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European Economic Review 44 (2000) 1701}1726
The optimal size of a bank:
Costs and bene"ts of diversi"cation
Vittoria Cerasi *, Sonja Daltung
Dipartimento di Statistica, Universita% degli Studi di Milano, Bicocca } Via Sarca 202,
20126, Milan, Italy
Sveriges Riksbank, S-103 37 Stockholm, Sweden
Received 1 March 1997; accepted 1 September 1998
Abstract
This paper provides a theory of diversi"cation and "nancial structure of banks. It shows
that by diversifying the bank portfolio and "nancing it with debt, the bank can commit to
a higher level of monitoring. By linking the bene"ts of diversi"cation to the costs, the paper
derives an optimal size of the bank, which is bounded. The costs of diversi"cation lie in
the higher overload costs with which the banker is faced by monitoring more projects. The
bene"ts of diversi"cation lie in increasing the bank's owner's incentives to monitor the
lenders. 2000 Elsevier Science B.V. All rights reserved.
JEL classixcation: D82; G21; G32
Keywords: Financial intermediaries; Financial structure; Diversi"cation; Monitoring
1. Introduction
In the literature on "nancial intermediation there are several explanations as
to why banks should be diversi"ed and large. Firstly, in credit markets where
* Corresponding author. Tel.: #39-02-64487321; fax: #39-02-6473312.
E-mail address: [email protected] (V. Cerasi).
0014-2921/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved.
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information is asymmetrically distributed, banks evaluate projects and monitor
borrowers. As a consequence, bank assets are indivisible and illiquid. Diversi"cation enables the bank to "nance illiquid assets with liquid liabilities. Hence,
through diversi"cation the bank can provide liquidity services to investors.
Secondly, diversi"cation on the asset side reduces the variance of the returns
that accrue to claimholders of the bank, as bank portfolio gets closer to the
market portfolio; this is bene"cial when claimholders are risk-averse or if there
are bankruptcy costs. On the other hand, because of indivisibility of bank assets,
increasing diversi"cation implies increasing the size of the bank.
This suggests that there should be economies of scale in banking and that
"nancial intermediation should tend to be a natural monopoly. However,
although banking systems tend to be concentrated, in some developed countries
they are quite fragmented. Furthermore, in the empirical literature on banking
there is no sharp evidence in favor of economies of scale above a certain
threshold. On the contrary, there is often evidence that large banks su!er a cost
disadvantage.
One explanation, as we see it, for the discrepancy between the theory of
"nancial intermediation and the empirical evidence is that the theory has not
considered the issue of a bank's internal organization. Loans are monitored not
by the bank as such, but by people working for the bank. Since there is a limit to
the number of projects one person can monitor, monitoring more loans entails
overload costs.
In this paper we provide a theory of the optimal size of a bank, the function of
which is to monitor loans on behalf of small investors. We also provide a theory
of the "nancial structure of banks. The starting-point is that the outcome of
monitoring depends on the e!ort of the person performing the task, and
therefore delegation of monitoring involves an incentive problem for a monitor
protected by limited liability. We prove that diversi"cation improves the incentive of the banker to monitor, if the bank is debt "nanced. If the bank is "nanced
with equity only, diversi"cation does not have this e!ect.
This result can be intuitively explained as follows. The portfolio performance
depends, among other things, upon the level of monitoring by the banker. With
debt, the share of pro"ts that accrues to the banker, who in this case is the bank's
sole owner, depends on the performance of the bank's portfolio; with equity,
however, this share is invariant. Moreover, diversi"cation augments the impact
of increased monitoring on the banker's pro"t share. Hence, by changing the
banker's marginal incentive to monitor, diversi"cation acts as a commitment to
There is evidence that bank loans are not perfect substitutes for public debt, as for instance in
James (1987) and Lummer and McConnel (1989), and that banks reduce managerial discretion (see
Yafeh and Yosha, 1995).
See for instance Clark (1988).
V. Cerasi, S. Daltung / European Economic Review 44 (2000) 1701}1726
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increase the monitoring e!ort in the bank. This in turn improves bank performance, which reinforces the positive e!ect on incentives.
Another way of looking at this is to say that diversi"cation increases the
probability that the bank will be able to repay its debt, and reduces the expected
shortfalls on debt, that is, the expected loss of debtholders when the portfolio of
the bank performs poorly. This means that the standard debt contract approaches the full liability (&net residual claimant') contract, which, as we know
from the agency literature, provides the correct incentives.
Our results are complementary to previous research on "nancial intermediation in that we provide an additional reason for why banks should be diversi"ed and debt "nanced. In our model, the bank does not provide any liquidity
service, the claimholders of the bank are risk-neutral, and there are no bankruptcy costs, but diversi"cation improves the incentive of inside equity holders
to monitor, if the bank is debt "nanced.
Contrary to the banking literature centered on the asset substitution problem
of debt, we focus on the incentive problems arising from lack of inside capital, as
we believe that these are at least as important, while attracting much less
attention. It follows directly from the role of the bank as delegated monitor that
the bank will have very little inside capital, and we provide one explanation for
why the bank is still able to operate. One important implication of the analysis is
that the incentive problem arising from lack of inside capital cannot be solved
through capital constraints. On the contrary, our analysis suggests that for
a diversi"ed bank, outside equity is a costly source of "nance.
While the benexts of diversixcation derive from improving the banker's
monitoring incentives, the costs of diversixcation arise from the growing size of
the portfolio needed to achieve diversi"cation. The idea is that an individual's
monitoring capacity is limited, that is, it is increasingly costly for one person to
monitor an increasing number of projects. The cost of increasing the size of the
bank portfolio causes &overload' in terms of the number of projects the banker
has to monitor. By linking the bene"ts of diversi"cation to the costs, the paper
derives an optimal size of the bank, which is bounded.
Finally, in contrast to the corporate "nance view, this paper suggests that
conglomerates can be valuable. Our result shows that, when projects need to be
monitored and externally "nanced, managing a large portfolio of non-correlated
projects "nancing them with debt allows the "rm to commit to a higher level of
monitoring.
Thus, this suggests a trade-o! between the asset substitution problem and the monitoring
incentive problem, which implies that capital constraints migth be costly for banks. Boyd and
Gertler (1993) observe that for the US banking industry the equity to assets ratio has been steadily
declining since 1980.
See for instance Brealey and Myers (1991, pp. 854}856).
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In the next section we relate this paper to the literature. Section 3 presents the
model and the main assumptions. In Section 4 we analyze the banker's incentive
problem. Section 5 contains the main result on diversi"cation and discussion of
the role of external "nance. Section 6 concludes.
