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January/February 1990

Vol. 72, N o .l

3 M a r k e t D iscipline o f Bank Risk:
T h e o r y and E viden ce
19 O n the Use o f O p tion P ric in g
M o d els to A n a ly z e Deposit
In su ran ce
35 W h a t D o W e K n o w A b o u t the
Long-R un Real E xch an ge Rate?

t iii:

iin u u i

A RESERVE
RANK of
A r S T .I/H IS

F e d e ra l R e s e rv e B a n k o f St. L o u is
R e v ie w
January/February 1990

In This Issue . . .

Policymakers in the federal governm ent are considering various
changes in deposit insurance in response to the large number o f bank
and thrift failures in recent years and the associated large losses by the
insurance funds. Some proposed reform s w ould reduce the govern
m ent’s insurance coverage to increase the effectiveness o f market forces
in limiting the risk that banks assume. Other reform s w ould base in
surance premiums on the risk that banks assume.
In the first article in this Review, "M arket Discipline o f Bank Risk:
Th eory and Evidence,” R. Alton Gilbert investigates the implications o f
deposit insurance reform proposals based on market discipline. Gilbert
uses a simple theoretical exercise to illustrate how market forces could
limit the risk assumed by banks under different approaches to reform
ing deposit insurance. He also summarizes the empirical studies o f the
effectiveness o f market discipline to determ ine w hether market forces
actually could be expected to limit such risks.
Under the current arrangements, the cost o f resolving bank and thrift
failures is borne largely by the taxpayer through the federal deposit in
surance agencies. In the policy debate, which considers the flaws and
potential alternatives to the present system, a num ber o f economists
have utilized a set o f theoretical tools — called option pricing models —
fo r analytic purposes. In the second article o f this issue, "On the Use o f
Option Pricing Models to Analyze Deposit Insurance,” Mark D. Flood
outlines the basic theory o f option pricing, which was originally
developed to assign dollar values to the option contracts traded on
financial exchanges. The author then illustrates how an option model
can be usefully em ployed to analyze the claims o f bankers, depositors
and insurers on the assets o f a bank or thrift by applying the model to
several insurance arrangements. Finally, Flood considers some o f the
limitations o f this approach.
* * *
Exchange rates have been at the center o f numerous economic policy
discussions in the 1980s. In the third article in this Review, “ W hat Do
W e Know About the Long-Run Real Exchange Rate?” Cletus C. Coughlin
and Kees Koedijk review what is known about movements in one type
o f exchange rate, the long-run real exchange rate. Despite much research,
there is no consensus on which variables cause changes in the real ex
change rate. Coughlin and Koedijk review the literature to provide an
elem entary understanding o f the three prim ary approaches and the
variables thought to influence the long-run real exchange rate. Using a
data set covering the current floating-rate period, the authors compare
the three approaches empirically and conclude that none provides an
adequate explanation o f movements in the long-run real exchange rate.

JANUARY/FEBRUARY 1990

R. Alton Gilbert
R. Alton Gilbert is an assistant vice president at the Federal
Reserve Bank of St. Louis. David H. Kelly provided research
assistance.

Market Discipline of Bank
Risk: Theory and Evidence
D

ECAUSE o f the many failures o f banks and
thrift institutions in recent years and the high
cost o f liquidating or reorganizing the bankrupt
savings and loan associations, policymakers are
now considering major changes in the w ay they
supervise and regulate depository institutions in
the United States. The Financial Institutions Re
form, Recovery and Enforcem ent Act o f 1989,
which provides the funds fo r closing bankrupt
savings and loan associations (S&Ls), calls fo r
several governm ent agencies to study the issues
involved.1 The federal budget document fo r fis
cal 1991 discusses the basis fo r reform o f
deposit insurance and the advantages o f various
reform s.2
To some extent, the unusually high failure
rate o f depository institutions (hereafter called
banks) can be attributed to developments in the
econom y such as declines in the prices o f oil
and farmland in the early 1980s. Some studies
conclude that fraud and mismanagement ac
count fo r many o f the bank failures.3 The gen
eral consensus, how ever, is that deposit insur
ance creates an incentive fo r banks to assume
higher risk than they w ould without it. Such
risk may be gauged in terms o f the variance o f
'Title X of the act directs the Secretary of the Treasury and
the Comptroller General, in consultation with various
federal government agencies and individuals from the
private sector, to prepare reports on issues related to the
reform of deposit insurance, including the implications of
policies that would enhance the effectiveness of market
discipline.

