A New Perspective on Bank-Dependency: TheLiquidity Insurance Channel*

Viral Acharyaa, Heitor Almeidab, Filippo Ippolitoc, Ander Pérez-Orived

a New York University, CEPR & NBER, 44 West Fourth Street, New York, NY 10012-1126, USAb University of Illinois & NBER, 4037 BIF, 515 East Gregory Drive, Champaign, IL, 61820, USAc Universitat Pompeu Fabra, CEPR & BGSE, Ramon Trias Fargas, 25-27, 08005 Barcelona, Spaind Federal Reserve Board, 20th St. and Constitution Ave., Washington, DC 20551, USA

June 2016

Abstract

We provide a new perspective on bank-dependency —the (bank) "liquidity insurance channel" —

based on the predominance of large, high credit quality firms among credit line users. Our model

can match the pattern of bank dependency in the cross-section of firms, and predicts when shocks

to bank health will affect primarily low or high credit quality firms. Our framework can also explain

why credit line origination is more cyclical than loan origination. Overall, we uncover an important

link between bank health and economic activity through the provision and effectiveness of bank

credit lines to large, high credit quality firms.

Keywords: Liquidity management, cash holdings, liquidity risk, investment, lending channel, creditlines.JEL classification: G21, G31, G32, E22, E5.

* We thank seminar participants at Boston University, University of Amsterdam, Bank ofPortugal, Catolica University in Lisbon, BI Oslo, Central Bank of Norway, Nova School of Businessand Economics (Lisbon), Euro Financial Intermediation Theory Conference @ London BusinessSchool, Faculté des Hautes Etudes Commerciales (HEC) de l’Université de Lausanne (UNIL), the2015 Adam Smith Asset Pricing Conference, and the 2015 Econometric Society World Congress forhelpful comments. Filippo Ippolito acknowledges support from the Beatriu de Pinos post-doctoralprogramme grant. Financial support from the Banco de Espana - Excelencia en Educacion yInvestigacion en Economia Monetaria, Financiera y Bancaria grant is also gratefully acknowledgedby Filippo Ippolito and Ander Perez-Orive.

Disclosure Statement – Viral Acharya

New York - June 14, 2016

I have nothing to disclose.

Viral Acharya (New York University, CEPR & NBER)

Disclosure Statement – Heitor Almeida

Urbana-Champaign, Illinois - June 14, 2016

I have nothing to disclose.

Heitor Almeida (University of Illinois & NBER)

Disclosure Statement – Filippo Ippolito

Barcelona, Spain - June 14, 2016

I have nothing to disclose.

Filippo Ippolito (Universitat Pompeu Fabra & CEPR)

Disclosure Statement – Ander Perez-Orive

Washington, D.C. - June 14, 2016

I have nothing to disclose.

Ander Pérez-Orive (Federal Reserve Board)

1 Introduction

The corporate sector has exhibited a remarkable increase in its demand for liquidity in

recent decades. An important instrument through which corporations manage their liquid-

ity needs are credit lines provided by banks. We seek to understand in this paper what

determines financial intermediaries’ability and willingness to satisfy corporate demand for

liquidity through the provision of credit lines, and how the supply of liquidity through credit

lines varies over the business cycle and interacts with the provision of term loans.

Credit lines create an important and intriguing channel of bank-dependency for the cor-

porate sector. While bank-dependency is usually associated with low credit quality, bank

credit lines are more commonly used by large, profitable, high credit quality firms (Sufi

(2009), Acharya et al. (2014)). Small, financially constrained firms tend to rely on cash for

their liquidity management. Bank credit lines are distinct from standard term loans in that

they provide liquidity insurance, allowing firms to access bank financing in states of world

in which their financial performance deteriorates. As a result, firms that rely on credit lines

may not draw down on credit lines often,1 but at the same time credit line access can be very

important for them in bad states of the world. Since negative shocks to bank health affect

their ability to honor credit line drawdowns, high credit quality firms may also be affected

by such shocks. We call this link between bank health and the real economy that operates

through liquidity insurance provision the (bank) "liquidity insurance channel".

We provide a model of the liquidity insurance channel. The model considers both the

standard motivation for bank lending to financially weak firms, and a framework for liquidity

management. The bank lending model is based on Holmstrom and Tirole (1997), while the

liquidity insurance framework is similar to that in Holmstrom and Tirole (1998). The key

innovation is to consider both the decision to borrow from a bank or not, and the decision

of how to hold liquidity, simultaneously.

In the model, the benefit of bank lending is to increase pledgeable income through moni-

toring. Since monitoring is costly, firms borrow from a bank only when the marginal benefit

of increasing investment is high. Besides funding current investment, firms must also plan for

a future liquidity shortfall, by holding pre-committed financing (liquidity insurance). Using

cash to insure liquidity is costly because of a liquidity premium, while relying on credit lines

exposes the firm to the risk of credit line revocation.

1Acharya and al. (2014) provide evidence on the frequency of credit line drawdowns.

2

The key intuition of the model is that firms with high credit quality have both a lower

benefit of bank monitoring, and a low cost of credit line revocation. These firms are less

financially constrained and thus invest at higher levels. Under decreasing returns to scale,

the marginal benefit of using bank monitoring to increase investment is lower. At the same

time, these firms also have lower liquidity risk, either because their probability of facing a

liquidity shock is lower, or because the losses associated with not having suffi cient liquidity

are smaller. Thus, high credit quality firms rely on credit lines for liquidity insurance but

do not benefit from using bank debt for regular borrowing. Low credit quality firms rely on

bank debt, and are also more likely to choose cash for their liquidity management.2

We then use the model to study banks’optimal liquidity allocation between term loans

and credit lines. Term lending to low quality firms against their future cash flows increases

their investment. Saving bank liquidity instead increases banks’ability to honor credit line

drawdowns in adverse future states, which in turn reduces expected bankruptcy costs for

high credit quality firms. This analysis also allows us to predict when shocks to bank health

will affect primarily low or high credit quality firms. For example, if expected bankruptcy

costs are large, banks find it optimal to allocate liquidity to the provision of credit lines, so

that shocks to bank capital or liquidity primarily affect high credit quality firms through our

liquidity insurance channel. To our knowledge, this channel of bank-dependency is new to

the theoretical literature and not hitherto explored in empirical work.

Our third contribution is to analyze how the allocation of bank liquidity between loans

and credit lines changes with the state of the economy. Because term loans are used by

relatively weaker firms, when bad times hit the marginal value of allocating bank liquidity

to term loans increases relative to the marginal value of saving cash to honor future draw-

downs on credit lines. Aggregate evidence on the dynamics of the stocks of undrawn credit

lines and total business loans outstanding are consistent with this novel testable prediction.

We provide additional evidence consistent with the counter-cyclicality of the share of bank

lending that goes to term loans versus credit lines using firm-level data from Capital IQ.

Taken together, the results of our theoretical work and descriptive empirical analysis shed

light on how the choice of banks to allocate liquidity between term loans and credit lines

affects liquidity and investment by high quality and low quality (or large versus small) firms,

2If a firm relies on credit lines for liquidity management, there will be states of the world in which it willdraw on the credit line and use bank debt. But for high credit quality firms these states of the world do nothappen very often. Thus, at any point of time they will have less bank debt than lower credit quality firmsthat rely on banks for their regular financing.

3

and differentially so over the economic cycle. Our framework has the potential to assess

the macroeconomic consequences of regulatory initiatives such as bank capital requirements

or liquidity coverage ratios in a setting where there is heterogeneity among firms in the

corporate sector in the nature of their bank-dependency.

An important strand of the theoretical banking literature has focused on the role of banks

in improving resource allocation by creating pledgeable income through a reduction in pri-

vate benefits, improved project screening, and other mechanisms (Fama (1985), Houston and

James (1996) and Holmstrom and Tirole (1997)) and analyzed the macroeconomic implica-

tions of bank financial constraints that limit their ability to perform this role (Brunnermeier

and Sannikov (2014), Gertler and Kiyotaki (2011), Adrian and Shin (2013)). Another strand

has instead addressed the liquidity provision by banks to the corporate sector through credit

lines (Boot, Greenbaum, and Thakor (1993), Kashyap, Rajan and Stein (2002) and Gatev

and Strahan (2006)). Our theory contributes to these literatures by bringing together both

aspects of financial intermediation and considering firms’external financing and liquidity

management problems in the same framework, and also by exploring the macroeconomic

implications of banks’liquidity provision role.

