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Working Paper 95 15
BANK CAPITAL AND EQUITY INVESTMENT REGULATIONS
by Joiio Cabral dos Santos
Joiio Cabral dos Santos is an economist at the Federal
Reserve Bank of Cleveland. The author thanks John Boyd,
Russell Cooper, Joseph Haubrich, Laurence Kotlikoff,
George Pennacchi, James Thomson, and especially Douglas
Gale for useful comments and suggestions. Financial
support from Faculdade de Economia da Universidade
Nova de Lisboa, Fundaq"a Luso Americana para o
Desenvolvimento, and Junta Nacional de Investiga~iio
Cientifica e Tecnolbgica-Programa Ciencia is gratefully
acknowledged.
Working papers of the Federal Reserve Bank of Cleveland
are preliminary materials circulated to stimulate discussion
and critical comment. The views stated herein are those of
the author and are not necessarily those of the Federal
Reserve Bank of Cleveland or of the Board of Governors of
the Federal Reserve System.
December 1995
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ABSTRACT
This paper uses an intermediation model to study the efficiency and welfare implications of
both banks' required capital-asset ratio and the regulation that limits, and in some countries
forbids, banks' investments in equity to a certain proportion of each firm's capital. There are
two sources of moral hazard in the model: one between the bank and the provider of deposit
insurance, and the other between the bank and the entrepreneur who demands funds to
finance an investment project. Among other things, the paper shows that capital regulation
irnproves the bank's stability and can also be Pareto-improving. Equity regulation is never
Pareto-improving and does not increase the bank's stability.
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1 Introduction
T h e risk-shifting effect caused by deposit insurance, when the insurance preiniuin does not
reflect banks' risk, has long been recognized.' It is usually identified by the banks' decision
to finance risky instead of safe investment projects. Nonetheless, regulators have responded
not by modifying the deposit insurance contract, but instead by introducing a wide range
of restrictions on banks7 activities designed t o limit their motivation and ability t o choose
very risky asset portfolios. Although these restrictions tend to vary substantially across
countries, there are some common patterns, such as regulations on banks! capital and on
their investments in the equity of nonfinancial firms.
The 1987 Basle Agreement on Capital Standards, reached by the GI0 countries, and the
1993 introduction by the European Community of the Banks' Own Funds and the Solvency
Ratio Directives, both in line with the Basle Agreement, were the main regulations that
implemented the international harmonization of capital requirements."
With respect to banks7 investments in the equity of nonfinancial firms, of particular
interest is the regulation that limits each of these investments to a certain percentage either
of the firm's capital or of its voting rights.3 For example, this limit is 50 percent in Norway;
25 percent in Portugal; 10 percent in Canada and Finland; 5 percent in Belgium, Japan,
the Netherlands, and Sweden; and zero percent in the United States, because U.S. commercial banks are not allowed t o invest in equity. Germany and Switzerland are examples of
countries where banks7 investments in equity are not limited by that form of regulation.
T h e objective of this paper is t o study the efficiency and welfare implications of both
'Kareken and Wallace (1978) and Dothan and Williams (1980) are examples of works that have used
state preference models to prove this result; Merton (1977, 1978) pioneered the use of options to show it.
'For a presentation of the capital regulations, see Cordell and King (1992).
31n most countries, such investments are also subject to the prudential limits of the banks' capital
regulation. This regulation, in general, limits these investments to a certain percentage of the bank's capital.
For a characterization of these limits in the OECD countries, see Pecchioli (1987) and Schuijer (1992).
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banks' capital and equity investment regulations, and the impact of these regulations on
banks' stability and on the contracts they use to finance firms.
This study is conducted in an intermediation inodel where banks are the only source
of external funds to firms. A crucial feature of the model is the existence of two sources
of moral hazard. One is in the relationship between the bank and the provider of deposit
insurance, because the bank assunles that the insurance premium it pays does not reflect
the risk of its assets. The other source of moral hazard is in the relationship between the
bank and the entrepreneur (firm) who demands funding, because the investment's return
depends on the entrepreneur's effort, which is not observable. Furthermore, because one
of the regulations studied here involves banks' investments in equity, the project held by
the entrepreneur is designed so that the optimal financing contract can be replicated by a
(unique) combination of debt and equity.
In this model, the bank must choose its capital structure and the contract it uses to
finance the entrepreneur. As a result, the risk-shifting effect due to deposit insurance is
translated here not in the bank's decision to finance risky instead of safe investment projects,
as is common in the literature, but in its choice of a contract that motivates the entrepreneur
to adopt a riskier behavior, which in turn increases the risk of the bank's assets. Therefore,
the firm's capital structure becomes dependent on the conditions under which the bank
operates, namely the presence of regulations and the existence of deposit insurance.
Using this framework, the paper shows that an increase in the required capital-asset
ratio improves the bank's stability, can be Pareto-improving, and in some cases, can even
increase the model's efficiency, in the sense of making its solution closer t o the first-best
outcome. This policy has a cost, because it forces the bank to use relatively more of its
most expensive source of funding (capital). However, it also has a positive effect. By
increasing what the bank's equityholders have a t stake in case of bankruptcy, an increase in
the required capital-asset ratio decreases the bank's incentives to motivate risky behavior
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by the firm to which it supplies funds. This is implemented through a reduction in the part
of the contract used by the bank that is more risk-motivating (debt). As a consequence, the
bank's risk of failure declines, which in turn reduces the inoral hazard costs due to deposit
insurance.
Soine of these results have not been captured by studies of banks' capital regulation,
because the approach usually adopted does not endogenize the financing contracts, and
because the inain focus has been to study the implications of that regulation on banks' risk.
