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Faculty Working Paper 92-0100
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A Theory of Optimal Bank Size
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Stefan Krasa
Department of Economics
University of Illinois
Anne P. ViUamil
Department of Economics
University of Illinois
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Bureau of Economic and Business Research
College of Commerce and Business Administration
University of Illinois at Urbana-Champaign
BEBR
FACULTY WORKING PAPER NO. 92-0100
College of Commerce and Business Administration
University of Illinois at Urbana-Champaign
January 1992
A Theory of Optimal Bank Size
Stefan Krasa
Anne P. Villamil
Department of Economics
A Theory of Optimal Bank Size
Stefan Krasa Anne P. Villamil*
First Draft: May 1991 This Draft: December 1991
Abstract
This paper provides a theory of optimal bank size determination
with implications for the size distribution of banks in a model with
asymmetric information and costly state verification. Production is
subject to minimum scale requirements and two types of risk: diver-
sifiable idiosyncratic project risk, and imperfectly diversifiable aggre-
gate "macroeconomic" risk. We first show that delegated monitoring
with two-sided simple debt contracts dominates direct investment if
the cost of monitoring the intermediary is bounded and if the variance
of the non-diversifiable macroeconomic risk is sufficiently small. We
next show that: (i) banks are of finite size; (ii) bank size is inversely re-
lated to the bank's exposure to macroeconomic risk, and (iii) multiple
banks co-exist with the same size within a locale but with (possibly)
different sizes across locales.
'Address of the authors: Department of Economics, University of Illinois, 1206 South
Sixth Street, Champaign, IL 61820.
We gratefully acknowledge useful comments from Anthony Courakis and financial sup-
port from the National Science Foundation (SES 89-09242).
1 Introduction
Recent research has studied how the structure of the financial system affects
the transmission of business cycle shocks in an economy. See Gertler (1988)
for an excellent survey of this emerging literature. An equally important but
quite different problem is the following: How do business cycle fluctuations
affect the structure of the financial system? This problem is important be-
cause all developed countries are subject to regular and recurrent business
cycle fluctuations. Further, banks in most countries operate under restrictive
regulations which limit their ability to insure fully against macroeconomic
risk. For example, in the U.S. there are branching restrictions which limit
the geographic operation of banks and portfolio restrictions which limit the
types of assets that banks may hold (e.g., Savings and Loan Institutions have
been subject to such restrictions). It is obvious that these restrictions can
lead to non-diversifiable portfolio risk, but little is known about the impli-
cations for financial structure. This problem is of particular interest at the
present time. The European Community is currently involved in a transition
toward an economic and monetary union (EMU). To date, most discussions
of the EMU have focused on the creation of a European central bank and its
supervision of a single currency. However, once an administrative structure
is in place, it will undoubtedly impose u pan European" restrictions. What
are the likely implications of such restrictions for the future European bank-
ing system? This paper proposes a theoretical model that can be used to
analyze this important policy question.
We propose a theory of optimal bank size determination which has im-
plications for the size distribution of banks. We consider a costly state ver-
ification model with finitely many borrowers and lenders where production
is subject to: (i) minimum project scale requirements, (ii) diversifiable id-
iosyncratic project risk, and (iii) a non-diversifiable aggregate (i.e., macroe-
conomic) risk. Agents can undertake production in two ways. Borrowers and
lenders can write direct bilateral investment contracts, or they can engage in
intermediated investment by contracting with a bank that accepts deposits
from lenders and grants loans to borrowers. Because there is non-trivial de-
fault risk in the economy, the lenders must monitor either the borrowers (in
the direct investment problem) or the bank (in the intermediated investment
problem) in default states which occur with strictly positive probability.
The minimum project scale requirement implies that it takes multiple
2
lenders to finance the project of a single borrower. It is well known (cf.,
Diamond (1984) or Williamson (1986)) that "delegated monitoring" may
be optimal under this requirement because it allows lenders to economize
on monitoring costs. However, in an economy with non-trivial default risk
lenders must "monitor the monitor" (i.e., bank) because the bank may mis-
report the state to minimize its payments to lenders. Thus, the essential
problem that lenders face is to provide the bank with an incentive to report
truthfully to them. Krasa and Villamil (1991a) study this problem in a fi-
nite economy that is subject to diversifiable default risk. They show that a
particular type of contract commonly issued by banks (i.e., two-sided simple
debt) solves this monitoring problem optimally. In contrast, in this paper
we are concerned with the optimal investment arrangement (given two-sided
simple debt contracts) when there is both diversifiable project risk and non-
diversifiable aggregate or "macroeconomic" risk. When a bank is subject
to non-diversifiable project (and hence default) risk, costly monitoring will
necessarily occur in some states — regardless of the bank's size.
In choosing an optimal portfolio size (i.e., scale of operation) a bank faces
the following tradeoff for most monitoring cost structures. Increasing the size
of the bank's portfolio (i.e., contracting with additional borrowers) given
some initial bank size generally decreases the bank's default probability, but
increases the lenders' cost of monitoring the bank. Thus, the crucial question
that the bank faces in choosing an optimal portfolio size is: Under what
circumstances do the gains from decreased default risk dominate the losses
from increased monitoring costs when the bank compares its current scale of
operation with an increased scale of operation? We begin our analysis of this
question with Theorem 1 which shows that delegated monitoring with two-
sided simple debt contracts (i.e., intermediated investment) dominates direct
investment if the lenders' cost of monitoring the intermediary is bounded and
the variance of the non-diversifiable macroeconomic risk is sufficiently small.
We interpret the variance of the macroeconomic risk as the magnitude of
business cycle fluctuations. For the U.S., Prescott (1986) reports that the
standard deviation of output for the period 1872 to 1985 is only 1.8 percent.
Theorem 2 provides the main result of the paper: A theory of optimal
bank size with implications for the size distribution of banks. To our knowl-
edge, this has been a neglected branch of the literature on financial inter-
mediation. Of course, the problem of firm size distribution has been studied
extensively in industrial organization theory. Panzar (1989, p. 33) summa-
rizes the findings from this literature by noting that firm size "is determined
in large part by the . . . cost function," while industry structure (i.e., limits
on the number and size distribution of firms that are present in equilibrium)
"is determined by the market demand curve." Our Theorem 1 implies that
this traditional industrial organization analysis is inadequate for a theory of
financial structure. In Theorem 2 we develop a measure of the rate of port-
folio diversification accruing from increases in bank size, and then show that
both risk and cost considerations are essential determinants of bank size.
The theory has three predictions that in principle are empirically testable.
First, banks will be of finite size with the precise scale dependent upon the
structure of monitoring costs and the degree of portfolio diversification that
the bank can attain. Second, banks that are better able to diversify risk (e.g.,
because they are subject to less stringent portfolio restrictions) will be larger
in size than banks which are less able to diversify risk. Third, multiple banks
with similar risk and cost characteristics may co-exist. The first prediction
pertains to firm size, while the latter two are industry structure predictions.
Clearly, firm size and industry structure are related; however, we shall return
to a more precise discussion of this relationship in Section 6.
2 The Model
Consider an economy with finite numbers of two types of risk-neutral agents,
borrowers and lenders. Each borrower i = 1, . . . ,n is endowed with a risky
investment project which transforms one unit of a single input at time zero
into x, units of output at time one, where x, is the realization of a random
variable X{ on the probability space {Q,A, P). 1 For simplicity assume that
borrowers have zero endowment of the input. Every lender j — 1, . . . ,m is
endowed with a < 1 units of a homogeneous input, 2 but has no direct ac-
cess to a productive technology. Thus, the project of a borrower cannot be
financed by a single lender which implies that in the absence of intermedia-
tion more than one lender would have to verify a single borrower. The total
available supply of investment is larger than the input required by all borrow-
ers, so m lenders can be accommodated by the h borrowers (i.e., rha > h).