Our work is also related to the empirical literature that shows that a deterioration of

the financial health of banks affects its bank dependent borrowers through a contraction in

credit supply (Kashyap and Stein (2000), Khwaja and Mian (2008), Paravisini (2008), and

Jimenez, Ongena, Peydró and Saurina (2012)). We contribute to this literature by showing

that the liquidity insurance channel is an additional reason why financial health of banks

may matter for corporate finance and the real economy.

Our work also builds on the growing empirical literature on the role of credit lines in cor-

porate finance (Sufi (2009), Yun (2009), Campello, Giambona, Graham, and Harvey (2011),

Acharya, Almeida and Campello (2013), and Acharya, Almeida Ippolito and Perez (2014)).

The most relevant for our paper are Sufi (2009), who shows that firms with low profitability

and high cash flow risk are less likely to use credit lines and more likely to use cash for

liquidity management because they face a greater risk of covenant violation and credit line

revocation, and Acharya, Almeida, Ippolito and Perez (2014), who show that credit line re-

vocation following negative profitability shocks can be an optimal way to incentivize firms to

not strategically increase liquidity risk, and also provides incentives for the bank monitoring

that can contain the illiquidity transformation problem. We contribute to this literature by

exploring the role of bank financial health and general economic conditions in determining

4

the ex-post access to credit lines and the ex-ante liquidity management policy of firms.

The paper is organized as follows. In Section 2 we introduce the evidence that guides

our theoretical work. In Section 3 we provide a model of the liquidity insurance channel,

which we use in Section 4 to show how it matches the cross sectional distribution of bank

dependency observed in the data, and in Section 5 to study the impact of bank health shocks.

Section 6 analyzes how the allocation of bank liquidity between loans and credit lines changes

with the state of the economy. Finally, Section 7 concludes.

2 Evidence

We start by documenting two empirical facts regarding bank dependence in the cross

section of firms and the business cycle behavior of the origination of loans and credit lines.

First, we provide evidence that in the cross section the distribution of credit lines and

loans is strongly correlated with the creditworthiness of firms. Existing research has shown

that bank credit lines are more commonly used by large, profitable, high credit quality firms

(Sufi, 2009, Acharya et al., 2014). Small, financially constrained firms tend to rely on cash

for their liquidity management. On the other hand, small firms, which mostly have limited

access to capital markets, are highly dependent on bank financing (Petersen and Rajan

(1994), Berger, Klapper and Udell (2001)).

We provide additional evidence using the Capital IQ database of firm-level balance sheet

data, which includes detailed information on firms’debt structure and access to credit lines.

Our database covers the Compustat universe of publicly listed firms between 2002 and 2011.

We regress several measures of bank loan dependence and access to credit lines on three

widely used proxies for firm financial constraints, and control for factors that have been

shown to be important in empirical studies of capital structure and cash holdings (Graham

and Leary (2011), Bates, Kahle, and Stulz (2009)). The results are in Figure 1. As proxies

for firm creditworthiness we follow the literature and use firm size, age, and access to bond

financing (Hadlock and Pierce (2010)). Bank dependence, measured as total bank debt over

either total assets or total debt, is significantly stronger in young and small firms with no

access to bond financing. Access to lines of credit for liquidity management, measured as

total undrawn credit lines over either total assets or total liquidity, is significantly greater for

older and larger firms with access to bond financing. Taken together, the evidence suggests

5

(1) (2) (3) (4) (5) (6)Bank

Debt/TotalAssets

BankDebt/Total

Assets

BankDebt/Total

Debt

UndrawnCredit/Total

Assets

UndrawnCredit/Total

Assets

UndrawnCredit/Total

Liquidity

Firm Size ­0.0039*** ­0.0028*** ­0.0463*** 0.0116*** 0.0098*** 0.0385***(­54.7326) (­19.9069) (­5.1725) (9.9308) (3.7228) (6.6585)

Bonds ­0.0038*** ­0.0124*** ­0.4140*** 0.0127*** 0.0086 0.0598***(­3.0885) (­4.9766) (­14.0243) (2.8071) (1.2304) (3.8261)

Age ­0.0020*** ­0.0041 0.0006 0.0034**(­10.5056) (­1.5654) (0.8522) (2.1545)

ControlsProfitability 0.0806*** 0.0649*** 0.3106*** 0.1696*** 0.1230*** 0.3871***

(251.1217) (20.6191) (3.9174) (14.3433) (4.4340) (6.9971)M/B ­0.0128*** ­0.0186*** ­0.0560*** ­0.0054*** ­0.0028 ­0.0453***

(­148.6553) (­25.0415) (­5.0134) (­4.3168) (­0.8991) (­7.0527)Leverage 0.3096*** 0.3544*** 0.1924*** 0.0203** 0.0086 0.1816***

(1,203.2120) (83.9387) (3.6870) (2.1849) (0.6128) (4.8064)Tangibility 0.1276*** 0.1268*** 0.2394*** 0.0170* 0.0558*** 0.3668***

(51.3606) (23.1241) (3.3459) (1.9037) (2.8956) (7.5247)Beta ­0.0075*** ­0.0304*** ­0.0054*** ­0.0278***

(­9.4408) (­4.0675) (­2.7974) (­7.2577)R&D/Sales ­0.0065*** ­0.0304*** ­0.0124* ­0.0294**

(­13.2639) (­3.1180) (­1.9566) (­2.0336)C­F Vol ­13.7843*** ­85.2914 ­11.6363 ­21.0408

(­71.5983) (­1.2823) (­0.7661) (­0.5464)

Year FE YES YES YES YES YES YESIndustry FE NO YES YES NO YES YESClustering(Firm)

YES YES YES YES YES YES

Observations 29,374 11,666 8,712 28,828 11,377 11,371

Figure 1: Bank Loan and Credit Line Dependence: This table presents Tobit regressionresults to study the relation between bank debt usage, credit line usage, and firm creditquality. The entire sample consists of non-utility (excluding SIC codes 4900-4949) andnon-financial (excluding SIC codes 6000-6999) U.S. firms covered by both Capital IQ andCompustat from 2002 to 2011. We have removed firm- years with 1) negative revenues, and2) negative or missing assets. All control variables are lagged. Robust standard errors arereported in parentheses. ***, **, and * denote statistical significance at the 1%, 5%, and10% levels, respectively.

6

2004 2005 2006 2007 2008 2009 2010 2011­40

­20

0

20

40

60

80

100

YEARS

Gro

wth

 rate

 (%)

Loan and Credit Line Growth (Aggregate, Compustat Universe)

Loan GrowthCredit Line Growth

2003 2004 2005 2006 2007 2008 2009 2010 201110

15

20

25

30

35Loan to Total Credit Ratio (Aggregate, Compustat Universe)

YEARS

Loan

s/To

tal C

redi

t (%

)

Figure 2: Time Series of Aggregate Bank Loan and Credit Line Originations: Thistable presents annual data on credit line and loan origination and stocks, by aggregatingfirm-level data from Capital IQ. The entire sample consists of non-utility (excluding SICcodes 4900-4949) and non-financial (excluding SIC codes 6000-6999) U.S. firms covered byboth Capital IQ and Compustat from 2002 to 2011. We have removed firm- years with 1)negative revenues, and 2) negative or missing assets.

7

that firms with low (high) credit quality rely on banks (markets) for their regular funding,

but rely on cash (credit lines) for liquidity insurance.

Second, we document that the share of bank lending that goes to term loans versus

credit lines varies with the state of the economy. Gilchrist and Zakrajsek (2012) and Bas-

sett, Chosak, Driscoll and Zakrajsek (2014) show using aggregated Call Report data of U.S.

commercial banks that unused credit lines contract strongly during the early stages of eco-

nomic downturns, unlike loans, which contract at a later stage. However, the sharp decrease

in unused credit lines can be driven by both lower issuance and higher drawdowns. In addi-

tion, since drawdowns become loans, an increase in drawdowns can generate the sustained

pattern of aggregate lending.