Examples of the research conducted in this area are Kahane (1977), Koehn and Santoinero
(1980)' Kim and Santomero (1988), Furlong and Keeley (1989)' Keeley and Furlong (1990),
Rochet (1992), and Campbell, Chan, and Marino (1992).
With respect t o the equity investinent regulation, we will see that the introduction of a
liinit on the percentage of the firm's capital that a bank can hold has mixed effects on the
efficiency of the model, does not increase the bank's stability, and is never Pareto-improving.
Furthermore, we will see that it is even possible to observe an increase in the bank's risk
of failure due to the introduction of this form of regulation. The intuition for these results
is based on the following argument: Limiting the bank's ability to finance a firm through
an equity contract forces the bank to siinultaneously supply part of the funds needed by
the firin through a different financial instrument that might be more risk-motivating than
equity. As a result, the gains in stability that might occur from the reduction of the bank's
stake in the capital of the firm are offset, and in some cases outweighed, by the risk effect
of the financial instrument that the bank uses instead to finance the firm.
Regulations on banks' investments in equity have not been a p r i ~ n ecandidate for research. In work done independently, John, John, and Saunders (1994) show that when
the bank cannot control the firm's investment decisions, the efficiency of the investment
is higher and the bank's risk is lower if the bank uses equity in conjunction with debt to
finance the firm. However, when the bank can veto the firm's investment decisions, there
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is a trade-off between the increase in investment efficiency and the increase in the bank's
risk.
When the bank cannot control the firm's investment decisions, both the John et al.
model and the model presented here produce the same conclusion: Limiting or forbidding
the bank to finance a firrn using equity in addition t o debt deteriorates both the investment's
efficiency and the bank's risk (John et al. rely on the variance of the bank's cash flows as
their measure of risk, while here this tneasure is given by the bank's probability of failure).
Despite the conlmon assumption that banks are the only source of external funds to
firms, there are important differences between the two approaches. For example, the model
adopted here incorporates the bank's capital structure and the existence of deposit insurance, both of which are absent from their model. This allows the study of the banks' capital
and equity investment regulations in the same framework, which also happens to sustain
the optimality of debt and equity contracts.
The paper proceeds as follows: Section 2 introduces the model, characterizes the firstand second-best solutions, and shows the optimality of debt and equity contracts. Section
3 studies the capital and equity investment regulations. Final remarks are presented in
section 4, followed by two appendices, one containing the proofs and the other a numerical
example.
2
The Model
The model adopted in this paper comprises four elements. First, there is an entrepreneur
(firm) with an investment project, but without the necessary funds to finance it. (In subsection 3.3, I discuss the implications for the results of this model if there were many firms.)
Second, there is a bank, which chooses its capital structure and the contract used t o finance the project. Third, there is the deposit insurance provider, which charges the bank
a premium and commits t o reimburse depositors if the bank fails. Finally, there is the
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public, which is willing to supply any amount of deposits as long as it receives the certainty
equivalent to the risk-free interest rate.
The crucial features of the model are the nature of the relationships between the bank
and the entrepreneur on the one hand, and between the bank and the provider of deposit
insurance on the other hand. The relationship between the bank and the entrepreneur
is characterized by a principal-agent problem, where the bank is the principal and the
entrepreneur the agent. T h e moral hazard in this part of the model is generated by the
dependence of the project's returns on the entrepreneur's effort, which is not observable.
With respect to the relationship between the bank and the provider of deposit insurance,
it is assumed that the bank does not take into account, ex ante, how its actions affect the
insurance premium it must pay. A possible explanation for this behavior is that the bank
views itself as a small unit of the banking sector, and it assumes the insurance premium t o
b e determined by the risk of the whole banking sector rather than by the risk of its own
assets.
The assumptions of the model are as follows:
Assumption 1 There is a risk-neutral entrepreneur with an investment project, but with-
out the necessary funds to finance it. The project requires a fixed initial investment equal
to 7, and produces one period later the total return y; with probability p;. The number of
possible returns of the project is finite. I n particular, I assume that it has only three possible
returns:
Yo
< Y1 < 512,
where yo = 0 and y = {y,, y2).
T h e probability distribution of the project's returns is assumed t o be an endogenous
variable because it depends on the entrepreneur's effort. Moreover, it is also assu~nedthat
the entrepreneur incurs a cost for each level of effort he chooses. One way t o model this
situation would be to take the entrepreneur's effort as the choice variable. In that case, it
5
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would be necessary to specify a functional relationship between the probabilities and the
effort, and t o define a cost function with effort as an argument. Instead, I use the approach
where the choice variables are the probabilities themselves. Because the entrepreneur incurs
a certain cost for each probability distribution he chooses, the cost function depends on the
probabilities (p,,p2) with the convention that p,
= 1 - C?==,
p;. T h e inain advantage of this
approach is that it avoids some of the technical difficulties that frequently appear when
solving a principal-agent p r ~ b l e m .In~ addition, the choice of a cost function that is strictly
convex and strictly increasing in its arguments makes it possible t o use convex p r o g r a ~ ~ l ~ ~ l i n g
theory in solving the model.
Assumption 2 Let C(.) denote the cost function.
Then C(p) : p -+ R+, where p =
{pI,p2), and C(.) is C< strictly increasing, and satisfies the condition C(0) = 0. 171 partic-
ular, I use the following cost function:
Assumption 3 The bank's equityholders are risk-neutral.
The opportunity cost of the
bank's capital ( r ) is assu~z~ed
to be larger than the risk-free iwterest rate (i) because of, for
example, a tax on the bank's profits. I n accordance with the actual capital regulation, the
bank must satisjjj a minimum capital-asset ratio, where the assets are weighted accordi7~g
to their risky5 that is:
where K is the bank's capital and 8 is the required capital-asset ratio.
