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Working Paper 94 17
BANK DIVERSIFICATION: LAWS AND FALLACIES OF LARGE NUMBERS
by Joseph G. Haubrich
Joseph G. Haubrich is an economic advisor at the Federal
Reserve Bank of Cleveland. The author thanks Steve
Zeldes for usefbl discussions.
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 not necessarily those of the Federal Reserve
Bank of Cleveland or of the Board of Governors of the
Federal Reserve System.
December 1994
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ABSTRACT
The conventional wisdom on bank diversification confuses risk with failure. This
paper clarifies that distinction and shows how increasing bank size may increase bank
risk even though it lessens the probability of failure and lowers the expected loss. The
key result is an application of Samuelson's "fallacy of large numbers."
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Introduction
Conventional wisdom states that large banks are safer than small banks because
they can diversify more. This conventional wisdom, however, confuses risk with
probability of failure. While the law of large numbers does imply that a large bank is
less likely to fail than a small bank, equating this tendency to less risk falls into what
Samuelson termed the fallacyof large numbers. A $10 billion bank may be less likely
to fail than a $10 million bank, but it may also saddle the investor with a $10 billion
loss.
In this paper, I hope to clarify what this distinction means for banks. Banks
diversify by growing-- by adding risks-- something distinctly different from the
subdivision of risk behind standard portfolio theory. A simple mean-variance example
will make the point that a bank's owner need not value diversification. After that, I
take a regulator's perspective and consider how a bank guarantee fund, such as the
FDIC, views bank growth and diversification. After a short review of why
diversification by adding risks decreases the probability of bank failure, I look at how
such diversification alters the expected value of FDIC payments, and then
diversification's impact on the FDIC's expected utility, using recent results from the
theory of standard risk aversion.
To concentrate on the cleanest example, this paper stays with the case of
independent and identically distributed risks. This admittedly ignores the alleged
ability of large banks to diversify regionally' or the possibly adverse incentives of
deposit insurance (Boyd and Runkle [ I 9931, Todd and Thomson [ I 99 I]).
' Compare Haubrich (1990)with Kryzanowski and Roberts (1993). Even small banks
may diversify, however, by selling loans or participating in mortgage pools or other
forms of securitization.
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I. A Simple Example
Probably the easiest way to understand the effects of diversification by adding
risks is to consider a bank financed exclusively by an owner/investor who cares only
about means and variances. With no debt, failure disappears as an issue, and instead
the question becomes the utility-maximizing portfolio for the bank's owner.
The owner and sole equity holder has, conveniently for us, sunk his entire
wealth W into the bank. He faces the problem of dividing his portfolio between
holding Ssafe government bonds with a certain r e l r n of zero and investing in some
number Kof risky, independent bank loans with r e l r n s ri normally distributed
as N(jI,S). If each loan costs a dollar, the investor's budget constraint is W S + K
These bank loans are indivisible--the bank cannot diversify by spreading one dollar
across many loans. Then the r e l r n on the portfolio is
K
ri is distributed N (@,&?S)
Since
,standard techniques (Fama and Miller [ 19721,
i=l
chapter 6, section 111) imply that
In mean-standard deviation space, equation (I) defines a portfolio opportunity
set illustrated by figure I (for W 5 ) . The opportunity set is disjointed, since the
decision to add another loan is discrete. Depending on the shape of the indifference
curves, the bank owner may put none, all, or some of his wealth into bank loans.
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Figure I shows a typical case with an interior solution. This illustrates quite clearly
that the bank does not always wish to diversify. Stated another way, the portfolio
return is distributed N(-
K p,-(T)
- J E , so that as the bank invests in more loans, the
W
W
standard deviation as well as the expected return increases. Which one matters more
depends on preferences.
An all-equity bank offers a nice illustration, but does not provide a very
representative case. Even a stylized bank should have deposits.
2. Does the FDIC Want Banks to Diversify?
Allowing banks to take in deposits means allowing banks to fail. The return on
assets may not cover the payments promised to the depositors. In the U.S., this liability
devolves upon the Federal Deposit Insurance Corporation. This provides a natural
focal point for our discussion. Actual banks raise money in many different ways, using
several types of preferred stock, subordinated bonds, and commercial paper. What
happens in bankruptcy is at best complicated and at worst unknown, as the courts
must determine the validity of claims as diverse as offsetting deposits and the sourceof-strength doctrine. A detailed consideration of how each class of investors views
diversification, then, is beyond the scope of this paper. Instead, to make what is
admittedly a simple point, I concentrate on the FDIC, which ultimately bears the
liability for bank failures.