We again make use of our Capital IQ data to tease out loan issuance from drawdowns,

as the database distinguishes between term loans and drawdowns of existing credit lines.

We aggregate all data at the annual level, and plot in the top panel of Figure 2 the growth

rate of credit line and loan originations. Credit line originations in year t are calculated

as undrawn credit lines at the end of year t minus undrawn credit lines at the end of year

t− 1, plus all credit line drawdowns during year t, to adjust for decreases in undrawn creditdriven by drawdowns. Consistent with the aggregate evidence discussed before, credit line

originations drop strongly in the early stages of the recent financial crisis, while loans fall

later and less strongly. In the bottom panel, we plot the ratio of the stock of total loans

to total credit, defined as the sum of loans and drawn and undrawn credit lines. This ratio

increased substantially during the recent crisis, consistent with previous evidence.3

3 A model of bank lending and liquidity insurance pro-

vision

Here we describe a framework which incorporates liquidity management through banks,

and bank monitoring in the same framework.

3Le (2013) shows that banks that suffered liquidity shocks during the 2007-2009 financial crisis were morelikely to reduce their exposure to credit lines than to reduce their lending. To the extent that economicdownturns are associated with a weakening of banks’financial strength, this would be additional indirectevidence in support of the idea that in downturns banks shift the allocation of liquidity to the provision ofloans.

8

3.1 Firms

There is a measure 1 of firms, and each of them can invest I at date 0. Entrepreneurs’

date-0 wealth is A > 0. Investment produces a payoff equal to R(I) with probability p if it

is continued until the final date, where the function R(.) exhibits decreasing returns to scale

(R′> 0, R

′′< 0). With probability 1 − p, the project produces nothing. The probability

p depends on entrepreneurial effort. High effort produces a probability pH , while low effort

produces pL < pH but also produces private benefits BI.4

Given this set up, the entrepreneur will only put high effort if her share of the cash flow

(call it RE) is greater than a minimum amount:

pHRE ≥ pLRE +BI, (1)

so that the project’s pledgeable income is:

ρ0(I) = pH

[R(I)− BI

pH − pL

]. (2)

Firms can raise (I − A) at date-0 from a bank, or directly from individuals (call this

“market financing"). There is no discounting (the required rate of return is one).

The investment opportunity also requires an additional investment at date 1, which

represent the firms’liquidity need at date 1. The date 1 investment can be either equal to

ρI, with probability λ, or 0, with probability (1− λ). If the date-1 investment is not madethe project is liquidated (no partial liquidation). Liquidation at date-1 produces a payoff

equal to τI ≥ 0. We assume that it is effi cient to continue the investment in state λ:

pHR(I)− ρI > τI, (3)

in the relevant range for I. However, firms will not have suffi cient pledgeable income to

continue the project in state λ when:

ρ0(I) < ρI. (4)

In this case, to continue the project in state λ firms need to bring liquidity from the good

state of the world 1− λ. We denote the firm’s total demand for liquidity by l(I), for a given4Bank monitoring will reduce private benefits to b < B, at a cost ϕI, as we model below.

9

investment level I:

l(I) = ρI − ρ0(I). (5)

The firm can hold liquidity by buying treasury bonds at date-0 (e.g., cash), or by securing

a bank credit line. Credit lines require a firm to make a payment to the bank in state 1− λ,in exchange for the right to draw down in state λ. We describe the frictions associated with

these liquidity management choices below.

Firms are potentially heterogeneous with respect to their initial wealth A and liquidation

payoff τ . We introduce specific assumptions about the distribution of these variables below.

3.2 Banks

There is a single bank in this economy. The bank plays two distinct roles:

3.2.1 Monitored lending

First, the bank provides monitored financing to firms, as in Holmstrom and Tirole (1997).

By paying a cost ϕI, which is proportional to the size of the investment, the bank can reduce

managerial private benefits from BI to bI.5 Because monitoring is costly, the bank must

retain a stake Rb in the project:

pHRb − ϕI ≥ pLRb, or (6)

Rb ≥ϕI

pH − pL.

This constraint will generally bind, and thus the income that can be pledged to investors

other than the bank is now:

ρb0(I) = pH

[R(I)− (b+ ϕ)I

pH − pL

]. (7)

The bank is endowed with initial capital equal to K0, which it uses to make loans ib. The

bank’s ex-ante budget constraint for a given loan level ib is:

ib + ϕI ≤ pHRb =pHϕI

pH − pL. (8)

5This formulation follows Holmstrom and Tirole (1997), with the main difference being that we considerdecreasing returns to scale.

10

An increase in ib transfers the bank’s monitoring rent pHRb−ϕI to the firm, and allows thefirm to invest more. This link between ib and I is the channel through which bank capital

affects the equilibrium in the Holmstrom and Tirole (1997) framework. Notice that bank

lending cannot be higher than ( pHpH−pL − 1)ϕI. We denote this value by:

imaxb = (

pHpH − pL

− 1)ϕI (9)

3.2.2 Liquidity insurance provision

Second, the bank provides liquidity insurance to firms through credit lines. In order to

explain banks’advantage in providing insurance to firms, we put additional structure on the

distribution of liquidity shocks. At date-1, an aggregate state realizes, which determines the

probability that a firm faces a liquidity shock (which is also the fraction of firms that face

a liquidity shock). This probability is λθ in state θ, and λ1−θ < λθ otherwise. Thus, the

unconditional date-0 probability of a liquidity shock (λ) must obey:

λ = θλθ + (1− θ)λ1−θ. (10)

We assume that in the aggregate state θ the corporate sector is not self-suffi cient to

meet liquidity needs, because the aggregate demand for liquidity is greater than aggregate

pledgeable income:6

λθρI > ρ0(I), (11)

in the relevant range for I. Thus, one should think of state θ as a state with an aggregate

liquidity shock.

Banks’ advantage in providing credit lines arises from a contingent source of outside

liquidity in state θ, which it can use to honor credit line drawdowns. Let the amount

of contingent liquidity be denoted by D1. This outside liquidity can arise from the bank’s

deposit-taking activities, as in Kashyap, Rajan, and Stein (2002). The bank must hold excess

cash to honor deposit drawdowns, and can use this cash to honor credit line drawdowns

as well. Alternatively, contingent liquidity in a bad aggregate state can arise from the

mechanism in Gatev and Strahan (2006), who show that cash may flow into the banking

sector following negative aggregate shocks.7 Credit lines enable firms to access this contingent

6The reverse is true in state 1− θ.7Pennacchi (2006) highlights the central role that government guarantees and deposit insurance play in

11

liquidity.

Because there is limited liquidity in state θ, it is possible that the bank will revoke access

to credit lines in this state. For example, in a situation in which all firms use credit lines,

revocation is necessary when D1 < λθρI − ρ0(I). We denote the probability of credit line

revocation in state θ by µθ. This probability will be computed in equilibrium below.

Finally, in state (1 − θ) the bank does not require excess liquidity to meet credit line

drawdowns. In particular, this means that the probability of credit line revocation is always

zero (µ1−θ = 0).

3.2.3 Trade-off between liquidity insurance and lending

The values of µθ and ib will be determined in equilibrium, as we discuss below. Essentially,

the bank faces a trade-offbetween using capital to provide loans (increasing ib), or to support

credit line drawdowns (reducing µθ).

To increase ib, the bank can choose to borrow against future contingent liquidity D1, and

use the proceeds to make loans. Given the assumptions above, D1 can generate up to θD1

in date-0 cash. Borrowing against contingent liquidity increases µθ.

The bank can also carry date-0 cash into date-1 to support additional drawdowns. In this

case, the bank must also pay the liquidity premium q0. Thus, K0 can generate K0

q0in date-1

cash. Saving cash into date-1 requires the bank to reduce date-0 lending, ib. We explore how

the bank decides between lending and saving in the discussion of the equilibrium below.

3.3 Optimal decisions

The firm must decide how much to invest I, how much liquidity to hold, and whether to

use the bank for monitored lending and/or liquidity insurance provision.

We assume for now that there are no additional frictions in the bank’s optimization

problem, such that the bank implements the optimal solution. Given this, the bank allocates

capital K0 and liquidity D1 to loans and credit lines, in order to maximize the aggregate

welfare of all firms and the bank.