We will implicitly refer to this probability space when writing P for probability and
E for expected value.
2 For technical purposes assume that \/a is an integer.
Also, assume there is a riskless alternative investment project available to all
lenders that yields return r with probability one.
All borrowers and lenders are fully informed about the distribution of
Xi at time zero, but asymmetric information exists about the state of the
project's actual realization ex-post: Only borrower i costlessly observes the
realization x, of his/her project at time one. Let F,(x) denote the distribution
of borrower Vs project and assume that the F{ are identical (i.e., all X, have
the same distribution). 3 We now depart from the standard intermediation
framework by considering an economy with non-trivial correlation among
the X(. 4 Assume that each X t can be decomposed into independent random
variables Yi, i — l...,n, and Z, where Y t is an idiosyncratic risk associated
with borrower z's project and Z is a non-diversifiable "macroeconomic" risk
common to all borrowers. Thus,
X t = Y, + Z, 5 (1)
Assume that the distributions of Y, and Z have continuous density functions,
Xi > for every i (because borrowers can never produce "negative output"
no matter how bad the macroeconomic shock), and that each Xi is bounded
from above.
Let a technology exist which can be used by agents other than borrower i
to verify at time one the realization x t of project X t . Assume that this state
verification technology is costly to use, and that when verification occurs,
x, is privately revealed only to the individual who requests (deterministic)
verification. Assume that the verification cost is comprised of both a pecu-
niary component and an indirect "pecuniary equivalent" of a non-pecuniary
cost. 6 These costs may be thought of as the money paid to an attorney to
file a claim (a pecuniary cost) and the monetary value of time lost when
3 This assumption simplifies the analysis but is not essential for the results.
4 See Dowd (1991) for an excellent survey of the literature on financial intermediation.
5 To simplify the analysis we make the following technical assumption: Let Q = fii x Q 2
and let P = Pi x P 2 , where Pi is a probability on Qi for i = 1,2. Assume that the
random variables Yj are independent of Q21 > e -, f° r every u>i G fii the mapping u> 2 t— ►
Yi(u>i } u> 2 ) is constant on Q 2 . Similarly, assume that u>\ ■— ► Z(wi,w 2 ) is constant on Qi.
This condition implies independence of Z and A", for every i € ^V, but is stronger than
independence. In our analysis, standard independence would require conditions on Q which
imply the existence of a regular conditional probability P{- \ Z) (cf., Parthasarathy (1977,
Proposition 46.5)).
6 The non-pecuniary costs permit negative utility but rule out negative consumption.
visiting the attorney (a pecuniary equivalent). Because agents have asym-
metric information, a key problem is to ensure that the borrower reports
the realization truthfully. We, like Williamson (1986), use the costly state
verification framework to solve this problem. This model was introduced by
Townsend (1979). However, unlike in Townsend's model where x t is publicly
announced after verification occurs, in our model x, is privately revealed only
to the agent who requests verification. 7 This assumption is essential for our
analysis since if all information could be made public ex-post, there would
be no need to verify the bank in default states. However, it also appears to
accurately describe the privacy and institutional features which characterize
most lending arrangements. For example, Diamond (1984, p. 395) observes:
"Financial intermediaries in the world monitor much information about their
borrowers in enforcing loan covenants, but typically do not directly announce
this information or serve an auditor's function."
3 Contract Arrangements
There are two types of basic investment arrangements, notably those involv-
ing direct contracts between "primary" borrowers and lenders, and those
involving an intermediary. In Section 3.1 we consider the direct investment
problem. In Section 3.2 we consider intermediated investment where lenders
and borrowers write contracts with a bank, and the bank is subject to non-
trivial default risk.
3.1 Direct Investment
Let all direct, bilateral interactions between lenders and borrowers be regu-
lated by a contract whose general form is defined as follows.
Definition 1. A one-sided contract between lenders and borrowers is a
pair (R(-),S), where R(-) is an integrable, positive payment function on M + ,
while pecuniary equivalents of non-pecuniary costs ensure that the costs can be shared by
the contracting parties.
7 See Krasa and Villamil (1991b) for an analysis of an economy with multiple het-
erogeneous agents, costly state verification, public announcement, and deterministic or
stochastic verification.
such that R(x) < x for every x E 1R+ and S is an open subset of JR+ which
determines the states where monitoring occurs.
The contract (R{-), S) describes the total claims against the borrower
by all lenders. If a lender invests 6 units of capital in a borrower's project,
then his/her claim against the borrower is given by bR{x), where x is the bor-
rower's announced wealth realization. Following standard practice in this lit-
erature, we restrict the universe of contracts to the set of incentive-compatible
contracts and denote this set by C = (/?(•), 5). Consequently, the realization
announced by each borrower is the true realization. The following condition
ensures that all contracts under consideration satisfy this restriction: There
exists R E M + such that S = {x: R(x) < R}. The imposition of this restric-
tion is without loss of generality because the Revelation Principle establishes
that any arbitrary contract can be replaced by an incentive-compatible con-
tract with the same actual payoff (cf., Townsend (1988, p. 416)). Therefore,
the set of all incentive-compatible contracts is fully specified by the tuple
(R(-),R).
We study a particular type of contract, called a simple debt contract,
which is defined as follows:
Definition 2. (R(-),R) is a simple debt contract if: R(x) = x for x E S =
{x < R} and R{x) = R if x E S c = {x > R}.
The payment schedules in Definition 2 resemble simple debt because:
(i) When verification occurs the payment to the lender is state contingent
(i.e., the borrower pays the entire realization for all outcomes below a
cutoff level), where the verification set S is viewed as the set of bankruptcy
states,
(ii) When verification does not occur the payment to the lender is constant
(i.e., the borrower pays a fixed amount R for all realizations of the state
above the cutoff), where S c is the set of all realization where verification
does not occur.
Townsend (1979) proved that debt contracts are optimal responses to
asymmetric information problems in economies with deterministic costly
state verification technologies because such contracts minimize verification
costs. Agents verify only low realizations of X{ and accept fixed payments
(which do not require monitoring) in all other states. Gale and Hellwig (1985)
and Williamson (1986) showed that simple debt is the optimal contract
among all one-sided investment schemes. In contrast to debt contracts, where
the borrower is the residual claimant in the default state (and hence may re-
ceive a non-zero payment if the bankruptcy is not "too severe"), a simple
debt contract requires the borrower's entire project realization to be trans-
ferred to the lender in default states (i.e., see (i) above). This result will be
useful in the analysis that follows, thus we state it formally.
Theorem GHW. Simple debt is the optimal contract among all one-sided
investment schemes.
The strategy of the proof of Theorem GHW is as follows. 8 Consider two
optimal contracts. Let (R(-),R) be a simple debt contract and (A(-),A)
be some alternative contract. Since both contracts are optimal, both must
provide borrowers with the same expected payoff. Under the simple debt
contract lenders request costly state verification if x < R, and under the
alternative contract lenders request verification if x < A. Clearly A > R
(otherwise the contracts cannot have the same expected return to borrowers),
thus the expected verification costs must be less for the simple debt contract.
In our economy with correlation among project realizations, the direct
investment problem is identical to the investment problem in an economy
without correlation among projects. This follows from the fact that when
borrowers and lenders write direct bilateral contracts, every lender must ver-
ify the borrower with whom he/she contracts in default states. Hence, non-
trivial correlation among projects is irrelevant (and simple debt contracts
remain optimal). Note that even though lenders have the opportunity to in-
vest in more than one project (i.e., contract with more than one borrower), it
is not optimal for them to do so because they reap no gain from diversifying
idiosyncratic risk while they incur higher expected monitoring costs. For ex-
ample, suppose that an agent invests in two projects. Assume that the total
outstanding debt of borrowers i = 1,2 is given by the contract {Rt{-), Ri),
and let a, be the capital the lender invests in each project. If both projects
do not fail, then the payoff is simply given by a x Ri + a 2 R2- Thus, there is
no gain in the good state from investing in multiple projects. However, the
probability that at least one of the two projects fail is strictly higher than
the probability that only a single project fails. 9
8 See Gale and Hellwig (1985) or Williamson (1986) for a formal proof.
9 For example, consider the random event to be the toss of a fair coin, where "head"
The direct investment problem between a borrower and lenders can now
be stated. Let c denote the lenders' cost of monitoring a borrower.