4For a discussion of the advantages of this approach, see HolmstrGm (1979).
51n line with the Basle Agreement, I assume that bank's investment in is treated, for the purpose of
capital regulation, in the same way as the loan it gives to the firm. However, some countries have adopted
the requirement that banks finance such investments solely with capital.
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Assumption 4 The public i n this economy is risk-averse.
It is willing to supply any
arnount of deposits, provided it is paid the certainty equivalent to the risk-free interest rate.
One way to satisfy this condition is to have the bank pay this interest rate o n deposits and
to hold deposit insurance, which will compensate depositors i f the bank fails.
2.1
The First-Best Outcome
The first-best outcoine would be the solution to the model if there were no sources of inoral
hazard, or if the entrepreneur had enough funds to finance the project. In both cases, this
outcome would be the solution to the following problem:
where
nibis the entrepreneur's first-best
profits and e is a vector of ones.
Because the objective function is C 2 and strictly concave in p, and the feasible set is
convex and compact, we are in the presence of a convex programming problem. In this
case, we know that there is a unique optiinu~nand that the Kuhn-Tucker conditions are
necessary and sufficient for a solution.
Assuming that the entrepreneur is not able to completely eliminate the risk of failure of
the project, that is, pib > 0, then the first-best outcome to this model is
H IIfEb = --1(1
2
+ r),
This solution implies some restrictions for the parameters of the nod el. First, because
we are working with probabilities, it is necessary to make sure that their values are positive
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and smaller than one. Second, because of the initial assunlption that the entrepreneur is not
able to eliminate the risk of failure of the project, we need to have pfb
+ pib < 1. Finally, in
order for this project to be undertaken, it is necessary that its first-best profits are positive,
that is, IILb > 0.
These restrictions are summarized in the following assumption:
Assumption
5
YI
< al,
a1 ( ~ 2 a,)
2.2
0
< y 2 - a 3 < a,,
+ a2y, < a1a2,
The Second-Best Outcome
When the entrepreneur gets the necessary funds to finance his project from an outside source
(in this model, a bank), the following question can be raised: What are the characteristics
of the optimal contract that will rule their relationship? Given that the effort chosen by the
entrepreneur is not observable, and that at the beginning of the period the bank will supply
a fixed amount of funds equal to
7 (because by assumption the entrepreneur has no funds),
then the only thing left t o be defined by the contract is the payment that the entrepreneur
will make t o the bank a t the end of the period. This payment will be contingent on the
observable inforination at that time, that is, the income of the project.
Let r; be the payment required by the bank contingent on the return y;. Due to the
limited liability condition, we have ro = 0, since by assumption yo
= 0, and
r; 5 y; for
i = 1,2. Based on this definition, the contract between the two parties can be written as
( I ,r ) , where I
is the number of monetary units supplied by the bank and r is the vector of
(non-negative) contingent payments made by the entrepreneur.
The optimal contract and the bank's optimal capital structure are given by the solution
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to the following problem:
y2 - as - r,
- a2p2= 0
0lr;ly;
for
rr 2 e 1
K+B=I
p,Max (0, - Q B )
where Q
i=1,2
+ C : = l p ; ~ a x (0, r; - Q B ) - K ( l + r ) 2 II,,
= [(I + i) + q];q is the insurance premium, which the bank assumes independently
of its actions; B are the bank's deposits; and I I B are the profits demanded by the bank to
finance the project, which can be any value between zero (representing the case where there
is perfect conipetition ainong banks t o finance this firm) and IIga" (representing the case
where the bank behaves like a monopolist).
The two linear constraints included in that problem are the entrepreneur's incentive
constraints. They are the first-order conditions to the following problem:
Max
P
s.t.
T I E = C:=l p;(y;
-
r;) - C(p)
pe 5 1
P>O'
where
r;
are the payments demanded by the bank. The importance of these constraints
results from the impossibility of observing the entrepreneur's effort, which determines p.
Through them the bank motivates the entrepreneur to choose (voluntarily) the proper
probability distribution.
In the process of finding the optimal contract and the bank's capital structure, the
following observations are taken into account. First, the problem is solved under the assumption that the bank's probability of failure is p,, that is, r; > &*B* for i = 1'2. As
a result, once the solution has been found, it is necessary t o ensure that it satisfies this
condition.
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Second, since it was assumed that the bank considers the insurance premium it pays
t o be independent of its actions, the model is solved assuming q fixed at a certain level, q.
Once the solution has been found,
q is replaced with its fair price in order t o determine the
equilibrium.6
Proposition 1 The optimal contract to the problem defined here is
with: First, Q*
[(l
( I ,r*), where
+ i) + q*], and q* being the smaller root to the following
equation:
where V2 >O, V,:O a n d & > 0.
1 -p*
, where p* is the Lagmnge multiplier associated with the bank's
Second, f (p*) =
1 - 2p*
participation constraint. It is defined a s p* = - - 2 2 w(q.12
w (q*) =
[ Y ~- &*B*I2
+
[y2 - a,
Wk*)
4[1r'*(i
+ T) + n,]
with
- Q*B*]
a1
a2
Finally, the bank's capital structure is
For a sketch of the proof of this proposition, the values of V,, and the equilibrium insurance
premium (q*), see appendix A.
Using the results in this proposition, it is possible to compute the second-best probability
'The fair price is the premium t h a t the deposit insurance provider must charge in order to get zero
profits. With probability p,*, the bank fails. In this state, the insurance provider gets nothing and must pay
B * ( l + i ) . With roba ability ( p : + p : ) , the insurance provider gets pB* and pays nothing. In equilibrium its
+
profits are zero if p,*B8(l i ) = ( p :
P*
+ P : ) ~ * B *which
,
implies a fair price equal to L(l
+i).