2. Relation to the literature
On the one hand, this paper is related to the literature on "nancial intermediation, and then especially to Diamond (1984), where "nancial intermediation
arises as an e$cient way of delegating monitoring. In Diamond diversi"cation
of the bank portfolio is bene"cial under debt "nancing, because it reduces the
probability that the bank will go bankrupt, and thereby the costs connected to
bank failure. We show that, even when there are diseconomies of scale in
monitoring and there are no costs of bankruptcy, the banker is able to raise
money from investors committing through diversi"cation of the portfolio to
monitor each project enough to avoid bad performances by entrepreneurs.
Finally, given that monitoring has diseconomies of scale, the optimal size of the
bank is limited, contrary to Diamond result of monopolization of the banking
sector. In a recent paper, Hellwig (1998) discusses the trade-o! between scale
and diversi"cation e!ects, by relaxing the assumption about "xed size of loans in
Diamond, and showing that in equilibrium the intermediary forgives diversi"cation opportunities.
On the other hand, this paper is related to the literature on incentives to
monitor. We exploit the idea that the "nancial structure may a!ect the monitoring level in the "rm. This idea is not new and it has been applied to banking by
Holmstrom and Tirole (1997) and Dewatripont and Tirole (1994).
Our vision of the main activity of banks is very similar to Holmstrom and
Tirole (1997), in that they assume that banks principally monitor entrepreneurs
and that banks can do this more e$ciently than credit markets. However, they
The paper has other technical di!erences with respect to Diamond (1984): in particular we have
ongoing monitoring instead of ex-post monitoring, and thus the delegated monitor must be
provided ex-ante with the incentives to monitor; furthermore, we do not have costs of bankrupcty,
while in Diamond it is because bankruptcy costs go to zero that delegation of monitoring to the
bank becomes viable with perfect diversi"cation.
Another way to question Diamond result is to introduce competition among intermediaries, as
done by Yanelle (1997).
Therefore, the paper is also related to the corporate "nance literature, but we defer the discussion
of this to Section 5.2.
Besanko and Kanatas (1993) also analyze bank moral hazard in monitoring, but they do not
consider the impact of the bank's "nancial structure on monitoring incentives.
V. Cerasi, S. Daltung / European Economic Review 44 (2000) 1701}1726
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focus on inside capital as an incentive mechanism, while we are more concerned
with diversi"cation of the bank portfolio.
Dewatripont and Tirole (1994) focus on the con#ict between the manager and
the bank's claimholders and show how the "nancial structure of the bank can be
used as an incentive mechanism for the manager, while we focus on how the
"nancial structure a!ects the incentive of the banker to monitor.
Aghion and Tirole (1997) analyze the incentive of the owner to monitor the
manager, when monitoring by the owner is not always good as it reduces the
manager's initiative. They suggest, among other things, the introduction of some
overload in order to reduce monitoring by the owner. The idea is that agents
have a limited capability to monitor, so by increasing the span of control the
owner can commit to monitor less. In this paper, monitoring by the owner is
always good, but there is not enough inside capital to incentivate the owner.
Instead incentives are provided by diversi"cation, so that increasing the span of
control, although it increases the cost of monitoring, may induce more monitoring by the owner.
The assumption that agents have limited capability to monitor allows us to
explain why the optimal size of intermediaries is bounded. This contrasts with
the result of Krasa and Villamil (1992), who examine the e!ect of increasing
monitoring costs incurred by depositors to monitor the bank. They "nd
that even if monitoring costs of depositors increase with the size of the bank,
there are increasing returns to scale in "nancial intermediation. We introduce
increasing monitoring costs due to limited capability to monitor by each single
banker and show that there may still be bene"ts from increasing the size of the
bank, but only up to a certain threshold where these bene"ts are overcome by
the costs of diversi"cation.
Our idea of the design of the bank's internal organization is in#uenced by
Qian (1994) and more generally by Williamson's (1967) view of factors a!ecting
the size of organizations. A "rm will be limited in size because of agency costs
arising from the owner's limited capability to monitor. We depart from that
literature in that we relax the assumption that the owner has no incentive
problem at all, always exerting the maximum monitoring e!ort. The di!erent
ingredient in our paper is that the owner has somehow to be given incentives to
exert this e!ort.
Holmstrom and Tirole (1997) exclude diversi"cation by assuming perfect correlation between
projects. The rationale is that in order to justify the need for bank capital, one has to exclude the case
where banking becomes a perfectly safe business. We too exclude this case by assuming limited
capability to monitor.
The reason is that, if the bank fails each depositor has to monitor ex-post all the loans of the
bank, therefore monitoring costs are increasing more than proportionally with respect to the size of
the bank. However the rate at which the probability of failure of the bank decreases is faster than the
rate at which monitoring costs increase.
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Finally, the paper is related to Innes (1990), who shows, in a framework
similar to ours, that is with ex-ante moral hazard, that debt is optimal for
incentives to monitor when the repayment schedule to creditors is constrained
to be monotonic. In our case, however, the debt contract alone does not provide
su$cient monitoring incentives, and we show that incentives to monitor can be
strengthen further through diversi"cation.
3. The model
Consider an economy consisting of two types of risk-neutral agents: entrepreneurs and investors. Each entrepreneur has access to a speci"c investment
project with stochastic return: if it succeeds it returns R; if it fails, it returns
nothing. Each project requires one unit of capital, which the entrepreneur has to
borrow from investors as he has no own capital. Project outcomes are observable and independent across entrepreneurs. The probability of success depends
on the entrepreneur's behavior, which, on the other hand, is not directly
observable. If the entrepreneur behaves well, the probability of success is p (&the
&
entrepreneur chooses the good project'). However, the entrepreneur may misbehave, which renders him a private bene"t B, but reduces the probability of the
project succeeding, p (p (&the entrepreneur chooses the bad project').
*
&
The investors have access to an alternative investment that yields a safe return
y, which is lower than the project's expected return when the entrepreneur is
behaving, p R, but higher than the expected return when the entrepreneur is
&
misbehaving, p R. For this reason we refer to the former type as good projects
*
and the latter as bad projects.
All agents are protected by limited liability, so no one can end up with
negative consumption. This means that the entrepreneur can repay the loan
only if the project succeeds, which depends on his behavior. If r is the loan
rate, the entrepreneur prefers the good project if and only if
p (R!r)5p (R!r)#B.
&
*
As we see, the incentive of the entrepreneur to behave depends on the loan rate.
We will assume that there is no loan rate that would convince the entrepreneur
Since, diversifying a bank portfolio implies that the banker has to allocate her monitoring e!ort
among several projects, the paper is also related to the multi-task literature; for reference see
Holmstrom and Milgrom (1991), and for a recent application to banking see Rochet and Tirole
(1996).
We refer to the contract between the entrepreneur and the lender as a debt contract, but,
because of the assumptions about the returns of the project, a debt contract is equivalent to an
equity contract. This framework allows us to reinterpret our results for the case of a conglomerate.
V. Cerasi, S. Daltung / European Economic Review 44 (2000) 1701}1726
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to behave and at the same time provide investors with an expected return of at
least y, that is
Assumption 1. p R!y!(p /Dp)B(0, where Dp"p !p .