Problem 3.1. Choose an incentive-compatible contract (/?(•), 5) to:
rT
max
subject to:
/ [x-R(x)]dF{x)
Jo
a I R{x)dF{x)- I cdF{x)>ra. (2)
Jo Js
In Problem 3.1 (the direct investment problem), the expected utility of a rep-
resentative borrower is maximized subject to a constraint that the lenders'
expected return, net of monitoring costs (c), be at least as great as some
reservation level (r). The first term in the lenders' constraint is multiplied
by a in order to account for each individual lender's capital investment a.
Without loss of generality we assume that each lender invests all of his/her
endowment in a single project. Finally, Problem 3.1 reflects the assumption
that credit markets are competitive. There are more lenders who wish to
invest than investment opportunities. Thus, the supply of loans is inelas-
tic, and the level of return necessary to attract lenders is driven down to
the reservation level r, the return available on the alternative investment
opportunity.
3.2 Intermediated Investment
Now consider an intermediated borrowing and lending problem. In the previ-
ous section (i.e., the one-sided problem), lenders and borrowers wrote direct
bilateral contracts and correlation among projects (as long as it was not
trivial) was irrelevant. However, duplicative monitoring is inherent in the
is non-default, and "tail" is a default. Clearly, for a single coin toss the probability of a
default is 0.5. If the agent "invests" in two coin tosses, the probability that at least one
of the two projects fails is 0.75. Thus expected monitoring costs will be higher. This is
true even if there is some correlation among projects, if idiosyncratic risk is non-trivial.
If the idiosyncratic risk is trivial (i.e., X{ = so only macroeconomic risk matters), then
every agent could monitor only a single project (since the realization of all projects can
be determined by the outcome of any one project) and the expected payoff from investing
in one project or many projects is the same.
direct investment problem because each lender must verify each borrower
with whom he/she contracts in certain states of nature. Thus, there may ex-
ist gains from "delegated monitoring" (cf., Diamond (1984)), where lenders
elect a monitor to perform the verification task and thereby eliminate some
of the duplicative monitoring associated with direct investment. In contrast
to previous delegated monitoring studies, our economy has an important
feature which significantly complicates the "standard" delegated monitoring
problem. The intermediary faces non-trivial default risk for two reasons:
(i) Since there are only finitely many borrowers, it is not clear that the
intermediary can completely diversify idiosyncratic risk,
(ii) Even if the intermediary can eliminate idiosyncratic risk, its portfolio is
still subject to non-diversifiable "macroeconomic" risk.
Thus, in our economy the probability that the bank may default is non-zero
(because at least the macroeconomic risk is non-diversifiable), so lenders must
verify the bank with strictly positive probability (i.e., in some states).
We begin our analysis of the delegated monitoring problem by considering
how agents select an intermediary. Since the loan market is competitive, any
lender who wishes to act as an intermediary must offer contracts which max-
imize the expected utility of the borrowers and assure the remaining lenders
of at least the reservation level of utility (r), which is determined by the
riskless rate of return on the alternative project. Otherwise, agents would
trade directly or another intermediary would offer an alternative contract
(i.e., there is free entry into intermediation) with terms that are preferable
to the n borrowers and/or the remaining m — 1 lenders. Let (R(-), S) denote
aspects of the two-sided contract which pertain to the borrower-intermediary
relationship and (/?*(•), 5*) denote aspects of the two-sided contract which
pertain to the intermediary-lender relationship. 10 The intermediary's prob-
lem clearly embodies optimization by all agents in the economy.
We next derive random variables which describe the income from the
intermediary's portfolio. Recall that R t (x) denotes the payoff by borrower i
to the intermediary if output x is realized, X{ is the random variable which
describes the output x of a particular borrower i in state x, and X{ = \\ + Z
from equation (1), where the Y t are independent random variables but the
l0 (R(),S) is also used in the direct investment problem in Section 3.1. We do not
introduce additional notation in this Section because the structure of the problem is the
same regardless of whether borrowers report to the lenders or to the bank.
10
Xi are not independent for Z/0. The intermediary's income from borrower
z, given transfer R(-), can now be defined by
Gi(R(-);u) = R(X { (u)), (3)
where u> G H denotes the state of nature. Because the X, are not inde-
pendent, it follows that in general the random variables G, are not inde-
pendent. If the intermediary contracts with i = 1,2, ...,n borrowers, 11 its
average income per borrower under payment schedule R(-) is: G n ( R{-); u) =
£ E?=i <?,-(#(•); u>)- Denote the distribution function of G n (-) by F n (-).
The two-sided contract between the intermediary and each borrower, and
the intermediary and the lenders, can now be defined.
Definition 3. A two-sided contract is a four-tuple ((#(•), S), (R m (-), S*))
with the following properties:
(i) R(-) is an integrable positive payment function from a borrower to the
intermediary such that R(x) < x for every x G M + , and S is an open
subset of JR+ which determines the set of all realizations of a borrower's
project where the intermediary must monitor;
(ii) R*(-) is an integrable positive payment function from the intermediary
to the lenders such that R*{x) < x. For every realization x of G n (-),
the payment to an individual lender is given by ^-^R"(x); 12 and S" is
an open subset of 1R + which determines the set of all realizations of the
intermediary's income from the borrowers the lenders must verify.
We now derive the set of all incentive-compatible two-sided contracts.
Each borrower will announce an output which minimizes its payment obli-
gations to the intermediary. Let x = arg min x65 R(x) be the output that
minimizes this payoff over all non-monitoring states, and recall that x is ob-
served directly in the monitoring states S. Consequently, the announcement
by a borrower is given by argmin ie r xf i R(x). A similar condition holds
for the intermediary-lender portion of the contract (i.e., R*(-),R*). As in
11 Note that n need not equal n. In fact, this paper shows that it in general it will not
be optimal for a bank to become as large as possible.
12 /?*() is the total payment by the intermediary to lenders per borrower. Since the
intermediary has a positive initial endowment, rn — 1 lenders are sufficient to finance the
m projects. Thus, to derive the payment to an individual lender, multiply this amount
with -2-7.
m— 1
11
the one-sided problem, the following condition ensures that all contracts are
incentive-compatible. There exist R, R' 6 IR+ such that S = {x: R{x) < R}
and S* = {x:R'(x) < R m }. The set of all incentive-compatible two-sided
contracts is fully specified by the four-tuple {R(-), R), {R*(-), R").
A two-sided simple debt contract is then defined as follows:
Definition 4. A contract (R(-),R), (R m (-),R m ) is a two-sided simple debt
contract if:
(i) R(x) = x for x € S = {x < R} and R{x) = R if x G S c = {x > R}; and
(ii) R"{x) = x for x <= 5* = {x < R*} and R*(x) = R' if x € S' c = {x > R*}.
We will often denote two-sided simple debt contracts by (R, R").
The intermediary's two-sided optimization problem can now be stated.
Let c denote the intermediary's cost of monitoring the borrowers, and let c*
denote the lenders' cost of monitoring the intermediary. In Section 4 these
monitoring costs shall be discussed in more in detail.