P: + P:
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distribution, p* = (p,",pT,p:), where
Comparing these results with the first-best outcome, we observe that, as expected, the
entrepreneur now chooses a lower level of effort in both states where the project produces
a positive outcome, that is, p:
< p;'b and pl < pib. As a result, the project's probability
of failure, which is also the bank's probability of failure, is now larger than the equivalent
first-best value (p," > pib).
2.3
Debt and Equity as Optimal Contracts
The optimal contract found in the previous section was not characterized in terms of the financial instruments known in the corporate finance literature. However, taking into account
the equityholders' limited liability condition, it is possible t o prove that such a contract can
be spanned by a combination of debt and equity, which shows the optimality of these financial instruments in the model adopted here.
Suppose the bank uses debt and/or equity to finance the investment project. Then the
new contract can be written as ( I , a , d ) , where
I is
the amount of funds supplied by the
bank t o the entrepreneur at the beginning of the period, a is the proportion of the firm's
equity held by the bank, (1 - a ) is the proportion held by the firm's entrepreneur, and d is
the face value of debt borrowed by the firm. As usual, I assume that equityholders are the
residual claimants and that they are protected by limited liability.
Proposition 2 Tlze optimal contract for the problem presented here can be replicated by a
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urzique conzbination of debt and equity, that is, by the contract
and
( I ,cr*,d*), wlzere
Q* and B* are equal to the values defined i n proposition (1).
For proof of this proposition, see appendix A .
Note that, in order to make econo~nicsense, the results in proposition (2) require the
following additional assumption on the parameters of the model.
Assumption 6
Y2 -
a3
- Y1
> 0.
The intuition on why a combination of debt and equity spans the optimal contract is
detailed in Santos (1995). It is based on the following explanation: The debt co~rlponent
of the contract is explained by the constant marginal cost of the entrepreneur's effort in
state 2, that is, a,. Its existence is necessary in order to avoid penalizing the entrepreneur
relatively more in this state than in state 1. With respect to the equity component of the
contract, its presence is justified by the difference of the project's returns across states. Its
existence is important so that the entrepreneur is not relatively more penalized in state I ,
which has a lower return.
Looking a t the contract defined in proposition (2), we see that the financial instruments
used by the bank to finance the firm (debt and equity) depend on the bank's capital structure
(mix of deposits and capital). In other words, the moral hazard due to deposit insurance
eliminates the known result of the separation between the bank's asset composition and
its capital structure.' The nature of the relationship existing between the two sides of the
o or a discussion about the separation of the bank's asset composition and its capital structure, see Klein
(1971), Hart and Jaffee (1974), Szego (1980), and Sealey (1985).
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bank's balance sheet and its impact on the firm's capital structure will become clear in
subsection 3.1.
3
Bank Capital and Equity Investment Regulations
As mentioned before, the efficiency costs of the two sources of moral hazard present in
the model are translated in a lower level of effort chosen by the entrepreneur, implying a
higher probability of failure for the project. Regulators cannot affect the efficiency costs
caused by the moral hazard existing in the bank-entrepreneur relationship, because these
costs are originated by an asymmetric information problem, which cannot be alleviated
by regulation. But what about the efficiency costs originated by the moral hazard due to
deposit insurance? Is it possible t o reduce them through regulation?
This is the subject of the remainder of the paper. In particular, two pieces of regulation
are addressed here: banks7 capital-asset ratio requirement, and the regulation that limits,
and in some countries forbids, banks7 investments in the equity of nonfinancial firms to a
certain proportion of the firms' capital.
Looking at the results in proposition (I), we see that the revenue of the contract used
by the bank t o finance the entrepreneur depends on the relative market power held by each
of these elements in its own sector. At one extreme, we have the case where there is perfect
competition among banks t o finance the project, that is, IIB = 0, and the entrepreneur
gets all the surplus of the project. At the other extreme, we have the case where the bank
behaves like a monopolist, that is, Il, = IIga",and the entrepreneur gets at least the
minimum he requires to undertake the project, which by assumption is zero.
Before we move to the analysis of the regulations in each market structure, it is important
t o take into account the following observation. Using the incentive constraints derived in
1
1
2
subsection 2.2, it is possible t o write the entrepreneur's profits as Il, = - a l p :
5a2p2.
2
Given that it is not optimal for the bank to motivate the entrepreneur to choose p, = p, = 0
+
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(because in this case the bank would get no revenue from the entrepreneur), then when the
bank behaves like a monopolist, the entrepreneur's participation constraint will not be
binding, that is,
II, > 0. As a result, the problem can be solved without imposing such a
constraint.
Because this result simplifies the problern studied here-in
particular, it allows the
finding of explicit analytic solutions for all of the endogenous variables-in
the next two
subsections the regulations are studied for the case where the bank behaves like a monopolist. In subsection 3.3, the same regulations are studied for the case where there is perfect
competition, but this time using a numerical example. As we will see, the major conclusions regarding the impact of both regulations on the bank's stability and their welfare
implications do not depend on the market structure existing in the banking sector.
3.1
Bank Capital Regulation
When the bank behaves like a monopolist, it captures the surplus of the project and, as
was explained in the previous subsection, the entrepreneur's participation constraint is not
binding. In this case, it is possible t o show that the equilibrium is given by the results in
proposition (1) when p* = -m, which implies in the limit f(p*) =
1
-,
and8
2
As stated before, the efficiency costs due to both sources of rnoral hazard are translated in
lower probabilities of positive outcomes (p;'
< pib,for i = 1,2), implying a higher probability
of failure for the project (p,* > p,lb), which is also the bank's probability of failure.
