Optimal Credit Policy¤
Roman Indersty
Holger M. Müllerz
April 2002
Preliminary draft – please do not quote or cite
Abstract
We consider optimal contracting between a borrower and a lender, where the lender
can perform a credit test prior to the loan decision. We show that, under fairly general
conditions, the lender’s credit policy will be too conservative, i.e., he will reject too many
borrowers. This holds even if the credit test is perfectly informative – in fact, increasing the
informativeness of the test may worsen the credit rationing problem. The optimal contract
between the borrower and lender minimizes the problem of excessive project rejection. In a
general setting with a continuous cash-‡ow distribution, we show that the unique optimal
contract is a standard debt contract. Collateral and multiple lending can alleviate the credit
rationing problem but do typically not eliminate it. In the case of multiple lending, we show
that the party performing the credit test must have a claim that is senior to that of other
lenders. Our model provides a new rationale for the separation of information production
and lending that is not based on economies of scale or scope in the production of information.
¤
We thank Tony Saunders for helpful comments and discussions.
y
London School of Economics, FMG, and CEPR. Address: Department of Economics & Department of Ac-
counting and Finance, Houghton Street, London WC2A 2AE, UK. Email: r.inderst@lse.ac.uk.
z
New York University and CEPR. Address: Department of Finance, Stern School of Business, New York
University, 44 West Fourth Street, Suite 9-190, New York, NY 10012, USA. Email: hmueller@stern.nyu.edu.
1
1
Introduction
Credit is rarely provided on the basis of blind trust. Before making a loan the lender typically assesses the borrower’s credit risk, which involves establishing the borrower’s ability to make timely
interest and principal repayments. While credit risk analysis has always been an important part
of any lending relationship, its importance has increased over the past years.1 Moreover, due
to advances in information technology and the development of historic loan databases, credit
risk analysis and the loan approval processes have become more and more standardized. The
following quote from James and Houston (1996) describing small business lending at Wells Fargo
illustrates this trend.
“Wells Fargo, recognized by American Banker as the “runaway leader in smallbusiness lending,” has taken great strides to develop an expertise in the application
of technology to small-business lending. Wells has used technology to centralize
large amounts of information about its customers. [...] Creating large data bases to
develop credit-scoring models is not new. Wells’ distinctive accomplishment has been
to do this on a massive scale, and to centralize this information, thereby expanding
the level of general knowledge throughout the bank. [...] Individual lenders are now
able to go out in the …eld armed with a laptop computer loaded with information
from the …rm’s large data base. This enables the lender to “plug in” the borrower’s
information into the computer model – and, in many cases, to approve loans on the
spot that in the past might have taken a great deal of time and a great number of
people to process.”
In this paper we consider optimal contracting between a borrower and a lender, where the
lender can perform a (standardized) test to assess the borrower’s credit risk prior to the loan
decision. Our testing technology closely resembles that of actual credit-scoring systems used in
practice. Credit scoring can be found in virtually all types of credit analysis, from consumer
credit to commercial loans. In the Federal Reserve’s January 1997 Senior Loan O¢cer Opinion
Survey on Bank Lending Practices, 70 percent of the respondents indicated that they use credit
scoring in their small business lending (Mester 1997).2 Lenders who do not not have su¢cient
1
Allen and Saunders (2002) name the following reasons: the number of bankruptcies during recessions has
increased signi…cantly, collateral value has declined and become more volatile, interest margins and spreads have
become thinner and the proposed New Capital Accord by the Bank for International Settlements (BIS) has
strenghtened the link between capital charges and credit risk exposures.
2
For similar evidence, see Cole, Goldb erg, and White (1997), Treacy and Carey (1998), English and Nelson
(1998), and Frame, Srinivasam, and Woosley (2001).
2
volumes to build their own small-business scoring systems can purchase scoring models with large
databases from commercial providers such as Fair, Isaac, CCN-MDS, or Dun & Bradstreet.
In our model the credit test generates new, valuable information about the risk of the borrower’s project that has not been available previously.3 Based on this information the lender
either approves or rejects the borrower. The lender thereby faces the usual problem of trading
o¤ type-1 errors (misclassifying bad projects as good) against type-2 errors (misclassifying good
projects as bad). We show that, under fairly general conditions, the lender’s credit decision will
be too conservative. This result is not driven by the fact that the credit score is a noisy signal
about the borrower’s risk – it holds equally if the credit test is perfectly informative. In fact,
we show that an increase in the test’s informativeness may worsen the credit rationing problem.
Rather, the result is driven by the fact that the lender is not the full residual claimant to all of
the project’s cash ‡ows. In a sense, the problem is one of lender moral hazard.
The optimal contract between the borrower and lender minimizes the probability of excessive
project rejection. In a general setting with continuous cash-‡ow distribution we show that the
unique optimal contract is a standard debt contract. Intuitively, a debt contract minimizes the
di¤erence in payo¤s to the lender from a bad and a good project. Hence a debt contract makes
the lender’s claim as insensitive to the borrower’s credit risk as possible, thereby alleviating the
problem of excessive project rejection. Our result implies that the common folk wisdom whereby
giving banks equity makes them less cautious in their lending practices is generally not correct.4
It is debt, not equity, that minimizes the ine¢ciency from being too cautious. Finally, we show
in a comparative statics exercise that the lender’s credit policy becomes more conservative the
greater the share of the proceeds, or rents, that he must leave to the borrower. One reason why
borrowers might receive rents is that the capital market is competitive. In this case, our results
imply that an increase in capital market competition leads to more credit rationing on the level
of an indivdual lender.
We then explore – still in an optimal contracting framework – the role of collateral and
multiple lending as ways to make the lender’s credit policy more lenient. If the borrower can
pledge private wealth as collateral the lender’s claim becomes less sensitive to the project’s risk.
3
This distinguishes our model from private information settings where the sole purpose of screening is to
uncover the borrowers’ private information (e.g., Bester 1985, Besanko and Thakor 1987). The notion that
lenders may p ossess or generate new, valuable information is not new and app ears in, e.g., Garmaise (2001) and
Manove, Padilla, and Pagano (2001). For empirical evidence supp orting this notion see Cressy and Toivanen
(1997) and the references in Garmaise (2001).
4
E.g., Berlin (2000) asks: “Do restrictions against U.S. banks holding equity make a di¤erence for banks’
behavior? Are U.S. banks’ borrowers at a disadvantage because their lenders are too cautious when evaluating
project risk [...]?” Berlin does not share this commonly found view, albeit his argument for why it is incorrect is
di¤erent from ours.
3
Unless the collateral is so high that the lender’s claim becomes riskless, however, the problem
of excessive project rejection remains. The unique optimal contract is again a standard debt
contract, now backed by collateral. Another way to mitigate the underinvestment problem is
multiple lending. For instance, the borrower might bring in a partner or sell a fraction of his
business to a private investor. Unlike the lender, this third party does not have access to a
testing technology. It will thus have to rely on the lender’s decision to provide …nance, which is
only a crude signal about the borrower’s quality, however. Introducing a third party (partially)
allows to disentangle the two tasks of testing and providing …nance, which is at the heart of the
problem. Our model thus provides a new rationale for the separation of information production
and lending, without appealing to economies of scale or scope. The lender again receives a
unique optimal contract, which is standard debt secured by collateral (if available). He …nances
only a fraction of the investment while the rest is …nanced by the third party. The third party
holds unsecured claims which are junior to that of the lender.
We provide several testable implications. As lending becomes more competitive, we predict
that total credit volume decreases, that collateral requirements increase, and that with multiple
lending the …rst lender’s share of …nancing decreases.
The probability of being ine¢ciently
denied a loan decreases with the value of the borrower’s own wealth and with the availability of
a second lender who has no access to the testing technology.
While our framework is su¢ciently general to have implications for all kinds of lending
relationships, it is probably most descriptive of transactions-based lending (as opposed to relationship lending), where the lending decision is primarily based on “hard” information that is
available at the time of the loan origination. There seems to exist no study examining the relative
importance of transactions-based vs. relationship lending (Berger and Udell 2002). There is,
however, su¢cient evidence suggesting that transactions-based lending plays an important role
in business …nance, in particular for small businesses (Feldman 1997, Mester 1997; see also the
Senior Loan O¢cer Opinion Survey cited above). (In other areas of lending such as mortgages,
consumer loans, or credit cards, transactions-based lending – and credit-scoring in particular –
plays a dominant role). Small business loans account for about one third of banks’ total business
loans (Mester 1999). Moreover, small businesses rely much on bank loans, which makes their
investment expenditures a crucial component of the bank lending channel (see the overview in
Walsh 1998, Chapter 7).
Credit tests have been studied previously in the literature. Following Broecker (1990), a
sizeable literature has examined the role of pool externalities arising from the fact that rejected
borrowers worsen the average quality of loan applicants in the market (e.g., Riordan 1993, Gehrig
1998, Sha¤er 1998, Dell’Arricia, Friedman, and Marquez 1999, Cao and Shi 2001). The focus of
this literature is clearly di¤erent from ours. In particular, it does not study optimal contracting
4
or the e¤ect of credit tests on …nancial contracts. Manove, Padilla, and Pagano (2001) study a
model in which high-type borrowers post collateral to avoid paying for testing costs. The form
of …nancial contract (debt) is taken as given. In our model the marginal cost of testing is zero,
the idea being that – once the credit-scoring system is in place – all the loan o¢cer has to do
is feed the data into his computer.5 Hence there is no ine¢ciency from too little testing, as in
Manove, Padilla, and Pagano (2001). Rather, the ine¢ciency is that, given that credit tests
are used, the cuto¤ level above which borrowers are accepted is ine¢ciently high. In Garmaise
(2001) investors receive a noisy signal about project quality prior to bidding for the …nancing
of the entrepreneur’s project in a …rst-price auction. Bids take the form of …nancial contracts.
As in our model, investors’ signals are informative in the sense that they contain information
that has not been available previously. A pecking order is derived whereby the entrepreneur
accepts equity contracts over debt whenever both types of contracts are bid. Unlike our model
all projects are pro…table. Hence underinvestment is not an issue in Garmaise’s model.
Our model suggests that a standard debt contract is optimal in settings where lenders can
perform a credit test prior to making a loan. Various other rationales for debt (or debt-like)
contracts have been o¤ered in the literature. Debt minimizes the cost of ex-post verifying the
borrower’s cash ‡ow (Townsend 1979, Gale and Hellwig 1985) or in‡icting a non-pecuniary
penalty on the borrower (Diamond 1984). If there is ex-post moral hazard on the part of the
borrower, debt provides the borrower with incentives to work hard (Innes 1990) or repay funds
(Bolton and Scharfstein 1990, Hart and Moore 1994). Finally, debt may be optimal if the issuer
has private information either before (Nachman and Noe 1994, who formalize the intuition
in Myers and Majluf 1984) or after the security design (DeMarzo and Du¢e 1999, Biais and
Mariotti 2001).
The rest of the paper is organized as follows. Section 2 introduces the model and testing
technology. Section 3 contains our main results. We show that, unless the lender obtains all
the proceeds from the project, his credit policy based on the credit test is too conservative.
Given this ine¢ciency, we show that the unique optimal contract is a standard debt contract.
We …nally consider the possibility that the borrower can pledge private wealth as collateral
and derive the optimal collateral policy. Section 4 considers the case where the borrower can
bring in a third-party investor who does not have access to a testing technology. We derive the
optimal contract, lending volume, collateral policy, and seniority rule. Section 5 summarizes
the empirical implications. Section 6 examines the role of the quality, or informativeness, of the
5
See the ab ove quote about small business lending at Wells Fargo. The use of credit scoring has drastically
reduced screening costs. Traditional loan approval processes average 12.5 hours p er small business loan. Credit
scoring can reduce this time to well under an hour (Mester 1997, Saunders and Thomas 2001, p.288). Also,
scoring systems themselves are rather cheap: the price p er loan of a commercially available credit scoring model
averages about $1.5 to $10 per loan, dep ending on volume (Muolo 1995).
5
credit test. Section 7 concludes. All proofs are in the Appendix.
2
The Model
An entrepreneur (“the borrower”) has a project requiring an investment k: The entrepreneur
has private wealth w, where 0 · w < k. For the moment we shall assume that w = 0: The case
where the entrepreneur has positive wealth is considered later.
The project can be of two types denoted by µ 2 £ := fl; hg. A type-µ project generates a
random payo¤ x drawn from the distribution function Gµ (x): Both types have the same support
X := [x; x]: The density function corresponding to Gµ (x) is denoted by gµ (x), where gµ (x) > 0
for all x in the interior of X. The high-type project dominates the low-type project in the sense
of the Monotone Likelihood Ratio Property (MLRP):
(A.1) gh(x)=gl (x) is strictly increasing in x for all x 2 X.