3.3.1 Feasible choices

Given the menu of choices that firms face, there are four distinct possibilities.

the role of banks as providers of liquidity simultaneously through deposits and credit lines.

12

Market financing and liquidity insurance through cash In this case, the firm does

not use the bank. As in Holmstrom and Tirole (1998), holding cash entails a liquidity

premium, which can be thought of as a date-0 price for treasury bonds which is greater

than one. Thus, if the price of treasury bonds is q0, we have q0 > 1. Given this, the firm’s

optimization problem is:

max pHR(I)− (1 + λρ)I − (q0 − 1)c(I) s.t. (12)

A+ ρ0(I) ≥ (1 + λρ)I + (q0 − 1)c(I)

The optimal investment level is given by:

pHR′(Icmax) = 1 + λρ+ (q0 − 1)c

′(Icmax), (13)

if the constraint does not bind at Icmax, and:

A+ ρ0(I′) = (1 + λρ)I

′+ (q0 − 1)c(I

′) (14)

if the constraint binds at Icmax. The firm’s payoff is:

U c = pHR(Ic)− (1 + λρ)Ic − (q0 − 1)c(Ic), (15)

where Ic is the optimal investment level. Clearly, the payoff U c decreases with the liquidity

premium in this case.

Market financing and liquidity insurance through credit lines In this case, the

firm uses the bank only for liquidity management. This means that the firm faces the risk

of credit line revocation in the negative aggregate state θ (µθ> 0). Given µθ, the ex-ante

probability of credit line revocation is θλθµθ, which we denote by µ to economize notation:

µ ≡ θλθµθ. (16)

We take this probability as given for now, it will be derived in equilibrium below.

13

The firm’s optimization problem is:

max(1− µ)pHR(I) + µτI − [1 + (λ− µ)ρ]I s.t. (17)

A+ (1− µ)ρ0(I) + µτI ≥ [1 + (λ− µ)ρ]I

As above, we define the optimal investment as ILC , and the associated payoff:

ULC = (1− µ)pHR(ILC) + µτILC − [1 + (λ− µ)ρ]ILC . (18)

The payoffULC decreases with the probability of credit line revocation µ given assumption

3, and it increases with the liquidation payoff τ .

Bank financing and liquidity insurance through cash As explained above, the so-

lution in this case depends on the amount that the bank allocates to monitored lending,

ib. We take this amount as given for now, and derive it in equilibrium below. The firm’s

maximization problem is:

max pHR(I)− (1 + λρ)I − (q0 − 1)c(I)−pH

pH − pLϕI + ib s.t. (19)

A+ ρb0(I) + ib ≥ (1 + λρ)I + (q0 − 1)c(I),

where ρb0(I) is defined above in equation 7.

We define the optimal firm investment by Icb , and the associated payoff by:

U cb = pHR(I

cb )− (1 + λρ)Icb − (q0 − 1)c(Icb )−

pHpH − pL

ϕIcb + ib. (20)

The investment level Icb and the payoffUcb both increase with ib, up to the maximum possible

level imaxb which is defined in equation 9.

The payoff to the bank in this case is equal to:

U cbank =

pHϕI

pH − pL− ib − ϕI ≥ 0.

This payoff is strictly positive whenever bank loan supply is constrained and ib < imaxb .

Bank financing and liquidity insurance through credit lines If the firm uses bank

monitoring and the credit line, the solution will depend both on the amount that the bank

14

allocates to monitored lending (ib), and the probability of credit line revocation µ. The firm’s

maximization problem is:

max(1− µ)pHR(I) + µτI − [1 + (λ− µ)ρ]I − (1− µ) pHpH − pL

ϕI + ib s.t. (21)

A+ (1− µ)ρb0(I) + µτI + ib ≥ [1 + (λ− µ)ρ]I

We define the optimal firm investment by ILCb , and the associated payoff by:

ULCb = (1− µ)pHR(ILCb ) + µτILCb − [1 + (λ− µ)ρ]ILCb − (1− µ) pH

pH − pLϕILCb + ib. (22)

The payoff to the bank in this case is equal to:

ULCbank = (1− µ)

pHϕILCb

pH − pL− ib − ϕILCb ≥ 0.

This payoff is strictly positive whenever bank loan supply is constrained and ib < imaxb , where

imaxb in the case of liquidity insurance through credit lines is given by:

imaxb =

((1− µ)pHpH − pL

− 1)ϕI. (23)

3.4 Solution

Firms choose the solution that maximizes their payoff, given their characteristics and the

key variables q0 (liquidity premium), ib (the amount of funds that they can borrow from

the bank), and µ (the probability of credit line revocation). Crucially, these three variables

also depend on firms’actions and thus must be determined in equilibrium. For example,

if the bank allocates additional funds to meet credit line drawdowns and reduce µ, ib may

decrease.

In order to characterize the solution, we proceed as follows. First, we discuss some

intuitive properties of the solution, taking the bank variables µ and ib as given. Second,

we provide a definition of the equilibrium in our framework. Third, we show how the bank

will choose µ and ib to implement equilibria. Finally, we provide a numerical solution of a

benchmark example that matches the empirical distribution of the usage of credit lines and

bank financing.

15

3.4.1 Properties of the solution

Consider first the choice of whether to use bank monitoring. As in Holmstrom and Tirole

(1997), the main benefit of using bank monitoring is to increase the firm’s pledgeable income.

Consider for example equations 15 and 20. By choosing bank financing, the firm reduces its

payoff for a given investment level, by an amount equal to pHpH−pLϕI

cb− ib, which is positive by

equation 8. Thus, the firm will only choose bank monitoring if it leads to a higher investment

level (Icb must be higher than Ic). It follows then that the benefit of using bank monitoring

will be higher when financial constraints are tight and the investment level is low. Since

there are decreasing returns to scale, highly constrained firms (for example, firms with very

low A) will benefit the most from bank monitoring.

Consider now the choice between cash and credit lines. The obvious trade-off is between

the cost of credit line revocation, and the liquidity premium. Take the difference between

the payoffs in 18 and 15 to obtain:

ULC − U c = pH[R(ILC)−R(Ic)

]− (1 + λρ)(ILC − Ic) + (q − 1)c(Ic)− (24)

−µ[pHR(ILC)− (ρ+ τ)ILC ].

The credit line reduces cash holdings, and thus avoids the liquidity premium (q−1)c(IC).However, the credit line also introduces a risk of credit line revocation which is associated

with value loss µ[pHR(ILC)− (ρ+ τ)ILC ], which is positive by assumption 3. In particular,notice that the value loss with liquidation µ[pHR(ILC)− (ρ+ τ)ILC ] is decreasing with the

parameter τ . Firms with high τ have lower liquidity risk, since they lose less value conditional

on credit line revocation.8

When firms are financially constrained, increases in A also reduce the loss of value con-

ditional on credit line revocation. Since the production function is decreasing returns to

scale, an increase in A that raises the firm’s ability to invest lowers the marginal return to

investment. The benefits of liquidation (ρ+ τ)ILC increase linearly with investment, which

means that the loss of value µ[pHR(ILC) − (ρ + τ)ILC ] decreases with A when firms are

financially constrained. In sum, high τ and high A both increase the incentive to choose

credit lines for liquidity management.

8A similar result is derived by Acharya et al. (2014).

16

3.4.2 Equilibrium definition

An equilibrium can be defined as follows:

• Firms pick the highest payoff among U c, ULC , U cb and U

LCb , given their type (A, τ),

and given µ = µ∗ and ib = i∗b .

• The bank’s optimization problem results in µ = µ∗ and ib = i∗b , given bank endowments

K0 and D1, and given optimal choices by all firms in the economy.

• The market for treasury bonds clears at price q∗0, that is, the demand for treasurybonds at q∗0 is less or equal to the supply.

To simplify the problem, we will generally assume a constant liquidity premium q∗0 > 1,

and a supply of treasury bonds that is large enough so that all firms that demand cash can

access bonds at a price q∗0.