Most of the literature that has studied the impact of mispriced deposit insurance has
identified banks' risk-shifting effect with their decision to finance risky instead of safe in'The same results would be found if the bank's profits were maximized subject to the entrepreneur's
incentive constraints, the bank's budget constraint, and the capital-asset ratio requirement.
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vestment projects. In the present model, as the results above indicate, this substitution
effect occurs through a different channel. I t is inanifested in the bank's adjustinent of the
contract it uses to finance the entrepreneur in a way that motivates hiin to adopt a riskier
behavior, which in turn implies an increase in the risk of the bank's assets. Thus, even if
banks do not change their portfolio of customers, they can still take advantage of the deposit
insurance subsidy by changing the way they do business with their current customers.
What is the impact of the capital regulation under these circumstances? If the bank
had taken into account how the insurance premium it pays is determined, that is, if there
were no moral hazard due to deposit insurance, the bank's profits would be
II,
H
= - - [(I - 8 ) ( 1 + i )
4
+ 8 ( 1 + r ) ]7.
Under these circumstances, since the bank's asset coinposition is independent froin its
capital structure, an increase in the required capital-asset ratio (8) does not affect the
contract used by the bank. As such, it does not reduce the inefficiency of the model caused
by the moral hazard in the relationship between the bank and the entrepreneur, and it does
not affect the entrepreneur's profits. However, since this policy forces t h e bank to substitute
capital for deposits (that is, it forces the bank to use relatively more of its most expensive
source of funding-capital),
it implies a reduction in the bank's profits.
When we consider the moral hazard due t o deposit insurance, t h e bank's profits in
equilibrium are
Comparing (1) with.(2), we see that now an increase in the required capital-asset ratio will
have different implications.
Proposition 3 A n increase i n the bank's required capital-asset ratio implies
(a) A n improvement i n the eficiency of the model, because its equilibrium gets closer to
the first-best outcome.
15
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(b) A n improvement i n the bank's stability, because its probability of failure decreases.
(c) A n increase i n the entrepreneur's profits and, within a certain range, an increase i n
the bank's profits, in which case the regulation is Pareto-improving.
For proof of this proposition, see appendix A.
The results in proposition (3) can be explained in the following way: When the capitalasset ratio is increased, forcing the bank to substitute capital for deposits, there is an
increase in the value of what the bank's equityholders have at stake in case of bankruptcy.
Moreover, given that the bank assumes the insurance premium it pays is independent of its
risky behavior, it does not internalize any potential positive effects arising from this policy.
As a result, in order to minimize its costs in case of failure, the bank adjusts the contract it
uses to finance the entrepreneur in order to rnotivate him to choose a safer behavior. This
is implemented through a reduction in the importance (value) of the financial instrument
that is generally more risk-motivating-debt.
This is why the bank's asset corrlposition
becomes dependent on its capital structure, invalidating the separation result.
The reduction in the payments demanded t o the entrepreneur explains the increase
in both his profits and his effort. This, in turn, explains the reduction in the project's
probability of failure and in the bank's risk of failure, which was the reason for the bank to
adjust its financing contract in the first place.
Finally, the increase in the bank's stability implies a reduction in the equilibriurrl insurance premium. The savings to the bank of this reduction are what differentiate this
situation from the case where there was no moral hazard due to deposit insurance. If they
outweigh the costs imposed on the bank, because it must substitute capital for deposits,
then an increase in the required capital-asset ratio also implies an increase in the bank's
profits (this relation is clear in the proof to proposition [3]). Note that, because the reduction in the equilibrium insurance premium due to increases in the required capital-asset
ratio occurs at a decreasing rate, eventually after a certain level of capital has been reached,
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further increases in the capital requirement will imply a decrease in the bank's profits.
3.2
Bank Equity Investment Regulation
As explained in the introduction, countries in which banks are allowed t o invest in the
equity of nonfinancial firms frequently have a regulation limiting each of these investments
in terms of either the firm's capital or its voting rights. In the extreme, banks in some
countries are not allowed t o invest in such financial instrument at all.
A frequent argument used t o justify this form of regulation is that through it, the
bank's involvement with each firm is limited, reducing the bank's exposure to any major
disturbances caused by a firm's bankruptcy and thus improving stability. This argument is
problematic, because it does not consider all the implications of this form of regulation for
the bank's role as a financial intermediary. In particular, it does not take into account that
by limiting the bank's ability to finance a firm through equity, it also forces the bank to use a
different financial instrument t o supply funds that might be even more risk-motivating than
equity. As a result, the gains originated by the reduction of the bank's stake in the capital
of the firm might be outweighed by the costs of using the alternative financial instrument,
in which case the final outcome of the regulation would be a perverse effect.
In a similar procedure to that adopted for the study of capital regulation, this subsection.
studies the implications for the model's efficiency and for the bank's stability of introducing
a limit on the bank's equity investment defined in terms of the firm's capital. That is,
the bank is not allowed t o hold more than ii percent of the firm's capital, where 15 = 0
represents the countries in which banks are not allowed to invest in equity.
According to proposition (2), if there were no equity investment regulation, the optimal
decision for the bank to finance the firin would be t o choose the combination ( a * ,d*), with
Proposition 4 The introduction of a limit o n the bank's investment i n equity, defined in
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terms of the firm's capital, implies:
(a) A reduction in tihe bank's profits. As such, the regulation is not Pareto-improving.
(b) No improvemertt in the bank's stability.
For proof of this proposition, see appendix A.
