources available at t = 1 in both banks is! M c M 1 + d M 1. However, one bank has a fraction! H of early consumers so that its demand for liquidity is! H c M 1 + d M 1, which results in an excess demand of (! H! M ) c M 1. At the same time, the other bank has a fraction! L of early consumers so that its demand for liquidity is only! L c M 1 + d M 1, which results in an excess supply of (! M! L ) c M 1. Given that (! H! M ) = (! M! L ) = (! H! L ) =2; 8 Notice that we are not explicitly considering the incentive contraints c s 1 c s 2 that prevent late consumers from pretending to be early consumers. This omission is however immaterial as the solution to the planner s problem automaticaly sati es such incentives constraints. This means that the rst-best allocation is also incentive e cient (see Proposition 1). 10

the excess demand can be cleared up with and excess supply at t = 1. At t = 2, resources move in the opposite direction in state M to clear up the bank excess demand and excess supply, while in states H each bank must satisfy its own demand with its own resources. 4 Interbank Deposit Market Consider now the decentralized economy in which each bank directly o ers a risk-sharing contract to its depositors and investors. We would like to know whether optimal risk sharing can also be achieved in this case. In the decentralized economy an allocation can only be achieved if it is feasible for each bank, separately. The rst-best consumption levels would not entail any feasibility problem in state H as, in this case, each bank demand for consumption is entirely satis ed using internal resources. 9 However, in state M both at t = 1 and t = 2, one bank has an excess demand for consumption while the other bank has an excess supply of exactly the same amount. One way to overcome this problem is to allow banks to exchange deposits at t = 0. To verify if this is feasible, assume that each bank o ers the rst-best allocation and deposits the amount! H! M with the other bank, under the same conditions applied to individual depositors. This means that when the fraction of early consumers in bank i is! H, bank i will behave as an early consumer and withdraw its interbank deposit at t = 1. In this case the bank obtains nothing at t = 2; while at t = 1 it gets (! H! M ) c M 1 if the fraction of early consumers in the other bank is! L (i.e., if the state is M), and (! H! M ) c H 1 otherwise (i.e., if the state is H). If the fraction of early consumers in bank i is! L, bank i will behave as a late consumer by holding its interbank deposit until t = 2, when it will nally withdraw it. In this case the bank obtains zero at t = 1 while at t = 2 it gets (! H! M ) c M 2 as the fraction of early consumers in the other bank is! H (i.e., the state is M for sure). We can now verify that the rst-best allocation is feasible in the decentralized economy 9 Notice that the rst-best allocation assigns a contingent consumption stream to the agents in each region. In state H both regions have a large fraction of early consumers but there is no liquidity shortage as the promised level of consumption in this case, c H 1, is the lowest possible (see proposition 1). We also allow for contingent consumption plans in the decentralized economy and we therefore abstract from problems of nancial distress and default. In any case, the state H represents a situation of economic distress at t = 1, with a strong pressure for immediate consumption, which however nds a frictionless (and e cient) solution in a reduction of per-capita consumption levels. 11