Since there are no frictions for now on the bank’s optimization problem, one can charac-

terize this problem as choosing the best possible allocation of bank endowments to loan and

credit line provision, given firms’expected reactions to changes in µ and ib. Effectively, the

bank plays the role of "social planner" in this version of the model.

3.4.3 Bank’s optimization problem

In order to better understand the bank’s problem, it is useful to work with a specific

assumption for the distribution of firm types.

Take for example the simplest case in which all firms are identical (same A and τ), and

thus make the same choice of financing or liquidity management in equilibrium. The main

goal of this example is to illustrate how the equilibrium is determined in the model.

Suppose that A is low enough, so that it is effi cient to use monitored lending as we

explained above (U c and ULC are low). We must still determine whether the bank will also

provide credit lines in equilibrium or not (whether ULCb is higher than U c

b ). are low. We now

show what is the trade-off that drives this choice.

If initial capital K0 < imaxb , then the bank can increase U c

b by borrowing against its date-1

contingent liquidity, and making additional loans at date-0. If we denote bank borrowing by

W0, we have:

ib(W0) = 2 (K0 +W0) , (25)

17

for W0 ≤ θD1.

The trade-off is that moving funds to date-0 will increase the probability of credit line

revocation in state θ. If firms choose to use credit lines, the total demand for drawdowns by

firms that face a liquidity shock is λθ(ρI − ρ0(I)), which by assumption 11 is larger than the

total pledgeable income generated by firms that do not face a liquidity shock, (1− λθ)ρ0(I).

The bank also has external liquidity equal to D1 − W0

θ. If external liquidity is not enough,

the bank will need to revoke access to credit lines with positive probability, such that:9

[1− µθ(W0)]λθ(ρI − ρ0(I)) = (1− λθ)ρ0(I) +D1 −W0

θ(26)

and thus µθ increases with W0. The ex-ante probability of revocation is then µ(W0) =

θλθµθ(W0).10

The bank picks W0 to maximize the aggregate payoff of firms and the bank. U cb + U c

bank

is maximized by making W0 as high as possible, W0 = θD1. Let W ∗0 be the value that

maximizes ULCb + ULC

bank. If

U cb (W0 = θD1) + U c

bank(W0 = θD1) > ULCb (W ∗

0 ) + ULCbank(W

∗0 ),

then the optimal bank action is to borrow against future liquidity up to the maximum

amount θD1, and make loans at date-0. If in contrast

ULCb (W ∗

0 ) + ULCbank(W

∗0 ) > UC

b (W0 = θD1) + U cbank(W0 = θD1),

then the bank borrows up to W ∗0 , and firms use the bank both for lending and for liquidity

insurance provision.

4 Cross Sectional Distribution of Bank Dependency

We now introduce an example that characterizes an equilibrium with firm heterogeneity.

Our goal is to show how the model can deliver an equilibrium that is consistent with the

empirical fact discussed in the introduction. Profitable firms, firms with low cash flow vari-

9We assume that when the bank does not have enough liquidity to satisfy all credit line drawdowns, thereis a random sequential servicing so that some firms are fully revoked and the rest get access to their entirecredit line.10It can also save additional funds into date-1, but since the bank pays the same liquidity premium that

firms do (q∗0), this action is equivalent to increasing cash holdings at the firm level.

18

ance and high credit ratings are more likely to use credit lines for their liquidity management

(Sufi, 2009). But firms with high credit-quality (high credit rating, profitable firms) are less

likely to be bank-dependent for their regular financing. In our model, this would correspond

to an equilibrium in which low credit quality firms choose UCb , and higher credit quality firms

choose ULC .

The model is able to generate such an equilibrium under fairly general conditions. The

only requirement is that the two sources of firm heterogeneity are not strongly correlated in a

particular way. Specifically, firms with low net worth (low A) must not also have significantly

lower liquidity risk (high τ). As discussed above in Section 3.4.1, low A increases the benefit

of bank monitoring by raising the marginal productivity of investment. Low A also increases

the value of cash by raising the marginal product of capital and the bankruptcy costs of

foregone output. And high liquidity risk (low τ) increases the value of cash by mitigating

the risk of credit line revocation.

The empirical evidence is consistent with a positive correlation between A and τ . Using

the Compustat-Capital IQ sample of publicly listed U.S. firms for the period 2002-2011, we

find that firm size and cash-flow volatility, proxies respectively for A and τ , are positively

correlated. The average volatility of quarterly cash-flows, measured as a share of total assets,

is 5.8% for firms in the lowest quartile of firm size and 1.2% for firms in the top quartile.11

Another proxy for τ is the likelihood that the firm needs to draw on its liquid reserves.

To capture this, we define a liquidity event as a year in which firm profitability is negative

and the sum of decreases in cash holdings and increases drawn credit lines is positive. The

likelihood of a liquidity event is 23.1% for firms in the lowest quartile of firm size, and only

1.3% for firms in the top quartile of firm assets. Other studies have suggested that large

firms are on average more diversified (Rajan and Zingales (1995), Zhang (2006)), and have

a lower likelihood of bankruptcy (Griffi n and Lemmon (2002)).

To show this result, consider an example in which half of the firms have high A and high

τ , and the other half has low A and low τ . We want to characterize an equilibrium in which

the first type of firm (high) chooses ULC , and the second type (low) chooses UCb . Let I

LChigh

and ULChigh represent the investment and payoff of the first type of firm, and I

cb,low and U

cb,low

represent the corresponding variables for the second type.

Given these choices, we can characterize the demand for loans and credit line drawdowns

11Cash-flow volatility is calculated as the standard deviation of operating income (Compustat item 13)over the previous 12 quarters, scaled by total assets (6).

19

from firms. The aggregate demand for date-0 loans is:

ib(Icb,low) = (1 + λρ)Icb,low + (q0 − 1)c(Icb,low)− A− ρb0(Icb,low). (27)

As long as ib(Icb,low) < imaxb (Icb,low), ib is an increasing function of I

cb,low. The bank can then

increase Icb,low by allocating funds to make date-0 loans:

1

2ib(I

cb,low) ≤ K0 +W0, (28)

for W0 ≤ θD1. This equation shows that Icb,low (and thus Ucb,low) increases with W0.

Alternatively, the bank can reduce W0 to support credit line drawdowns:

1

2[1− µθ(W0)]λθ(ρI

LChigh − ρ0(I

LChigh)) =

1

2(1− λθ)ρ0(I

LChigh) +D1 −

W0

θ. (29)

This equation implies that µθ(W0) increases with W0, and thus ULChigh decreases with W0. It

follows that increasingW0 will increase the payoff for low quality firms, but reduce the payoff

for high quality ones. The bank will choose the optimal W ∗0 such that:

W ∗0=argmax

[1

2

(U cb,low(W

∗0 ) + U c

bank,low(W∗0 ))+1

2ULChigh(W

∗0 )

]. (30)

Finally, for this situation to be an equilibrium, firms must not have an incentive to change

their optimal choices:

U cb,low(W

∗0 ) = max{U c

b,low(W∗0 ), U

clow(W

∗0 ), U

LClow(W

∗0 ), U

LCb,low(W

∗0 )}, (31)

ULChigh(W

∗0 ) = max{U c

b,high(W∗0 ), U

chigh(W

∗0 ), U

LChigh(W

∗0 ), U

LCb,high(W

∗0 )}. (32)

4.1 Excess capital

Consider first an example in which there is suffi cient capital K0 in the economy to satisfy

the maximum demand for date-0 loans:

1

2imaxb (Icb,low) ≤ K0. (33)

In this case the bank’s decision is simple because there is no benefit of borrowing against

future liquidity and because there is no benefit of saving additional cash in the bank. This

20

means that W ∗0 = 0, minimizing the probability of credit line revocation.

Given this result, it suffi ces to show that U cb,low(W

∗0 = 0) is the highest payoff for low

quality firms, and ULChigh(W

∗0 = 0) is the highest possible payoff for high quality firms. We do

this in the numerical example below.

4.2 Limited capital

Suppose now that capital is limited, such that:

1

2imaxb (Icb,low) > K0 (34)

Now, the bank must choose the optimal level of W ∗0 . As we characterized above, the trade-

off is that increasing W0 increases the probability of revocation (hurting high quality firms),

but relaxes financial constraints for low quality firms (increasing Icb,low). As equation 30

suggests, the bank’s choice will depend on the marginal impact of W0 on the payoffs of high-

and low-quality firms. We characterize the solution below.