The FDIC steps in if the realization of bank assets Y is too low to repay the face
value of the debt I; that is, if Y<E This is a fairly general formulation in that the assets
producing Ymay be funded by means other than deposits, but it is not completely
general because it ignores the possibility that the FDIC may have priority over some
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investors. For the rest of the paper, however, I will restrict myself to purely depositfinanced banks.
What is the face value of the debt, F? With no capital, if the bank funds n
projects each requiring funds 4 the face value is the sum of the deposits, F= n f . The
payout of bank assets is likewise the sum over the different projects,
where n indexes the number of projects in which the bank has invested.
A. f i e Probability of Bank Failure
How likely is it that this bank will fail? The answer is Pr(Yn < n . f)or
Assume the x,'s are independent and identically distributed (i.i.d.) with mean E(xi)=p
and further assume that Rp, SO that the face value of the debt is smaller than the
expected payout of the assets.
We can rewrite expression (2) as
because the set
{y:y < n .f)is the same as the set {y:-Y < f).
n
The weak law of large numbers (Shirayev [ I 9841, theorem 2, p. 323) tells us that
provided E I XI <= and EX= p ,then for all E>O,
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In particular, since f<p, Pr(-yn < f ) < Pr(l--r, - ~1.12p - f ) . That is, we can represent
n
n
yn below f as values more than p - f away from the mean p . Thus, as
the values of -
n
Diamond ( 1 984) explicitly states, the weak law of large numbers implies that
diversification by adding risks reduces the probability of bank failure.
B. fie Expected Value of the fWIC-5Liabilities
As Samuelson points out, a rational utility maximizer maximizes expected
utility, not the probability of success. The probability of each outcome must be
weighted by the utility of that outcome. As mentioned before, a $10 billion bank that
does fail may cost the FDIC more to resolve than a $10 million bank.
In the simplest case of risk neutrality, expected utility corresponds to expected
value. The first question, then, equivalent to assuming risk neutrality on the part of
the agency, concerns the expected value of the FDIC's payout. Though the calculation
is not particularly difficult, I have not seen it before in the literature. The expected
payout value becomes a question of finding the expected value of a particular function.
The FDIC must pay
0 if Y, 2 F .
F-Y, ifY,<F.
Figure 2 plots the function along with a typical density function.
It is worth noting that the expected value of ( 4 ) is not a conditional expectation.
If the set A = { Yn: Y,<F),then the expected value of ( 4 ) is P(A) E( YnIA) rather than
E ( K (A). A simple example will make this clear. Take a four-point distribution,
+ +
P ( I ) = P ( Z ) = P ( ~ > = P ( ~ ) = ~Then
.
E ( X ) = 1(1+2 3 4) = -. Now define the function
4
4
2
J
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1
3
g(x) as g(x)={O, if x 2 2.5 and x, if x < 2.5). Then E[g(x)]=-(1+ 2) = -, while
4
4
The question before us is what happens to the expected value of the FDIC's
n
payments as the bank diversifies. Recall that the FDIC pays off if x x i < n -f or,
i=l
equivalently, -< f . By the strong law of large numbers, the mean of the partial
n
Sums L converges to a mass point on E(x), and intuition suggests that the expected
n
value of anything below the mean (and a fortiori anything below f ) will have very
little importance, that is, an expected value approaching zero.
To establish this rigorously and to understand what diversification does to the
expected value of the FDIC's payments requires a more formal approach. Let each
random variable be defined on the probability space ( a , F, P) and identify
with R,
the real numbers, without loss of generality. The random variables are then functions
on this space, &(a), and define Z,(o) as
z,(W) =
2- n
Xi(o) . Next, define the function g(o) as
i=l
Note that we can think of the expectation E(X(o)) as a random variable, and so
g(E[X(o)])=
p).
=0 since R p . Further define g,(o)as g(Zdo)).
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The value of diversification can then be expressed as saying that as n
approaches infinity, the expected value of g(ZJ approaches zero, or
To prove (5), we use the Lebesgue dominated convergence theorem (Royden [1968],
p. 229), which says that if h(o) 2 0 is integrable, if Ign(o)ll~ ( c o )and
, if
The theorem first requires that we prove g, ( a )
a's'
> g(p) . To do this we use
the strong law of large numbers. The strong law of large numbers for i.i.d random
variables (see Breiman [1992], p. 52, theorem 3.30) says that for i.i.d. X1,Xz, X3 ..., if
Xxi
E 1 X I J<- then n
a.s.
> E(X,) ,where
a.s'
> denotes almost sure convergence,
that is, convergence on all but a set of measure (probability) zero.
Hence, given an o, except for a set of measure 0, we have that for any
there exists an Nsuch that if n>N, IZ, (a) - pI< E . Choose
E
E
> 0,
< p - f ,which implies
that if IZn(0) - pI< E ,then Z, ( o ) > p - E > f . This, with the definition of g, in turn
implies that gn( a ) = 0. For this 0, then, gn( a ) = g(p) = 0 , and a fortiori
Ig, ( a ) - g(p)I< E . Since g, (61) + g(p) for each o where Zn + p ,the almost sure
convergence of the strong law implies the almost sure convergence g, ( o )
a.S.