a bank’s return on assets as a percentage o f its
capital. The logic that underlies this consensus
is that without deposit insurance, banks that
choose portfolios o f assets with higher variance
in their rates o f return, or low er ratios o f capi
tal to total assets, w ould have to pay higher in
terest rates on deposits. Deposit insurance
blunts this penalty. The relatively high failure
rate and losses o f the deposit insurance funds
reflect, to some extent, the banks' response to
the incentives to assume risk created by deposit
insurance. Thus, a major issue in the debates
over financial reform is the future role o f de
posit insurance.
Some recent proposals to reform deposit in
surance are designed to increase the effective
ness o f market forces in reducing the risk
assumed by banks. Under these proposals, bank
owners and creditors w ould be exposed to
larger losses if their banks fail. The theory is
that if bank ow ners and creditors have greater
exposure to losses, they w ill limit the risk as
sumed by their banks. In some proposals, this
influence w ould complement the efforts o f bank
supervisors. In others, market discipline would
replace governm ent supervision.
2Budget (1990), pp. 246-53.
3Graham and Horner (1988) and Office of the Comptroller
of the Currency (1988).

JANUARY/FEBRUARY 1990

This paper describes some o f these proposals
fo r enhancing the effectiveness o f market disci
pline and illustrates how they w ould affect the
banks’ incentive to assume risk. The paper also
examines the empirical evidence on the effective
ness o f market discipline. Proposals fo r the re
form o f deposit insurance that rely on market
discipline assume that market participants can
differentiate among banks on the basis o f risk,
and that market yields on bank debt reflect that
risk. Th e paper lists the results o f several em
pirical studies and draws conclusions about the
potential effectiveness o f these proposals in re
form ing deposit insurance.

THE OBJECTIVES OF D EPO SIT
INSURANCE
Various approaches to enhancing the e ffe c
tiveness o f market discipline o f bank risk are
presented in table 1. Choosing one approach
over another depends in part on which basic
objective o f deposit insurance is considered to
be most important.
The follow ing are the prim ary objectives o f
deposit insurance:
1. T o protect depositors with small accounts,
2. T o prevent widespread runs by depositors on
banks, and
3. T o protect the insurance fund from losses
that w ould bankrupt it.4
Th ere are tradeoffs among these objectives.
The policy that provides the greatest protection
against runs by depositors is complete coverage
o f all deposit accounts. That policy, how ever,
eliminates any incentive fo r depositors to exert
their discipline over the risk assumed by their
banks, leading perhaps to an increase in the in
surance fund's losses.
The dollar limit on the amount in each ac
count that is insured, currently $100,000,
reflects an attempt to balance these objectives.
Total coverage o f accounts less than $100,000
protects small depositors. The limit on the in
surance coverage per account is designed to in
duce the depositors w ith large accounts to
m onitor their banks and require that riskier
banks pay higher interest rates on their de
posits. Those w ith relatively large accounts are
assumed to be better able to impose such m ar
ket discipline. Th e limit on insurance coverage

“ Federal Deposit Insurance Corporation (1983), pp. viii-xiii.

Digitized forFEDERAL
FRASER RESERVE BANK OF ST. LOUIS

Table 1
Proposals to Increase the
Effectiveness of Market Discipline
of Bank Risk______________________
(1) Phase out federal deposit insurance to facilitate the
development and use of private deposit insurance.
Short and O ’Driscoll (1983), Ely(1985), England (1985)
and Smith (1988).
(2) Lower the ceiling on federal insurance coverage per
account. Council of Economic Advisers (1989), pp.
203-4.
(3) Co-insurance: limit federal deposit insurance to some
fraction of each account. Boyd and Rolnick (1988).
(4) Place a ceiling on federal deposit insurance per in
dividual at all depository institutions. England (1988).
(5) All institutions must maintain subordinated debt
liabilities that are some fraction of their total assets.
Cooper and Fraser (1988), Keehn (1989) and Wall
(1989).
(6) Early closure: close or reorganize depository institu
tions when their capital ratios, reflecting the market
value of assets and liabilities, are low but still positive.
This proposal is designed to enhance the effectiveness
of market discipline by closing or reorganizing banks
whose shareholders have weak incentive to limit risks.
Benston and Kaufman (1988).