Hence a high cash ‡ow is more likely to be generated by a high-type project than by a low-type
project. Denote the expected payo¤ of a type-µ project by xE
µ :=
R
x2X gµ (x)xdx.
It follows from
E
(A.1) that xE
h > xl : We shall assume that only the high-type project is pro…table:
E
(A.2) xE
l < k < xh :
To …nance the project the entrepreneur turns to an investor (“the lender”). The lender can
perform a costless credit test generating for each type µ a signal, or score, s 2 S := [0; 1] drawn
from the distribution function Fµ (s): Hence the score provides a noisy signal about the borrower’s
type. The density function corresponding to Fµ (s) is denoted by fµ (s); where fµ (s) > 0 for all
s in the interior of S: The borrower and lender have common priors denoted by ¼0µ > 0: Given
a score s, the lender then has posterior beliefs that the borrower is of type µ with probability
¼0µ fµ (s)
:
0 0
µ 0 2£ ¼ µ0 fµ (s)
¼ Sµ (s) := P0
We make the natural assumption that the posterior probability of the high type ¼Sh (s) is
increasing in the score s: This is equivalent to saying that the signal distribution of the high
type Fh(s) dominates that of the low type in the sense of MLRP. Finally, we assume that the
test score is perfectly informative at the boundaries of S: The assumptions regarding the testing
technology are summarized by
(A.3) fh(s)=fl (s) is strictly increasing in s for all s 2 S. Moreover, fl (0) > 0, fh(1) > 0; and
fh(0) = fl (1) = 0:
The contracting environment, testing technology, and timing of events closely resemble actual
lending practices, in particular for small business lending. We comment on the setup below. We
begin with the sequence of events.
6
The lender posts a (conditional) contract o¤er, which stipulates a repayment t(x) based on
the realized project cash ‡ow in return for …nancing the investment k: The o¤er is conditional
upon subsequent approval by the lender. The approval decision is based on a (costless) credit
test using the testing technology described above. The test score s is only known to the lender.
Finally, we assume that the fact whether the borrower applied for a loan is nonveri…able. Hence
if a borrower does not obtain a loan, this could be either because he was rejected or because he
did not apply. As a consequence, the contract cannot stipulate a penalty for the lender in case
a borrower is rejected.6
Since w = 0 the repayment scheme must satisfy t(x) · x: Note that this allows negative
transfers, i.e., payments from the lender to the borrower. Moreover, we invoke the standard
assumption that
(A.4) t(x) is nondecreasing.
If (A.4) did not hold the borrower could make an arbitrage pro…t by borrowing money to boost
the project cash ‡ow ex post. Since this loan is riskless any third party would be willing to
provide the required funds.7
Finally, while we assume that the lender makes the contract o¤er we do not have in mind a
situation where the lender is a monopolist who can extract all the surplus. We take into account
the possibility that there are other potential lenders, or sources of …nance, by assuming that
the contract must grant the borrower at least V R ¸ 0: One interpretation of V R is that it is a
measure of capital market competition. The more competitive the capital market, the greater
the borrower’s bargaining power, and hence the greater the surplus V R which must be granted
to the borrower.8 On the other hand, an increase in switching cost might reduce V R: In what
follows we treat V R as a parameter, thus leaving open the possibility that the borrower can
extract a large fraction of the surplus.
Let us …nally comment on the model setup. The setup is extremely simple: the lender posts
a contract o¤er, performs a credit test, and then either accepts or rejects the borrower. This
is how small business …nance (but also consumer and mortage loans) typically works. lenders
o¤er …xed conditions subject to approval by the loan o¢cer. Speci…cally, the loan conditions
6
Instead of assuming that the loan application is nonveri…able, we could equally assume that there is a large
pool of borrowers with bad projects, or “‡y-by-night operators” (Rajan 1992, von Thadden 1995) who would
apply for a loan, get rejected, and cash in the penalty. In either case the contract will not stipulate a penalty for
the lender in case a borrower is rejected. See App endix A for details.
7
(A.4) is standard in the literature (e.g., Nachman and Noe 1994, DeMarzo and Du¢e 1999). Another standard
assumption is that x ¡ t(x) is nondecreasing. Since this holds anyway under the optimal contract in our model,
we do not have to make this assumption.
8
See Inderst and Müller (2001) for a model endogenizing V R as a function of capital market competition.
7
do not depend on the subsequent test score; the only choice to be made is whether to accept
or reject the borrower. Saunders and Thomas (2001, p.323) describe retail lending as follows:
“[L]oan decisions made for many types of retail loans are reject or accept decisions. All borrowers
who are accepted are often charged the same rate of interest and by implication the same risk
premium. [...] In the terminology of …nance, retail customers are more likely to be sorted or
rationed by loan quantity restrictions rather than by price or interest rate di¤erences.”
Interestingly, this uniformity in loan conditions occurs even in relationship lending, where
one would think that the terms of the loan are “tailor made” to the risk pro…le of the borrower:
“In fact, the lenders have increasingly centralized this process, using centralized criteria to
determine lending and interest rates. However, as far as we could tell, they do little to discourage
SMEs (small and medium-sized enterprises, the authors) from the belief that a good relationship
helps, even though the discretion of relationship managers within the guidelines for assessing
risk, agreeing loans and setting terms under which he or she operates is normally very limited.”
Similarly, “relationship managers [...] had become largely a salesforce with limited discretion to
depart from centralized criteria on loan decisions” (UK Competition Commission 2002, Ch. 2,
p.39 and p. 47, respectively).
The reason why small business loans are typically not tailor made is often a simple cost
argument: “A $50,000 loan yielding three percent over cost of funds provides only $1,500 p.a.
gross revenues, before provisions of credit losses and allocation of overheads. This level of gross
revenue can pay for very little of the loan o¢cer’s analytical and monitoring time” (Saunders
and Thomas, p. 288). This cost argument, in particular, is the reason why credit-scoring models
involving simple accept/reject decisions have become so popular. There is a second, more subtle
argument why the contract terms might not depend on the test score. The argument is laid out
in detail in Appendix A. While the lender cannot o¤er a contract conditioning on the test score
since the test score is private information,9 he could …rst perform the credit test and then o¤er
a contract based on this information (or choose a contract from a previously speci…ed menu).
In the Appendix we show that such …ne-tuning is not optimal.
9
Brunner, Krahnen, and Weber (2000, p.4) remark: “ Typically, banks do not inform their customers of
the internal ratings or the implied PODs [estimated default probability], nor do they publicize the criteria and
methods used in deriving them.” See also the recent rep ort by the UK Comp etition Commission (2002), where it
is reported that “banks are reluctant to share their credit assessment criteria with the SME” (Ch. 2, p. 39)
8
3
Optimal Debt Contracts with Ex-Ante Credit Tests
3.1
No Private Wealth
The …rst-best credit policy is to reject a project only if its expected payo¤ is not su¢cient to
cover the required initial …nance. This decision can be based on the obtained score and the
corresponding updated beliefs. The …rst-best policy prescribes a threshold for the score, which
we denote by s¤¤, such that the project is rejected if the score lies below this threshold. At the
threshold s¤¤ the project just breaks even, given the updated beliefs:
X
¼Sµ (s¤¤ )xE
µ = k:
(1)
µ2£
Note tyat by (A.2) and (A.3) s¤¤ is indeed uniquely determined and satis…es 0 < s¤¤ < 1. We
denote the surplus obtained under the e¢cient credit policy by W ¤¤, which is given by
W ¤¤ :=
X
µ2£
h
i
¼0µ [1 ¡ Fµ (s¤¤)] xE
µ ¡k :
(2)
Note that 1 ¡ Fµ (s¤¤ ) is the ex-ante probability with which type µ passes the threshold
s¤¤ . With probability Fh(s¤¤ ) a high-type project is wrongly rejected, while with probability
1 ¡ Fl (s¤¤) a low-type project is wrongly accepted. The …rst-best policy optimally trades o¤
these two types of error.
We turn next to the credit policy that is actually implemented by the lender. Given some
repayment scheme t, we denote the lender’s expected payo¤ from a type-µ project by
Uµ (t) :=
Z
x2X
t(x)gµ (x)dx ¡ k:
As the low-type project has negative expected payo¤, the lender can only break even in case
(3)
Uh(t) > k > Ul (t)
holds. Using (3) it is immediate that the lender’s optimal policy is to set a threshold on the
required score, which depends on the speci…ed repayment scheme t. We denote this threshold by
s¤(t). It will be frequently convenient to omit the argument by writing simply s¤. From (A.3)
and (3) the threshold is uniquely determined by the requirement that the lender breaks at this
score:
X
¼Sµ (s¤)Uµ (t) = k:
(4)
µ2£
Under the lender’s optimal credit policy the realized surplus is given by
W ¤(s¤) :=
X
µ2£
h
i
¼0µ [1 ¡ Fµ (s¤ )] xE
µ ¡k :
9
(5)
We argue now that the lender’s policy is always too strict, unless the lender obtains all
proceeds. To see this, suppose the lender has granted a loan to some borrower who obtained
the score s¤ . From the lender’s perspective this is equivalent to acquiring a portfolio of low- and
high-type projects, where the fraction of each type is equal to the respective posterior probability
¼ Sµ (s¤ ). Suppose now we reduce the lender’s payo¤ t(x) over a positive measure of states x 2 X .
As the lender’s “portfolio” realized just k before this transformation, the new portfolio will now
realize strictly less than k. To compensate for this shortfall, we must by (3) increase the fraction
of high-type projects in the lender’s portfolio, i.e., the required posterior probability for the high
type. By (A.3) this is …nally equivalent to increasing the required threshod s¤. By this reasoning
we obtain the following result.
Lemma 1. If w = 0; then s¤¤ < s¤ holds unless the lender obtains all proceeds.
Proof. See Appendix.
Lemma 1 suggests a natural approach to …nd the optimal contract. Subject to the borrowers’
participation constraint, the optimal contract minimizes the loss implied by excessive rejection.
We show that this is uniquely achieved by a debt contract. Note that a debt contract is fully
characterized by the repayment requirement (or face value) R, such that t(x) = x holds for
x · R and t(x) = R holds for x > R.
At this point it is useful to set up more formally the lender’s program. The lender’s expected
payo¤ is given by
U (t) :=
X
µ2£
¤
¼0µ [1 ¡ Fµ (s¤)] Uµ (t);
where s satis…es (4). Similarly, a borrower’s expected payo¤ is given by
V (t) :=
X
µ2£
where we use Vµ (t) :=
R
x2X [x
¼0µ [1 ¡ Fµ (s¤)] Vµ (t);
¡ t(x)]gµ (x)dx. The lender chooses t to maximize U (t) subject
to the monotonicity requirement (A.4) and the participation constraint V (t) ¸ V R. If V R is
not too large, a non-trivial contract under which the loan is granted with positive probability is
feasible.
Observe now that the borrower’s participatoin constraint will surely bind, implying that all
rents exceeding V R accrue to the lender. By Lemma 1 we also know that the implemented
credit policy will be too conservative. Consequently, the lender wants to design a contract that
makes a subsequent approval most likely. Why is this accomplished by a debt contract? Recall
again our picture that granting a loan after observing s¤ is equivalent to acquiring a portfolio
where the fraction of type-µ projects is equal to the respective posterior probability ¼Sµ (s¤). For
a given value of this portfolio, i.e., k by de…nition of s¤ , the rquired fraction of the high-type
projects must be larger the higher the di¤erence in payo¤s Uh(t) ¡ Ul (t). This di¤erence in the
10
lender’s payo¤s from the two projects is now minimized by using debt. To see this, note that
under the monotonicity constraint (A.4) the debt contract makes the maximum transfers to the
lender for low outcomes and the minimum transfers for high outcomes. By (A.1) this increases
the payo¤ from the low- and decreases the payo¤ from the high-type project.
Proposition 1. If w = 0, the lender o¤ers a unique standard debt contract.
Proof. See Appendix.10
By Proposition 1 we can restrict consideration to standard debt contracts. Depending on
the distribution of rents, which is determined by V R, the lender will o¤er a lower or higher
repayment level R. The more rents must be shifted to the borrower by way of decreasing the
stipulated repayment, the more conservative becomes the lender’s credit policy.
Corollary 1. If w = 0, a rise in V R increases s¤ ,which exceeds s¤¤ for all values V R > 0.
Proof. See Appendix.
In Section 5 we derive further implications. Moreover, in Section 6 we allow for changes
in the test’s informativeness. We also show that our results do not depend on the assumption
that the test is not perfectly informative. As a …nal observation note that under the optimal
contract the lender will always receive positive rents. Put di¤erently, even under the contract
that maximizes borrowers’ payo¤ the lender makes striclty positive pro…ts. It is not optimal to
lower these rents by, for instance, requiring the lender to make an initial payment exceeding the
loan size k. This requirement would make the lender’s policy more conservative and thereby
reduce borrowers’ payo¤.