with interbank markets. To this end, notice that at t = 0 the net ow of funds between the two banks is zero so that the rst-best level of capital e and liquidity y are still compatible with the rst-best level of investment in the long asset given by 1 + e y. Thereafter, at t = 1 in state H the two banks withdraw their deposits at the same time so that the net ow of funds between banks is zero at both t = 1 and t = 2. First-best consumption levels are feasible within each bank in state H and will therefore remain so also in the presence of the interbank deposits market. In state M the two banks receive asymmetric liquidity shocks so that one bank will withdraw its interbank deposit at t = 1 (the bank with the high shock), while the other will withdraw at t = 2 (the bank with the low shock). For concreteness, let A be the bank with the high liquidity shock. In this case in both banks the amount of the short asset at t = 1 is y! M c M 1 + d M 1 but bank A needs! H c M 1 + d M 1 at t = 1 to cover its withdrawals and pay the promised amount to investors. Bank A redeems its interbank deposit at t = 1 and receives the amount (! H! M ) c M 1. Therefore it is able to satisfy its budget constraint:! H c M 1 + d M 1 =! M c M 1 + d M 1 + (! H! M ) c M 1 y + (! H! M ) c M 1 : Bank B faces withdrawals from both its depositors and from bank A, and pays d M 1 investors. Hence, the total amount of resources needed at t = 1 by bank B is to! L c M 1 + d M 1 + (! H! M ) c M 1 : However, it is also able to satisfy its budget constraint:! L c M 1 + d M 1 + (! H! M ) c M 1 =! M c M 1 + d M 1 y: Budget constraints are also satis ed at t = 2; and the case in which bank B receives the high liquidity shock is similar. Let m s 1 = (! H! M ) c s 1 and m s 2 = (! H! M ) c s 2 denote the amount that banks can withdraw in state s 2 fh; Mg at t = 1 and at t = 2, respectively. Table 2 below summarizes the net ow of funds between banks, as well as their net interbank positions, denoted by s t at time t and state s. A bank net position is positive when it is a net borrower (a debtor), and negative when it is a net lender (a creditor). 10 Notice that the interbank net position can only be di erent from zero at t = 1. Indeed, interbank deposits capture a market for liquidity at t = 1 and we will mainly refer to s 1 in what follows. 10 Notice that at t = 0 the two banks exchange exactly the same amount of resources and, therefore, the net interbank ows and positions are both equal to zero. 12

Table 2: Net interbank ows and positions State A B S S 0 ows s t=1 s 1 ows s t=2 s 2 ows s t=1 s 1 ows s t=2 s 2 HH H m H 1 m H 1 = 0 0 0 0 m H 1 m H 1 = 0 0 0 0 HL M m M 1 m M 1 m M 2 0 m M 1 m M 1 m M 2 0 LH M m M 1 m M 1 m M 2 0 m M 1 m M 1 m M 2 0 5 First-Best Allocation In this section we further characterize the rst-best allocation and we study how both bank capital and interbank deposit markets play a role in achieving the optimal risk sharing. In a nutshell, interbank markets can only work when bank liquidity needs are asymmetric, that is in state M, but are of little help when both banks are hit by the high liquidity shock. The existence of undiversi able liquidity uncertainty (i.e., the possibility of liquidity shocks that cannot be diversi ed away through the interbank market) creates a scope for bank capital. In fact, by raising bank capital part of this undiversi able risk can be transferred to the risk-neutral investors. We now have Proposition 1 Assume p < 1 and consider the rst-best allocation. We have c H 1 < c M 1 c M 2 < c H 2 : Moreover, d M 1 d H 1 = 0; d H 2 d M 2 = 0; and positive rollover either occurs in state M, in which case c M 1 = c M 2, or it never occurs, in which case c M 1 < c M 2. This result is proved in the appendix and clari es that as bank capital is costly, undiversi able uncertainty makes it impossible for banks to o er full insurance to risk-averse depositors. In particular, rst-period (second-period) consumption tends to decrease (increase) with the overall fraction of early consumers. Risk-neutral investors can bear the uncertainty more e ciently. Banks can partially transfer the undiversi able uncertainty to investors by collecting part of their resources at t = 0, in the form of bank capital, in exchange for a contingent payout at t = 1; 2. The optimal way of arranging this form of risk sharing is to avoid any bank capital remuneration (i.e., payout to investors) when the 13