Problem 3.2. Choose incentive-compatible contracts (R(-), R),(R"(-), R')
to:
max/ [x — R(x)]dF(x)
Jo
subject to:
?— I R*(x)dF n (R(-),R)(x) - I c n dF n (R(-),R)(x) > r (4)
— 1 Jo Js*
m
I [x - R"{x)]dF n {R(-),R)(x) - I cdF(x)
> r. (5)
Problem 3.2 states that the intermediary maximizes the expected utility of
each ex-ante identical borrower subject to two constraints. (4) states that
the expected payoff to the m — 1 remaining lenders (i.e., those who did not
become intermediaries) must be at least r, the level of utility available from
the alternative project. (5) states that the profit from intermediation (i.e.,
net payoffs from the borrowers less the payoff to the lenders) must also be at
least r. Note that the bank's decision variables are the loan contract /?(•),
the deposit contract /?*(•), and the number of projects n. The number of
lenders is determined by the choice of n.
12
4 Optimal Investment Arrangements
The structure of the optimal investment arrangement will depend crucially
on the nature of the monitoring costs, c and c*, because default risk is non-
trivial and monitoring will occur with positive probability. We now proceed
to prove Theorem 1 which establishes that delegated monitoring is optimal
when the lenders' costs of monitoring the intermediary are bounded and the
variance of the nondiversifiable macroeconomic risk is sufficiently small. The
proof of Theorem 1 depends on continuity of the constraints of Problem 3.2
in the face values R and R* of the two sided debt contract, which follows from
Lemma 1 in Krasa and Villamil (1991a). The strategy of the proof of the
Theorem is as follows. Let R be the simple debt contract which is optimal
among all one-sided schemes described by Theorem GHW in Section 3.1.
We show: (i) there exists an alternative two-sided debt contract (R, R") such
that (4) is satisfied and binding; (ii) (5) is fulfilled but does not bind under
(R, R*); and (iii) by increasing the face value of the lenders 1 debt (say to
B' > R") the payoff to the lenders increases. 13 Then by continuity of the
constraints in the face value of the lenders' debt, a two-sided contract (R, B")
can be found such that both constraints are slack. Finally, by continuity of
the constraints in R, the face value of the borrowers' debt, R, can be lowered,
with both constraints still satisfied. Thus, the delegated monitor offers better
contracts to agents than the best feasible direct investment contract, which
proves the Theorem. The argument requires n, the number of borrowers, to
be sufficiently large; a more precise characterization of n shall be provided
in Section 5.
Theorem 1. Assume that c* is bounded and that the variance of Z is
sufficiently small. Then delegated monitoring with two-sided debt contracts
dominates direct investment.
Proof. Consider first the investors' cost of monitoring the bank. Recall
that the bank faces two types of risk: a diversifiable, project-specific risk
y,, and a non-diversifiable macroeconomic risk Z. Thus, the banks default
probability will in general not converge to zero (even if it contracts with a
I3 In general, the lenders' payoff does not increase monotonically with R' because the
probability that lenders must verify the intermediary is an increasing function of/?". This
is also true for one-sided schemes (cf., Gale and Hellwig (1985, p. 662)).
13
large number of borrowers). By Lemma 1 of the Appendix, the average payoff
from borrowers to the intermediary converges to the expected conditional
return E[R{X X ) \ Z]: 14
±JTR{Y i + Z)-*E[R{Y 1 + Z)\Z}, (6)
as n — » oo. Further, by Lemma 2 in the Appendix, the probability that the
return from borrowers is less than the lenders' fixed payment converges to
the probability that the expected return E[R(X\) | Z] is less than the face
value of the lenders' debt:
P ({^ £ R ( y i + Z )< R '}) - P {{ E i R ( y i +Z)\Z]< R')}) ■ (7)
Now choose z{R') such that E[R{Y X + z(R m ))] = R\ Note that z(R*) is
independent of the distribution of Z. Furthermore,
E [R{Y x + z(R*)j\ = E [R{Y l + Z)\Z = z(BT)\ . 15
Note that z(-) is a "cutoff value" in the distribution of Z which separates
solvency states from insolvency states in the limit. Since R(-) is monotonic
the right-hand-side of (7) equals P (<Z < z(R*)\), which we shall show is
the bank's default probability in the limit.
Next, consider the bank's payoff to a lender when its portfolio is large.
From (6) and from continuity of R'(-) it follows that
Rm [^ E R ( y i + z )) - R ' (E[R(Y l + Z) | Z\)
asn-> oo. Lebesgue's dominated convergence Theorem therefore implies
Hm JR* (^JTRiY^ + Ziu))^ dP(u)
= J R'(E[R(Y l + Z)\Z](u))dP(u) (8)
14 For all random variables X, Y denote by E[X | V] the conditional expectation of X
with respect to Y (which is a random variable, measurable with respect to the information
contained in Y). In particular, let w € ^ be an elementary event for which Y(u) = y.
Then E[X \ Y = y] = E[X \ Y](u). If A' and Y have only a countable number of different
values then this corresponds to the elementary definition of a conditional expectation.
15 This follows from our strong independence assumption. See footnote 5 and the proof
of Lemma 1.
14
Substituting the distribution of - Y^?=i R{Y t (uj) + Z(u)) and the distribution
of Z for P in (8) yields
lim / R*(x) dF n {R(-)){x) = I R' (E[R(Y 1 + Z) \ Z = z}) dH(z) (9)
n— ►oo J J
where H denotes the distribution of Z. For example, if there is no macroeco-
nomic risk (i.e., Z = 0), the right-hand-side of (9) is given by R* (E[R(X X )])
so when R* < E[R(Xi)] the expected return in the limit is given by R* and
the lenders receive the face value of the debt with certainty.
A two-sided contract which dominates the one-sided, direct investment
contract can now be constructed. Let e > be some arbitrary constant.
First, choose B" such that r < B* < E[R(Xi)]. Then (7) implies that the
bank's default probability is less than e for large n, if P({Z < z(B")}) < £,
i.e., the probability that the realization is in the "tail" of the distribution of
Z is sufficiently small. Note, that z(B') < 0. 16 Thus, there exists a 6 >
such that whenever var(Z) < 8 we get P({Z < z(B*)}) < e. 17 Thus, the
lender's expected costs of monitoring the bank are bounded above by £c* for
large n. For similar reasons, the lenders' payoff is bounded from below by
^"(1 — e) for sufficiently large n.
If £ is sufficiently small, (7) and (9) imply that constraint (4) is fulfilled,
but does not bind for the two-sided contract (R,B*). By continuity of (4)
with respect to B* (see Krasa and Villamil (1991, Lemma 1)), there exists a
face value R* < B" such that (4) binds for the two-sided contract (R,R*).
We next show that (5) is fulfilled under contract (R, R'), but does not bind.
Recall that the bank's default probability is less than e. Thus, J 5 . c* dF n <
£c* for the contract with face value B' . Since R* < B* , the bank's default
probability is lower with R*. Thus, by choosing e sufficiently small and since
c* is bounded we can ensure that f s cdF > f s . c* dF n for all sufficiently large
n. 18 This and the fact that (4) binds implies
it
( R'{x)dF n <( m -l)(r+ / cdF). (10)
Jo Js
16 We normalize the mean of the macroeconomic shock Z to zero, so :(B') < denotes
a recession.
17 This is possible since z(B') is independent of the distribution of Z .
18 The inequality indicates that the intermediary's expected cost of monitoring the bor-
rowers is higher than the lenders' expected cost of monitoring the intermediary.
15
Consequently,
j [x-R-{x)\dF n - I cdF
Jo Js
> nE[R(Xi)] -ml cdF - (n - l)r > mr - (m - l)r = r.
Js
The first inequality follows from (10) and from the fact that / x dF n =
E[G n ] = E[R{X\)\, which is the expected value of an aggregate version
of equation (3). The second inequality follows because R must fulfill (2) by
assumption. Now increase R' slightly. Then by the continuity of the con-
straint and by the construction of R" the lenders' payoff increases and thus
both constraints can be made slack. This proves Theorem 1 because there
exists some surplus that can be redistributed to borrowers by lowering the
face value of their debt R.