&
&
&
*
Hence, unless monitored, the entrepreneur always chooses the bad project.
The idea is that the interests of the lenders and of the entrepreneur di!er so
much that they cannot be aligned through a "nancial contract. The only way to
resolve this con#ict is through monitoring. Given this, we can assume, without
loss of generality, that the loan rate r is such that the lender extracts all the
surplus from the entrepreneur, namely that r"R.
Through costly monitoring the investor can "nd out whether the entrepreneur is misbehaving. However, each investor is assumed to have a negligible
amount of capital, and therefore has no incentive to monitor. Therefore direct
lending is not feasible.
As a response, investors form intermediaries (banks) which borrow from
investors and lend to entrepreneurs. For simplicity, each bank will have just one
investor as inside equity holder, named the banker. Bankers can, by monitoring, induce the entrepreneur to behave (&choose the good project').
As for the monitoring technology, we assume that by choosing the level of e!ort
E, with 04E41, the monitor is able to discover with probability E that the
entrepreneur is misbehaving and can then intervene to insure that the good
project is chosen instead. Each agent has limited capability to monitor in the
sense that the marginal cost of monitoring increases with the number of projects.
Bank monitoring has the e!ect of reducing the private bene"t of the entrepreneur to zero. If the
banker gives an in"nitesimal amount e to the entrepreneur in case the project succeeds, the
monitored entrepreneur will choose to carry out the project anyway. In what follows we derive,
without loss of generality, all results for e"0. The working of bank monitoring is similar to that in
Holmstrom and Tirole (1997), in that it reduces the private bene"t of the entrepreneur.
This assumption simpli"es the formulas, but for the results it is su$cient that each investor has
a small amount of capital compared to the capital required by the project. The idea is to capture that
banks collect funds from small investors, who in contrast to su$ciently large investors are not able
to lend directly to entrepreneurs.
More generally, a bank could be established by a group of investors, if the free-rider problem
could be solved within this group. Although such a bank could have a non-negligible amount of
inside capital, the group of inside equity holders has to be small in order to avoid free-riding so that
presumely the amount of inside equity would not be su$cient to eliminate the incentive problem of
the inside equity holders.
Yafeh and Yosha (1995) present evidence for Japan that the presence of a bank in a group
reduces managerial moral hazard in the "rm belonging to the group. In particular they identify a list
of balance sheet items that may entail greater managerial discretion, such as R&D and advertising
expenditure, or entertainment expenses, travel expenses, corporate pensions, etc.
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More speci"cally we assume that monitoring m projects, that is, "nding out with
probability E the behavior of entrepreneur i, implies the following private cost:
G
c K
c K
c(E )" E# EE ,
(1)
G
G H
2
2
G
G H$G
where E is a vector of monitoring e!orts, and c and c are positive parameters.
cost function, increasing monitoring of
With this
one project has two e!ects: not
only increases the marginal cost of monitoring this project, but also the marginal cost of monitoring all the other projects. The idea is that monitoring
a project takes time and each person's time is limited. The probability of "nding
out the behavior of the entrepreneur depends not only on the time spent on
monitoring, but also on the monitoring intensity. Thus increasing the time spent
on one project, leaving less time to the others, implies that the monitor has to
work harder in order to keep the quality of monitoring with respect to these
projects constant. In other words, when a monitor is &overloaded' monitoring
costs increase more than proportionally for each additional unit of e!ort in
a speci"c project, whenever c '0. This cost function allows to capture two
e!ects at the same time. First, given that the disutility of e!ort increases with the
level of monitoring in a particular project, it implies diseconomies of scale in
monitoring. Second, since the disutility of e!ort increases with the number of
projects to be monitored, it embodies also diseconomies of scope in monitoring.
We assume that the good project is su$ciently pro"table to cover the
marginal cost of fully monitoring one project, that is,
Assumption 2. p R!c 'y'p R.
&
*
To summarize, there are good projects which can be pro"tably "nanced.
However, direct lending is not feasible, since the entrepreneur will always
misbehave, as no investor has the incentive to monitor him. Delegation of
monitoring to a banker solves the free-rider problem of investors, but because
the banker does not have enough money to "nance the entrepreneurs herself, she
needs to raise money from outsiders. For investors to be willing to "nance the
banker, they need to be convinced that the banker will exert a su$ciently high
e!ort, so that the expected return is at least as high as the alternative return.
The timing of the game is the following: "rst, the number of projects m to be
"nanced is chosen, then the "nancial structure of the bank is determined and
"nally, the monitoring e!orts for each project are determined. We assume that
investors observe the number of projects before investing, whereas monitoring
e!orts are neither observable nor contractible.
The crucial ingredient for our results is that the banker can commit to a certain size of its
portfolio, before collecting funds from depositors. In other words, depositors can observe the size of
the bank and moreover know that a large bank will not suddenly shrink its portfolio and become
V. Cerasi, S. Daltung / European Economic Review 44 (2000) 1701}1726
1709
We will focus on the size of a bank as an incentive mechanism and assume
that the demand for loans does not place any restriction on size. This assumption is reasonable when for instance the number of banks is restricted by entry
regulation or when the number of entrepreneurs is very large. Hence, we can
consider the problem in terms of a representative banker.
4. The incentive problem of the banker
We want to show that there is scope for diversi"cation in "nancial intermediation even though there are overload costs in monitoring. The idea is that
diversi"cation may work as a commitment device when there is moral hazard in
monitoring. In this section we illustrate the moral hazard problem of a banker,
which arises because monitoring is not contractible. In the next section we will
show how diversi"cation can mitigate this moral hazard problem.
In order to create a benchmark we relax the assumption that the banker has
only a negligible amount of wealth and consider the case in which the banker
has enough capital to "nance the projects herself. This means that, even if the
banker "nances the investment with a loan, she can be made fully liable for the
loan. Because all agents are risk-neutral, this case is equivalent to external
"nancing with contractible e!orts.
E The benchmark: full liability. The banker "rst decides how many projects to
"nance, and then how much monitoring e!ort to allocate to each project. By
exerting a monitoring e!ort E , the banker "nances a good project with probG
ability E . Hence, the expected return to the banker from "nancing m projects is
G
K
P"
p R!my!c (E ),
(2)
G
G
a small bank. This would for instance be the case if the bank has a geographical monopoly or if
borrowers face switching costs. Another motivation is valuable customer-relationships. The commitment is important, because it allows a larger bank to attract deposits at a lower deposit rate. As
argued in Winton (1995), if the bank cannot lower its deposit rate, it may not be willing to increase its
size, because depositors would get too much of the gain from the change.
When the e!ort is contractible, the return to the investor can be made contingent on the
e!ort levels. Hence, because of perfect competition among investors, the cost of funding is equal
to y per unit of capital, independently of the level of e!ort. Notice that in our paper monitoring
is good, thus a monitor who is residual claimant, by internalizing the full returns from monitoring,
has incentive to supply the optimal level of monitoring. There are other papers, Burkart et al.