marginal utility of depositors is high, that is, in state H at t = 1, and in state M at t = 2. In principle, banks could raise enough capital to completely insure depositors against the liquidity uncertainty, but this turns out to be suboptimal because the use of bank capital is costly, in the sense that investors have a strong preference for time-zero consumption. In fact, when c H 2 = c M 2, the marginal value of insurance is zero but the marginal cost of capital is positive, as investors incur a marginal cost 0 > R to postpone consumption to t = 2; and a marginal cost 0 = 1 > 1 to postpone consumption to t = 1. In any case, the cost of capital is higher than the returns of the available investment opportunities (see Allen and Gale [4]) and this makes the use of bank capital costly. To conclude this section notice that we cannot exclude that the rst-best level of capital is zero. This trivial case emerges for example if 0 is too large with respect to 1, and bank capital becomes too costly to be used for risk-sharing purposes. In what follows we therefore abstracts from this case. 5.1 Bank Capital The optimal amount of bank capital clearly depends on the scope of the interbank market as measured by p. Let us use the notation e(p) to make this relationship explicit. The parameter p can be interpreted in a variety of ways. (1) At the level of a single nancial institution, p re ects the degree of connectedness to the overall interbank network; (2) At the country level, p is a ected by the external position of the banking system; (3) at the level of the overall economy, it re ects the relative importance of regional (and diversi able) shocks versus systemic shocks. Intuitively, if p increases, the interbank market can be used more often to smooth the liquidity shocks and, as a consequence, the incentive to raise bank capital should be smaller. This intuition is indeed correct when we consider the extreme case of p = 1. In this case, an allocation can be simply thought of as an array (y; e; c M 1 ; c M 2 ; d M 1 ; d M 2 ), as whatever happens in state H has zero probability and is therefore irrelevant. In this case, the optimal allocation has e 0, d M t 0; and solves max! M u(c M 1 ) + (1! M )u(c M 2 ) (5) subject to! M c M 1 + d M 1 y; (6) (1! M )c M 2 + d M 2 (1 + e y)r + y! M c M 1 d M 1 ; (7) 1 d M 1 + d M 2 0 e: (8) 14

Notice that (6)-(8) must all bind at the solution, and it is possible to verify that the rst-order conditions imply e(r 0 )u 0 (c M 2 ) = 0: (9) Clearly, as 0 > R and u 0 (c M 2 ) > 0, (9) implies that e = 0. Hence, with no aggregate uncertainty, the interbank market is su cient to smooth away liquidity shocks, and there is no need for bank capital. A continuity argument now immediately implies Proposition 2 If p 0 > p and p 0 is su ciently close to one, whenever e(p) > 0 we also have e(p 0 ) < e(p). In other words, whenever there is some scope for bank capital for risk-sharing purposes, a substantial reduction in undiversi able uncertainty also reduces the optimal level of bank capital. Figure 1 shows a numerical example in which bank capital is decreasing for all values of p 1=2, not only for su ciently high values. The example assumes R = 1:8, 0 = 2, 1 = 1:75,! H = 0:6,! L = 0:4, and depositors have constant relative risk aversion = 2. From panel (a) we can see that bank capital over total asset is indeed decreasing for all values of p 1=2. Panel (b) shows that investors receive a payout at t = 2 in state H for any p 2 (1=2; 1), while a payout at t = 1 in state M is only realized when p is below approximately 0.68. [FIGURE 1] The negative relationship between the level of bank capital and p is not a general property of the model though. Indeed, Castiglionesi et al., [8] show that under general conditions a reduction in undiversi able liquidity uncertainty (i.e., here an increase in p beyond its minimum level of 1=2) can induce higher consumption volatility due to a reduction in the bank liquidity ratio. The same e ect shows up in this case and can induce banks to increase their amount of bank capital when p increases. Clearly, bank capital eventually decreases with p as it approaches one (i.e., as the overall liquidity uncertainty tends to vanish). Figure 2 shows a numerical example with R = 1:2, 0 = 1:35, 1 = 1:30;! H = 0:6,! L = 0:4, and in which depositors have constant relative risk aversion = 2. From panel 15