3.2
Private Wealth
We assume now that the borrower has wealth w > 0. In principle this allows to make a given
transfer t(x) out of both wealth and proceeds. Likewise, the borrower’s wealth could be used
to reduce the loan size L. In what follows we resolve this ambiguity in the following way. As
we focus on the lender’s credit policy, we assume that implementing the most e¢cient credit
policy is still the overriding objective. However, if this can be accomplished in various ways,
we assume that it is optimal to choose the contract that minimizes the expected transfer made
10
It should b e noted that, though similar in ‡avor, Proposition 1 and its proof are not analogous to the result
in Innes (1990). First, Innes (1990) analyses borrower moral hazard after the loan has been granted, while we
study the lender’s decision to provide …nance. On top of this, there is also an important technical distinction.
While it is only the di¤erence b etween low and high output states that incentivates a borrower in Innes (1990),
the bank’s credit policy is a¤ected both by the payo¤ di¤erence b etween high- and low-type projects as well as
by the resp ective payo¤ levels. As a consequence, for V R > 0 the bank’s credit policy can not b e made e¢cient,
even if we were to neglect the monotonicity constraint (A.4). In Innes (1990) non-monotonic transfer schemes
can implement the e¢cient level of b orrower e¤ort.
11
out of borrower’s wealth. This choice is particularly natural if the borrower’s wealth is stored in
physical assets, which may fetch a lower price on the market compared to their intrinsic value
to the owner. An immediate implication is that using wealth to reduce the loan size is never
optimal. Instead, the lender should …nance the full investment, while increasing its payout by
the same amount in all states. This additional transfer is only paid out of borrower’s wealth in
case the project’s proceeds are su¢ciently low.11
To distinguish between transfers made out of the project’s proceeds and those made out
of borrower’s wealth, we denote the former by tP (x) and the later by tC (x). These transfers
must satisfy t P (x) · x and 0 · tC (x) · w for all x 2 X. As previously the total transfer
t(x) = t P (x) + t C (x) must satisfy the constraint (A.4).
In what follows, we restrict consideration to the case where the safe payment that can be
promised to the lender, i.e., x + w, is not su¢cient to cover the investment size k.
(A.5) x + w < k.
Under (A.5) it will not be possible to implement the e¢cient credit policy if the borrower
receives a su¢ciently large share of realized rents. In contrast, if (A.5) did not hold, this would
always be possible. Apart from this di¤erence, our results extend to the case where (A.5) does
not hold.
Consider now the way how the project’s proceeds x 2 X are distributed. We …nd that tP
takes on the shape of a debt contract. By our previous arguments this ensures the most lenient
credit policy. Moreover, even if the e¢cient policy could be implemented with various contracts
by transferring su¢cient wealth, choosing tP as debt allows to minimize the necessary transfer
out of wealth. Likewise, wealth should be put to its most e¤ective use. By previous arguments
this is ensured if it is transferred predominantly in states with bad outcomes. The precise way
how wealth is used depends now on the distribution of rents. For low values of V R we can
achieve the e¢cient credit policy without ever transferring all wealth. The corresponding use of
wealth is depicted in Figure 1.
11
Neither the derived optimality of debt nor our qualitative implications hinge on making the credit policy the
overriding objective. As is also stated below, a debt contract both implements the most lenient credit policy for
a given amount of transferred wealth and minimizes the amount of transferred wealth required to implement a
given credit policy. As a consequence, debt remains optimal even if there exists a true trade-o¤ between making
the credit policy more e¢cient and reducing the costs incurred by liquidating and transferring a larger fraction
of the borrower’s wealth.
12
w
x
Use of collateral for low values V R
Note that the monotonicity requirement (A.4) imposes a restriction on the slope of t C . A
slope of minus one, as depicted in Figure 1, represents the lowest feasible value. Given that the
distribution of proceeds speci…es t(x) = x, the total transfer to the lender remains constant in
this interval.
As V R increases, all wealth must be transferred in bad states. While this still allows to
implement the e¢cient credit policy for intermediate values of V R, this is no longer the case for
su¢ciently high values of V R. The respective use of wealth is depicted in Figure 2.
w
x
Use of collateral for high values V R
13
We next summarize our results on optimal contracts.
Proposition 2. If w > 0, the lender lends the full investment sum L = k. The distribution
of proceeds -as described by tP - and the use of the borrowers’ wealth -as described by tC - are
uniquely determined. Moreover, tP takes the form of a debt contract, whereas tC takes the form
described in Figures 1-2.
Proof. See Appendix.
In Appendix B we show how the respective distribution of proceeds and wealth can be
obtained by using collateralized debt. The following implication follows from Proposition 2 and
our previous discussion.
Corollary 2. Under the conditions of Proposition 2 the credit policy is e¢cient for low
values V R and too conservative for high values V R. In the latter case the threshold s¤ strictly
increases in V R.
In Section 5 we derive more implications.
3.3
Example
Suppose the properties of the test are given by the densities fl (s) = 2(1 ¡ s) and fh (s) = 2s.
The respective distribution functions are Hl (s) = s(2 ¡ s) and Hh(s) = s2. In this example we
only allow for two possible outcomes such that X = fx; xg. Furthermore, the high-type project
realizes x with probability one and the low-type project realizes x with probability one. The
investment requires k = 1. We choose the prior distribution ¼0h = ¼0l = 1=2, implying together
with the speci…cation of the testing technology that the posterior distribution satis…es ¼Sh (s) = s.
The …rst-best credit policy is to set the threshold s¤¤ such that s¤¤x + (1 ¡ s¤¤)x = 1. This
yields s¤¤ = (1 ¡ x)=(x ¡ x) and generates the …rst best-best surplus
W ¤¤ =
i
1h
1
1 ¡ (s¤¤)2 [x ¡ 1] + [1 ¡ (s¤¤ ) (2 ¡ s¤¤)] [x ¡ 1] :
2
2
Substituting s¤¤ we obtain W ¤¤ = [x ¡ 1]=[(x ¡ x)2]. From our results it is straightforward that
the optimal contract extracts all proceeds x in the bad state and x < R · x in the good state.
Additionally, some amount c of wealth is transferred in the bad state. As the lender makes a
loan of size k = 1, the optimal threshold s¤ satis…es s¤R + (1 ¡ s¤)(x + c) = 1. This yields
s¤ =
1¡ c¡ x
:
R¡ c¡ x
(6)
We ask for which values of V R the e¢cient credit policy can be realized. Requiring that
s¤ = s¤¤, we obtain that collateral c must satisfy
c = (1 ¡ x)
14
x¡ R
:
x¡ 1
(7)
Note next that the borrower’s payo¤ is given by.
V =
i
1h
1
1 ¡ (s¤)2 [x ¡ R] ¡ [1 ¡ (s¤ ) (2 ¡ s¤¤)] c:
2
2
(8)
Substituting s¤ = s¤¤ and the collateral (7), we obtain from (8) that a given payo¤ V = V R
is realized if we set R = x ¡ 2V R(x ¡ x)=(x ¡ 1). Using this value of R and substituting it
into the collateral required to implement the e¢cient credit policy, we obtain from (7) that the
borrower’s wealth constraint c · w is only satis…ed if
V R · W ¤¤
w
:
1¡ x
Hence, we can only obtain the e¢cient credit policy as long as the rent obtained by the
borrower does not surpass the fraction w=(1 ¡ x) of the …rst-best surplus. This fraction rises if
either the borrower has more wealth at his disposition or if the project’s safe payo¤ x increases.
4
Multiple Lenders
4.1
Preliminary Discussion
The major obstacle to obtain an e¢cient credit policy is that the same party, i.e., the lender,
serves the twin function of providing …nance and conducting the credit test. To see this more
clearly, suppose we could disentangle these functions as some external credit agency, which has
no …nancial stake in the project, can perform the test. In this case the borrower and the lender
could condition their contract on the report of the agency and thereby always implement the
e¢cient credit policy.
If other investors have su¢cient funds, they could likewise employ the lender only for conducting the test for a …xed fee without taking a …nancial stake. Reducing the lender’s role to
testing is, however, no longer feasible or optimal if other investors have insu¢cient funds or
higher opportunity costs. As we focus on small business …nance this case seems more realistic.
A second lender could be a wealthy individual who has to be compensated for his lack of diversi…cation or for the foregone opportunity of putting his scarce resources into another (start-up)
investment. What is now interesting is that despite this preference for lender …nancing, one may
still want to use a second lender in order to achieve a more e¢cient credit policy.
A …nancial arrangement prescribes now the investment of L1 by the lender and of L2 by
a second lender whose funds are restricted to z. In return the lender receives the transfers
(tP ; tC ) out of proceeds and out of the borrower’s wealth. Note that we do not admit for direct
transfers from the lender to the lender and vice versa. We argue brie‡y why this does not impose
restrictions. Instead of reducing the lender’s loan by setting L2 > 0 the second lender’s funds
could be used to provide the lender with a guarantee that stipulates additional payments for bad
15
outcomes. In this case the second lender’s funds would be used in the same ways as we previously
used the borrower’s wealth, which was pledged as collateral. The use of the borrower’s wealth
as collateral was optimal as this reduced the need to liquidate wealth. The major di¤erence is
now that the second lender’s funds are assumed to be already liquid. To honour any guarantee
to the lender the second lender will have to set aside the maximum possible transfer stipulated
under this guarantee. As this sum is no longer available for alternative investments, the second
lender has to be compensated for the foregone opportunity on this full amount. Consequently,
it is now e¢cient to use this amount fully for the initial investment L2 .
We still give the lender the opportunity to initiate the contract design. It is constrained
by the requirement to leave the other parties, i.e., the borrower and the second lender, with
the rents V R . Our results are independent of how these rents are distributed and we therefore
neglect the speci…cation of the second lender’s share in the project’s proceeds.
4.2
Analysis
Suppose for a moment that the borrower has no wealth. We know from our previous results that
relying solely on the lender does not allow to implement the e¢cient credit policy unless the
lender can appropriate all rents. In contrast, being able to draw on the funds of a second lender
allows now to implement the e¢cient credit policy even if the lender must share some of the
rents. This is accomplished by reducing the lender’s loan. In many instances the e¢cient policy
could be implemented in various ways by changing the size of the lender’s loan and adjusting
its claims accordingly. We resolve this ambiguity by specifying that, in case the e¢cient policy
can be implemented, it is optimal to do so in a way which leaves the lender’s share of …nancing
L1 as large as possible. As argued already above, this choice follows naturally from our focus
on small-business …nancing where the second investor might be a wealthy individual.12
As previously we con…ne the analysis to the case where the safe payment that can be promised
to the lender is not su¢cient to cover the required investment outlay.
(A.6) It holds that x + w + z < k.
Under (A.6) it will not be possible to implement the e¢cient policy for high values V R. In
contrast, if (A.6) did not hold this would always be feasible. Apart from this di¤erence our
results extend to the case where (A.6) does not hold.13
12
As previously implementing an e¢cient credit policy is thus the prime ob jective. Again neither the derived
optimality of debt nor our qualitative implications hinge on this speci…cation as, in particular, debt will now
additionally allow to minimize the use of the second lender’s funds for a given credit p olicy. As a consequence,
debt remains optimal even if there exists a true trade-o¤ between making the credit policy more e¢cient and
reducing the costs of drawing on the second lender’s capital.
13
If (A.6) does not hold, another case distinction applies. For x + z ¸ k it is possible to implement the e¢cient
16
Our …rst result is that t P takes again the shape of a debt contract. As argued earlier,
debt allows to implement a more e¢cient credit policy and to save on using the borrower’s
wealth. Additionally, it now allows to make the lender’s share of …nancing as large as possible
without a¤ecting its credit policy. To see this, observe …rst that the lender’s threshold s¤ is now
determined by the requirement
P
S ¤
µ2£ ¼µ (s )Uµ (t)
= L1. For given claims the lender is clearly
more conservative the higher its investment L1 . We also know that this shift is less pronounced
the less sensitive the lender’s payo¤ is to the di¤erence between projects and that debt minimizes
this sensitivity. As a consequence, debt allows to implement a given credit policy while keeping
the second lender’s investment L2 as low as possible.
If the lender must leave the other parties with a share of the rents, we are now still able
to implement the e¢cient policy by drawing on the second lender’s funds but not on the less
liquid wealth of the borrower. The optimal investment of the second lender, i.e., the minimum
investment at which this is feasible, strictly increases in V R. For higher values V R implementing
the e¢cient credit policy requires to draw additionally on the borrower’s wealth. Finally, the
e¢cient policy is no loner feasible if the rents that must be ceded by the lender are too high.