The results associated with the introduction of the limit on the bank's investlnent in
equity are better understood if the following two effects of this form of regulation are
considered: First, because in this model equity is one of the optimal financial instruments
the bank uses to finance the firnl, restricting its use creates in itself a distortion. Second,
as was explained before, when the limit is introduced, in order to channel the same amount
of funds to the firm, the bank must increase its use of another financial instrument. In this
model, it increases the loan given to the firm, which is a contract that tends to be more
risk-motivating than equity.
For the case where the regulation is not very restrictive (that is, G is not substantially
smaller than a*),after its introduction the bank will hold the maximum investment in equity
allowed by the regulation (G) and will demand a loan repayment
& with d* < 2 < y,. The
impact of the equity investment regulation here is, in addition to the results in proposition
(4), an increase in the entrepreneur's profits and an improvement in the efficiency of the
model according to a first-order stochastic dominance criterion. There is an increase in the
probability of the project's highest outcome (p,), a decrease in the probability of its lowest
positive outcome (p,), and no change in its probability of failure (p,).
However, as the regulation becomes more restrictive (as ti becomes smaller), the higher
are the chances of observing the bank completely drop its use of equity and finance the
firm through a loan with a face value larger than y,, in which case the regulation not only
decreases the entrepreneur's profits and the model's efficiency (p, decreases, p, is now zero,
and p, increases), but it also has the perverse effect of increasing the bank's risk of failure.
In sum, it is clear from this set of results that the form of equity investment regulation
18
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studied here is never Paretwimproving and, contrary to what is usually claimed, does not
improve the bank's stability.
These results are substantially different from those obtained in subsection 3.1 regarding
the capital regulation. The fundamental reason for such a difference is that by increasing
what the bank's equityholders have a t stake in case of bankruptcy, capital regulation decreases their incentive to take advantage of the deposit insurance subsidy. However, t h e
form of equity regulation addressed here not only lacks this effect, but it also creates a
distortion against one of the optimal financial instruments used by the bank to finance the
firm, which also happens t o be the instrument that is less risk-motivating.
As a conse-
quence, i t is difficult to justify the introduction of such a regulation on equity investments
in a scenario where firms are strongly dependent on banks to raise external funds, and where
the optimality of debt and equity contracts is driven by incentive effects.
3.3
Additional Results
The analysis conducted here assumes that there is only one investment project, and that
the bank behaves like a monopolist. This subsection studies the importance of these assumptions for the results found in subsections 3.1 and 3.2.
When we have a model with only one investment project, what results are missing
compared to the case of multiple projects? Potentially, we could miss the analysis of the
substitution effect, and we surely won't be able to study the scale effect of the regulations.
These effects are particularly relevant for the analysis of the capital regulation. For
example, the literature that has studied this regulation in multiple-project frameworks has
shown that banks substitute safe for risky investments in response t o an increase in the
required capital-asset ratio.
In the model adopted here, even though the bank finances only one investment, we are
still able to capture the substitution effect originated by the regulations, because of the
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endogeneity of the contract used by the bank t o finance the firm. This effect, instead of
being achieved through changes in the portfolio of the bank's investments, is implemented
through the adjustment of the contract used by the bank, but the final result is identical.
As observed in subsection 3.1, an increase in the capital regulation makes the bank change
the financing contract in such a way that the entrepreneur is motivated t o decrease the risk
of his investment, implying a decline in the risk of the bank's assets.
With respect to the scale effect, the assumption of having only one project that requires
a fixed investment imposes some limitations. However, it is still possible to use the results
of this framework in order t o understand the scale effect of the capital regulation. I t seems
clear that the impact on the number of projects financed by the bank due t o an increase in
the capital-asset ratio requirement will depend, on the one hand, on the bank's flexibility
in raising additional capital and, on the other hand, on the relative size of the gains that
this regulation brings to the bank (the decrease in the insurance premium it has to pay)
versus its costs (the cost of having t o use relatively more of its most expensive source of
funding-capital).
Therefore, an increase in the capital regulation will not always imply
a negative scale effect. For example, it is possible to show through the model presented
here that when there are many identical projects, if the cost of capital does not rise rapidly
along with an increase in its demand, then we might observe an increase in the number of
projects financed by the bank in response to an increase in the capital regulation.
What about the assumption of the bank's behaving like a monopolist? Will the results
change if we assume that the bank behaves as if there were perfect competition in the
banking sector?
T h e main advantage of using the monopoly assumption was the possibility of finding
explicit analytic solutions for all of the endogenous variables of the model. This was possible because the entrepreneur's participation constraint was not binding and, as a result,
was ignored in solving the model. When there is perfect competition among banks, this
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simplification can no longer be used and, as a consequence, it becomes impossible to find
explicit analytic solutions for all of the endogenous variable^.^
This precludes the study of the regulations in a way identical to that adopted in subsections 3.1 and 3.2. Thus, the analysis had to be conducted through a numerical example. T h e
parameters of this example and the effects of both regulations are in appendix B. Comparing them to the results found under a monopoly framework, we see that capital regulation
improves the bank's stability and can be Pareto-improving in both market structures. In
addition, when the bank behaves like a monopolist, we see that efficiency is clearly increased
because of the approximation of the second-best solution to the first-best outcorne. But,
when there is perfect competition among banks, due to the decrease in the probability of
the highest state ( p : ) , such approximation is only partial.
Regarding the equity investment regulation, we see that in both market structures it
is not Pareto-improving, it does not improve the bank's stability (in fact, when there is
perfect competition, the perverse effect is dominant), and it has the same mixed effects in
terrns of its impact on the efficiency of the model.
This set of comparisons confirms that the main conclusions about the impact of the
bank's capital and equity investment regulations hold both when there is a monopoly and
when there is perfect competition in the banking sector.