Theorem 1 establishes optimality of delegated monitoring schemes if n is
sufficiently large and if the variance of the macroeconomic shock is sufficiently
small. However, it does not follow that it is optimal for the bank to be as large
as possible. Indeed, Theorem 1 suggests that an optimal bank size may exist
because as the bank increases its portfolio size there are gains from default
risk reduction but losses from increased monitoring costs. In Theorem 2 we
characterize these gains and losses more precisely. However, before doing so
we first relate Theorem 1 to the previous literature on delegated monitoring.
Specifically, we focus on the bank size and industry structure predictions
implicit in previous models.
Diamond (1984) and Williamson (1986) use a law of large numbers ar-
gument to prove the optimality of delegated monitoring (i.e., financial inter-
mediation) in an economy with bounded costs and no macroeconomic risk.
Because the probability that a bank fails is zero in the limit in their mod-
els, the lenders' expected costs of monitoring the bank are zero. Clearly, a
bank that operates in such an environment can always reduce the expected
monitoring costs borne by lenders by increasing its size. Thus, "big banks
are always better," and the model predicts banks of large but indeterminate
size. 19 This size prediction is implicit in the Diamond and Williamson mod-
els, and stems from the fact that increasing returns to scale are inherent in
19 Because the set of borrowers is infinite, it is possible to get multiple banks that are
indeterminately large in this framework. However, the argument requires that the infinite
16
the framework they consider. Specifically, in their models a bank can al-
ways both decrease the riskiness of its portfolio and reduce monitoring costs
by contracting with additional borrowers. An obvious question, therefore,
is: Does delegated monitoring in an economy with non-diversifiable portfo-
lio risk also give rise to increasing returns to scale in intermediation, and
hence indefinitely large banks? The answer to this question depends on the
specification of monitoring costs.
Consider first a "best case 1 ' situation where lenders face a fixed cost of
monitoring a bank in default states. Specifically, let c* = k, where k is a
positive constant which is independent of the size of the bank. In this case,
(7) from Theorem 1 implies that the bank's default probability converges
to P{{Z < z}). This follows from the fact that a bank's default prob-
ability decreases (in general) as it contracts with more borrowers because
idiosyncratic risk is diversified away. The non-diversifiable macroeconomic
risk obviously remains. 20 To make this argument more precise, let p„ de-
note the bank's default probability when it has a portfolio of size n, where
n < n. The lenders' expected costs of monitoring the bank under this cost
structure are PhC' n = Pnk. Since the bank's default probability when its port-
folio size is h is at least as great as its default probability in the limit (i.e.,
Pn > lim n _ 00 p n ), the lenders' expected costs of monitoring the intermediary
are lower for larger banks. It follows from this observation that the delegated
monitoring model with non-diversifiable portfolio risk and constant monitor-
ing costs will generally also display increasing returns to scale in intermedi-
ation (because increasing the bank's portfolio size does not raise monitoring
costs but it may lower the bank's default probability). Consequently, like
Diamond and Williamson, the optimal bank size under constant monitoring
costs is indeterminately large.
Now consider a polar opposite "worst case" situation, where the lenders'
set of borrowers be partitioned into an infinite number of subsets where each infinite
subset of borrowers contracts with a particular delegated monitor. This argument does
not appear to be a plausible explanation of the observed co-existence of multiple banks.
Of course, the models were not designed to explain this observation.
20 Convergence in equation (7) need not be monotonic. Thus, there may exist points
of non-monotonic convergence where even under constant monitoring costs the optimal
bank size is finite if the macroeconomic risk is non-trivial. We therefore assume without
loss of generality that the bank's probability of default is always bounded from below by
P{{Z < z}), which is the default probability of a bank of infinite size with macroeconomic
risk but no idiosyncratic risk.
17
monitoring costs are unbounded. Krasa and ViUamil (1991, Theorem 1) an-
alyze this problem when there is no macroeconomic risk. They show that
even if costs are unbounded but do not increase at an exponential rate, dele-
gated monitoring with two-sided simple debt contracts still dominates direct
investment. However, the non-trivial macroeconomic risk which we consider
in this paper complicates this problem considerably. Specifically, in the limit
the lenders' expected monitoring costs are given by lim n _^oo p n c*. Clearly, if
c* is unbounded this product converges to infinity when the bank's portfolio
is subject to non-diversifiable macroeconomic risk (because lirr^^oo p n > 0).
Thus, constraint (4) from Problem 3.2 is violated for sufficiently large n, and
this implies that delegated monitoring is not feasible with unbounded costs,
non-diversifiable macroeconomic risk, and a sufficiently large portfolio size.
This argument suggests that an optimal portfolio (or bank) size may exist
because the feasibility of delegated monitoring depends on n.
Consider now the problem of whether or not a bank of a given size (n)
should contract with additional borrowers, thus increasing its scale. Suppose
that monitoring costs are bounded but not constant. Theorem 1 establishes
that delegated monitoring is optimal if the variance of the macroeconomic
risk is sufficiently small, but this does not imply that the bank should be as
large as possible. When there is non-trivial macroeconomic risk and the bank
has a portfolio of size n, the bank must consider two factors when deciding
whether or not to increase its scale.
(i) Adding additional projects to its portfolio decreases the bank's default
risk. This decrease is given by the difference between the bank's proba-
bility of default in the limit (i.e., lin^oo p n ) and its probability of default
at portfolio size h (i.e., p^); but for h sufficiently large, the gains from
reducing default risk by adding additional projects are essentially zero,
(ii) Adding additional projects to the banks' portfolio raises the lenders' costs
of monitoring the bank.
Thus, the crucial question is: For what cost structures do the gains from
reduced default risk dominate the losses from increased monitoring costs
when additional projects are added?
18
5 Optimal Intermediary Size
We now obtain the main results of the paper: analytic predictions for both
the optimal size of a financial intermediary and for the size distribution of
banks in a Pareto efficent industry. In order to analyze the problem in
detail (and answer the question posed above), we must provide a precise
quantitative characterization of the gains from diversification as the size of
the bank increases. To this end we construct a rate function which measures
the speed at which the bank's idiosyncratic portfolio risk is eliminated when
the size of its portfolio is increased. The Theory of Large Deviations (cf.,
Varadhan (1984)) provides the formal structure.
Consider first the bank's portfolio diversification problem for the case of
a fixed realization z of Z. The argument will be generalized to permit any
z by integrating over the distribution of all possible realizations of z. Then
XI = Y t + z are independent random variables (given that z is fixed) since the
Yi are independent. Since the random variables X* are independent, the law
of large numbers holds. The large deviation principle gives a rate function
(cf., Varadhan (1984), Theorem 3.1) which provides a measure of the speed
of convergence in the law of large numbers. The rate function implies that
for every z < E Z [R(X*)], the probability that a realization is in the tail of
the distribution converges exponentially to zero. Formally
^({^E A 7<«})<^ (a)n , (ii)
where l z (a) > is the rate function which gives the speed of convergence of
the distribution (i.e., a measure of how rapidly contracting with additional
borrowers reduces the bank's default risk).
The rate function T z {-) is derived from the moment-generating function
of a random variable. Let M z (9) denote the moment- generating function of
the distribution of R(X*), where p. z denotes the distribution. Then
M z {6) = J e 6x d^(x). 21
The rate function is found by solving the following maximization problem:
1(a) = max0a- log M z {0). (12)
9£ #t
21 M z (0) is called the moment generating function since the fcth derivative of M 2 (0)
evaluated at 9 = gives exactly the fcth moment of /i.