Proposition 3. Suppose there is an additional lender without access to the testing technology. The respective loan sizes and the lender’s claims are uniquely determined. The payout
to the lender (tP ; tC ) is uniquely determined, where tP takes the form of debt. Starting from
V R = 0, the lender’s loan size decreases from L1 = k to L1 = k ¡ z. As V R increases further,
the borrower’s wealth is additionally used according to Figures 1-2.
Proof. See Appendix.
It is important to note that the nature of the lender’s claim makes it senior to any claim held
by the second lender. In particular, if the second lender also received debt, this would have to
be junior to that of the lender. There is indeed ample evidence that lender …nancing is typically
senior and secured (e.g., Mann 1997 and Schwartz 1997.)14
From our previous remarks we obtain once more the following implications for the optimal
credit policy. More implications from Proposition 3 are derived in Section 5.
Corollary 3. Under the conditions of Proposition 3 the credit policy is e¢cient for low
values V R and too conservative for high values V R. In the latter case the threshold s¤ strictly
increases in V R.
Proof. See Appendix.
policy without using the borrower’s wealth, while for x + z < k this is no longer feasible for high values V R .
14
An often quoted exception are trade credits.
17
4.3
Example
We make the same speci…cations for the production and testing technologies as in the example
presented in Section 3.3. Hence, there are two outcomes, where the high outcome x is realized
for sure by the high-type project and the low outcome x is realized for sure by the low-type
project. Moreover, the speci…cation of the testing technology and of prior beliefs gives rise to
the posterior ¼ Sh (s) = s and the …rst-best threshold s¤¤ = (1 ¡ x)=(x ¡ x).
If the lender makes a loan of size L1 and receives all proceeds in the bad state and R in
the good state, its optimal policy is to set s¤ such that s¤ R + (1 ¡ s¤ )x = L1 . This yields the
threshold s¤ = (L1 ¡ x)=(R ¡ x), which is only e¢cient if
R=
(1 ¡ L1)x + x(L1 ¡ x)
:
1¡ x
(9)
The second lender invests the residual amount 1 ¡ L1, where we speci…ed that k = 1. If the
e¢cient policy is implemented, the rent received jointly by the borrower and the second lender
is given by
i
1h
1
1 ¡ (s¤¤)2 [x ¡ R ¡ L2] ¡ [1 ¡ s¤¤ (2 ¡ s¤¤)] L2:
2
2
R
Substituting R from (9) and setting V = V , we obtain that the e¢cient policy is implemented
V =
if the lender’s loan satis…es
VR
;
W ¤¤
implying for the second lender the investment of L2 = (1¡x)V R=W ¤¤. The investment required
L1 = 1 ¡ (1 ¡ x)
from the second lender is strictly increasing in the rents ceded by the lender. Adjusting the
second lender’s investment in this way is only feasible as long as it holds that
V R · W ¤¤
z
:
1¡ x
If this threshold is obtained or surpassed, all of the second lender’s funds are invested, i.e., it
holds that L2 = z. For higher values of V R we then have to draw additionally on the borrower’s
wealth to implement the e¢cient policy. The borrower pledges now the collateral c · w, which
is additionally paid to the lender in the bad state. The lender’s credit policy is now determined
by the requirement s¤ R + (1 ¡ s¤ )(x + c) = L1, where L1 = 1 ¡ z. To implement the e¢cient
policy, the pledged collateral must satisfy
c = (1 ¡ x)
x¡ R
x ¡x
¡z
:
x¡1
x¡ 1
(10)
Using (10) and the borrower’s wealth constraint, we obtain in analogy to the analysis in
Section 3.3 that the e¢cient policy is only feasible as long as
V R · W ¤¤
18
w+z
:
1¡ x
5
Implications
Our previous results focused mainly on the optimality of debt in governing the lender’s credit
policy. Additional results on the bene…ts of using collateral or of introducing a second lender
were only brought in as much as they shed more light on the contractual problem. We next use
our results to derive a series of wider implications.
Distribution of rents
While the lender has the power to design contracts, we speci…ed that it must leave the other
parties, i.e., the borrower and possibly the second lender, with a share of the realized rents,
which we denoted by V R . As noted above the distribution of rents may be determined by various
factors. One factor, on which we focus in what follows, is the availability of alternative sources
of …nancing or, more broadly, competition among potential lenders. The following implication
is then a restatement of Corollaries 1-3.
Corollary 4. As lending becomes more competitive, the credit volume is ine¢ciently reduced.
While this result holds always strictly if the lender is the single lender and the borrower has
no wealth, we know from Corollaries 2-3 that it only holds strictly for high values of V R. A
similar quali…cation will apply to most of the other implications derived next.
A recent empirical literature has documented the potential that increased competition between lenders may reduce small business loans, in particular if the respective industry is young
and risky (e.g., Petersen and Rajan 1995, Jackson and Thomas 1995, Cetorelli and Gambera
2001).15 To explain this phenomenon the literature has mainly relied on the conjecture that
competition reduces the scope for relationship banking (e.g., Petersen and Rajan 1995). By
Corollary 4 a reduction in lending can also occur in a static context. As the lender has to cede
more rents, its credit policy becomes increasingly conservative. This hold despite choosing the
optimal contract, which we showed to be debt. Incidentally and in contrast to an often heard
view, allowing banks to hold equity would therefore not alleviate borrowers’ credit constraints.
We consider next the impact of V R on the use of collateral. In empirical work it may only
be possible to measure the value of pledged collateral, i.e., the maximum amount the lender can
take possession of. Focusing on this measure, we obtain the following result.
Corollary 5. As lending becomes more competitive, the borrower will post more collateral.
Proof. See Appendix.
For the case of multiple lenders the fraction of the investment …nanced by the lender provides
an additional observable variable. We already know that the second lender’s stake increases
15
There is related evidence that following consolidation in the US banks cut back on their small-business loans.
Stein (2001) provides an explanation which builds on banks’ internal agency problems.
19
strictly in V R until his funds are exhausted. We restate this result as follows.
Corollary 6. As lending becomes more competitive, the fraction of the investment …nanced
by the lender decreases.
Availability of wealth
By Propositions 2 and 3 we can go a longer way in implementing the e¢cient credit policy
if the borrower has more wealth that can be pledged as collateral. And it is also intuitive that,
even if the e¢cient credit policy is no longer feasible, a rise in the borrower’s wealth allows to
obtain a more e¢cient policy.
Corollary 7. Increasing the value of the borrower’s wealth that can be posted as collateral
makes the lender less conservative and increases credit volume.
Proof. See Appendix.
Corollary 7 may point to interesting macroeconomic implications. Focusing on the initial
…nancing of small business, w may re‡ect, in particular, the value of property, which forms a
major part of households’ wealth. Hence, one implication would be that a decrease in house
prices may ine¢ciently reduce credit volume as lenders become too conservative.
Corollary 7 also provides an alternative view how …nancial constraints depend on the value
of borrowers’ existing assets, i.e., in our case the private wealth of the small business founder.
The more standard view is based on incentive problems of the borrower, who must exert noncontractible e¤ort or can hide, steal, or mis-allocate funds. The incentive problems of the
borrower reduce the ability to borrow against the future revenue stream, unless the lender’s
investment is safeguarded by the (sales) value of existing assets.16
Availability of a second lender
Our …nal implication follows immediately from the results of the analysis with multiple
lenders. If a second lender becomes available or if we increase his funds, we know from Proposition 3 that it becomes more feasible to implement the e¢cient credit policy. Moreover, it is
more often possible to draw on the second lender’s liquid funds instead of pledging the borrower’s
wealth as collateral in order to obtain the e¢cient policy. Finally, it can also be shown that,
even if the e¢cient credit policy is still not feasible, an increase in the second lender’s funds will
allow to implement a more e¢cient policy.
Corollary 8. If the borrower can get additional funds from a second lender, the lender’s
credit policy becomes more e¢cient and the collateral requirement decreases.
Proof. See Appendix.
16
For instance, b orrower moral hazard is assumed in the models of the lending channel in Bernanke and Gertler
(1989), of liquidity provision in Holmstrom and Tirole (1996), or of credit cycles in Kiyotaki and Moore (1997).
20
6
Informativeness of the Test
In this section we focus on the role of the test’s informativeness. We derive two results. We show
that our main results do not depend on the assumption that the test is imperfect. Moreover,
we show that a more informative test may even ine¢ciently reduce the credit volume.
6.1
Tests with Imperfect Information
To introduce the notion of the test’s informativeness, suppose that the test only provides an
informative signal with probability m. In this case the signal is again drawn out of the respective
distributions Fµ , which satisfy (A.3). With the residual probability 1 ¡ m the signal is fully
uninformative and represents a random draw from a uniform distribution over s 2 S. With this
speci…cation the posterior distribution after observing some score s becomes
¼ Sµ (s) = P
¼0µ [mfµ (s) + (1 ¡ m)]
:
0
µ 0 2£ ¼ µ0 [mfµ0 (s) + (1 ¡ m)]
(11)
We start by considering the …rst-best world. We restrict consideration to the case where
…nancing will not break down completely if the test is rather uninformative. This holds if the
expected value of the investment is strictly positive under the prior distribution.17
(A.7)
P
E 0
µ2£ xµ ¼µ
> k.
Suppose …rst the test’s informativeness is rather low, which implies that for any s 2 S the
posterior distribution is close to the prior distribution. By (A.7) the project should thus be
…nanced for any realization of the score, in particular also for the worst outcome s = 0. In other
words, if the test is rather uninformative, the e¢cient credit policy does not condition on its
outcome. As the informativeness increases, the posterior distribution becomes more and more
informative, in particular for extreme values of the score. By (A.3) this makes it optimal to reject
a project with a su¢ciently low score. Hence, there exists a threshold on the informativeness of
the test, which we denote by m¤¤ , such that the score shall be used to make a more informed
decision only if the informativeness exceeds this threshold.
To derive the threshold m¤¤, denote the posterior distribution at which the project just
breaks even by ¼ ¤¤
µ , i.e., it holds that
P
¤¤ E
µ2£ ¼ µ xµ
= k. At the threshold m¤¤ a borrower ending
up with the worst score s = 0 is expected to just break even. Formally, m¤¤ is thus de…ned by
¤¤
substituting ¼ Sh (0) = ¼ ¤¤
h into (11). If the informativeness exceeds m , it is e¢cient to apply a
threshold 0 < s¤¤ < 1.
17
Given (A.7) we can discuss when it is e¢cient and when it is optimal for the bank to use the testing technology.
If (A.7) did not hold, there would be no …nancing at all if the test was not su¢ciently informative. Hence, in the
latter case there would be no meaningful distinction b etween lending with or without using the test.
21
Lemma 2. There exists a threshold on the informativeness 0 < m¤¤ < 1 such that the
…rst-best credit policy does not condition on the test’s outcome for m · m¤¤ and sets a threshold
s¤¤ exceeding zero for m > m¤¤ .
Let us now return to the second-best world. Moreover, we focus on the case where the lender
is the single lender and where the borrower has no wealth, while the lender must cede some rents
due to V R > 0. By our previous arguments (Lemma 1) the lender will thus be too conservative,
i.e., it will only …nance the investment if its updated beliefs put on the high-type project a
probability that strictly exceeds ¼ ¤¤
h . It is again intuitive that the lender will not condition
its decision on the test’s outcome if this is rather uninformative. As the test’s informativeness
increases, there will be level m¤ from which on the lender will reject the borrower if the score is
too low.
Appealing now to Lemma 1, the lender’s conservative credit policy has two implications.
First, the lender will use the test to reject borrowers with a low score before it is e¢cient to do
so. Second, even if it is e¢cient to use the test to make a more informed decision, the lender’s
required threshold will again be too high. To mitigate the resulting ine¢ciency, a debt contract
will again be chosen.
Lemma 3. For V R > 0 the lender’s credit policy does not condition on the test’s outcome if
m · m¤, where 0 < m¤ < m¤¤ , and sets for all values m > m¤¤ a threshold s¤ > s¤¤ . Moreover,
for all values m ¸ m¤ a unique debt contract is again optimal.
Proof. See Appendix.
Note that if the test’s informativeness is rather low, a debt contract may no longer be
uniquely optimal. As the lender ignores the test’s outcome there is some scope for choosing
contracts that are more sensitive than debt to the project’s outcome.