per account, how ever, tends to undermine the
objective o f preventing runs by depositors on
the banking system.
O f course, a run by depositors on an in
dividual bank does not create a serious problem
fo r the banking system, because these depositors
simply rem ove their cash from one bank and
deposit it in another in which they have m ore
confidence. If the bank subject to the run can
not meet its depositors’ demand fo r currency, it
will have to close. Its depositors w ill be paid as
the assets o f the failed bank are liquidated. A
run on a bank can serve a useful purpose—a
mechanism fo r closing a bank in which deposi
tors have lost confidence.
Runs become a problem fo r the banking
system, however, w hen depositors w ithdraw
currency and, hence, reserves from banks as a
group. Banking history in the United States and

the United Kingdom prior to their central banks
acting as lenders o f last resort indicates that
runs on banking systems have occurred, al
though they tended to be separated by many
years.5
Some argue that deposit insurance is not
necessary to avoid the adverse social effects o f
banking system runs. They maintain that, as long
as the central bank acts as an effective lender
o f last resort, the liquidity it provides would
limit any damage that runs on individual banks
could do.6 An alternative vie w emphasizes the
dangers o f relying on the central bank to
operate as the lender o f last resort to a banking
system without deposit insurance. A central
bank might respond inappropriately in a finan
cial crisis, as the Federal Reserve did in the ear
ly 1930s, leading to rapid declines in the assets
o f the banking system and widespread bank
failures. Deposit insurance reduces the role o f
the central bank in maintaining stability in the
operation o f the banking system. Thus, the
choice among the potential reform s o f deposit
insurance rests on view s about the vulnerability
o f the banking system to runs and the effe c
tiveness o f a lender o f last resort in dealing
w ith runs.
I f the prim ary objectives o f deposit insurance
are to protect small depositors and to protect
the insurance funds from large losses, a logical
change w ould be to reduce the insurance cover
age per account. This was proposed by the Pre
sident’s Council o f Economic Advisers in 1989.
Those w h o consider the possibility o f banking
system runs a serious threat to the stability o f
the banking system w ould oppose a large reduc
tion in the insurance coverage on bank deposits.

THE EFFECTS OF REFORM P R O
PO SALS ON BA N K IN G RISK
The reform proposals are designed to reduce
the incentives fo r banks to assume risk. In
evaluating their effectiveness, it is useful to con
sider three indicators o f the banking system’s
perform ance that reflect this risk: the expected
loss by depositors due to the bank failure, the
5Gilbert and Wood (1986) and Dwyer and Gilbert (1989).
6See Kaufman (1988) and Schwartz (1988).
7See Bernanke (1983), Calomiris, Hubbard and Stock
(1986), Grossman (1989), and Gilbert and Kochin (1989).
8To illustrate the need for such a model, consider a basic
reform of eliminating deposit insurance. That change
would increase the interest expense of a bank with a given

expected loss o f the Federal Deposit Insurance
Corporation (FDIC), and the probability that a
bank w ill fail. The expected loss by depositors
and the FDIC are considered separately since
proposals that reduce the FDIC’s expected loss
tend to increase the expected loss by depositors.
Focusing on only one o f these measures o f per
form ance misses some o f the reform proposals’
implications. The third measure, the probability
o f bank failures, is o f interest because o f evi
dence that bank failures have adverse effects on
economic activity in addition to the wealth
losses b y depositors and ow ners.7 The studies
that find adverse effects o f bank failures on
economic activity attribute those effects to the
constraints on the availability o f credit created
by bank failures.
Th e proposals’ implications fo r the effective
ness o f market discipline can best be derived by
using a m odel o f the behavior o f banks and
their creditors.8

Nature o f the M od el
The implications o f the various reform pro
posals are derived by examining the effects o f
proposed changes in deposit insurance on the
optimal choice o f risk by a representative bank
er. Several assumptions are made to simplify
the model.
R a te o f R e t u r n o n A ssets — The only ran
dom variable in the model is the rate o f return
on assets o f the representative bank, which has
the same probability distribution under all as
sumptions about the nature o f deposit in
surance. Bank regulators are assumed to deter
mine the probability distribution o f the rate o f
return by restricting the types o f assets the
bank may hold. Th e only choice fo r the rep re
sentative banker in this m odel is the level o f the
bank’s total assets. The capital o f the bank is
held constant at $100 in each case. W ith a given
level o f capital and a given probability distribu
tion o f the rate o f return on assets, the proba
bility o f failure (losses exceeding capital) is posi
tively related to the total assets o f the bank.
Management is assumed to choose the level o f
portfolio of assets, thereby tending to increase its pro
bability of failure. The penalty of higher interest expense
imposed by depositors on those banks that assume more
risk would induce banks to assume less risk in their
choice of assets. The net effect of eliminating deposit in
surance on the probability of bank failure must be derived
from a theoretical model that specifies the risk preferences
of depositors and bank managers.