(1997) and Pagano and Roell (1998), where full liability is not optimal as overmonitoring harms
manager initiative and the "nancial structure is used to "netune manager incentives without
reducing his initiative.
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V. Cerasi, S. Daltung / European Economic Review 44 (2000) 1701}1726
where p "p #E Dp is the probability of success of project i, and Dp"p !p
G
*
G
&
*
is the marginal increment in the probability of success. The opportunity cost of
funds is given by the alternative return y. The "rst-order conditions (FOCs) for
the optimal e!ort choices are
*P
"DpR! c E #c
E 50, i"1,2, m.
(3)
G
H
*E
G
H$G
Note that for this particular cost function the FOCs allow us to "nd a unique
symmetric solution for the optimal e!ort level, denoted by EH.
The solution could be either an internal e!ort, when DpR4c #c (m!1), or
the corner solution EH"1. From Assumption 2 follows that EH"1 for m"1.
However the LHS of Eq. (3) is strictly decreasing in m due to the overload costs
and for m large enough, EH(1. When the solution is internal, the optimal
average e!ort level is decreasing in m.
E The one-project bank. Since the banker has a negligible amount of capital, she
has to raise one unit from investors to "nance the project. The banker promises
to pay a share to outsiders in case the project returns R, or 0 otherwise,
protected by limited liability.
The banker chooses the optimal level of her e!ort, given the share
outsiders, by maximizing pro"ts:
to
c
P"p (R! )! E,
2
where p"p #EDp is the probability that the project succeeds. The FOC with
*
respect to e!ort is
DpR!Dp !c E50.
(4)
The investors will be willing to "nance the banker, only when the amount of
monitoring induced by the banker is large &enough' to make their investment
worth at least as much as the alternative return y. This is only possible if the
banker monitors the entrepreneur, but monitoring is not contractible. We
exclude the one-project bank by assuming that the marginal cost of monitoring
is su$ciently high for there not to be any share which would provide incentives
to the banker to monitor and at the same time provide investors with an
expected return of at least y, that is
In the one-project bank there is no di!erence between equity or debt, because of the simple
project structure.
V. Cerasi, S. Daltung / European Economic Review 44 (2000) 1701}1726
1711
Assumption 3. p R!y!( p /Dp) c (0.
&
&
In other words, there is no rational expectation equilibrium in which investors
are willing to deposit at the one-project bank.
5. Diversi5cation and incentives
We have shown that the need for external "nance gives rise to an incentive
problem for the banker and that this incentive problem could be su$ciently
severe for the bank not to be viable when "nancing only one project. In this
section we will show that diversifying the bank portfolio by "nancing more than
one project may mitigate the banker's incentive problem, if the bank is debt
xnanced.
Why would diversi"cation of the bank portfolio improve the banker's
monitoring incentive under debt "nancing? From the agency literature we know
that the incentive problem could be resolved by making the banker the residual
claimant of the bank's net pro"ts, that is, if investors were to receive a return y in
all states of nature, while the residual return accrues to the banker. This is not
possible when the banker is "nancing only one project, since, having no initial
wealth, the banker cannot pay y when the project fails.
However, if the banker is "nancing many projects, according to the Law of
Large Numbers the return on the loan portfolio will be almost certain, so that, if
the expected return is su$ciently high, the banker will almost always be able to
repay depositors. Therefore the deposit rate will be approximately equal to y,
and virtually all the marginal bene"t of monitoring accrues to the banker. In
other words, the debt contract will be close to the full liability contract. This is
not true for the equity contract. Independently of the size of the bank, outside
equity holders always get a share of the bene"t from increased monitoring.
The banker's incentive to monitor, however, depends also on the marginal
cost of monitoring, which increases with the number of projects. As a matter of
Furthermore, the banker is protected by limited liability and cannot be punished for not being
able to repay the debt; in other words, there are no private bankruptcy costs here.
This argument is related to the solution to multi-task problems, for reference see Holmstrom
and Milgrom (1991). In a multi-task problem agency costs can be reduced by letting one agent
performing several tasks instead of having one agent performing each task. The debt contract does
something similar to the optimal multi-task contract; with debt the banker gets nothing if the bank
fails to pay the deposit rate, and the probability of bank failure depends on all e!ort levels. This
could be compared to the equity case, in which the return to the banker from a project depends on
the monitoring e!ort exerted on this project only. If the returns to depositors do not need to be
monotonic, incentives could be improved further by letting the banker get all the return in the
highest return states (see Innes, 1990), however, as the bank becomes more diversi"ed the debt
contract approaches the optimal contract.
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fact, if overload costs are high, a bank with a well diversi"ed portfolio will never
be able to repay depositors, as the deposit rate will be higher than the expected
portfolio return, since the high monitoring costs spoil the incentive to monitor.
What we want to show is that diversi"cation can resolve the banker's incentive
problem when monitoring costs increase suzciently slowly with the number of
projects to be monitored.
5.1. Debt and diversixcation: How to reduce the incentives to exploit depositors
The debt contract could be described as a promise by the debtor to pay r in
"
every state of the world; however, the contracting parties are aware that in some
states this promise will not be kept. If the promise is not kept, the bank is
declared bankrupt and depositors recover whatever there is. Hence, the expected
return per unit of deposit can be written as
1
r ! S ,
" m K
where S are the expected shortfalls on the total debt, that is the di!erence
K
between the promised amount mr and the amount recovered by the creditors in
"
expected terms. For instance, in the case the average return of the bank
portfolio, z"(1/m) K z (z is the repayment by entrepreneur i ), has a contiG G
Gg(z),
nuous density function
the expected shortfalls are given by
S "m
K
P"
(r !z)g(z) dz.
(5)
"
Thus, what distinguishes the risky debt contract from the full liability contract is
the expected shortfalls on debt.
The distribution of the portfolio returns depends on the number of projects
and the degree of monitoring by the banker. A higher monitoring level, shifts the
distribution to the right, while an increase in the number of projects reduces
the variance of the average portfolio return.
The banker's expected returns on "nancing m projects by issuing deposits at
the interest rate r are
"
K
P"
p R!(mr !S )!c (E ).
(6)
G
"
K
G
For each project, the banker chooses the e!ort so as to maximize pro"ts:
*S
DpR# K! c E #c
E 50, i"1,2, m.
G
H
*E
G
H$G
(7)
An increase in the monitoring e!ort will also a!ect the variance of the distribution. In our case,
the variance is reduced if the probability of success is larger than 0.5.
V. Cerasi, S. Daltung / European Economic Review 44 (2000) 1701}1726
1713
Notice again that the FOCs might be ful"lled with inequality, since E
G
is constrained to be less than one. As in the full liability case, there is only a
symmetric solution to the system of FOCs, which we will denote by EK .