19
We now show that 1(a) > for every a < E(X-). For fixed a let f(a,6) =
9a — log M z (0). Since /(a,0) = 0, it is sufficient to show that ^/(a,0) ^ 0,
which can be easily verified:
d fxe ex dfi z (x)
da- f{a ' 9)=a - fe*dp,(x)-
Consequently, £/(a,0) = a - EX? ^ 0. 22
We have shown that the probability P ({jE?=i R(Yi + z) < #*}) con-
verges exponentially to E[R(Y t + 2)] for fixed z such that z > z(R") (i.e.,
if the macroeconomic shock is not too severe). We now make the conver-
gence argument independent of z. The primary technical problem is that the
behaviour of the rate function must be analyzed as z comes close to z(R*).
Recall that for z = z(R"), the face value of the lenders' contract, R', is ex-
actly the expected value of R(Y t + z). Clearly, the probability of observing
realizations less than or equal to the expected value of independent random
variables does not converge to zero. However, in Lemma 3 we show that for
z > z(R*)i and 2 sufficiently close to 2, the rate function I Z (R") is bounded
from below by k(z — z") 2 , where k > 0. In particular, from (A. 15) and (A. 16)
in the Appendix it follows that we can choose for k any number smaller than
2v*rR(Y +z) - 23 Thus, the smaller the variance of the idiosyncratic risk the
faster convergence. Thus, the rate function converges to zero at the speed
(z — z) 2 for z — > z. Integrating over 2, the probability of default by a bank
of size n, given that Z > z, is:
I
00
e- TAR ' )n dH(z). (13;
An important technical result, which is essential for proving that an op-
timal bank size exists (Theorem 2), can now be stated.
Proposition 1. There exist constants k, > 0, i ' = 1,2 such that the proba-
bility of default by a bank of size n, given that Z > z, is bounded from above
by^ + e-^.
22 This follows from the fact that f xe 8x d^i z (x) evaluated at 6 — is the expected value
of X* , since fx z is the distribution of X*. Furthermore f e 0x dfi z (x) evaluated at 9 = is
one
23
Apply Lemma 3 as in Proposition 1 and choose X a — R(Y, + z).
20
The proof of Proposition 1, which provides a speed of portfolio conver-
gence result, is in the Appendix. Note that in the bound in Proposition 1,
the term e~ nk2 converges to zero much faster than ^h. By the law of large
numbers, a bank of infinite size will never default if Z > z. Thus, the bound
implies that a bank of size n can lower its default probability by only ap-
proximately -h if it becomes infinitely large (given that Z > z) because the
second term is approximately zero for large n. Consequently, H~ is our de-
sired measure of the gain from default risk reduction arising from additional
diversification.
The main result of the paper can now be stated.
Theorem 2. Let c" n denote the cost of monitoring a bank of size n. Let
c*^ — lim n _ 00 c* , 24 and assume that c^ — c* converges to zero at a slower
rate than -4-. Then it is never optimal for a bank to become infinitely large.
Thus, there exists an optimal size for the bank.
Proof. Assume by way of contradiction that it is optimal for the bank to
become infinitely large. By Proposition 1, the bank's probability of default
is bounded above by
p n = i^ + e-*»» + P({Z<z})» (14)
\Jn
By Lemma 2, the bank's default probability in the limit is given by P{{Z <
z}). Let n be arbitrary. Now compare the expected monitoring costs for a
bank of size n with those of a bank of infinite size. By (14), the lenders'
expected costs of monitoring a bank of size n (i.e., p n c* n ) are at least
-j^ + e
\Jn
c' n + P({Z<z})c' n . (15)
The expected costs of monitoring a bank of infinite size are at most
P({Z < z})^. (16)
24 If c* is unbounded then c^ = oo, and clearly Theorem 2 holds.
25 The first two terms are the bound given by Proposition 1 for all states where Z > z.
Assuming that the probability of default in all states Z < z is one (which is clearly an
upper bound for these states), we get the third term.
21
If it were optimal for the bank to become infinitely large, then at least for
large n the expected costs of monitoring the bank per lender must decrease
if the size of the bank is increased from n to infinity. By (15) and (16) the
expected monitoring costs will decrease if 26
4= + e-** n
<> P({Z < z}) (<£,-<) . (17)
By the assumption of the Theorem, c^ — c* converges to zero at a slower rate
than -4jj. Thus, for every M > there must exist an h such that c^—c^ > -t-
for all n > n. This, and equation (17) yields
c' n >P({Z<z})^=. (18)
Since M can be chosen arbitrarily, there exist values such that inequality
(18) is violated. 27 The bank cannot be infinitely large, and the Theorem is
thus proved.
Theorem 2 establishes that when the lenders' monitoring costs depend on
the bank's portfolio size, an optimal determinate bank size can be computed,
under the assumption that the rate of increase of the lenders' monitoring costs
converges to zero at a sufficiently slow rate. This assumption is fairly weak.
In particular, it is fulfilled if the rate of increase ofc* is of the order -y-r. 28
Most economically plausible cost structures will satisfy this assumption. An
alternative way to interpret Theorem 2 is that the case of constant lender
monitoring costs (on which others have relied) is rather special. In particular,
we argued in Section 4 that under constant monitoring costs a bank's size is
indeterminate (because of increasing returns to scale in delegated monitor-
ing). However, with a slightly changed (i.e., size dependent) cost structure
an optimal bank size exists. Thus, the indeterminacy (and hence increasing
returns to scale for all n) result does not seem to be very robust. Theorem 2
26 This follows from (16) - (15) < 0.
"Multiply both sides of (18) by y/n to obtain [ki + v/ne - ^"] c' n > P{{Z < *})M. Let
n — ► oo. Then, kyc*^ > P{{Z < z})M, which cannot hold for every M . Thus (18) must
be violated for all sufficiently large n.
Note that c^ — c* is approximately -j=. Differentiating with respect to n (ignoring
that n is an integer) of course yields — fr-
22
also shows that even when the bank's portfolio is subject to non-diversifiable
macroeconomic risk (as long as the variance of the risk is not too large so
intermediation remains optimal), a bank of size n can only improve upon its
default probability by at most 4jj. Increasing bank size after some critical
h is not optimal because it leads to increased monitoring costs, but there
are very limited gains from default risk reduction. Thus, even very "small"
banks (given Z) may improve upon direct investment when intermediation
is optimal.
6 Testable Implications of the Theory
Theorem 2 has the following testable implications. First, bank size is deter-
minate and inversely related to the bank's exposure to macroeconomic risk.
In a large economy like the U.S. where aggregate macroeconomic shocks have
different effects on different regions of the country, our model predicts that
banks of different sizes will coexist across locales. In particular, if different re-
gions of the country have different effective macroeconomic shocks (i.e., Z's),
the model predicts that across regions both large "money centre banks" that
are very well diversified and smaller "local banks" that are less well diversi-
fied will co-exist. "Money centre banks" may have been able to lower their
exposure to macroeconomic risk, by evading portfolio restrictions via hold-
ing companies or because they operate in regions of the country with better
diversified economic bases. Our model predicts that these better diversified
(i.e., low Z) banks will be larger than "local banks" (with higher Z's). This
follows from (18) in the proof of Theorem 2 because P({Z < z}) is lower for
a better diversified bank, since the bank's exposure to macroeconomic shocks
is effectively less severe. Therefore, the bank size (n) that violates (18) is
necessarily higher. Williamson (1989) provides an interesting discussion of
stylized facts regarding the structure of U.S. versus Canadian banks. Histor-
ically, Canadian banks have had fewer portfolio (e.g., branching restrictions)
and hence lower Z's than U.S. banks. As our theory predicts, there have been
fewer banks in Canada of larger size (adjusted for population differences).