4.3 Numerical analysis

Assume that the production function is represented by:

R(I) = kIα,

where k = 8. We also assume that pH = 0.8, pL = 0.4, B = 1.8, and ρ = 1.4.12 The date-0

price of cash is q0 = 1.05. The high credit-quality firm has high net worth (Ahigh = 4),

and high liquidation value (τhigh = 0.3). The low credit-quality firm has Alow = 0, and

τ low = 0. Our assumptions about the states are: θ = 0.1, λθ = 1, and λ1−θ = 0.20. The

bank parameters are as follows. The cost of monitoring ϕ is 0.4, and b = 1.

4.3.1 Excess capital

We assume initially that capital is large so that date-0 loans are not constrained. Under

our benchmark calibration this requires that

K0 ≥1

2imaxb (Icb,low) = 0.731, (35)

12This calibration exercise is only for illustrative purposes. We do not seek to match real world quantities.

21

Liquidity Management Financing Ilow Ulow

cash bank 3.65 7.31credit line bank 3.68 7.17cash market 1.86 6.70credit line market 1.86 6.54

Table I: Optimal Investment Levels and Payoffs of Different Liquidity Management andFinancing Alternatives for Low Credit Quality Firms

so we set K0 = 0.8. Finally, D1 = 1.5.

We first compute U cb,low, the payoff for low quality firms, given that capital is large (so

that ib = imaxb ). Under our calibration, U c

b,low = 7.31 and Icb,low = 3.65. Table I shows that

U cb,low is the highest possible payoff for low credit quality firms.

13

Firms are acting optimally by choosing to manage liquidity through cash and access

external financing through banks. Accessing market financing reduces significantly their

ability to pledge output, obtain external finance, and invest. Managing liquidity through

lines of credit exposes them to costly liquidation and low expected returns to investment due

to their low liquidation value.

Turning to high quality firms, ILChigh and µθ depend on each other so must be determined

jointly through the equations:

1

2[1− µθ]λθ(ρILChigh − ρ0(I

LChigh)) =

1

2(1− λθ)ρ0(I

LChigh) +D1 −

W0

q0

(36)

µ ≡ θλθµθ (37)

and the firm’s optimization problem in 3.3.1.14 It is either the first-order condition if the

budget constraint doesn’t bind:

(1− µ)pHR′(ILChigh) + µτhighI

LChigh − [1 + (λ− µ)ρ] = 0.

or:

A+ (1− µ)ρ0(ILChigh) + µτILChigh = [1 + (λ− µ)ρ]ILChigh

13For this exercise we assume that the probability of revocation is at its equilibrium level of µ∗ = 2.46%.14Note that in equation (36) W0 is not divided by θ, unlike in equation (26), because when the bank

saves liquidity between date-0 and date-1 it does not do so in a state-contingent way. Also, the bank pays aliquidity premium q0 when saving cash, just like firms.

22

Liquidity Management Financing Ihigh Uhigh

credit line market 3.58 8.59cash market 3.47 8.47credit line bank 6.04 7.79cash bank 5.75 7.68

Table II: Optimal Investment Levels and Payoffs of Different Liquidity Management andFinancing Alternatives for High Credit Quality Firms

if the constraint binds. These two equations determine µ∗ and ILChigh, and thus ULChigh.

Under our benchmark calibration, we get that the date-0 probability of revocation in

date-1 is µ∗ = 2.46%, and that ILChigh = 3.58 and ULChigh = 8.59. Table II shows that U

LChigh is

the highest possible payoff for high credit quality firms when µ∗ = 2.46%.

High quality firms are acting optimally by choosing to manage liquidity through credit

lines and access external financing through arms length finance. Accessing bank financing

increases their ability to pledge output, obtain external finance, and invest, but at a very

high cost in terms of monitoring costs, which is not optimal given their lower marginal

return to investment. Managing liquidity through lines of credit exposes them to the risk of

revocation, but their costs of liquidation are small and do not justify incurring costly cash

accumulation to prevent liquidation.

We provide an analysis of firm optimal liquidity management and financing for a range

of values for A, τ . The results are in Figure 3. Low credit quality firms (Alow = 0, τ low = 0)

choose cash for liquidity management and bank financing. If only A increases (vertical

axis), firms first move away from bank monitoring but still use cash. As A grows enough,

firms also move out of cash for liquidity management and into credit lines. Increasing τ ,

on the other hand, provides the incentive for firms to switch into credit lines. Higher A

also increases the incentives to shift into credit lines. This is because the overall costs of

liquidation (which include the foregone date-2 output) are lower on the margin for high net

worth firms whose investment scale is larger. These firms are thus less worried about facing

a positive probability of bankruptcy in state θ. So high A firms are relatively more likely to

switch to credit lines when tau increases.

23

Optimal Firm Financing Choices

Liquidation Value: τ

Net

 Wor

th (A

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.5

1

1.5

2

2.5

3

3.5

Credit Line ­ Bank Financing

Cash ­ Bank Financing

Credit Line ­ Market Financing

Cash ­ MarketFinancing

Figure 3: Optimal liquidity management and financing for a range of values for A, τ .

4.3.2 Limited capital

We now analyze the case in which capital K0 is limited. Specifically, we assume that

K0 = 0.5. The rest of the parameters are calibrated as in the previous section. The relevant

range for W ∗0 is between 0 and the value that causes i

∗b to reach i

maxb . Call this maximum

value Wmax0 , its value is

Wmax0 = 0.5imax

b −K0 = 0.223.

Under our current calibration, Wmax0 < θD1 = 0.25, which means it is feasible. We will

evaluate aggregate welfare in the range [0,Wmax0 ]. We compute the payoffs of low quality

firms, high quality firms, and the bank, respectively U cb,low, U

LChigh and U

cbank. For each value

of W0, we restrict lending to low quality firms to be:

i∗b = 2 (K0 +W ∗0 ) .

We proceed to solve the equilibrium value for ILChigh and µθ jointly as in the previous section,

and also check that ULChigh and U

cb,low are the highest possible payoffs for the levels of µ

∗ and

i∗b for all values of W∗0 considered.

The results of this exercise are plotted in Figure 4, where in the horizontal axes of the

24

0 0.1 0.27.7

7.8

7.9

8Aggregate Welfare (date­1)

0 0.1 0.20

50

100Prob of Revocation Date­1 State θ (µ

θ)

%0 0.1 0.2

8.2

8.4

8.6

8.8Welfare ­ High Credit Quality

0 0.1 0.27.1

7.2

7.3Welfare ­ Low Credit Quality + Bank

0 0.1 0.23.5

3.6

3.7

3.8Investment ­ High Credit Quality

W0

0 0.1 0.23

3.5

4Investment ­ Low Credit Quality

W0

Figure 4: Optimal Bank Choice of Liquidity Provision when Bank Capital is Limited

six panels we plot the range of values of W ∗0 considered. For values close to W0 = 0, there is

enough liquidity in state θ in date-1to meet all credit line drawdowns, and the probability

of revocation µθ = 0. Increasing borrowing at that point is welfare improving as it increases

date-0 lending to low quality firms without affecting liquidity provision in date-1. Beyond a

certain point, increasingW0 can only be done at the cost of increasing µθ.15 The bank trades

off an increase in investment by all low quality firms, against a decrease in the probability of

revocation by high quality firms in the bad aggregate state. The latter turns out to be more

important under our current calibration because bankruptcy costs, which include foregone

15Increases in µθ are associated with increases in investment by high credit quality firms because pledgeableoutput increases due to higher expected revenues from liquidation and lower liquidity costs.

25

0 0.1 0.27.94

7.95

7.96Aggregate Welfare (date­1)

0 0.1 0.20

50

100Prob of Revocation Date­1 State θ (µ

θ)

%0 0.1 0.2

8.65

8.7

8.75Welfare ­ High Credit Quality

0 0.1 0.27.1

7.2

7.3Welfare ­ Low Credit Quality + Bank

0 0.1 0.23.4

3.6

3.8

4Investment ­ High Credit Quality

W0

0 0.1 0.23

3.5

4Investment ­ Low Credit Quality

W0

Figure 5: Optimal Bank Choice of Liquidity Provision when Bank Capital is Limited - Casewith High Liquidation Value (τhigh) for High Credit Quality Firms

output and capital destruction, are large. As a result, the bank finds it optimal to limit

borrowing to the maximum amount consistent with µθ = 0, which is W∗0 = 0.0480.