K
Comparing the FOCs in Eq. (7) with the FOCs for a fully liable bank given by
Eq. (3), we see that the di!erence arises from the e!ect of a change in the
monitoring level on the expected shortfalls. This term is negative, as increased
monitoring shifts the distribution of the portfolio return to the right. When the
banker exerts more monitoring e!ort, she increases the probability of her being
able to repay the debt and reduces the expected shortfalls on debt. Since
depositors cannot observe the e!ort provided by the banker, higher e!ort does
not lead to a lower deposit rate. Consequently a part of the bene"t from
increased monitoring would accrue to depositors, which reduces the banker's
incentive to monitor. This incentive to exploit depositors arises because the
deposit rate does not depend on the monitoring level in the bank, which is not
observable to outsiders, and thus is related to the well-known risk incentive of
"xed-price deposit insurance.
Since the expected shortfalls depend on the deposit rate, so do the FOCs. In
the absence of interbank competition for funds, the banker sets the lowest
deposit rate at which investors are willing to deposit at the bank. Investors
cannot observe the banker's e!ort choice but have rational expectations about
her behavior. The banker takes into account that the deposit rate has to ful"l the
IR condition for the investors given by
1
r ! S "y.
" m K
(8)
If depositors expect the monitoring level to be low, they require a higher deposit
rate to compensate for the expected shortfalls. A higher deposit rate in turn
implies a higher probability of bank failure, and therefore a lower incentive to
monitor. It therefore may not exist a solution to conditions (7) and (8). In fact
there is no solution if m"1. However, given that monitoring costs rise slowly
with the number of projects to be monitored, diversi"cation improves incentive
to monitor and we have the following result.
As a matter of fact, assuming that deposits are fully insured and that the insurer charges a fair
premium given his expectation about the bank's behavior, would not change the results if the insurer
has the same information as investors, as in Daltung (1994). The insurer, of course, may have both
stronger incentives and greater possibilities to monitor banks than depositors do have. This is for
instance the motivation of Dewatripont and Tirole (1994) for deposit insurance. In this case, as in the
case with a large shareholder, the incentive problem of banks is less of an issue. However, monitoring
banks is a di$cult task and the outcome is likely to be far from perfect, why we believe that
alternative incentive mechanisms still are important.
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V. Cerasi, S. Daltung / European Economic Review 44 (2000) 1701}1726
Proposition 1. There exists a c , such that for all c 3[0, c ], a suzciently diversix
ed bank can raise debt although a one-project bank cannot.
Proof. We will assume that there exists a deposit rate r , lower than the average
"
expected portfolio return, z , at which depositors are willing to deposit their
means at the bank, and then show that there are c 3[0, c ] and m'0, for which
this indeed will be true.
From the Law of Large Numbers it follows that the distribution of z becomes
more and more concentrated around its mean, z , as m increases. This means that
G(a), the probability that z will take a value lower than a, approaches zero as
m goes to in"nity ∀a(z . Moreover, for r (z , the average expected shortfalls,
"
(1/m) S , are approaching zero as m goes to in"nity, which is easy to see by
K
applying integration by parts to the expected shortfalls in Eq. (5):
S "m
K
P"
G(z) dz.
From this follows that for an m su$ciently large the IR condition for investors is
approximately equal to r "y.
"
Also the derivative of the expected shortfalls with respect to E approaches
G
zero as m goes to in"nity. The amount by which an increase in E could reduce
G
the probability of bank failure is limited by the size of the probability of bank
failure, as the probability cannot be less than zero. Hence, as the probability
approaches zero, so must the derivative of the expected shortfalls. In the
Appendix, this is proved for the normal distribution. This means that for r (z
"
there is an m for which the FOCs in Eq. (7) are approximately equal to the FOCs
in the full liability case given by Eq. (3):
p R!p R! c E #c
G
&
*
E 50, i"1,2, m.
H
H$G
According to Assumption 2, p R!p R!c 'p R!y!c '0. Hence, inde&
*
&
pendently of m, there are c 3[0, c ] for which the LHS in the equation above is
strictly positive. It follows that, for a su$ciently large bank there are values of
c '0 for which (p #E Dp)R is larger than r ∀i. So indeed r (z and the
*
G
"
"
bank is able to raise deposits although a one project bank is not, since Assumption 3 is independent of c and thus holds true. 䊐
By increasing the size of the bank the incentive to exploit depositors is
reduced, but the marginal monitoring cost is increased. If overload costs are
su$ciently small, the "rst e!ect will dominate and the monitoring level in the
bank will approach the highest possible as the size of the bank is increased.
Positive overload costs, however, imply that eventually a larger size will reduce
the monitoring level; when the bank is perfectly diversi"ed, increasing the size of
the bank only results in increasing overload costs.
V. Cerasi, S. Daltung / European Economic Review 44 (2000) 1701}1726
1715
5.1.1. Optimal size of the bank
What bank size will the banker choose? We show that, in order to become
viable an externally "nanced banker typically needs to choose a larger size than
a fully liable banker.
E Fully liable bank. By applying the Envelope Theorem, the impact of "nancing
an additional project on the pro"ts of the bank, given by Eq. (2), can be written
as
dP
c
c
"pHR!y! # (2m!1) (EH),
2
dm
2
(9)
where pH"p #EHDp. From Assumption 2 follows that dP/dm'0 for m"1.
*
Multiplying Eq. (9) by m, and comparing the result to the equilibrium pro"ts,
gives that pro"ts must be positive for the banker to be willing to increase further
the size. Let us denote by mH the optimal size for a fully liable banker. It could
be that this size is larger than m"1, because the projects are pro"table enough
to cover the additional monitoring costs. Hence, the banker may want to "nance
more than one project in order to extract as much surplus from projects as
possible. The banker will trade-o! the increased monitoring cost and lower
monitoring e!ort against surplus of the project.
E Debt xnanced bank. The banker takes into account that investors are rational
and demand an expected return equal to y, and that the size of the bank a!ects
the rational expectation equilibrium. In order to be active the banker must
choose m su$ciently large to convince investors that she will do enough
monitoring. When investors are willing to deposit at the bank, we can substitute
the IR condition for investors into the pro"t function of the banker given by
Eq. (6), and derive the optimal size from
dP
c
c
" p( R!y! # (2m!1) EK K
K
dm
2
2
dEK
#m[DpR!(c #c (m!1))EK ] K ,
K dm
(10)
where p( "p #EK Dp is the equilibrium probability of success. The "rst term
K
*
K
captures the scale e!ect, present also in the fully liable bank, namely the surplus
Given that *P/*EH"m(*P/*E ), the Envelope Theorem applies also to the symmetric e!ort
G
level. Note also that Eq. (9) is true even if EH"1, since in that case dEH/dm"0.
In fact, dP/dm(P/m for any m for which P'0. This implies that the pro"ts are concave in m,
whenever c '0, and thus, if there is an incentive to choose m'1, this size is bounded, due to
overload costs.