The second testable implication of our theory pertains to industry struc-
ture, i.e., limits on the number and size distribution of firms that are present
in equilibrium. Theorem 1 establishes that intermediation (banking) im-
proves social welfare. In other words, there is a some level of intermediation
23
services in an economy that is socially optimal (given preferences, costs, prob-
ability distributions, and alternative opportunities). In contrast, Theorem 2
proves that there is a specific bank size that is optimal. Although the notions
of bank size and industry structure are closely related, they need not be iden-
tical. For example, suppose the socially optimal "industry" (i.e., economy)
level of welfare improving intermediation services is twenty units of input
capital and the optimal size for all banks is to provide two units of capital.
Clearly, the optimal industry structure is then ten banks. What causes some
banks to be of similar size in our model? The result that within a region (or
among banks with similar effective portfolio restrictions) banks with similar
"local" idiosyncratic risk characteristics will have a common size (given their
Z) follows immediately from equation (18), because fci, M and P{{Z < z})
will be similar for such banks. Why might many moderately sized local banks
coexist in the same region? This again follows from Theorem 2 because the
Theorem establishes that after some critical size — increasing a bank's scale
of operation further is not profit maximizing. Finally, why might we expect
to observe many small "local banks" and fewer large "money centre banks?"
The high frequency of smaller local banks (relative to large money centre
banks) stems from the fact that banks with higher Z's cannot achieve suffi-
cient portfolio diversification in their locale to justify the additional monitor-
ing costs associated with increasing their scale of operation. Multiple banks
operating at the efficient scale within a particular locale provide welfare im-
proving intermediation services optimally (given the risk and cost structure
in the economy). Differences in bank size and distribution across locales
stem entirely from different effective Z's in (18) (i.e., different exposure to
macroeconomic risk in local and money centre banks).
7 Concluding Remarks
In this paper we develop a theory of bank size distribution with testable
implications. We regard the theoretical model to be useful for two reasons.
First, there is a longstanding debate in the banking literature about the mar-
ket structure of the financial industry. Some have argued that the banking
industry is inherently non-competitive and hence must be regulated to pro-
tect consumers from the evils of banks' market power. In contrast, others
have argued that the banking system is fragile and hence must be protected
24
from the destabilizing forces of competition. Because we derive optimal fi-
nancial contracts from Pareto problems, the allocations that we obtain are
necessarily Pareto efficient. An interesting problem for future research is to
compare the structure of a banking industry where firms have market power
with the Pareto efficient structure implied by Theorem 2 (given parametric
specifications for preferences, costs, probability distributions, and alternative
assets).
Second, the model may prove useful in understanding changes in the Eu-
ropean banking system that will undoubtedly result from the EMU. In the
previous Section we discussed bank size and industry structure predictions
for the U.S. economy. For example, our model predicts that largely agricul-
tural sections of the U.S. like the Mid-west will have many moderate or small
sized banks because this region is subject to marcoeconomic shocks that are
difficult to diversify (especially given portfolio restrictions imposed by bank
regulators). However, the model predicts money centre banks that operate
in more economically diversified regions of the country (and that have been
better able to evade portfolio restrictions) will be larger. In 1999 when the
EMU begins, twelve countries will form a "federal Europe." What will be
that nature of the EMU banking regulations imposed by the central bank
or by the governments of the individual countries? Are the banks of some
countries (specifically, those with better diversified economies) destined to
become "money centre" banks while the banks of other (less well diversified)
economies are destined to become small "local" banks? This question is
important because — although our model permits existence of both multiple
banks of the same size, and perhaps more interestingly the co-existence of
banks of different sizes — larger (better diversified banks) have a lower de-
fault probability than smaller banks. Are members of the EMU with less
well diversified economies, that might wish to encourage their banks to lend
domestically for development reasons, destined to become problematic mem-
bers of the Union from the outset?
8 Appendix
We first derive a "law of large numbers" for random variables with correla-
tion. The argument works essentially as follows: For any fixed realization z
of Z we can apply the law of large numbers to the random variables \\ + z
25
and show that - X^"=i Y x + z converges to z except for a set N z which has
measure zero with respect to the probability Pi (recall that P is the product
of the probabilities P x and P 2 ). We then use Fubini's Theorem (i.e., integrate
with respect to P 2 ) to show that \J Z £rN z has measure zero with respect to
the original probability P. We now state Lemma 1.
Lemma 1. Let Y t , i E W and Z be independent random variables. Assume
that the Y t are identically distributed. Let X t = Y x + Z , for every i and let R(-)
be a simple debt contract. Then limn^oo J2?=i R{X t ){u) = E[R{Y\) \ Z] (uj)
for almost every u) € 0.
Proof. Let 2 be a realization of Z. Let E[R{X\) \ Z] be the conditional
probability of X\ with respect to Z. Let u 2 £ 2 . Define the probability
space ( Sl W2 , A W2 , P^ ) as follows. Recall that Q, = fij x Q. 2 . Then let Q W2 =
Q,i x {u> 2 }. Let A u , 2 be the set of all events Ax{w 2 }, where A is an event in Cl\.
Finally, let P^{A x {u 2 }) = P X {A). Let z = Z{u 2 ). Then R(X t ) = R(Yi + z)
are independent random variables on (Q Z ,A Z ,P Z ). We can therefore the
apply the law of large numbers to the X- , i £ JN. Thus,
Jim£/2(X.-M)= / R(X l (u) + z)dP^ 2 (u) (A.l)
for all u> £ fl^ except for a set A^ £ A W2 with Puj 2 {N W2 ) — 0. Further,
R{X 1 (u))dP U2 (u;) = £[/2(X0 I Z](w), (A.2)
for almost every u 6 £l W7 , because Yi and Z are independent. 29
•/n«, 2
29 Note that u h-> / n /?(Xi(u;)) dF W2 (oi) is measurable with respect to Z. Thus, to
show that we have a version of the conditional expectation it is sufficient to prove that
/ / R(X 1 (w))dP u ,(u)= I R(X x (u))dP{u) y
J a Jn„2 •**
for every event A which is measurable with respect to Z (i.e., which is of the form Z~ l (B)
where B is a measurable subset of IR). However, since Z is constant on Qi, all such sets
are of the form Q\ x C, where C is a measurable subset of Q 2 - The result then follows
from Fubini's Theorem since integrating over f2i and then over Q W7 (which is essentially
Q\) is the same as integrating over Q{ once.
26
Let H be the distribution of Z. It remains to show that N = Uu/ 2 €n 2 ^2
has measure zero. This follows from Fubini's Theorem. The set D of all u
where ^ Yl?=i R{X t ) does not converge is given by
D = I u:\immt -Y.R(Xi) ^ f and limsup - Y R(X t ) ^/l,
where / = E\R(X\) \ Z\. Since the limsup and the liminf of a sequence of
measurable functions is measurable, it follows that D is measurable. Further,
D D Q W2 — ^2 smce A^ ls exactly the set of all u E ft W2 where the sequence
does not converge. Thus D = N and Fubini's Theorem implies
P(N)= I P 1 (N U2 )dP 2 (u) = 0.
Jn 2
This concludes the proof of the Lemma.
We next prove a technical result necessary for the proof of Theorem 1
(i.e., a convergence result for the bank's probability of default).
Lemma 2. Let X{, Y t and Z be as in Lemma 1. Assume that the distribution
of Z is non-atomic, i.e., P{{Z = z}) = for every z (£ IR. Let ft* be the
face value of the lenders' simple debt contract. Then
P ( " E R (Yi + Z) < R'\ ^ P (E\R(Y X + Z) I Z] < ft"
Proof. Let f n = i£?=i R{Y t + Z), and let / = E[R{Y 1 + Z) \ Z\. By
(6) we get / n (w) — * f(u) for almost every to. Thus, f n converges to / in
distribution 30 by Proposition 24.12 of Parthasarathy (1977). Let h be an
indicator function of the interval (—00, .ft*), i.e., h(x) = 1 for every x G
( — 00, .ft") and h(x) = 0, otherwise. Note that h is only discontinuous at R* .
