A New Perspective on Bank-Dependency: The Liquidity Insurance … · 2020. 1. 7. · why credit...
Transcript of A New Perspective on Bank-Dependency: The Liquidity Insurance … · 2020. 1. 7. · why credit...
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.
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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
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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
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(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.
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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.
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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.
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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.
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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.
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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
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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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