Consider now an increase in the informativeness. Precisely, suppose that the original informativeness m1 and the new informativeness m2 satisfy m1 < m¤ < m2 < m¤¤ . By Lemma
2 it is in both cases e¢cient to make the loan regardless of the borrower’s score. In contrast,
if the lender can use the test with the higher informativeness, it will start to reject borrowers
with a low score. Likewise, if previously no test was available, the introduction of a test with
informativeness m2 ine¢ciently reduces credit volume.
Proposition 4. Introducing a credit test or improving its informativeness may lead to an
ine¢cient reduction in credit volume.
It is instructive to brie‡y rehearse the intuition for this result. If the lender must share some
of the rents, we know from Lemma 1 that the lender’s credit policy is too conservative in case the
test is su¢ciently informative. Its threshold on the required score strictly exceeds that applied
under the e¢cient policy. The comparison of the thresholds on the informativeness m¤ and m¤¤
22
in Lemmas 2-3 is simply another re‡ection of this result. In order to sort out borrowers with a
rather low score, the lender is willing to apply a test even if its low informativeness would not
warrant its use under the e¢cient credit policy. In this case the lender would be better o¤ if it
could commit not to screen the borrower at all. We have assumed throughout our analysis that
such a commitment is not feasible.
We next illustrate our results by means of an example. We explicitly derive the respective
thresholds m¤ and m¤¤ , thus con…rming Lemmas 3-4 and providing an example for Proposition 4.
Moreover, we can also show that even the introduction of a testing technology that is su¢ciently
informative to warrant its use under the e¢cient credit policy may lead to a reduction in surplus
under the lender’s optimal credit policy.
Example
The production technology is again chosen as in Sections 3.3 and 4.3. Hence, the high-type
project realizes for sure the high outcome x and the low-type project realizes for sure the low
outcome x. The prior probabilities are given by ¼0µ = 1=2 and the investment volume equals
k = 1. To satisfy (A.7) it must therefore hold that x + x > 2.
With probability m the score is generated by the testing technology used in the previous
examples, where we speci…ed that fl (s) = 2(1 ¡ s) and fh(s) = 2s. Using (11) we obtain for a
given score s that the posterior probability put on the high type satis…es
2ms + 1 ¡ m
¼ Sh (s) =
:
(12)
1+ m
At m = 0 the posterior is equal to 1=2 irrespective of the score. At m = 1 the posterior equals
just s. Note also that, as the test becomes more informative, the di¤erence ¼Sh (s)¡ ¼Sl (s), which
equals 2m(2s ¡ 1)=(1 + m), strictly increases for all values s 6= 1=2. Hence, unless the score is
just 1=2, in which case the posterior is always equal to the prior, a more informative test will
lead to a more precise posterior.
When is it now e¢cient to use the test? The project should always be …nanced if the
posterior distribution puts at least probability ¼¤¤
h = (1 ¡ x)=(x ¡ x) on the high type. By (12)
this threshold is surpassed even at the worst score s = 0 whenever
x¡ 1
m > m¤¤ =
;
1 + x ¡ 2x
where 0 < m¤¤ < 1.
(13)
We turn next to the second-best world. We …rst want to con…rm Proposition 4 by showing
that m¤ < m¤¤. To derive m¤, suppose that for a given value of V R the lender’s optimal credit
policy is indeed to ignore the outcome of the test. As the lender gets all proceeds in the bad
state and R in the good state, the borrower realizes V = (x ¡ R)=2, which by setting V = V R
transforms to
R = x ¡ 2V R:
23
(14)
The lender will only ignore the test if it is optimal to grant the loan even for the worst score s = 0.
In this case we obtain from (12) that the posterior puts probability ¼Sh (0) = (1 ¡ m)=(1 + m)
on the high type. Substituting R from (14) shows that the lender will only ignore the outcome
of the test in case
m · m¤ =
x ¡ 1 ¡ 2V R
:
1 + x ¡ 2x ¡ 2V R
(15)
Comparing …nally (13) with (15) con…rms that m¤ > m¤¤ holds whenever the borrower
receives a share of the rents.
Suppose …nally that m = 1. In this case the test should be clearly used under the e¢cient
credit policy. We construct now an example where the introduction of this testing technology
can nevertheless reduce surplus in the second-best world. First, note that without the test, i.e.,
for m = 0, the expected surplus equals [x+x¡ 2]=2. For V R > 0 we no longer obtain an explicit
solution for the realized surplus under the lender’s optimal credit policy. We therefore con…ne
ourselves to a numerical example and specify x = 0 and x = 8. Without the test the realized
surplus is then equal to W ¤¤ = 3. If the lender receives R in the high state, its policy is to set
s¤ = 1=R, which for R < 8 is strictly above the e¢cient threshold s¤¤ = 1=8. Substituting this
threshold, we obtain for the borrower the payo¤
V =
and the total surplus
1 (R2 ¡ 1)(8 ¡ R)
2
R2
(16)
3R2 ¡ 4 + R
:
(17)
R2
= 1:98, which by (16) generates the optimal repayment requirement R ¼
W¤ =
Suppose now V R
3: 734. Substitution into (17) yields the surplus W = 2:981, which is strictly smaller than the
surplus realized if no test was available.
6.2
Tests with Perfect Information
We argue now that our main implications survive if the lender can conduct a perfect test. This
follows immediately from our previous results if we change our set-up as follows. Suppose now
that there is a continuum of borrower types. We abuse our previous notation and denote a
borrower type by s 2 S. The payo¤ of type s is described as follows. The respective borrower
can be seen as holding a mixture of high- and low-type projects where the respective fraction of
project µ is given by ¼ Sµ (s). Hence, the fraction of the high-type project increases in the type s.
It is also equal to zero for s = 0 and equal to one for s = 1.
The borrower’s type is again not known a priori. In contrast to our previous speci…cation
the lender can now perfectly observe the borrower’s type when conducting the credit test. As
is easily seen all our previous results continue to hold, once we introduce the necessary changes
24
in terminology. In particular, the threshold on the testing score corresponds now to a threshold
on the perfectly observed borrower type. We state this result as follows.
Proposition 5. The obtained results extend to the case where the credit test generates
perfect information.
7
Conclusion
In this paper we focused on the case where testing loan applicants is the single function performed
by a lender, besides providing …nance. We showed that collateralized debt optimally governs
the lender’s credit policy. Building on the derivation of optimal contracts, we obtained a series
of implications concerning, amongst other things, the impact of competition between lenders or
of multiple lending on aggregate credit volume and the use of collateral.
Needless to say that the focus on credit tests must necessarily ignore other important tasks
performed by lenders, e.g., the monitoring of borrowers and the participation in corporate governance. For some lenders such as venture capitalists providing incentives to perform other
functions may have an additional important impact on the design of optimal contracts. Despite
these reservations, we feel that more theoretical work on the implications of credit testing seems
warranted. While credit tests have much gained in importance and while there is a substantial
literature on the statistical properties of (optimal) credit tests, the strategic aspects of credit
tests have been largely ignored. As noted in the introduction, a small literature focuses on the
resulting pool-externality if lenders conduct independent tests and do not share information
on rejected applicants. In this paper we focused instead on the contractual design problem,
which gave rise to various implications on the features of optimal contracts and the resulting
credit policy. The omissions of our paper suggest the following further avenues to investigate
the strategic aspects of credit testing.
First, we have treated the lender as a black box, characterized simply by its objective function
to maximize payo¤s. In reality loan decisions are made by self-interested individuals, whose
incentives are only aligned with those of the bank if either their actions are severely restricted
-as in our case- or if their compensation is structured accordingly. The (increased) use of credit
tests, which focus on hard information and constrain the decisions of loan o¢cers, may be part
of an optimal design to alleviate internal agency problems.
Second, we have not modelled competition between banks using more or less sophisticated
credit tests. While the literature has focused on the compliance with regulation or the accurate
determination of loss probabilities as the objectives governing the design of credit tests, it seems
worthwhile to explore more broadly the use of credit tests as a strategic tool in a competitive
environment. Will all banks see a comparative advantage in developing sophisticated tools to
25
crunch hard information before granting or extending loans? Or will (some) banks buck the
trend and focus on the development of relationship-banking with a particular clientele, thereby
relying heavily on information which does lend itself less as an input to a sophisticated test?18
8
Appendix A: Endogenization of Contractual Restrictions
We assume that the score s is the lender’s private information. Additionally, the lender has
always the option to reject the borrower without any further consquences. The latter assumption
follows courts cannot distinguish between the cases where the borrower has been rejected or
where he has not applied at all. Alternatively, even if the application was ver……able, the existence
of a large pool of fraudulent applicants would likewise rule out any penalty paid by the lender
in case of rejection, which is all we need.19
By standard arguments we can restrict the lender’s o¤ers to direct mechanisms of the following type. The lender o¤ers a menu of contracts. After acceptance by the borrower the lender
receives the signal s and is free to choose any contract in the menu or to reject the borrower.
As the two sides are risk neutral, we only consider deterministic transfers. A menu is then a
collection (t i)i2I of repayment schemes, where I denotes some index set and each contract t i is
chosen for some s 2 S and satis…es (A.4).
We argue next that the optimal menu contains only a single contract ti . As the lender must
break even for any implemented ti, it is immediate that Lemma 1 still holds, implying that the
optimal menu minimizes the ine¢ciency from a too conservative policy. In a slight abuse of
notation denote by s¤ the lowest score for which the lender does not reject the borrower. We
argue …rst that the contract chosen in this case, which we denote by t¤, must be debt.20 If this was
not the case, it follows from the argument in Proposition 1 that we can …nd some debt contract t^
with the following characteristics. First, given the realization s¤ and the respective beliefs ¼Sµ (s¤),
the payo¤ of both sides does not change. Second, if the lender does not change its choice of
contracts for any s > s¤, the expected payo¤ of both the lender and the borrower strictly increase
as the lowest score for which the borrower is accepted decreases. Third, the new contract satis…es
Uh(^t) < Uh(ti ) and Ul (^t) > Ul (ti ). The latter fact now implies that the lender does indeed not
adjust its behavior for scores s > s¤. To see this, note that ¼Sh (s¤) < ¼Sh (s) holds for s > s¤ ,
P
P
which together with µ2£ ¼ Sµ (s¤)[Uµ (^t) ¡ Uµ (ti )] = 0 implies µ2£ ¼Sµ (s)[Uµ (^t) ¡ Uµ (ti)] < 0 for
18
On a related theme see Boot and Thakor (2001).
19
Precisely, assume that the project of a fraudulent borrower yields zero return, but gives the borrower some
private bene…ts. A fraudulent applicant can be detected for sure. Any penalty paid to the borrower in case of
rejection would a attract the (by assumption su¢ciently large) pool of fraudulent applicants. If the penalty was
paid to a third party, a fraudulent applicant could still blackmail the bank and gain from an application.
20
In case of mixing t ¤ is in the support of the bank’s distribution if s¤ is realized.
26
s > s¤. We have thus shown that at the lowest score for which the applicant is accepted a debt
contract must be implemented.
Note next that the higher the borrower’s expected payo¤ for an s > s¤, the larger can we
choose the repayment requirement R ¤ in t¤ and the lower becomes s¤ (see Proposition 1). Hence,
if the menu contains any other set of contracts that is chosen with positive probability and leaves
the lender with a strictly higher payo¤ than under t¤, this is not optimal as eliminating all of
these contracts would rise the borrower’s payo¤ and allow us to adjust R¤ and s¤ accordingly.
Suppose that t0 6= t¤ is chosen for some s0 > s¤. By ¼Sh (s0 ) > ¼Sh (s¤ ) incentive compatiblity
requies Uh(t0 ) > Uh(t¤), Ul (t 0) < Ul (t¤ ), and
£
¤
£
¤
Uh(t0 ) ¡ Uh(t¤ ) ¼Sh (s0) + Ul (t0 ) ¡ Ul (t ¤) ¼Sl (s0 ) ¸ 0:
(18)
As ¼ Sh (s) increasing in s, it follows by (18) that for any s > s0 the lender’s payo¤ from choosing
t0 strictly exceeds that from implementing t¤. This rules out the case s0 < 1 and thus proves
our assertion that the menu must not contain any other set of contracts besides t¤ that is
implemented with positive probability.
9
Appendix B: Proofs
Proof of Lemma 1. For our following arguments it is helpful to restate (4) as a condition on
the posterior probability put on the high type:
¼ Sh (s¤) =
k ¡ Ul (t)
:
Uh(t) ¡ Ul (t)
(19)
Using (A.2) and (A.3), we obtain the e¢cient threshold s¤¤ in analogy to (19) by the requirement
¼Sh (s¤¤) =
k ¡ xE
l
E:
xE
¡
x
h
l
(20)
Compare now (19) with (20). The right-hand side of (19) is decreasing in Uh(t). To see that it
is also decreasing in Ul (t), note that the sign of the respective derivative is determined by the
di¤erence k ¡ Uh(t), which is strictly negative by (3). The assertion follows as ¼Sh (s) is by (A.3)
strictly increasing in s. Q.E.D.