(a) we can see that bank capital is indeed slightly increasing until about p = 0:65 and decreasing thereafter. Panel (b) shows that the liquidity ratio, de ned as y=(1 + e), is everywhere decreasing in p, both when bank capital is optimally set to the levels shown in panel (a), and when it is forced to zero. Panels (c) and (d) show the volatility of rstperiod and, respectively, second-period consumption both with (left scale) and without (right scale) bank capital. [FIGURE 2] Notice that in the absence of bank capital, consumption volatilities are higher. This con rms that bank capital is used to partially insure depositors against liquidity uncertainty. Notice also that, in the absence of bank capital, the consumption volatility both in the rst and in the second period increases with p, for values of p below some threshold. This e ect depends on the reduced liquidity ratio documented in panel (b), and induces banks to increase their capital ratio to deal with the tendency toward an increased consumption volatility. In the example of Figure 2, anytime that the undiversi able liquidity uncertainty decreases (i.e., p increases) the use of (possibly increasing levels of) bank capital allows banks to reduce the volatility of consumption in the second period (but not always in the rst period). 5.2 Short-Term Capital Variation and Interbank Markets The relationship between bank capital and p is intuitive but di cult to study empirically because of the unobservability of p. What we do observe is a bank s activity in the interbank market at t = 1 which is captured by s 1, the net interbank position at t = 1. As we are mainly interested in the level of liquidity coinsurance provided by the interbank market, it does not matter whether s 1 is positive or negative (i.e., whether a bank is a net lender or borrower). Hence, we take its absolute value as a measure of interbank activity. In order to develop a testable prediction of the model we can consider what happens to the value of bank capital at t = 1, thought of as the value of (expected) future payouts to investors. We interpret the change in the value of capital at t = 1 as short term, as it materializes on the same time horizon as liquidity shocks. Notice that, after the observation of the state s at t = 1, the uncertainty about future payouts is completely resolved, and the 16

value of bank capital (in terms of t = 1 consumption) equals the expected payout at t = 2 divided by 1. In this sense, the state s determines the change in bank capital between t = 0 and t = 1, which is denoted by Cap s. Recall that the state s also determines banks net position in the interbank market (see Table 2). 11 Table 3 displays the t = 1 net positions in the interbank market in absolute value j s 1j together with the value of the changes in bank capital Cap. Both variables are a function of the state. It also reports the optimal payout policy to investors established in Proposition 1. Since the net position in the interbank market is in absolute values, the distinction between bank A and B is immaterial. Table 3: Change in bank capital and net interbank position State Cap s t=0 Cap s t=1 Cap s j s 1j H e d H 2 = 1 0 d H 2 = 1 e 0 M e d M 2 = 1 = 0 d M 2 = 1 e m M 1 > 0 We are interested in the relationship that the net position in the interbank market at t = 1 has with the value of bank capital at t = 1, and with its short-term variation between t = 0 and t = 1. Clearly, as Cap and Cap t=1 only di er by a constant, they have the same relationship with j 1 j. In fact, from Table 3 it is immediate to obtain E [Cap s t=1 j j 1 j > 0] E [Cap s t=1 j j 1 j = 0] = d H 2 = 1 ; E [Cap s j j 1 j > 0] E [Cap s j j 1 j = 0] = d H 2 = 1 : which are clearly both negative. These results can be summarized in the following Proposition 3 The net position in the interbank market at t = 1, measured by j 1 j, has a negative relationship with (i) the level of bank capital at t = 1, and (ii) with the short-time change in bank capital, measured by Cap. We now turn to the empirical section of the paper where we use the testable predictions contained in Proposition 3 to investigate whether the risk-sharing role of bank capital described in this section is indeed a relevant concern for banks. 11 Notice that the value of capital at t = 0 is expressed in terms of t = 0 consumption. To obtain its value in terms of consumption at t = 1 we should simply multiply it by 0 = 1, but this would clearly not alter in any ways the results in proposition 3. 17