1716
V. Cerasi, S. Daltung / European Economic Review 44 (2000) 1701}1726
from monitoring an additional project, while the second term, which is new,
captures the incentive e!ect of diversi"cation. Denote the smallest m for which
the bank is viable by m . If, for a given m5m , the FOCs for optimal e!ort levels
are non-binding, dEK /dm"0 and the above derivative coincides with the
K
derivative in the full liability case. In this case, the incentive problem connected
to external "nancing is not constraining the banker's choice of the size. However, typically the incentive problem is binding at mH, either that m 'mH or
at least that dEK /dm'0 at mH, and we have the following result:
K
Proposition 2. If the incentive problem of the banker is binding at mH, the optimal
size of a debt xnanced bank is larger than the size of a fully liable bank.
Proof. If at mH the FOCs in Eq. (7) are binding, it follows that the term within
the second square brackets in Eq. (10) is positive for EK (1. Therefore, at
K
mH there is a further reason, with respect to the fully liable banker, to increase
the size of the bank. 䊐
Note that the banker will never choose a size such that dEK /dm(0, since the
K
term within the "rst square brackets is negative for m 'mH. We can conclude
that the optimal size of the bank is bounded. In other words, there is a trade-o!
between the scale e!ect, given by the "rst term in Eq. (10) and the diversi"cation
e!ect, given by the second term in the same equation. On the one hand, the scale
e!ect decreases with m as pro"ts of an additional project shrink, given our
assumption about diseconomies of scale in monitoring. On the other hand, there
are bene"ts of diversi"cation in terms of stronger incentives to monitor. The
solution to this is a bound to the optimal size of the bank.
5.2. The cost of equity: Inside vs. outside equity
In this section we show that inside equity plays a complementary role to
diversi"cation in that it reduces the incentive costs of monitoring and furthermore that this is not true with outside equity. Thus, we conclude that, on the one
hand, if the banker has enough inside capital on his own, there would be less
need for diversi"cation, and, on the other hand, if the banker has very little
inside capital and needs external "nance, he cannot exploit the bene"ts of
diversi"cation by raising outside equity.
Hellwig (1998) analyzes this trade-o! in a framework where intermediaries can choose between
lending to few entrepreneurs a larger amount or to many a smaller amount. He shows that, given an
investment technology which is not too convex, the banker will choose to diversify, although not
completely, that is, not all opportunities for diversi"cation will be exploited.
V. Cerasi, S. Daltung / European Economic Review 44 (2000) 1701}1726
1717
Inside equity is a substitute for diversi"cation in that both reduce the incentive to exploit depositors. The e!ect on incentives is stronger the more inside
capital the banker has. Therefore we can say that the more inside capital there is,
the smaller the need of the banker to diversify to attract external "nance. Quite
contrary, outside equity, by introducing outsiders who share the returns of the
bank in the good states, while not internalizing the cost of monitoring, worsen
the incentive to monitor of the banker.
We analyze the di!erent role of equity, inside vs. outside equity, by comparing
two di!erent cases: in the "rst the banker has some inside capital, w, which is
signi"cantly greater than zero, although not enough to avoid the need of
external "nance; in the second the banker, who has no inside capital at
all, raises the amount w from outside investors by issuing equity shares in the
bank.
E Inside equity. Assume that the banker has w units of capital. Thus, she needs to
raise (m!w) units of capital from outsiders to be able to "nance m projects.
External "nance is in the form of debt, hence the banker promises to repay
(m!w)r to investors. If the debt is not fully repaid, the bank fails. Bank failure
"
occurs whenever K z ((m!w)r . Depositors expected returns can be writG
"
ten as (m!w)r !SG(w), where S (w) are the expected shortfalls for an amount
"
K
K
of equity w. It is easy to see that the expected shortfalls decrease with the amount
of inside equity. The optimal level of e!ort in monitoring is set by the banker
in order to maximize her expected returns:
K
P"
p R![(m!w)r !S (w)]!wy!c(E ).
(11)
G
"
K
G
For each project, the banker chooses the e!ort so as to maximize pro"ts:
*S (w)
DpR# K ! c E#c
E 50, i"1,2, m.
(12)
G
H
*E
G
H$G
Furthermore, in equilibrium, investors should earn in expected terms at least the
alternative return y, that is
(m!w)r !S (w)5(m!w)y.
"
K
(13)
Notice that this allows to have continuity between the two cases in the paper, namely when the
banker is fully liable, w"m, and when the banker has very little inside capital, wP0.
In the case of a continuous distribution, the expected shortfalls are given by
S (w)"m
K
K\UKP"
m!w
r !z g(z) dz
"
m
where g(z) is the continuous density function of the average return of the bank portfolio.
1718
V. Cerasi, S. Daltung / European Economic Review 44 (2000) 1701}1726
The inside equity will not change the distribution of the portfolio return, but just
the breaking point at which the bank goes bankrupt. It is thus straightforward
to apply the results in the previous section to state that the banker can replicate
the full liability case by diversifying enough the bank portfolio, as the debt
contract converges to the riskless contract and the e!ort levels to the levels of
a fully liable banker. The role of inside capital is similar to that of diversi"cation
in that it reduces the incentive problem of the banker.
Proposition 3. To reach a given level of monitoring, a banker with more inside
capital needs to be less diversixed compared to a banker with less inside capital.
Proof. For a given number of projects, m, increasing inside capital reduces the
marginal impact of e!ort on the expected shortfalls, thus lowering the incentive
of the banker to exploit depositors. It follows that when the FOCs are ful"lled
with equality, as inside capital rises, the e!ort of the banker increases for a given
m. Thus to reach a given level of e!ort, there is less need to increase the amount
of diversi"cation. 䊐
That inside capital reduces the incentive problem of the monitor is already
known. What we provide is an explanation for why banks, that own little
inside capital, are nevertheless viable.
Given that inside equity has proven to be helpful in reducing the incentive
problem of the banker, one may wonder whether this is not an intrisic virtue of
equity, therefore of outside equity as well.
E Outside equity. To make this case comparable to the previous one, we assume
that the banker, who has no inside capital, raises w units of capital by issuing
shares to outsider investors, which entitle equityholders to a proportion a of the
return stream of the bank portfolio after repayments to debtholders. As before,
bankruptcy occurs whenever K z ((m!w) r , and so the expected shortfalls
G
"
Gcase.
are de"ned as in the previous
After repayments to debtholders, the
portfolio of the bank returns:
K
G
p R![(m!w)r !S (w)].
G
"
K
We presume that *S /*w*E '0. This is the case for the normal distribution (see the
K
G
Appendix). Since an increase in w will reduce the deposit rate, for a given e!ort level, the result then
follows.
In Holmstrom and Tirole (1997) bankers incentive to monitor correlated projects is indeed
provided through inside capital.