Since the distribution of Z is continuous, the point {R*} has probability zero.
Corollary 1 of Billingsley (1968, p. 31) therefore implies that h(f n ) converges
30 Convergence in distribution means the following: Let F n denote the distribution func-
tion of /„ for every n £ IN, and let F be the distribution function of/. Then /„ converges
to / in distribution, if f g(x) dF n (x) = f g(x) dF(x) for every bounded and continuous
function g on Wt.
27
in distribution to h(f). Let H(F n ) denote the distribution of h(f n ), let H(F)
denote the distribution of h(f), let F n denote the distribution of / n , and let
F be the distribution of /. Then
lim fh(x)dF n {x) = lim fxH(F n )(x)
n— »oo J n—KX J
= lim fxdH{F)(x) = lim [h{x)dF{x),
n— »oo J n—'oo J
where the second equality follows from the fact that h(f n ) converges to h(f)
in distribution. This proves the Lemma 2.
We next prove a technical result on the convergence of the rate function
Lemma 3. Let X a , a > a > 0, be a collection of random variables such that
a is the expected value of X a for every a > a. Assume that the moment gen-
erating function M a (a) is thrice continuously differentiate in a neighborhood
[a, a) of a. Let fi a be the distribution of X a . Assume that the support of (i a is
contained in the compact interval [T!,T 2 ] for every a > a. Then there exists
a constant k > such that X a {a) > k(a — a) 2 for all a which are sufficiently
close to a.
Proof. Define
f(a,0)=0a-\ogM a {0), (A3)
where M a {9) is the moment generating function M a (0) = / e 6x dp, a (x). Note
that /(a,0) = 0. Furthermore,
Let F a denote the distribution function of p, a (i.e., F a (t) = p* a {{— oo, t]).
Partial integration of M a (0) yields
M,
a (0) = I' e ex dfi a {x) =e Bx F a {x)\% - f ' e 9x F a (x) dx
-9 ( 2 e 9x F a {x)dx (A5)
;T 7
, e 0T 2
28
Furthermore, by partial intergration we also get
/ * F a {x)dx = xF a (x)\%- I ' x dF a {x) = T 2 - a, (A.6)
JTi ' JTi
and
QxF a {x)dx = l -x 2 F a (x)\% - l -j\ 2 dF a (x) = I (r 2 2 - E(X 2 a )) . (A.7)
Let m (a) denote the value of 6 which maximizes (A. 3) for fixed a. Thus,
from (A. 4) and the Implicit Function Theorem we get
— 6* a = d % V ' d if a . (A.8)
da - a JLM a (0)-^M a (6)
We now evaluate j^O m (a) at a = a. Thus, since 0*(a) = by (A.4), 31 we
must evaluate the right-hand side of (A.8) at = and a = a. Taking the
partial derivatives of M a (6) in (A. 5), evaluating them at 6 = and a = a
and using (A. 6) proves that J^A/ s (0) = 0. Furthermore,
d 2 ., _ d r T *
-M,(0) = -— / 2 F a (x)<fx = l,
a oa JT\
dOda da JTi
where the last equality follows from (A. 6). Thus, the numerator of (A.8) is
— 1. We next derive the denominator of (A.8). Note that (A. 5) implies,
— M a (0) = T 2 - I ' F- a (x) dx = T 2 - (T 2 -d) = a,
dO Jt x
where the second equality follows from (A. 6); and
do 2
M-M = T 2 - 2 / 2 xF- a {x)dx = T 2 - (T 2 - E(X 2 a )) = E(X 2 ),
JTi
where the second equality follows from (A.7). Thus, (A.8) implies
da a 2 - F(Xl) var(A a )
'This follows since ^M a (0) = EX a - a and M a (0) = 1.
29
By (A. 5) and differentiation we get
^M a (9*(a)) = (T 2 e°'W- j\™* F a (x) dx^J ±6*(a)
-0'(a)4- I 2 t 9 ' (a)x F a (x)dx (A10)
da JT\
Evaluating (A. 10) at a = a and using (A. 6) we get
±M d (9'(d)) = d^-9'(d). (All)
da da
Note that I a = l a (a) is the maximum of f{a,9) over 9 by (12) in Section 5.
(A. 3), (A. 10), (A. 11), and the fact that M(9*{d)) = Af(0) = 1 immediately
imply
i-I g = A/(a,r(a))=0, (A12)
aa aa
Next we show that ^-J d > 0. Using the fact that M(9"(d)) - 1, it follows
that
^/(a,^(a)) = a^r(a)-^M a (^(a))+^M a (r(a))J . (A13)
Furthermore, by differentiating (A. 10) one more time at a = a and by (A. 6)
and (A. 7) we get
^M & (0-(a)) = a^- 2 6'{d) + (^-0'(d)) E(X? a ) + 2^-9'(d). (AAA)
da* da 1 \da j da
Substituting (A.9), (A. 11), (A. 14) into (A. 13) we get
£/(«,*■(»)) = - (^-( 5 )) 2 va r .V a -2^-( S ) = -L-, (A.15)
where the last equality follows from (A.9). Recall that the rate function
T a — (a, 6* (a)). Thus, (A. 15) implies j^la > 0. Since by assumption, M a (9)
is three times continuously differentiate in a neighborhood of (a,0), (A. 8)
implies that 9*(a) is twice continuously differentiate for all a > a which are
sufficiently close to a. Thus, T a = f(a,6*(a)) is continuously differentiate
30
for all a > a which are sufficiently close to a. We can therefore find a constant
k > such that
d 2
—l a > 2k > 0, (4.16)
da z
for all a in a neighborhood of a. Thus, developing X a in a Taylor series at
a = a we get
d d 2 (a - a) 2
1 a = I& + -r l*{a - a) + -j-rla , (>4.17)
aa acr 2
for an a between a and a. The Lemma now follows from (A. 12), (A. 16) and
(A. 17) since X a = by (A. 4) and footnote 31.
We now use Lemma 3 to prove our main convergence result.
Proof of Proposition 1. Let Z > z. By (13) in Section 5.1, the bank's
probability of default is bounded above by
J~e-W*)*dH(z).
Choose z x > z. Since 1 Z (R*) is monotone increasing in z (cf., Stroock (1984,
Lemma 3.3.) we get
/•oo
/ e- J * {R ' )n dH{z) < e- J ^ {R ' )n P{{Z > z,}). (4.18)
Clearly, this expression converges to zero exponentially as n — ► oo, i.e., it
can by bounded above by a term e~ k2n . It therefore remains to give an
estimate for the rate of convergence if the realization of Z is between z and
Z\. We can get such an estimate by using Lemma 3. 32 Furthermore, note
that E[R(Yi + z)\ converges to E[R(Y l + z)] at the same rate as z — > z. This
and Lemma 3 implies that 2 Z (R*) > k(z — z) 2 . It therefore remains to derive
the rate of convergence of
/;
e
-k(z-z) 2 n
dH{z) (4.19)
32 Let X a be R(Y l + z) where a = E[R(Yi + *)], and let a = E[R{Y X + z{R'))}.
Then all assumptions of Lemma 3 are fulfilled. Differentiability follows since the func-
tion z •— ► e R(Y*+Z) i s differentiable except on a set of measure zero. Thus, Leibnitz's rule
of differentiation under the integral can be applied.
31
as n — > oo. Let h denote the density function for H . Substituting z by *j£
in (A. 19) implies
\ e - k{z ~' z)n dH{z) < — / e~ kz h(z + -=) dz. (A.20)
iz v/n J -00 y/n
2 2
Note that e -z /i(s + -4«) converges to e~ z h(z) as n — -> 00. Thus, there exists
a constant k\ such that the right-hand side of (A.20) is bounded above by
4^. The result follows now from (A. 18) and (A.20).
32
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33
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