Proof of Proposition 1 and Corollary 1
We show …rst that any payout t that is not debt cannot be optimal. We argue to a contradiction. Suppose thus that a contract t that is not a debt contract is optimal. Clearly, we
can restrict consideration to the case where V R > 0, implying from Lemma 1 that s¤(t) > s¤¤ .
We construct now a contract t~ which realizes a strictly higher payo¤ for the lender, while still
satisfying all constraints. Suppose for a moment that the lender’s credit policy is kept …xed at
27
the threshold s¤(t). Using this threshold, the lender’s expected payo¤ from a debt contract with
repayment R equals
X
µ2£
¼ 0µ [1 ¡ Fµ (s¤(t))]
Z
x2X
(21)
min fx; Rg gµ (x)dx;
which is continuous and strictly increasing in R. We next determine the unique repayment level
~ under which the lender’s payo¤ using (21) is equal to the payo¤ under the original contract
R
U (t). Denote the respective debt contract by t~.
Conditional on passing the threshold s¤(t), beliefs put on type µ the probability
¼
~ µ := P
¼0µ [1 ¡ Fµ (s¤(t))]
:
0
¤
µ 02£ ¼ µ0 [1 ¡ Fµ 0 (s (t))]
As ¼ Sh (s) is strictly increasing in s, it holds for s¤ < 1 that ¼
~ h > ¼Sh (s¤) and ¼
~ l < ¼Sl (s¤). By
construction the lender’s payo¤ under ~t is equal to that under t, given that the same threshold
s¤(t) applies. As this requires
X 0
X 0
£
¤
¼µ [1 ¡ Fµ (s¤ (t))] [Uµ (t) ¡ k] =
¼µ [1 ¡ Fµ (s¤ (t))] Uµ (~t) ¡ k ;
µ2£
(22)
µ2£
it must also hold that
£
¤
Ul (t) + ¼
~ h [Uh(t) ¡ Ul (t)] = Ul (~t) + ¼
~ h Uh(t~) ¡ Ul (~t) :
(23)
Expressing the shift from t to t~ by the di¤erence ¢Ul = Uh(t~) ¡ Ul (~t), the “true” credit
policy satis…es by (19)
¼ Sh (s¤(~t)) =
k ¡ Ul (t) ¡ ¢Ul
;
Uh(t) ¡ Ul (t) ¡ ¢Ul =~¼h
(24)
where we used (23). We argue next the true credit policy di¤ers from that under t in the way
that s¤(t~) < s¤(t) holds strictly. We proceed in two steps. We show …rst that the construction
of the debt contract together with (A.3) implies ¢Ul > 0. We then show that the right side in
(24) is strictly decreasing in ¢Ul .
For the …rst step note that t(x) must satisfy the monotonicity requirement (A.4). By the
construction of ~t this implies existence of a threshold x
~ such that ~t(x) · t(x) holds for x · x
~ and
~t(x) ¸ t(x) holds for x ¸ x
~ , where the strict inequality holds over sets of positive measure. Note
R
£
¤
next that another way of writing condition (22) is x2X [gl (x)~¼l + gh (x)~
¼h] t(x) ¡ ~t(x) dx = 0,
which transforms to
Z
x2X
£
¤
°(x)gl (x) t(x) ¡ ~t(x) dx = 0;
(25)
where we use °(x) = 1+ ¼
~ hgh(x)=[~
¼ l gl (x)]. As °(x) is positive and by (A.1) strictly increasing, it
R
£
¤
~
follows from (25) that the di¤erence ¢Ul = x2X gl (x) t(x) ¡ t(x)
dx is indeed strictly negative.
For the second step we di¤erentiate (24) with respect to ¢Ul . The sign of the derivative is
28
determined by the term [k ¡ Ul (t)] ¡ ¼
~ h[Uh(t) ¡ Ul (t)];which by the de…nition of ¼Sh (s¤(t)) and
by ¼
~ h > ¼Sh (s¤(t)) is indeed strictly negative.
Recall now that by construction of t~ the payo¤ of the lender and therefore also that of the
borrower would be equal to U(t) and V (t), respectively, if the credit policy s¤ (t) applied. As
~ < s¤(t), both sides must
we have, however, shown that the true credit policy satis…es s¤ (t)
be strictly better o¤ under the new contract. For the lender this follows immediately from
the optimality of s¤(~t). As the lender applies the threshold s¤(~t), which increases the ex-ante
~ < x the expected payo¤ of the borrower is strictly
probability of acceptance, and as from R
positive regardless of the obtained core, also the borrower’s payo¤ increases. Summing up, the
new contract satis…es all constraints and strictly realizes a higher payo¤ for the lender. We have
thus shown that any non-debt contract is dominated by a debt contract.
It remains to show that, restricting the set of contracts to debt, the lender’s problem has a
unique solution. Clearly, by optimality the borrower only receives V R. The borrower’s payo¤,
which equals
X
µ2£
¼0µ [1
¤
¡ Fµ (s )]
Z
x2X
max f0; x ¡ Rg gµ (x)dx
for a given repayment requirement R, is continuous in R and equal to zero at the two boundaries
R = x and R = x. We thus obtain for all feasible values V R a compact set of values R for
which the participation constraint binds. Denote this set by R¤ (V R). As the lender’s payo¤ is
strictly increasing in R, max R¤(V R) must be the unique solution. This completes the proof of
Proposition 1.
For the assertion in Corollary 1, note …rst that max R¤ (V R) is strictly decreasing. This
follows immediately from the fact that the borrower’s participation constraint must bind. Observe next that the threshold s¤ becomes strictly higher if the repayment requirement R decreases. This follows from the de…nition in (19) as di¤erentiating Ul (t) with respect to R yields
1¡ Gl (R) > 0, while di¤erentiating Uh(t) ¡Ul (t) with respect to R yields Gl (R) ¡Gh(R), which
by (A.3) is also strictly positive for all x < R < x. Hence, we have shown that s¤ is strictly
increasing in V R. Q.E.D.
Proof of Proposition 2 and Corollary 2
We use the decomposition of the transfer t into the distribution of proceeds tP and the
distribution of private wealth tC . A contract is thus fully described by the loan size L and
a pair (t P ; t C ). We …rst set up the problem more formally. Tthe lender’s payo¤ must be
maximized subject to the borrowers’ participation constraint and (A.4). This problem was
solved for Proposition 1, albeit with the restriction to w = 0 and therefore t C (x) = 0 for all
x 2 X . If the set of solutions to the this problem is not a singleton, we have to choose the
29
contract that minimizes the expected liquidated wealth, which we denote by
C :=
X
µ2£
¼0µ [1 ¡ Fµ (s¤ )]
Z
x2X
tC (x)gµ (x)dx:
(26)
We prove …rst two auxiliary results.
Claim 1. For all x 2 X, tC (x) > 0 implies tP (x) = x. Moreover, it holds that L = k.21
Proof. Suppose tC (x) > 0 and t P (x) < x over some set X 0 µ X. This implies existence of
a set X 00 µ X 0 and a value " > 0 such that tC (x) > " and tP (x) < x ¡ " for all x 2 X 00 . We
can then construct a new contract satisfying ~tP (x) = tP (x) + " and ~tC (x) = tC (x) ¡ " for all
x 2 X 00 and t~P (x) = tP (x) and t~C (x) = tC (x) for all x 2 XnX 00. This operation leaves total
transfers unchanged for all outcomes, but decreases the expected transfers from private wealth.
~ = k, ~t(x) = t(x) + k ¡ L,
Suppose next that L < k. We now transform the contract by setting L
©
ª
~tP (x) = max ~t(x); x and t~C (x) = ~t(x) ¡ ~tP (x). The new contract satis…es all constraints,
leaves payo¤s unchanged, and reduces the expected transfer from private wealth. Q.E.D.
Claim 2. tC (x) ¡ tC (x0 ) · x0 ¡ x holds for all x0 > x.
Proof. The assertion follows by combining (A.4) with Claim 1. We have to treat three cases
separately. Note …rst that by (A.4) and the de…nition of transfers it must hold that
tC (x) ¡ tC (x0 ) · tP (x0) ¡ tP (x):
(27)
The assertion follows immediately if tC (x) · tC (x0 ). For tC (x) > t C (x0 ) we distinguish
between two cases. Suppose …rst that tC (x) > 0 and tC (x0 ) > 0, implying by Claim 2 that
tP (x) = x and t P (x0 ) = x0. Substituting these values into (27) proves the assertion. Finally,
suppose that tC (x) > 0 and tC (x0 ) = 0, implying tP (x) = x, the right side of (27) transforms to
tP (x0) ¡ x, which is surely bounded from above by x0 ¡ x. Q.E.D.
We prove next that transfers (tP ; tC ) are uniquely determined and that it can be characterized
as follows. First, tP is a debt contract with some repayment RP . Second, tC follows the
description in Figure 1 and can thus be fully described by some value xP from which on it holds
n
o
that tC (x) = 0. (Precisely, for xC · w it holds that tC (x) = max xC ¡ x; 0 , while for xC > w
n
o
it holds that tC (x) = w if x · xC ¡ w and tC (x) = max xC ¡ x; 0 if x > xC ¡ w.) We denote
the respective set by (tP ; t C ) 2 T P £ T C .
Claim 3. The solution must be an element of T P £ T C .
Proof. Assume that a pair (tP ; tC ) 2
= T P £ T C solves the problem. We transform this
contract in the following way. First, we transform tP into t~P . Assuming that s¤ is kept constant,
21
In what follows, we use the assertion that some claim holds for all x 2 X interchangeably for the claim that
it holds for all x 2 X but a set of measure zero.
30
~tP 2 T P is uniquely determined by the requirement that payo¤s for both sides remain constant.
(Note that both payo¤s are continuous and strictly monotonic in the repayment level R~ P .)
Second, we transform tC into t~C . Assuming that s¤ is kept constant, t~C 2 T C is uniquely
determined by the requirement that payo¤s for both sides remain constant. (Note that both
payo¤s are continuous and strictly monotonic in the value x
~ C .)
From the argument in Proposition 1 we know that the …rst operation leads to a decrease in
the lender’s optimal threshold. (Recall that this followed from the MLRP condition (A.1) and
the fact that tP and t~P cross exactly once unless t P 2 T P .) We argue next that the same holds
for the second operation. To see this, note that by Claims 2 t C and ~tC cross exactly once, i.e.,
there exists a single value x
~ 2 X such that ~tC (x) ¸ ~tC (x) holds for x · ~x and t~C (x) · t~C (x)
holds for x ¸ x
~, unless tC 2 T C :
~P ,
If s¤(~t) < s¤¤ holds we have to perform another operation. In this case we have to adjust R
which characterizes t~P , and x
~C , which characterizes t~C , in the following way. Keeping again s¤
~ P and simultaneously adjust x
constant, i.e., at s¤ (t), we increase R
~C such that payo¤s remain
constant until the true threshold s¤ (t~) satis…es s¤(~t) = s¤¤ . By Lemma 1 a solution satisfying
x
~C > 0 exists. Note that this operation reduces the expected transferred wealth. Denote the
~ 22
respective realization by C.
We are now in the position to complete the proof, comparing t with the newly constructed
contract t~. Assume …rst that s¤(t) = s¤¤, implying by our operations that also s¤(~t) = s¤¤ (t) and
~ < C must hold, we obtain a contradiction
that payo¤s are therefore unchanged. As, however, C
to the optimality of t.
Assume next that s¤(t) > s¤¤. Note …rst that our construction implies s¤¤ · s¤ (~t) < s¤(t).
As the lender optimally readjusts its threshold following our change of contracts, it follows by
construction that the lender is strictly better o¤ under ~t. It thus remains to show that the
borrower is not worse o¤ such that his participation constraint is still satis…ed. Given our
construction, the borrower’s payo¤ would not be a¤ected if the lender stuck to its policy of only
accepting types s ¸ s¤ (t). Hence, it is su¢cient to prove that the borrower’s expected payo¤
~ s¤ (t)). We argue to a contradiction,
is non-negative after acceptance for realizations s 2 [s¤ (t);
implying that for some corresponding score s and respective beliefs ¼Sµ (s) the lender’s expected
payo¤ exceeds the expected total proceeds, i.e., that
S
E S
Uh(t~)¼ Sh (s) + Ul (~t)¼Sl (s) > xE
h ¼ h (s) + xl ¼ l (s):
(28)
At s = s¤(~t) it holds by construction of the optimal policy that
Uh (~t)¼Sh (s) + Ul (~t)¼Sl (s) = k;
22
(29)
Note that we do not claim for this operation that s¤ (~
t) is monotonic in x~C , which characterizes t~C . If there
~
are more than one intersections where s¤(~
t ) = s¤ ¤ holds, we may choose the one with the lowest realization of C.