> g(p) .
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All that remains to be shown is existence of the integrable bound h(0). For
this, use I X,( 0 ) + p - f I , which bounds g, and is integrable because E I XII<= is a
hypothesis of the strong law. Hence, the Lebesgue dominated convergence theorem
applies.
As a bank makes more loans, the expected value of FDIC payouts tends toward
zero. Diversification works.
3. A Risk-Averse FDIC
Strictly speaking, what Samuelson terms the fallacyof large numbers enters
only with risk aversion. Applying this to an agency such as the FDIC, rather than to an
individual, requires some justification. The FDIC obtains its funds by taxing people,
either indirectly through its assessment on banks, or directly by congressional
appropriation. Risk aversion by the FDIC may then reflect risk aversion on the part of
those taxed, or nonlinearities associated with distortionary taxation. Alternatively, the
risk aversion may result from the incentives, constraints, and information facing the
organization. (Of course, as Kane [I9891 points out, these may at times promote riskseeking behavior, as in the FSLIC case.)
A. Conditionsfor the Fallacy
Samuelson (1963) shows that if a consumer rejects a bet at every wealth level,
then he will always reject any independent sequence of those bets. Under the
Samuelson condition, if the FDIC found one bank loan too risky, it would find a
portfolio of any number too risky.
Samuelson posits a rather stringent condition. It rules out, for example,
constant relative risk averse (CRRA) utility, because CRRA exhibits decreasing absolute
risk aversion (DARA), and so some unacceptable gambles would become acceptable at
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higher wealth levels. Pratt and Zeckhauser (I 987) improve considerably on the
condition with their notion of proper risk aversion. The conditions for proper risk
aversion answer the question, "An individual finds each of two independent monetary
lotteries undesirable. If he is required to take one, should he not continue to find the
other undesirable?" (Pratt and Zeckhauser, p. 143). Proper risk aversion shares one
defect with Samuelson7scondition, however. It is difficult to characterize and difficult
to determine whether a particular utility function satisfies the condition.
A slightly stronger condition proposed by Kimball (1993) has a simple
characterization. Kimba117sstandard risk aversion implies proper risk aversion. It
thus applies a slightly stronger condition than is strictly necessary for the fallacy. If a
utility function displays standard risk aversion, then an investor disliking a bet will
also dislike a collection of such bets.
Kimball (1993) shows that necessary and sufficient conditions for standard risk
aversion are (monotone) DARA and (monotone) decreasing absolute prudence. If the
utility function in question has a fourth derivative, then these conditions become
A key point here is that the individual finds each independent risk undesirable.
(Kimball has a slightly weaker, more technical condition that he calls loss aggravation.)
This certainly applies to the problem as we have defined it, because the payoff to the
FDIC is nonpositive--at best it pays nothing. This is not the only way to structure the
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problem, however, because the FDIC collects deposit insurance premiums from banks.
A major strand in banking research has been to ascertain whether the insurance
premiums are fairly priced, that is, whether they represent a tax or a subsidy on the
bank (Pennacchi [ 19871, Thomson [ 19871). The empirical results are mixed, varying
by time period and by bank, and in any case assume risk neutrality so do not directly
answer the question most relevant here. It makes sense, then, to think about both
possibilities--the case where the FDIC finds insuring a single loan undesirable and the
case where it finds insuring a single loan desirable.
In the first case, where the FDIC dislikes insuring an individual loan,
expressions (6) and (7) provide sufficient conditions for the agency to dislike insuring
any portfolio of such loans. That is, diversification by adding risks does not work;
adding risks makes'the insurance agency (guarantee corporation) worse off.
In the second case, where the FDIC likes insuring an individual loan, equations
(6) and (7) do not help. Their derivation presupposes that the agency dislikes the risk
it bears. For favorable bets, Diamond (1 984) builds on Kihlstrom, Romer, and
Williams (1 98 1) to develop sufficient conditions for when the fallacy of large numbers
is not a fallacy.
Diamond poses the problem in terms of risk premiums and notes that adding
risks provides true diversification if it reduces the risk premium. That is,
diversification works if the incremental risk premium for adding the second risky loan
to the portfolio is lower than for adding the first, identical risky loan. Kihlstrom,
Romer, and Williams (1 991) show how to handle risk aversion with two sources of
uncertainty by defining a new utility function, given initial wealth Wo and initial risky
bet
Z1, as
(8) v(x,) = Eu(W, +
+ x,) .