6 Empirical Analysis 6.1 Data The interbank market is an Over the Counter market and its volume is not publicly available. A database that provides a proxy for the volume in this market is the quarterly Federal Financial Institutions Examination Council (FFIEC) Reports of Condition and Income (Call Reports), which all regulated commercial banks le with their primary regulator. We consider the Call Reports for banks with foreign o ces (FFIEC031) and for banks with domestic o ces only (FFIEC041). Call Reports contain detailed on-balance and o -balance-sheet information for all banks 12. Speci c to our study, we collect information on the amounts a given bank lends to, and borrows from other banks. This information comes from bank balance-sheets and has a quarterly frequency. This means that it does not allow to observe all the interbank ows throughout the quarter, nor it allows to distinguish interbank loans of di erent maturities 13, or the positions towards individual banks. Nonetheless, it gives a picture of the overall position a bank has vis-a-vis other banks at the time of the quarterly balance-sheet closure, that we take as a proxy of the interbank participation during the quarter. We build a quarterly panel dataset spanning from the rst quarter of 2005 to the third quarter of 2010 that includes 2954 commercial banks that report information about their interbank lending and borrowing positions. In order to measure a bank s activity on the interbank market we take the di erence of what a given bank lends to and borrows from other banks. We concentrate our analysis on three di erent types of borrowing: (a) Unsecured interbank lending and borrowing with more than one day maturity (Deposit from and Due to banks); (b) Securities purchased under agreements to resell and securities sold under agreements to repurchase, i.e., Repos and Reverse Repos with more than one day maturity; (c) Lending and borrowing on the one-day Federal Funds market 14 (Fed Funds sold and purchased). We take the absolute value of the di erence between lending and borrowing normalized by total assets as the empirical counterpart of j 1 j, that is, as a measure of activity in the interbank markets. We use the absolute value since we are rather interested in the 12 Call Report data are publicly available on the website of the Federal Reserve Bank of Chicago. 13 We could however distinguish between overnight versus longer maturities. 14 The accounting rules include also Overnight Repo in this cathegory. 18

overall bank s exposure to the interbank market than whether a bank is a net lender or a net borrower. As for capital, we consider a broad de nition that includes equity and reserves as well as subordinated debt and hybrid capital. Our model does not capture all the peculiarities and the di erent roles that bank capital may have. Instead, it focuses on its role as a bu er against liquidity shocks. For this reason any source of funding with a long maturity and no collateral could be considered as a good proxy for the capital variable included in our model. We consider bank capital in book values rather than in market values because of the liquidity bu er role bank capital plays in our model. We exclude banks from the sample that do not report interbank market exposure or capital. Proposition 3 (i) in Section 5 predicts a contemporaneous negative relationship (correlation) between a bank s activity in the interbank markets (henceforth referred to as interbank market participation) and bank capital. To test this relationship, we include a series of balance-sheets variables normalized by total assets as control variables. While the amount of liquid assets is endogenously determined in our model, we have no clear prediction on the contemporaneous relationship between interbank market participation and bank liquidity holdings. Bank liquidity holdings, which comprise cash and government securities and money deposited at the FED, however clearly a ects both the amount of bank capital and the interbank market exposure. Hence, we include it as a control variable in testing the relationship between the aforementioned variables. Earlier literature on bank regulation stressed the role of bank capital in increasing bank solvency. Hence, more capital leads to lower default risk. At the same time, the riskiness of the bank may also a ect its ability to borrow from the interbank market as shown by Afonso et al. [1]. Thus, to test the predicted substitution e ect between capital and interbank market exposure we need to control for bank risk characteristics. We consider two di erent variables: the amount of loans outstanding and risk weighted assets. The relationship between a bank s loan outstanding and its interbank market exposure in principle is ambiguous. A larger loan portfolio may result in a larger need to borrow in the interbank market. A larger loan portfolio may however render the borrower more risky, hence it may result in a lower supply of funds. As an additional control variable of bank risk we use the risk weighted assets reported by banks in the FFIEC report for determining capital requirements. To control for the impact of bank pro tability on the relation between capital and interbank market exposure we consider the return on assets (ROA). Pro tability clearly 19