31
while by s¤(~t) ¸ s¤¤ it holds at s = s¤ (~t) that
S
E S
xE
h ¼h (s) + xl ¼ l (s) ¸ k:
(30)
By (29) and (30) the inequality (28) clearly cannot hold at s = s¤(~t). Hence, to ensure that (28)
holds for higher values of s and thus higher values of ¼Sh (s), we obtain from di¤erentiating (28)
the requirement
E
Uh(~t) ¡ Ul (~t) > xE
h ¡ xl :
(31)
We show that this cannot hold. Observe that by construction of ~tC , which is non-increasing,
and by (A.1) it holds that Uh(t~C ) · Ul (~tC ). Then (31) implies the requirement
E
Uh (~tP ) ¡ Ul (~tP ) > xE
h ¡ xl :
(32)
^ which is characterized by the repayment level
To contradict (32) take some debt contract t,
^ and de…ne the function ¢(R)
^ = Uh (^t)¡ Ul (t).
^ ¡Gh(R),
^ which
^ Observe that d¢=dR^ = Gl (R)
R,
^ 2 (x; x). This together with ¢(x) = xE ¡ xE contradicts the
by (A.1) is strictly positive over R
h
l
~P .
claim that (32) holds for some ~tP and repayment level R
It remains to consider the case where s¤(t) < s¤¤ . (In contrast to the analysis for w = 0
such a contract is now feasible.) By our operations this implies s¤(t~) = s¤¤ . Hence, in contrast
to the original contract there is now no acceptance for scores s 2 [s¤(t); s¤¤). Recall that by
construction payo¤s are unchanged if the original threshold s¤(t) is kept …xed. As the lender
optimally adjusts its policy, its payo¤ is strictly higher under the new threshold s¤¤ . It thus
again remains to show that the borrower is not made worse o¤, which is surely the case if his
expected payo¤ is non-positive for realizations s 2 [s¤ (t); s¤¤ ). This follows from an argument
analogous to that used for the case s¤(t) > s¤¤ . Precisely, as it holds again at the lender’s
~ = s¤¤ that Pµ2£ ¼Sµ (s¤¤)Vµ (t)
~ = 0, it is su¢cient that the borrowers’
optimal threshold s¤(t)
expected payo¤ increases in s. This follows again by contradicting (31) and (32). Q.E.D.
By Claim 3 the lender’s program has only the two variables xC and RP , while (A.4) transforms to the requirement RP ¸ xc. As in Proposition 1 the borrower’s participation constraint
must again bind, which implies that the lender maximizes the joint surplus subject to the program’s constraints. It is then immediate that the optimal contract implements a credit policy
satisfying s ¸ s¤¤. If the …rst-best policy cannot be achieved, i.e., if s¤ > s¤¤ holds under an
optimal contract, it must hold that xC = RP , i.e., the constraint (A.4) must be binding. If this
did not hold, we could use xC < R P and by the arguments used in Claim 3 …nd a better contract
where a lower threshold s¤ is applied as we increase xC and decrease RP . If it is possible to
obtain the e¢cient threshold, i.e., if s¤ = s¤¤ holds under a contract satisfying the constraints,
we argue next that the optimal contract is still unique. This follows from the second objective to
32
minimize C. While there may be many pairs (xC ; RP ) implementing the e¢cient policy, there
is clearly a single pair with the lowest value xC . (Note that by Lemma 1 this satis…es xC > 0.)
We have thus obtained the following results.
Claim 4.
i) Under an optimal contract it must hold that s¤ ¸ s¤¤ .
ii) If s¤ > s¤¤, there is a unique optimal contract and it satis…es xC = RP .
iii) If s¤ = s¤¤ , there is a unique optimal contract.
To later prove Corollaries 4-8, it is helpful to introduce some additional notation. We claim
that there are two values 0 < V1R < V2R such that the e¢cient policy is obtained only for
V R · V2R, while additionally xC ¸ w holds for V R ¸ V1R. We turn …rst to V2R .
Claim 5. There exists a value V2R < 0 such that for V R > V2R it holds that s¤ > s¤¤, while
for V R · V2R it holds that s¤ = s¤¤.
Proof. By Claim 4 we know that an e¢cient credit policy can be implemented if and only
if this is possible by a contract t that can be written as
t(x) = w + max fx; Rg
for some value R. De…ne U µ (R) :=
R
x2X
(33)
min fx; Rg. Hence, an e¢cient credit policy requires
k ¡ w ¡ U l (R)
= ¼Sh (s¤¤);
U h(R) ¡ U l (R)
(34)
E
E
where the right side is by de…nition also equal to (k¡xE
l )=(xh ¡xl ). It follows that the left-hand
side of (34) is below ¼ Sh (s¤¤) at R = x, while it is immediate that it exceeds ¼Sh (s¤¤ ) as R ! x.
(Note that we use here (A.5) by which k ¡ w ¡ U l (x) > 0:)
Moreover, it is continuous and strictly decreasing in R, which follows as U l (R) is surely
increasing and as by the MLRP in (A.1) the di¤erence U h(R) ¡ U l (R) is increasing. We can
thus conclude that the requirement (34) has a unique solution for R, which we denote by R.
Substituting R into (33) we can calculate the borrowers’ expected payo¤, which we denote by
V2R and which is strictly positive for w > 0. Q.E.D.
Claim 6. For V R ¸ V2R it holds that s¤ increases in V R.
Proof. From Claim 4 we know that for V R ¸ V2R it holds that xC = RP = R. As
V R decreases, it is immediate that this implies an increase in R. Moreover, s¤ is by previous
arguments (see the proof of Proposition 1) strictly decreasing in R. Q.E.D.
Claim 6 together with the preceding results concludes the proof of Proposition 2 and Corollary 2. We …nally turn to V1R and prove the following result, which is used later for Corollaries
4-8.
33
Claim 7. There exists a threshold V1R satisfying 0 < V1R < V2R such that xC < w holds for
V R < V1R and xC ¸ w holds for V R ¸ V1R. Moreover, for V R < V1R it holds that xC increases
in V R.
Proof. Suppose the e¢cient credit policy is obtained for some level V R and a corresponding
payo¤ of the lender U = W ¤¤ ¡V R. Fixing the e¢cient credit policy, this payo¤ can be obtained
for a range of values (RP ; xC ). Precisely, starting from xC = 0 and the corresponding value RP ,
we can increase xC and decrease RP until we obtain U for values xC = RP . An increase in
xC and a corresponding decrease of RP shift the lender’s total payo¤ t in way such that the
transfer is higher for lower values of x and lower for higher values of x, implying by (A.1) that
the true credit policy s¤ (t) strictly decreases. If s¤ = s¤¤ is feasible (which by Claim 5 holds up
to V R = V2R), we thus obtain for given U a unique pair (RP ; xC ). Moreover, as we increase U
the respective value RP must decrease and the respective value xC increase. This proves the
monotonicity of xC and the existence of the threshold V1R. Q.E.D.
Proof of Proposition 3 and Corollary 3
We prove …rst the following results, where we apply by now standard arguments.
Claim. There exists V0R > 0 such that we can implement the e¢cient credit policy only for
values V R · V0R if we require additionally tC (x) = 0 for all x 2 X. Moreover, in these cases a
unique debt contract is optimal, while L1 strictly decreases to L1 = k ¡ z at V R = V0R.
Proof. Note …rst that the optimal threshold depends now both on the loan size L1 and the
transfer t. Precisely, we obtain
s¤(t; L1 ) =
L1 ¡ Ul (t)
:
Uh(t) ¡ Ul (t)
(35)
Suppose we can implement the e¢cient policy for a given value V R, while tC (x) = 0 holds
for all x 2 X. In this case our objective is to minimizes L2. We show that this gives rise to a
unique optimal contract, where t P takes the shape of debt. We apply a similar argument as in
Propositions 1-2.
Suppose thus that an optimal contract specifying t = tP and L1 is not debt. By Lemma 1 it
must hold for V R > 0 that L1 < k. Then we can again …nd a debt contract t~ such that, keeping
s¤ = s¤¤ and L1 …xed, payo¤s do not change, while the true threshold is strictly smaller than s¤¤ .
~ 1 > L1 and increase the repayment R
~
We propose now to simultaneously adjust the loan to L
until the true threshold is again equal to s¤¤, while payo¤s remain unchanged. If this is feasible,
we have contradicted the optimality of the original contract.
Keeping s¤¤ …xed, inspection of the lender’s payo¤ reveals that a change in L1, which we
denote by ¢L1, must be accompanied by changes in Uµ (~t), which we denote by ¢Uµ such
that ¢L1 =
P
~ µ ¢Uµ ,
µ2£ ¼
where ¼
~ µ was de…ned in the proof of Proposition 1 and denotes
34
the probability conditional on passing the …rst-best threshold s¤¤. Substituting into (35) and
denoting ¢ = ¢Uh ¡ ¢Ul reveals that the new threshold after a change in L1 is determined by
¼ S (s¤) =
L1 ¡ Ul (t~) ¡ ¼
~ h¢
.
~
~
Uh(t) ¡ Ul (t) ¡ ¢
(36)
Di¤erentiating the right side of (39) with respect to ¢ reveals that the sign is determined by
~ ¡ Ul (t)]
~ ¡ [L1 ¡ Ul (~t)];which is strictly positive by the de…nition of s¤(L1; ~t)
the term ~¼h [Uh(t)
and by ¼ Sh (s¤ ) < ¼
~ h, which holds by construction. This completes the proof that the optimal
contract must be debt if we can implement the e¢cient credit policy while setting tC (x) = 0 for
all x 2 X.
To determine the maximum value of L1 that is still compatible with an e¢cient policy, we
set s¤(t; L1) = s¤¤ and rewrite (35) to obtain
L1 = ¼ Sh (s¤¤) [Uh(t) ¡ Ul (t)] + Ul (t):
(37)
Substituting into the lender’s payo¤ function yields next
h
i
U= ¼
~ h ¡ ¼ Sh (s¤¤) [Uh(t) ¡ Ul (t)]
X
µ2£
¼0µ [1 ¡ Fµ (s¤¤)] :
(38)
As the lender receives a lower share of the rents, the requirement to implement the e¢cient
policy requires by (38) that the di¤erence Uh(t) ¡ Ul (t) decreases. The respective debt contract
has thus a lower repayment requirement, which also implies that Ul (t) decreases. Substitution
into (37) shows then that L1 must decrease. Proceeding like this up to L1 = k ¡ z yields the
threshold V0R. Q.E.D.
This claim together with the results of Propositions 1-2 proves now the assertions. In particular, for V R ¸ V0R we just have to replace k by the new loan size k¡z and proceed as previously.
Q.E.D.
Proof of Corollary 5
The assertion can be restated formally as showing that maxx2X tC (x) increases in V R (and
increases strictly for a non-degenerate subset). Using Proposition 3 we can restrict consideration
to the case where z = 0. Recall that the contract is described by two values (RP ; xC ). Moreover,
for V R < V1R, which was de…ned in Claim 7 of the proof of Proposition 2, it holds that xC < w,
while thereafter xC ¸ w implies max tC = w. Hence, we only have to show that xC strictly
increases over V R < V1R. This was proven in Claim 7 in the proof of Proposition 2. Q.E.D.
Proof of Corollary 7
Focusing again on z = 0, it remains by Proposition 2 to show that i) V2R increases in w
and that ii) a (marginal) increase in w reduces s¤ if V R > V2R . Regarding assertion i), note
that this follows immediately from the de…nition of V2R in Claim 5 in the proof of Proposition
35
2. Regarding assertion ii), note that by Claim 4 in the proof of Proposition 2 the lender’s total
transfer can be expressed as t(x) = w + max fx; Rg. Assertion ii) follows then from by now
standard arguments. Q.E.D.
Proof of Corollary 8
By Propositions 2-3 we have to prove the following results. The threshold V0R, from which
on collateral is used at all, must increase in ¼0h (assertion i). If we set then L1 = k ¡ z and
de…ne V1R and V2R from Proposition 2 accordingly, we must show that ii) V1R increases in ¼0h
and that iii) for values V R < V1R a (marginal) increase in ¼0h reduces xC . For assertion ii, we
make use of the Claim in Proposition 3. If the e¢cient policy is implemented, we can realize for
same investment L1 a particular value V R with a higher repayment requirement R. (Note that
Vh (t) > Vl (t) holds for the joint claims of the borrower and the second lender, while an increase
in ¼ 0h also increases the probability of passing the e¢cient threshold.) From the Claim in the
proof of Proposition 3 this allows to increase L1 without a¤ecting the e¢ciency of the credit
policy. This creates indeed scope for increasing V0R. Assertions ii and iii follow from the same
argument. Q.E.D.