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Now v(xd defined in equation (8) can be treated as a utility function, so Diamond's
question comes down to whether u(.) is more risk averse than 4.).
If it is, then the
risk premium for bearing the second risk is lower than for the first, and the fallacy of
large numbers is not a fallacy.
Diamond derives two sufficient conditions for u(.) to be more risk averse than
4.).
Using Jensen's inequality, he shows that
(9) u'~'> 0
(10) u'~'> 0
are sufficient conditions when the risk has zero expected value. When the risk is
freely chosen, he must append decreasing absolute risk aversion, equation (6). The
reason for this is that a freely chosen gamble increases mean wealth, which requires us
to augment the sufficient conditions.
Notice that inequalities (7)and (TO) cannot both hold: (7) demands a negative
fourth derivative, and (10) demands a positive fourth derivative. The inequalities
apply in different situations, however. Inequality (7) concerns unfavorable bets and
describes when bearing one such risk makes the agent less willing to bear another.
Inequality (10) concerns favorable bets and describes when bearing such a risk makes
the agent even more willing to bear another. The conditions really answer two quite
different questions. Since each inequality provides a sufficient but not necessary
condition, any contradiction between the answers is more apparent than real.
An important caveat is that this analysis is consciously partial equilibrium,
concentrating on the risk of a single bank. If the bank grows by absorbing smaller
banks, the total number of loans insured by the system does not change. In a bank
with many loans, the profitable loans may offset unprofitable loans and lessen the
guarantor's liability. Since the FDIC does not share in the positive profits, it cannot
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undertake a similar offset if the loans are in different banks. This is not the only
scenario, however. The bank may grow at the expense of nonbank intermediaries or
by making loans that would not be made without the guarantee. Either case results in
new liabilities for the deposit guarantor.
B. An Exponential E x m e
A simple example can serve to illustrate some of the subtleties involved. To
illustrate what can happen, I use an exponential utility function and an exponential
distribution. The exponential distribution keeps the algebra simple because sums of
exponentials are gamma distributions. Exponential utility exhibits constant, rather
than decreasing, absolute risk aversion. It does not satisfy the sufficiency conditions of
Kimball ( [ 6 ]and [7])or of Diamond ([9] and [lo]). Therefore, adding risks can
sometimes help and sometimes hurt the investor.
Whether diversification helps or hurts depends on the risk premium. If the risk
premium decreases as the investor adds i.i.d. risks, diversification helps. If the risk
premium increases, diversification hurts. The simplicity of the example allows us to
calculate the risk premium explicitly.
Recall from equation (4) that for one loan, the FDIC pays nothing if the loan's
payoff exceeds its face value, and otherwise pays the difference. Denoting this
function by g(x) (as in section II), the risk premium is defined as the n l that satisfies
(I I) u(Wo + E(g(?))
+ n 1) = Eu(Wo + g(?)) .
With xfollowing the simplest exponential distribution,
(I 1) becomes
+
Eg(?) = (f - 1) e-f.
e-",the expected value in
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Using exponential utility of the form
allows us to solve for a , :
e-aW
For two loans,g(x) is zero if xexceeds 2 f and 2f-xotherwise. The random variable x,
as the sum of two independent exponentials, has a gamma distribution,
Then the expected value becomes Eg(2) = (f - 1) + (f
premium implicitly defined by u(W,
+ l)e-2r.
Solving for the risk
+ E(g(2)) + n ,) = Eu(W, + g(2)) yields
To complete the example, set 4 the face value of the debt, to I, and risk aversion to 1
and 2, and evaluate (12) and (I 3).
risk aversion?a
face value
El
1
1
-0.0659
-0.0640
This example illustrates two points. First, diversification can work. For low risk
aversion, the required risk premium for two loans is lower than for only one.
Conversely, for higher risk aversion, adding risks does not help: The risk premium for
two risks exceeds twice the risk premium for one risk. Both points emphasize the
sufficiency of expressions ( 6 ) ,(7), (9), and (I 0),because the example satisfies neither
set and still illustrates both gains and losses from diversification.
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4. Conclusion
Discussions of banking have been obscured by a false analogy with portfolio theory. A
bank diversifies differently than does a mutual fund. The bank adds risks, rather than
subdivides risks. Using the weak law of large numbers to establish that diversified
banks have a lower expected failure rate neglects the deeper question of whether this
represents a decrease in economic risk. Clearly posing that question is the main point
of this paper.
Just because a bank is less likely to fail does not mean the bank is less risky. If
the insurer, or owner, is risk neutral, a more complicated argument shows that the
bank is less risky in the sense of expected value. With risk aversion, however, the
question become ambiguous. As a practical matter, sufficient conditions exist, and the
combination of exponential utility with exponential distributions provides a tractable
framework for further exploration.
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