If bankruptcy costs for high credit quality firms were lower, the bank might have an

incentive instead to borrow against its date-1 liquidity at the expense of increasing credit

line revocations. Consider a case in which τhigh = 1.35. Figure 5 displays the implications of

different choices for W0 in this case, and shows how the bank will find it optimal to borrow

as much as possible from date-1, resulting in µθ = 1 and W∗0 = 0.224.

26

0.5 1 1.5 2 2.52

3

4

5Investment by Low Quality Firms

0.5 1 1.5 2 2.53.2

3.4

3.6

3.8Investment by High Quality Firms

0.5 1 1.5 2 2.56

7

8

9Payoff ­ Low Quality Firms

0.5 1 1.5 2 2.58

8.5

9Payoff ­ High Quality Firms

0.5 1 1.5 2 2.57.7

7.8

7.9

8Aggregate Welfare

D1

0.5 1 1.5 2 2.50

50

100

Prob of Revocation Date­1 State θ (µθ)

%

D1

Inv­LC/MarketInv­Cash/MarketInv­Optimal

Payoff ­ LC/MarketPayoff ­ Cash/Market

Effect of a Shock to Contingent Liquidity D1 in the Case of Excess Bank Capital in date-1

5 Impact of Bank Health Shocks

The comparative statics in the case of excess capital (i∗b = imaxb ) are straightforward.

Since banks already allocate the maximum possible amount to date-0 loans, shocks to bank

health affect only credit line provision. In Figure 4.3.2 we plot the effect of a decrease in D1

from its benchmark value of 2.5 down to a value of 0. We observe the following results. A

decrease in D1 increases the probability of credit line revocation, conditional on a negative

aggregate shock (µ∗θ). This result follows directly from equation 29. Notice that in this

case, W0 = 0. In the model, the ex-post impact of the revocation of credit lines is absorbed

27

0.5 1 1.5 23.4

3.5

3.6

3.7Investment by Low Quality Firms

0.5 1 1.5 22

3

4

5Investment by High Quality Firms

0.5 1 1.5 2D1

6.5

7

7.5

%

Payoff ­ Low Quality Firms

Payoff ­ Cash/MarketPayoff ­ LC/Market

0.5 1 1.5 2D1

7

8

9

10Payoff ­ High Quality Firms

Figure 6: Effect of a Shock to Contingent Liquidity D1 in the Case of Limited Bank Capitalin date-1

entirely by continuation investment, which is ρILChigh. This effect holds for high-quality firms

that suffer a liquidity shock in the negative aggregate state θ. Thus, we have that a decrease

in D1 reduces investment by firms that rely on credit lines for liquidity management (high-

quality firms), conditional on a negative aggregate shock, and conditional on a firm-level

liquidity shock. The model also has implications for how shocks to bank health affect ex-

ante liquidity management. Starting from an equilibrium in which high quality firms rely

on credit lines, a decrease in D1 may cause these firms to switch from credit lines to cash for

their liquidity management, which under our calibration happens when D1 falls below 0.30.

This has negative consequences for investment by high quality firms because holding cash is

costly. Investment falls from 3.7 to 3.4 as firms switch to cash.

If there is limited capital (i∗b < imaxb ), then shocks to K0 and D1 have similar implications

28

because the bank optimally allocates funds to loans and credit lines. There are two possible

cases. If W ∗0 = Wmax

0 (which occurs when τhigh is suffi ciently high), then we are in a corner

solution in which the bank allocates all funds to make loans. Starting from such a situation, a

decrease in bank health (lower K0 or D1) reduces bank loans, and investment by low quality

firms. This case is displayed in Figure 6. High quality firms are not affected since the bank

does not use liquidity to support credit lines.

IfW ∗0 <Wmax

0 , which occurs when τhigh is suffi ciently low, then a shock to bank health can

affect both low- and high-quality firms. In our calibration, we only obtain corner solutions

in the choice of W ∗0 , so in our case W

∗0 = 0.048, so that the bank keeps the probability

of revocation at 0%. A decrease in D1 in this case has similar implications as in the case

with excess bank capital displayed in Figure 4.3.2: the probability of credit line revocation

increases, reducing these firms’payoffs, even though their investment increases. Loans to

low quality firms remain unchanged.

6 Loan and Credit Line Originations across the Busi-

ness Cycle

In this section, we analyze how the allocation of bank liquidity to loans and credit lines

changes with the state of the economy. We show that because term loans are used by

relatively weaker firms, when bad times hit the marginal value of allocating bank liquidity to

term loans increases, relative to the marginal value of saving cash to honor future drawdowns.

We introduce a slightly modified version of the model to make this point clearly.

There is a single bank endowed with initial capital equal to K0, which it uses to make

loans ib and save cash to support future credit line drawdowns. We denote cash by W1 and

assume the bank can save cash at zero premium. There are η firms with low credit quality

(A = Alow), which use bank financing and cash for liquidity insurance, and 1− η firms withhigh credit quality (A = Ahigh) that use market financing but rely on the bank for liquidity

management through credit lines. Total loans made to low quality firms are equal to ηib and

thus:

K0 = ηib +W1. (38)

29

The bank chooses

W ∗1=argmax η[Ulow + Ubank] + (1− η)Uhigh (39)

We assume that the budget constraint for low quality firms binds and thus:

Ulow = pHR(Ilow)− (1 + λρ)Ilow − (q0 − 1)c(Ilow)−pH

pH − pLϕIlow + ib (40)

Ilow = Alow + ρ0,low(Ilow) + ib − λρIlow − (q0 − 1)c(Ilow). (41)

Also:

Ubank =pHϕI

pH − pL− ib − ϕI. (42)

Thus we can write:

Ulow + Ubank = v(Ilow), (43)

where v(Ilow) is an increasing and concave function of Ilow defined as v(Ilow) = pHR(Ilow)−(1 + λρ)Ilow − (q0 − 1)c(Ilow)− ϕIlow, and Ilow is an increasing function of ib which we writeas Ilow(ib).

We can write Uhigh as:

Uhigh = pHR(Ihigh)− (1 + λρ)Ihigh − µ[pHR(Ihigh)− (τ + ρ)Ihigh]. (44)

We assume that Ihigh is fixed and does not depend on the bank’s liquidity choice. Thus this

utility can be written as:

Uhigh = U − µK, (45)

where U = pHR(Ihigh)− (1 + λρ)Ihigh and K = pHR(Ihigh)− (τ + ρ)Ihigh represents the costof financial distress for high quality firms (the payoff loss in case they are liquidated). The

bank’s allocation of liquidity will affect Uhigh through the probability of revocation µ. We

assume λθ = 1 (all firms face a liquidity shock in state θ) and thus:

µ ≡ θµθ, (46)

where θ is the probability of an aggregate liquidity shock and:

µθ = 1−W1

(1− η)[ρIhigh − ρ0(Ihigh)]. (47)

30

The bank needs to use cash W1 to support credit line drawdowns and (1 − η)[ρIhigh −ρ0(Ihigh)] is the aggregate demand from high quality firms.

The bank’s optimization problem can then be written as:

W ∗1 = argmax ηv(Ilow) + (1− η)(U − µK), s.t. (48)

K0 = ηib +W1 (49)

Ilow = Ilow(ib)

µ ≡ θ − θW1

(1− η)[ρIhigh − ρ0(Ihigh)].

An interior solution for W1 must obey:

θK

[ρIhigh − ρ0(Ihigh)]= v

′(Ilow)I

′

low(ib). (50)

The left hand side is the marginal benefit of saving cash in the bank, which is to reduce

liquidation losses for high quality firms in the bad aggregate state. The right hand side is

the marginal benefit of saving less cash and making additional loans to low quality firms

(this increases their investment and payoff).