Banks are heavily leveraged "rms. See for instance Dewatripont and Tirole (1994).
V. Cerasi, S. Daltung / European Economic Review 44 (2000) 1701}1726
1719
Given that the banker retains only (1!a) of the returns, we can now write the
pro"ts of the banker as
P"(1!a)
K
p R![(m!w)r !S (w)] !c (E ).
G
"
K
G
The optimal levels of e!ort are given by the following FOCs:
(14)
*S (w)
(1!a) DpR# K
! c E#c
E 50, i"1,2, m.
(15)
G
H
*E
G
H$G
To complete the equilibrium, we need to add to the IR condition for
debtholders, Eq. (13), the IR condition for equityholders, namely:
a
K
p R![(m!w)r !S (w)] 5wy.
(16)
G
"
K
G
By comparing the FOCs to the case with inside equity, one can conclude that,
for a given size m and a'0, outside equity "nance implies smaller monitoring
e!orts than inside equity. The intuition is that, while the banker has to share the
pro"ts of the bank with outsiders, that is, she is residual claimant of a smaller
amount, (1!a), when the bank does not fail, indeed she has to bear the full cost
of monitoring.
Increasing the amount of outside equity has two counteracting e!ects on
incentives. On the one hand, it, similarly to inside equity, reduces the impact on
the expected shortfalls, but on the other hand, contrary to inside equity, it may
worsen monitoring incentives as it increases the share promised to the outside
equity holders, a. The more diversi"ed the bank is, the more dominating the
negative e!ect on incentives will be.
When there is perfect diversi"cation, that is when mPR, the equilibrium
approaches the full liability case only if the banker substitutes all the shares of
outside equity with debt. In other words, if there is outside equity, namely for
any a'0, the banker's incentive problem cannot be eliminated through diversi"cation as the FOCs approach
(1!a)DpR! c E#c
G
E 50, i"1,2, m,
H
H$G
which are di!erent from the FOCs in the full liability case. In this case increasing
the amount of outside equity only has the e!ect to increase a, which reduces the
monitoring level and thereby the pro"ts of the banker. Thus there is indeed
a cost attached to outside equity and we can conclude that a perfectly diversi"ed
banker would prefer not to issue outside equity.
Discussion. As in Jensen and Meckling (1976), outside equity "nancing involves
agency costs. In our case, independently of its size, the bank is not able to "nance
1720
V. Cerasi, S. Daltung / European Economic Review 44 (2000) 1701}1726
its lending with outside equity only. Moreover, it is not possible to avoid the
agency problem by selling the "rm to the inside equity holder. Even if she
bought the outside equity claims by issuing debt, risky debt involves the same
type of agency costs as equity. As a matter of fact, in our model, because of the
simple project structure, when "nancing only one project, debt "nancing involves exactly the same agency cost as outside equity "nancing. Only when the
debt is risk-free does debt "nancing not introduce any distortion of incentives.
This is related to the result of Myers and Majluf (1984), that is, the "rm prefers to
issue risk-free debt, while risky debt is preferable to outside equity. In their case,
the "rm might refrain from pro"table investment opportunities when it has to
"nance the investment by issuing equity.
While risk-free debt does not involve any agency costs, there is no bene"t in
making the debt secure by splitting cash #ows, since agency costs would then
arise instead from outside equity "nancing. For instance, if the project returned
something in the bad state, full debt "nancing would be better than full equity
"nancing. If the bank is debt "nanced, introducing a small amount of outside
equity has no e!ect at all on the banker's monitoring incentive, while introducing an amount of equity which is so large that the debt becomes secure reduces
the monitoring incentive. The intuition is the following: while the incentive
problem with equity is worse than with debt, issuing equity reduces the incentive
problem of debt, but only when debt is not already secure. This suggests that
a capital requirement can be costly as it might reduce the monitoring level in the
bank. Of course, if debt can be made secure by injection of inside capital, the
incentive problem could be alleviated, and this could be a cheaper alternative to
diversi"cation.
To conclude, our analysis suggests that small banks should be closely held
"rms, while ownership of more diversi"ed banks could be more dispersed.
Moreover, capital requirements is most likely to be binding in large banks given
that diversi"cation is a sign of lack of inside equity.
6. Conclusions
In credit markets with asymmetric information, monitoring may be crucial to
"nance pro"table projects. Banks may arise as a response to the free-riding
problem of small investors. However, when the outcome of monitoring depends
on the monitor's e!ort, delegating the task of monitoring to a bank gives rise to
an incentive problem for the agent performing the monitoring task. Consider
a banker, who collects funds from investors to lend to entrepreneurs. This
banker may have too low an incentive to monitor as she has to share the gains
from monitoring with the investors, while bearing all the costs of monitoring
herself.
V. Cerasi, S. Daltung / European Economic Review 44 (2000) 1701}1726
1721
We show that, if the banker is the bank's sole owner, "nancing the bank's
lending with debt, then diversi"cation of the bank portfolio increases the
banker's incentive to monitor. Intuitively, when the bank is debt "nanced, the
gains from monitoring accrue to the banker as long as the bank does not fail,
and diversi"cation reduces the probability of bank failure.
Our results are complementary to previous research on "nancial intermediation in that we provide an additional reason for why banks should be diversi"ed and mainly debt "nanced. To put it di!erently, we o!er an explanation for
why large and diversi"ed banks can function with very little inside capital.
A large shareholder is believed to be an important device for controlling
managers, but the view also is that this would be a very costly way of controlling
managers of large "rms. We show that a very small stake of the large
shareholder is su$cient to provide her with the incentive to control managers in
the interest of all claimholders, if the other investors hold debt and the bank is
well diversi"ed.
There is no role for capital requirements in our model, as inside equity is
limited and the incentive problem is worse with outside equity than with debt. In
practice, one motive for capital regulation is the well-known incentive of
shareholders to exploit debtholders by taking on more risk. The incentive
problem in this paper is related to this risk incentive. The reason that the banker
has incentive to choose a too low monitoring level is that reduced monitoring
increases the probability that the bank will fail, and the banker does not carry
the costs of bad loans if the bank fails. Outside equity involves higher agency
costs than debt in our model, because monitoring involves a private cost. If the
agency cost instead was in the form of lower expected return only, the interests
of inside equityholders would be totally aligned with the interests of outside
equityholders, and the risk incentive would arise only from debt. Thus, our
result that debt "nancing of the bank is next best to inside equity depends on the
assumption that the bank only invests in assets that require monitoring. Of
course banks also have the opportunity to invest in assets which do not require
any monitoring e!ort, which then might motivate capital constraints, but
equity's negative e!ect on the monitoring incentive must be traded o! against
the risk incentive of debt. Moreover, it is shown in Daltung (1994) that the risk
incentive of debt is reduced by diversi"cation. This means that our conclusion
that capital constraints are more likely to bind in large banks with dispersed
See for instance Shleifer and Vishny (1997).