4
Final Remarks
The risk-shifting effect due to deposit insurance has usually been identified by banks' decision to finance risky rather than safe investment projects. In the model presented here, this
effect is manifested through a different channel. It occurs through the bank's adjustment of
the financing contract it uses in a way that motivates the firm t o adopt a riskier behavior,
'NOW
we need the entrepreneur's profits so that this function can be maximized subject to, among other
things, the bank's participation constraint.
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which in turn increases the risk of the bank's assets. Thus, the firm's capital structure becomes dependent on the conditions under which the bank operates-namely,
the existence
of deposit insurance and the presence of regulations.
Under these circumstances, capital regulation decreases the bank's incentives to iriotivate
risky behavior by the fir111 to which it supplies funds, because it forces the bank to hold
relatively more capital-the
funds it has at stake in case of bankruptcy. Hence, in addition t o
the cost that this policy imposes on the bank (because it requires the bank t o use relatively
more of its most expensive source of funding-capital),
it also reduces the moral hazard
costs due to deposit insurance (because the equilibrium insurance premium is reduced). As
a result, an increase in the capital requirement reduces the bank's risk of failure, can b e
Pareto-iinproving, and can also have a beneficial effect on the efficiency of the model.
Introducing a limit on one of the optimal financial instruments a bank uses to financ.e
a firm not only creates a distortion against this instrument, but it also forces the bank to
use alternative contracts in order to finance the firm. This is what happens when firms
depend largely on banks t o raise external funds, and when the regulation limits banks'
equity investments in nonfinancial firms t o a certain limit, defined in terms of each fir~n's
capital.
In the model presented here, this type of regulation is not Pareto-improving, and i t does
not improve the bank's stability: By limiting the bank's ability to use equity to finance a
firm, the regulator forces the bank to use more debt in order t o channel the necessary funds
to the firm. This offsets the effects of the reduction of the bank's stake in the capital of the
firm and, in some cases, it might even create the perverse effect of increasing the bank's
risk of failure because debt, in general, is more risk-motivating than equity.
How robust are these results? Conducting the analysis in a framework where both debt
and equity are optimal contracts introduced certain limitations because of the design of the
project held by the entrepreneur. Nonetheless, most of the results hold regardless of whether
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there is a rnonopoly or perfect coinpetition in the banking sector. In aclclition, the results
concur both with the literature on banks' capital regulation [see, for example, Furlong
and Keeley (1989)l and with the literature that has studied the risk effects associated
with different financial instruments-namely, that debt financing tends to be more riskmotivating than equity financing [see, for example, Pozdena (1991)].
It reinains a topic for future research t o study the impact of equity investment regulations
when firms have access t o capital markets, particularly to the stock market.
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Appendices
Proofs of Propositions
A
A.l
Proof of Proposition 1
For a given value of the insurance premium (q), the problem defined here can be solved
through the following steps. First, because of the assumption that capital is more expensive
than deposits, we know that the bank chooses the minimuin required capital, that is, K* =
91. Based on this and on the bank's budget constraint, we know its demand for deposits,
B* = (1 - 9)I. Second, using that information and the entrepreneur's incentive constraints,
we can write both the bank's and the entrepreneur's profits in the probabilities. Third, froin
the first-order conditions of the problem, we can find the optimal probabilities (p:,p:).
Since
this is a convex problem, there is no need to consider the second-order conditions. Fourth,
using the values of p f , through the entrepreneur's incentive constraints, it is possible t o find
r f , and through the definition of the fair insurance premium, one can derive the second-
-- {
degree equation in q* presented in the proposition, where
V,
v,
(1
+ i) { a,a2 -
[u~[YI
(a,+az)(l-s)f-
% = (a,
- (1 - O ) ~ I +a1[~2- s - (1 - 9)f1] f (P*)},
[a,[?,,
-(1-9)f1+a1[~,-a3-(1-~)f1]}f(P*),
+ a,)(l - B ) f f ( ~ * ) .
Note that Vo > 0 because of the conditions imposed by the first-best solution to the model
[assu~nption(5)], V,
> 0, and V, :o. In order to have an equilibrium, we need to have V, < 0.
In this case, the equilibrium insurance premium is the smaller root to the second-degree
equation referred to above, because this root Pareto dominates the larger one. Hence we
have
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A.2 Proof of Proposition 2
Taking into account the equityholders7 liittited liability condition, this proposition can b e
shown through a spanning argument. For a given percentage of the firm's capital held by
the bank (Ci), and a given face value of debt ( d ) , the entrepreneur must solve the following
problem:
Max
P
s.t.
TIE =
pe
c:=,
p;(l - Ci)(y;
-
2) - C(p)
51
P > 0.
This is a convex programming problem, so the usual results apply. For the case where
pi
> 0 for i = 0,1,2, the incentive constraints are
The idea of the proof is to show first that there exists a feasible combination of cr
and d that motivates each entrepreneur, through his incentive constraints, to choose the
probability distribution p*, in which case the initial conditions are verified (p;
>
0 for
i = 0,1,2), and second, that such a combination generates the same revenue to the bank as
r* does.
From the incentive constraints and the second-best probability distribution p*, it is
possible t o find a* and d*, that is, the values that the bank would have to choose in order
t o implement p*. Since these values satisfy the initial conditions, the last thing left to be
proved is that the coinbination (cr*,d*) generates the same revenue to the bank as r* does.
This is apparent immediately once we recognize that cr(y;
-
d)
+ d = r;, for i = 1,2, and
we take into account the equityholders7limited liability condition in order to explain why
in state 0 (the state where the project's outcome is equal to zero), the bank in a position
as debtholder receives no payment from the firm's equityholders.