a ects the need and the ability to participate in the interbank market, and at the same time the amount of bank capital a bank elects to hold. In our regression analysis, we also control for size of the bank that we measure with total assets. To capture the potential impact of the size of the interbank market that may a ect a bank s participation decision we also include as control variables two proxies of interbankmarket size. For each bank we calculate the total amount lent (borrowed) in the interbank market by other banks located in the same State and normalize this number by the total assets of these banks. The size of the interbank market could also be a ected by the amount of liquid assets held on the balance sheet of banks. For each bank we calculate the liquidity holdings by others banks located in the same State as the given bank and normalize it by total assets of these banks. Table 4 provides descriptive statistics for the main variables of interest and control variables we use in our empirical analysis. The sample exhibits considerable heterogeneity in the cross-section. We use two measures for a bank s interbank market activity. IabsTA1 is calculated as the absolute value of the di erence of unsecured interbank market lending and borrowing (Deposit from and Due to banks) normalized by total assets. IabsTA2 is de ned as the sum of unsecured interbank market lending and borrowing, Repos and FedFunds divided by total assets. The average interbank market exposure is 2.5% of total assets in our sample; this ratio is 5.93% if we also include Repo and FedFunds (IabsTA2), with a median of respectively 0.92% and 3.1%. However, the dispersion is rather signi cant. It ranges from 0.03% for the 5th percentile to 9.58% for the 95th percentile, and if we include Repos and FedFunds the dispersion is even larger (1.7% to 19.60%). The same applies to bank capital. On average bank capital is 10.99% of total assets but the standard deviation is 8.29%. [TABLE 4] Liquid assets, including cash and marketable securities, over total assets is on average 16.59%; bank balances at Federal Reserve Banks are on average 1.42%. Retail deposits and loans amount to 67.59% and 60.38% percent of total assets on average, respectively; risk weighted assets are 73.53%. The mean of the total book assets is US$ 5,59 billions and the median is US$ 549 million. The sample therefore includes both large, medium and small banks. The average 20

interbank market lending by the other banks in the same state is 1.11% (3.71% with Repo and FedFunds). The average interbank market borrowing by other banks in the same state is 0.58% (6.40% with Repo and FedFunds). The average liquidity holdings by banks located in the same state is 16.91% of total assets. Table 5 reports the pair-wise correlation between all the variables we consider. Correlations among the explanatory variables and control variables are not large, therefore we do not have to worry about collinearity in the analysis we perform. [TABLE 5] 6.2 Results The theoretical model presented in the previous section provides two testable implications. First, we predict a negative correlations between the level of capital banks choose to hold and their interbank market activity. Second, we predict a negative relationship between changes in bank capital and participation in the interbank market as banks delay remuneration to shareholders when the interbank market is less able to provide liquidity insurance as a consequence of banks facing common shocks. 6.2.1 First panel analysis: Interbank Markets We start by investigating the empirical relationship between a bank s interbank market activity and its level of capital. Our theoretical model does not predict any causality among these two variables but point to a negative relationship (Proposition 3 (i)). Therefore, our analysis is based on a regression panel approach that allows us to estimate the conditional correlation among these two variables. In fact, measuring a simple correlation among these variables would not be satisfactory as this correlation could be generated by other variables that contemporaneously a ect both interbank market participation and bank capital. In the model speci cation, interbank market participation is regressed on bank capital using the panel approach. We control for both rm xed e ects and time e ects so as to sweep out unobserved heterogeneity at the bank level as well as aggregate trends. The panel regression performed is: Y i;t = a + a i D i + a t D t + bcap i;t + cx + i;t, (10) 21