In the literature, the risk-shifting incentive is modeled as an incentive to increase the variance
without changing the mean of the distribution. This increases the returns to the shareholders,
although the probability of failure too, because returns are higher in the good states. In our model
instead a lower monitoring level reduces the mean of the distribution, but it increases the returns of
the banker in the good states, because she can save on monitoring costs.
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V. Cerasi, S. Daltung / European Economic Review 44 (2000) 1701}1726
ownership would still hold true when the asset substitution problem is taken
into account.
The paper also gives some insight into corporate "nance. It provides an
explanation for why shareholders usually do not want their "rm to diversify. In
the model, if the bank is equity "nanced, diversi"cation does not improve
incentives to monitor, but it does increase monitoring costs. Thus, this suggests
that if a "rm is able to "nd a few large investors who are willing to "nance the
"rm's projects, project "nancing is better than a conglomerate. What distinguishes the bank from the "rm is that the bank collects funds from small
investors, in which case project "nancing is not possible.
In the paper size and diversi"cation are perfectly correlated as the banker can
diversify only by taking on more projects. Actually banks have an alternative
choice to reduce monitoring costs, that is to specialise, thus gaining e$ciency, at
the cost of higher correlation among projects in the bank portfolio. This is an
interesting extension, that could explain the existence of an alternative trade-o!
between size and diversi"cation.
In this paper we have not addressed the issue of delegation inside the bank.
Since monitoring an increasing number of projects is increasingly costly for the
banker as time becomes more and more scarce, delegation of monitoring to
managers may be good from the owner's point of view as it reduces overload
costs in monitoring. Delegating the monitoring task to bank managers, however
introduces an incentive problem for the manager. To reduce shirking by managers, the banker must exert e!ort in supervising them. Thus, further delegation
of monitoring of borrowers to managers sometimes implies duplication of
monitoring costs. There will be less need for supervision, and therefore less
duplication of monitoring, if the manager gets a higher return on e!ort than he
would in alternative occupations. Furthermore, increasing the owner's monitoring incentive through diversi"cation is less costly when the owner has to spend
little time supervising each manager. This also implies that it is less costly to
increase the bank's overall monitoring level, since sharing monitoring costs with
the managers reduces diseconomies of scale in monitoring by reducing the
&overload' of the banker. As long as there is some need for supervision, however,
total monitoring costs will increase with bank size, and the optimal size of the
bank will be bounded. We leave the analysis of delegation of monitoring inside
the bank for future research.
Furthermore, we have restricted the access to the monitoring technology to
investors. It is possible to relax this assumption by giving also to entrepreneurs
We thank an anonymous referee for suggesting this possible extension.
In a di!erent framework Hellwig (1998) analyses this trade-o!, when the choice of concentration of risks is interpreted as an option to specialise.
We refer the interested reader to a previous version of this paper, Cerasi and Daltung (1996), for
a preliminary analysis of delegation of monitoring to managers in our framework.
V. Cerasi, S. Daltung / European Economic Review 44 (2000) 1701}1726
1723
access to the monitoring technology, for instance by letting entrepreneurs to set
up their own bank. On the one hand, this solution allows to save monitoring
costs, but, on the other hand, it gives rise to a costly con#ict of interest between
the banker-entrepreneur and outside investors, since, by not being monitored,
he may choose to "nance his bad project.
In this paper, "nally, we have not addressed the issue of interbank competition for lending, but analyzed the monitoring decision for a given loan rate. One
interesting issue for future research is whether competition worsens the banker's
incentive problem. In this model, it is true that the incentive to monitor
decreases with the loan rate. However, the question is to what extent bankers are
able to internalize this e!ect and, related to this question, what is the e!ect of
competition on the optimal size of the bank.
Acknowledgements
This paper was prepared for the conference &The Design of the Banking
System' jointly organized by COMIT and IGIER in February 1995. The views
expressed in the paper are those of the authors and do not necessarily re#ect
the views of Sveriges Riksbank. We would like to thank Patrick Bolton,
Arnoud Boot, Vincenzo Denicolo', Mathias Dewatripont, Paolo Garella,
Jean Charles Rochet, Raphael Repullo, Peter Sellin, David Webb, two anonymous referees and seminar participants at the Nordic Banking Research Seminar, IGIER, Bologna University, Uppsala University, the Stockholm School
of Economics, and the Norwegian School of Management for comments on
previous versions of the paper. We are particularly grateful to Jean Tirole
for a helpful conversation at a very preliminary stage of this project and to
Denis Gromb for very detailed comments. The remaining errors are our own
responsibility.
Appendix A
Proof of Proposition 1 (Normal distribution). According to the Central Limit
Theorem the distribution of the standardization of z tends to the standard
In Cerasi and Daltung (1998) we show the conditions under which this solution is e$cient,
given that diversi"cation reduces the con#ict of interest between the banker-entrepreneur and
depositors.
1724
V. Cerasi, S. Daltung / European Economic Review 44 (2000) 1701}1726
normal as m goes to in"nity. Thus, for a given m, su$ciently large, the expected
shortfalls are approximately
m
P"
(r !z)
"
z!z
1
(p (p
dz ,
(A.1)
where z "(1/m) K p R is the expected return of z, p"(1/m) K p (1!p )R
G
G G
G G
is the variance, and ( . ) is the density function of a standard normal variable. By
subtracting and adding z /(p the integral in (A.1) can be rewritten as
(r !z )
"
P"
z!z
(p (p
1
dz!
P" z!z
(p
z!z
(p
dz.
(A.2)
Since d (x)/dx"!x (x), (A.2) is equal to
r !z
!z
!U
(r !z ) U "
"
(p
(p
#(p
r !z
"
!
(p
!z
,
(p
(A.3)
where U is the c.d.f. of a standard normal variable. We have that lim
p"0.
K
Moreover, when p R'r , then not only !E(z)/(p, but also (r !E(z))/(p
"
G
"
approaches !R as mPR, and we have that lim
U(x)"0 and
V\
lim
(x)"0. Hence, the average expected shortfalls approach zero as
V\
m goes to in"nity.
For m, su$ciently large, the partial derivative of the expected shortfalls w.r.t.
E is approximately equal to m times the partial derivative of expression (A.3)
G
w.r.t. E , which is
G
r !z
z
!DpR U "
!U !
(p
(p
;
#r
"
r !z
"
!
(p
1
!
(1!2p )DpR
G
2(pm
z
!
(p
z
1
z
!
DpR!
(1!2p )DpR .
G
2pm
(p (p
(A.4)
Whenever p R'r for all i, the limit of the "rst two terms in (A.4) as mPR is
G
"
equal to zero for the same reasons as before. We also have that the limit of the
last term is zero, since lim
(!z /(p)(1/(p)"0, and the term within the
K
square brackets is a "nite number.
V. Cerasi, S. Daltung / European Economic Review 44 (2000) 1701}1726
1725
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