There are bounds to W1 from above and below. The maximum value of W1 is the value

that makes the probability of revocation equal to zero:

Wmax1 = min[K0, (1− η)[ρIhigh − ρ0(Ihigh)]]. (51)

The minimum value is such that ib reaches the maximum possible level (the point at

which all rents are transferred from the bank to firms):

Wmin1 = max[0, K0 − ηimax

b ]. (52)

An equilibrium in this economy is obtained by solving for Ilow, ib andW1, using equations

(41), (49) and (50). If we express the optimality condition for W1 (50) as a function of Ilow

31

and ib, we obtain:

θ[pHχI

αhigh − (τ + ρ)Ihigh

]ρIhigh − pH

[χIαhigh −

BIhigh∆p

] (53)

=pH

(b+ϕ)∆p− ϕ

1 + λρ+ (q0 − 1)ρ− q0

[αχ [Ilow]

α−1 − (b+ϕ)∆p

] − 1,where Ilow is given by (41). Notice that W1 does depend on η. Total lending ib is given by

solving (41) and (53), and is not a function of η. To calculate the corresponding W1, we

make use of (49). It becomes clear that W1 depends negatively on η. Intuitively, the size of

each loan and the revocation probability of credit lines do not depend on η, but the total

amount of cash that the bank needs to save to satisfy credit line drawdowns does.

A decrease in firm net worth A will decreaseW ∗1 . Notice that Ilow is an increasing function

of Alow. Thus, a reduction in A reduces Ilow and increases the marginal benefit of investment

by constrained firms (v′(Ilow) goes up). The effect of ib on Ilow is determined by the budget

constraint:

Ilow = Alow + ρ0,low(Ilow) + ib − λρIlow − (q0 − 1)c(Ilow). (54)

The function ρ0,low(Ilow) will inherit the concavity from R(Ilow), thus a reduction in Alowwill increase the marginal effect of Ilow on pledgeable income. Similarly c(Ilow) = ρIlow −ρ0,low(Ilow), so −c(Ilow) is concave. Given this, the effect of ib on Ilow should also increase asAlow goes down. The term v

′(Ilow)I

′low(ib) thus increases, increasing the marginal benefit of

date-0 loans.

In this model, the marginal benefit of bank cash θK[ρIhigh−ρ0(Ihigh)]

is independent of Ahighsince we assumed that investment Ihigh is fixed. Thus, a reduction in Ahigh will not change

the marginal benefit of bank cash, and thus bank cash goes down with A.

The marginal benefit of bank cash increases with the expected (systematic) financial

distress costs of high quality firms, θK. An increase in the probability of the aggregate

liquidity shock θ for example will increase bank cash and reduce loans to low quality firms.

Thus, if one interprets bad times as a situation when firm net worth is low, then bank

cash should go down in bad times. But if one interprets bad times as a situation when the

probability of an aggregate liquidity shock goes up, then bank cash should go up. In reality,

both the benefit and the cost of bank cash are likely to go up in bad times.

32

Comparative Statics: Productivity

To consider productivity shocks we introduce a productivity parameter χ in the firms’

production function. Investment produces a payoff equal to χR(I) with probability p if it is

continued until the final date, and 0 otherwise.

A decrease in χ has ambiguous effects onW ∗1 , but is likely to decreaseW

∗1 , which is given

by the optimality condition

θ [pHχR(Ihigh)− (τ + ρ)Ihigh]

[ρIhigh − ρ0(Ihigh)]= v

′(Ilow)I

′

low(ib). (55)

This is because, as we explain below, the marginal benefit of saving cash in the bank (the

left hand side of expression (55)) decreases unambiguously, even though the marginal benefit

of date-0 loans (captured by term v′(Ilow)I

′low(ib)) could increase or decrease with χ.

Ilow is an increasing function of χ because ρ0,low(Ilow) is an increasing function of χ, which

means that, as productivity increases, firms can borrow more. An increase in χ has a direct

negative impact on c(Ilow) = ρIlow−ρ0,low(Ilow), and liquidity needs could only increase with

χ through the indirect impact of an increase in Ilow. Thus, a reduction in χ reduces Ilowunambiguously. It does not however necessarily increase the marginal benefit of investment

by constrained firms (v′(Ilow)) because the marginal payoff χR′(I) also goes down.

The effect of ib on Ilow is determined by the budget constraint:

Ilow = Alow + ρ0,low(Ilow) + ib − λρIlow − (q0 − 1)c(Ilow). (56)

A reduction in χ will decrease the marginal effect of Ilow on pledgeable income and increase

its effect on liquidity needs. Given this, the effect of ib on Ilow should decrease as χ goes

down.

The marginal benefit of date-0 loans, captured by term v′(Ilow)I

′low(ib), could thus increase

or decrease with χ.

In this model, with constant Ihigh, the marginal benefit of bank cash, given by

θ [pHχR(Ihigh)− (τ + ρ)Ihigh]

[ρIhigh − ρ0(Ihigh)], (57)

is positively related to χ. When productivity decreases, the bankruptcy costs (the numerator

in (57)) decrease, because the expected payoff in the final date falls. The liquidity needs (the

denominator in (57)) increase, because pledgeable output ρ0(Ihigh) falls. A dollar of bank

33

0 0.005 0.01 0.015A

0.7

0.75

0.8

0.85

0.9

0.95

1

W1/K0

Firm Net Worth

0.5 0.55 0.6 0.65 0.7 0.750.55

0.6

0.65

0.7

W1/K0

Share of Low Quality Firms

0.05 0.1 0.15 0.2 0.250.5

0.6

0.7

0.8

0.9

1

W1/K0

Likelihood of Crisis

1.8 1.9 2 2.1 2.2 2.30.6

0.7

0.8

0.9

1

W1/K0

Productiv ity

Figure 7: Sensitivity of the bank’s optimal liquidity allocation to variations in economicconditions.

savings thus provides an unambiguously lower return when χ falls.

6.1 Numerical Exercise

We introduce a numerical analysis to study the sensitivity of the bank’s optimal liquidity

allocation to variations in economic conditions. Assume that the production function is

represented by:

R(I) = χIα,

where χ = 2. We also assume that pH = 0.7, pL = 0.55, B = 1, and ρ = 0.9.16 The date-0

price of cash is q0 = 1.05. The high credit-quality firm has high net worth (Ahigh = 5), and a

liquidation value τhigh = 0.25. The low credit-quality firm has Alow = 0, and τ low = 0. Our

assumptions about the liquidity shock probabilities are: λθ = 1, and λ1−θ = 0.20. The bank

16This calibration exercise is only for illustrative purposes. We do not seek to match real world quantities.

34

parameters are as follows. The cost of monitoring ϕ is 0.5, and b = 0.7.

Economic conditions are captured by the likelihood of a crisis (θ), the share of low credit

quality firms (η), the net worth of low credit quality firms (Alow), and the productivity of

the production function (χ). The range of values considered for these comparative statics

are in Figure 7.

Consistent with the intuition above, a decrease in firm net worth A or an increase in

the share of low credit quality firms η will both decrease W ∗1 . The productivity shock χ

operates in the same way as A or η, and low productivity is associated with a decrease in

W ∗1 . Thus, if one interprets bad times as a situation when firm net worth is low, the share of

low credit quality firms goes up, or productivity decreases, then bank cash should go down

in bad times.

7 Conclusion

Two empirical observations motivate this paper. First, financially weaker firms depend

on loans for funding, and stronger firms do not depend on loans but use banks for liquidity

insurance. Second, the share of bank lending that goes to term loans versus credit lines

increases in bad times. We develop a model that can generate both the cross sectional and

the time series result. In fact, the model shows that the two patterns are linked - because

term loans go to low quality firms, the marginal benefit of loans relative to credit lines tend

to go up in bad times.

The results of our theoretical work and descriptive empirical analysis shed light on how

the choice of banks to allocate liquidity between term loans and credit lines affects liquidity

and investment by high quality and low quality (or large versus small) firms, and differentially

so over the economic cycle. Our framework has the potential to assess the macroeconomic

consequences of regulatory initiatives such as bank capital requirements or liquidity coverage

ratios in a setting where there is heterogeneity among firms in the corporate sector in the

nature of their bank-dependency.

35

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