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A.3
Proof of Proposition 3
Given that an increase in the bank's required capital-asset ratio implies a reduction in the
equilibrium insurance premium (q*), it is straightforward to show that both probabilities
of the project's positive outcomes rise when 8 is increased. This in turn implies an increase
in the entrepreneur's profits ( T I E ) and a decrease in the project's probability of failure (p:).
The impact, in equilibrium, of the capital regulation on the bank's profits (TI:)
is given by
From here we see what happens to the bank's profits when there is an increase in the
required capital-asset ratio. This implies a cost for the bank because it must use relatively
> i by assumption (3)].
more of its most expensive source of funding-capital
[note that r
But it also implies a positive effect for the bank-the
reduction of the moral-hazard costs
caused by deposit insurance, which is given by dq*(e) < 0. Whether the bank's profits
de
increase with that policy depends on the relative magnitude of these two effects.
A.4
Proof of Proposition 4
One way of showing the results in this proposition is t o derive all endogenous variables as
functions of a, and then study the impact on these variables of a reduction in a.
The problem that the bank must solve is
There is no need to consider here the entrepreneur's participation constraint because, as
previously explained, this constraint is not binding when the bank behaves like a monopolist.
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We already know that the bank's capital structure is K* = fir and B* = (1 - 0)I. In order
to find the endogenous variables as functions of a, the bank's problem is solved in two steps.
In the first step, the optimal value of d is found assuming that a and q are constant. In the
second step, the first-order c.ondition of the first step is substituted in the bank's problem
so that the optimal value of a can be computed.
The first-order condition of first step is
Using (A.5) and the entrepreneur's incentive constraints, one can compute the project's
probabilities as functions of a. They are
1
= 2a1(a, a,)
+
+
[p1,1
,
~ 1 , 2 ~ ]
where
PI,,
= (a1 + a2)[~1- Q(1 - fl)J]
1
' ,2
=T
- a 1 ( ~ 2- a3 - Y,),
2a1(~2- yl),
P2,1= (al
+ a2)[y2
Pz,, =. 2a2(yz
- a,
-
&(I
- 8)f]
+ a2(y2- as - y,),
- -1).
Since P,,, > 0 and P,,,
> 0, the equity regulation implies an increase in p2 and a decrease in
p,. Furthermore, since p, does not depend on a, this regulation does not affect the bank's
probability of failure and, as a result, has no impact on the equilibrium insurance premium.
Through the same procedure, one can determine both the entrepreneur's and the bank's
profits as functions of a. They are
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1
n~(c-w)= 4ala2(a1+ a,)
[B,
+ B, a - B2a2]- Of (1 + r ) ,
where
Eo= {a2[yI - Q(1 - e)f]
El = 8ala2(~2- y1)(y2- a,
E
2
2
+ al[y2- a, - Q(1 - e)f]} + 4a,a2(y2- a,
-
-
yl)2 ,
y,),
= 4a1az(~z- Y I ) ~ ,
and
so= {a2[y, - Q(1 - ~ ) f+]a,[y2 - a, - Q(1 - e)i]} ,
2
B1 = 4ala2(~2- y1)(y2- a,
B2
4a,a2(~2- ~
-
y,),
1 ) ~ .
Because the profits of the entrepreneur are a decreasing, strictly convex function of a
in the relevant range, the equity investment regulation implies an increase in the value of
this function. With respect t o the bank's profits, note that they are an increasing, strictly
concave function of a with its maximum, as expected, at the point where c-w = a* with
1
f ( p * ) = -. Thus, the introduction of the equity limit implies a reduction in the bank's
2
profits.
The results derived so far in this proof assume that the optimal value of d, determined by
(A.5) for a given maximum equity that the bank can hold. ( 6 ) )is smaller than y,. However,
because there is an inverse relationship between d and 6, the smaller the value of 6 (the
more restrictive the regulation), the higher the chances that d becomes larger than y,. In
this case, depending on the parameters of the model, it may be optimal for the bank to
co~npletelydrop its use of equity and begin financing the entrepreneur using only debt with
a face value higher than y,. Under these conditions, the entrepreneur chooses to put forth
no effort in state 1, which i~npliesan increase in the bank's probability of failure and, as
a result, an increase in the equilibrium insurance premium. Note, however, that the bank
does not take this into account, because it assumes that the insurance premium it pays is
independent of the risk of its assets.
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B
Numerical Example
The set of parameters chosen for the numerical example was
The first-best solution t o the model with these parameters is
The results for the bank's capital and equity investment regulations, when there is
perfect competition among banks, are presented in the next two subsections. For the case of
the capital regulation, subsection B.1, the equilibrium is given by the results in propositions
(1) and (2) with IT, = 0. The values of the endogenous variables are plotted as functions
of 8, the required capital-asset ratio. The impact of an increase in the capital regulation is
given by the variation in those variables when 8 is increased.
With respect to the equity investment regulation, subsection B.2, the equilibrium for
each given value of 6 (the maximum stake of the firm's capital that the bank can hold) is
determined by the face value of debt that the bank has t o charge the firm, given the bank's
zero-profit condition, and its behavior with respect to deposit insurance.
The values of the endogenous variables are plotted as functions of 6 , for the range of
the nutnerical example where this limit is binding, that is, for 6
< a*.The impact
of the
equity investment regulation is given by comparing the second-best solution t o the model
with the values of these variables associated with a given value of 6 .
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B.l
Results of the Capital Regulation
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B.2
pi
0.4809
0.4786
0.4763
Results of the Equity Investment Regulation
I--
I
For 8 = 0.08 :
1
a
Represents the second-best so-
lution to the model.
-
Represents the equilibrium for
-----------
each maximum stake of the firm's capi-
I
I
I
tal that the bank is allowed to hold (2).
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