Proof of Corollary 9
The assertions follow from substituting L1 = k ¡ z in the proof of Proposition 2. Precisely,
regarding the credit policy it then holds that V2R increases in z and s¤ decrease for V R > V2R .
And regarding the use of collateral, we obtain that V1R increases in z and xC decreases for
V < V1R. Q.E.D.
Proof of Lemma 3
We …rst provide a restatement of the …rst-best and of the optimal credit policy. Instead
of focusing on the minimum required score, which depends on the test’s informativeness, we
express the credit policy in terms of the required posterior distribution. The …rst-best policy is
then to reject the borrower if his posterior distribution puts less than probability
¼¤¤
h =
k ¡ xE
l
E
xE
¡
x
h
l
(39)
on the high type. This threshold on the distribution was already introduced in the main text.
We proceed likewise for the lender’s optimal policy. Given a transfer t the optimal credit policy
requires for the high type the probability
¼ ¤h(t) :=
k ¡ Ul (t)
:
Uh(t) ¡ Ul (t)
(40)
By (A.2) it holds that 0 < ¼ ¤¤
h < 1. By (A.7) we can also restrict consideration to contracts such
that …nancing is provided with positive probability. This requires ¼¤h(t) < 1 and, in particular,
36
Uh(t) > k, implying from the argument of Lemma 2 that ¼¤h (t) > ¼¤¤
h holds strictly unless the
lender obtains all proceeds.
Using the de…nition of the posterior distribution ¼Sµ (s) in (11), we obtain for each m a
threshold on the score such that the borrower should be rejected under the …rst-best credit
policy for s < s¤¤(m). Moreover, by the arguments of the main text we obtain s¤¤(m) = 0 for
values m · m¤¤, where 0 < m¤¤ < 1. Proceeding likewise for the lender’s optimal credit policy,
we obtain a threshold on the score s¤(m; t) that depends both on m and t and a threshold on
the informativeness m¤ (t) that depends on t. The following assertions regarding the lender’s
optimal credit policy follow immediately from the construction of these thresholds and from the
characteristics of the debt contract used in Proposition 1.
Claim. Given V R > 0, the following results hold for the set of all contracts under which
borrowers realize V R.
i) The unique debt contract in this set minimizes m¤¤(t).
ii) If m · m¤(t) holds when evaluated at the unique debt contract in this set, this contract
is uniquely optimal.
Setting m¤ = m¤(t) for the unique debt contract realizing V R, this claim together with a
comparison of (39) and (40) proves also that m¤ < m¤¤. This completes the proof of Lemma 3.
Q.E.D.
10
Appendix C: Implementation with Collateralized Debt
We …rst describe how di¤erent combinations of debt contracts a¤ect the two sides’ payo¤s and
the use of private wealth. A typical contractual arrangement, which implements the payo¤s of
Proposition 2, takes the following form. A loan is made by signing three separate debt contract
denoted by i 2 f1; 2; 3g, which have the respective face values Ri. Contract i = 1 has the highest
priority, contract i = 2 has intermediate priority and is fully secured, and contract i = 3 has the
lowest priority and is again unsecured. For an illustration suppose R1 > x. Then for outcomes
x 2 [x; R1] only the most senior debt is served from the proceeds, while the obligation under
i = 2 is fully met out of wealth. For outcomes x 2 (R1 ; R1 + R2 ] all proceeds exceeding R1, i.e.,
x ¡ R1, are used to serve i = 2, which reduces the amount of transferred wealth to w ¡ R1 + x.
Only for outcomes x > R1 + R2 does the borrower start to serve the most junior debt i = 3.23
23
We could also obtain this payout without relying on di¤erent seniorities. Instead of making the loan i = 1
senior, it could be secured with “internal collateral”, e.g., the liquidation value of purchased assets. This ensures
that all (liquidation) proceeds x · R1 are used to repay this loan. If the borrower has some leeway how to use
proceeds exceeding the pledged internal collateral, he will try to …rst serve the loan secured by external collateral,
thereby rescuing more of his private wealth. Though this may con‡ict with standard pro-rata rules in serving
37
We obtain the following result, where use the thresholds V1R and V2R de…ned in the proof of
Proposition 2.
Claim. The following …nancial arrangements implement the optimal contract if w > 0.
- V R · V1R : Finance is provided by two debt contracts. The senior debt is fully secured.
- V1R < V R · V2R: Finance is provided by three debt contracts. Debt with intermediate
seniority is fully secured with collateral w.
- V R > V2R : Finance is provided by two debt contracts. The junior debt is fully secured with
collateral w.
Proof. The results follow from the proof of Proposition 2. More precisely, we obtain the
following construction.
V R · V1R: Recall that in this case the optimal contract speci…es xC < w < RP . The senior
debt i = 1 has the repayment level R1 = xC and is fully secured with collateral xC . The junior
debt i = 2 is unsecured and has the repayment level R2 = RP .
V1R < V R · V2R: Recall that in this case the optimal contract speci…es xC = w < RP . The
senior debt is unsecured and has the repayment level R1 = xC ¡ w. The debt with intermediate
seniority i = 2 is fully secured with all wealth, i.e., R2 = w. The junior debt is unsecured with
repayment level R3 = R P + w ¡ xC .
V R > V2R: Recall that in this case the optimal contract speci…es xC = RP > w. The senior
debt i = 1 is unsecured with repayment level R1 = RP ¡ w. The junior debt i = 2 is fully
secured with all wealth, i.e., R2 = w.
11
References
Berger, A. N. and Udell, G. F., 2002, Small business credit availability and relationship lending:
The importance of bank organizational structure, forthcoming Economic Journal
Berlin, M., 2000, Why don’t banks take stocks?, Federal Reserve Bank of Philadelphia,
Business Review, May/June, 3-13.
Bernanke, B. and Gertler, M., 1989, Agency costs, net worth and business ‡uctuations,
American Economic Review 79, 14-31
Besanko, D. and Thakor, A.V., 1987, Collateral and rationing: Sorting equilibria in monopolistic and competitive credit markets, International Economic Review 28, 671-689.
Bester, H., 1985, Screening vs. rationing in credit markets with imperfect information,
American Economic Review 75, 850-55
outstanding debt in case of liquidation, a more realistic dynamic setting should give the b orrower substantial
freedom in serving debt that is not ordered by seniority.
38
Biais, B. and Mariotti, T., 2001, Strategic liquidity supply and security design, mimeo
Bolton, P. and Scharfstein, D.S, A theory of predation based on agency problems in …nancial
contracting, American Economic Review 80, 93-106.
Boot, A. and Thakor, A. V., 1994, Moral hazard and secured lending in an in…nitely repeated
credit market game, International Economic Review 35, 899-920
Boot, A. and Thakor, A. V., 2000, Can relationship banking survive competition? Journal
of Finance 55, 679-713
Broecker, T., 1990, Credit-worthiness tests and interbank competition, Econometrica 58,
4129-452
Brunner, A., Krahnen, J. P., Weber, M., 2000, Information production in credit relationships:
On the role of internal ratings in commercial banks, mimeo
Cao, M. and Shi, S., 2001, Screening, bidding, and the loan market tightness, European
Finance Review 5, 21-61
Cetorelli, N. and Gambera, M., 2001, Banking market structure, Financial dependence and
growth: International evidence from industry data, Journal of Finance 56, 617-48
Cole, R. A., Goldberg, L. G., and White, L. J., 1997, Cookie-cutter vs. character: The micro
structure of small business lending by large and small banks, mimeo, Stern/NYU
Cressy, R. and Toivanen, O., 1997, Is there adverse selection in the credit market? mimeo,
Warwick Business School
Dell’Arricia, G., Friedman, E. and R. Marquez, 1999, Adverse Selection as a Barrier to Entry
in the Banking Industry, Rand Journal of Economics 30, 515-534
DeMarzo, P. and Du¢e, D., 1999, A liquidity-based model of security design, Econometrica
67, 65-99
Diamond, D., 1984, Financial intermediation and delegated monitoring, Review of Economic
Studies 51, 393-414
English, W. B. and Nelson, W. R., 1998, Bank risk rating of business loans, Federal Reserve
Board, Washington
Feldman, R., 1997, Banks and a big change in technology called credit scoring, The Region,
Federal Reserve Bank of Minneapolis, September, 19-25.
Frame, W. S., Srinivasan, A. S., and Woosley, L., 2001, The e¤ects of credit scoring on small
business lending, Journal of Money, Credit, and Banking, forthcoming
Gale, D. and Hellwig, M., 1985, Incentive-compatible debt contracts: The one period problem, Review of Economic Studies 52, 627-645
Garmaise, M., 2001, Informed investors and the …nancing of entrepreneurial projects, mimeo,
University of Chicago.
Gehrig, T., 1998, Screening, cross-border banking, and the allocation of credit, Research in
39
Economics 52, 387-407
Hart, O. and Moore, J., 1994, A theory of debt based on the inalienability of human capital,
Quarterly Journal of Economics 109, 1119-58
Holmstrom, B. and Tirole, J., 1996, Modelling aggregate liquidity, AEA Papers and Proceedings 86, 187-191
Inderst, R. and Müller, H.M., 2001, Venture Capital Contracts and Market Structure, CEPR
Discussion Paper 3203.
Innes, R., 1990, Limited liability and incentive contracting with ex-ante action choices, Journal of Economic Theory 52, 45-67
Jackson, J. and Thomas, A., 1995, Bank structure and new business creation: Lessons from
an earlier time, Regional Science and Urban Economics 25, 323-353
James, C. and Houston, J., 1996, Evolution or extinction: Where are banks headed?, Journal
of Applied Corporate Finance 9 ,8-23
Kiyotaki, N. and Moore, J., 1997, Credit cycles, Journal of Political Economy 105, 211-48
Mann, R., 1997, The role of secured credit in small-business lending, Washington University
School of Law, mimeo
Manove, M., Padilla, A.J., and Pagano, M., 2001, Collateral vs. project screening: A model
of lazy banks, Rand Journal of Economics 32, 726-44.
Mester, L., 1997, What’s the point of credit scoring?, Federal Reserve Bank of Philadelphia,
Business Review, September/October 1997, 3-16.
Mester, L., 1999, Banking industry consoldiation: What’s a small business to do?, Federal
Reserve Bank of Philadelphia, Business Review, January/ February 1999, 1-16.
Myers, S. and Majluf, N., 1984, Corporate …nancing and investment decisions when …rms
have information investors do not have, Journal of Financial Economics 13, 187-221
Nachman, D.C. and Noe, T.H., 1994, Optimal Design of Securities under Asymmetric Information, Review of Financial Studies 7, 1-44
Park, C., 2000, Monitoring and structure of debt contracts, Journal of Finance 55, 2157-95
Petersen, M. A. and Rajan, R. G., 1995, The e¤ects of credit market competition on lending
relationships, Quarterly Journal of Economics 110, 407-444
Petersen, M. and Rajan, R., 1994, The bene…ts of …rm-creditor relationships: Evidence from
small business data, Journal of Finance 49, 3-37
Rajan, R.G., 1992, Insiders and Outsiders: The Choice between Informed and Arm’s Length
Debt, Journal of Finance 47, 1367-1400
Riordan, M., 1993, Competition and bank performance: A theoretical perspective, in C.
Mayer and X. Vives, eds., Capital Markets and Financial Intermediation, Cambridge Univ.
Press
40
Saunders, A. and Allen, L., 2002, Credit Risk Measurement. New York: John Wiley & Sons.
Saunders, A. and Thomas, H., 2001, Financial Institutions Mangement, 2nd Canadian edition. McGraw-Hill Ryerson: Toronto
Schwartz, A., 1997, Priority contracts and priority in bankruptcy, Cornell Law Review 82,
1396-1419
Sha¤er, S., 1998, The winner’s curse in banking, Journal of Financial Intermediation 4,
158-187
Stein, J., 2002, Information production and capital allocation: Decentralized vs. hierarchical
…rms, forthcoming Journal of Finance
Treacy, W. F. and Carey, M. S., 1998, Credit risk rating at large US banks, Federal Reserve
Bulletin 897-921
UK Competition Commission, The supply of banking services by clearing banks to small
and medium-sized enterprises: A report on the supply of banking services by clearing banks to
small and medium-sized enterprises within the UK, March 2002.
von Thadden, E.-L., 1995, Long-Term contracts, short-term investment and monitoring,
Review of Economic Studies 62, 557-575.
Walsch, C.E., 1998, Monetary Theory and Policy, MIT Press, Cambridge/Mass.
41