JANUARY/FEBRUARY 1990

total assets that maximizes the expected profits
o f the bank.
W ith a given level o f bank capital, the condi
tions under w hich the bank fails can be derived
only with a specific probability distribution o f
return on assets. This paper uses the discrete
probability distribution presented in table 2.9
For each o f the seven possible outcomes, the
rate o f return on assets is net o f the operating
cost o f servicing the assets.
The rate o f return associated with each out
come is assumed to be inversely related to the
size o f the bank’s total assets. One reason fo r
this assumption o f an inverse relationship is
that, as the bank increases its total assets, it
must lend to b orrow ers beyond the local area
in which it has some market power. Another
reason is diseconomies o f scale in the operating
cost o f servicing assets. For each outcome with
a positive return on assets, therefore, the rate
o f return falls as total assets increase. This
feature o f the model yields a maximum ex
pected profit fo r the bank under each assump
tion about deposit insurance.10
B a n k C osts — For a given level o f total
assets, the bank's cost depends on the insurance
coverage on its liabilities. This paper considers
the four cases described below. If, as in case A,
all deposits are fully insured, the bank can at
tract an unlimited supply o f deposits by paying
the risk-free rate o f interest. Under each o f the
9The use of a discrete probability distribution, with a limited
number of outcomes, makes the presentation simpler than
if a continuous probability distribution was used. In using a
discrete probability distribution, there is a trade-off be
tween simplicity and continuity of the probability of failure
with respect to leverage. The smaller the number of possi
ble outcomes, the larger the jumps in the probability of
failure at certain asset levels. Increasing the number of
possible outcomes, however, increases the difficulty of il
lustrating the calculations. Thus, the probability distribution
in table 2 is arbitrary.
10The model abstracts from possible losses by our represen
tative bank on the deposits it holds at other banks. If a
reform proposal increases the probability of losses on in
terbank deposits, the effects of such reform proposals on
the probability of failure at our representative bank would
be understated.
This point about possible loss on interbank deposits is
most relevant in comparing the case with no deposit in
surance to the other cases examined below. The model in
this paper is not modified to reflect directly the effects of
possible losses on interbank deposits.
This model also ignores losses from runs on the bank
by its depositors in reaction to the failure of other banks. If
deposit insurance coverage is reduced or eliminated, the

FEDERAL RESERVE BANK OF ST. LOUIS

four assumptions about deposit insurance, the
costs o f servicing deposit accounts are offset by
fees charged to depositors.
For a given level o f total assets, the highest
expense occurs in case B, w ith no deposit in
surance. In this case, the interest rate that the
bank must pay on deposits is positively related
to its total assets. Depositors are assumed to be
risk-neutral and to know the probability distri
bution o f the bank’s return on assets. Hence,
the bank must pay the rate to depositors that
makes their expected return on deposits equal
to the risk-free rate.11
The interest rate that the bank pays deposi
tors is above the risk-free rate if the bank fails
in at least one o f the seven possible outcomes.
If it fails, the depositors receive the liquidation
value o f the bank’s assets. Liquidation value in
those outcomes reflects the probability distribu
tion o f the bank’s return on assets. Th ere is no
additional loss to depositors resulting from the
elimination o f the bank as a going concern.12
Th e equation fo r calculating the rate paid to
depositors in case B is presented in table 2.
Cases C and D reflect tw o methods o f enhanc
ing the effectiveness o f market discipline o f
bank risk, w hile retaining some form o f deposit
insurance. Co-insurance in case C limits deposit
insurance coverage to 90 percent o f each de
posit account.13 In case D, deposits are fully in
sured, but each bank is required to keep its
failure of some banks may induce depositors to run on
other banks to receive currency in exchange for their
deposits. Several such episodes occurred in the United
States prior to the establishment of deposit insurance in
the 1930s. See Dwyer and Gilbert (1989). To incorporate
the possible effects of runs, the probability distribution of
the return on assets at the representative bank would have
to be specified as a function of the number of bank
failures.
11The general nature of the comparisons among the four
cases would not be changed if depositors were assumed
to be risk averse.
12The assumption that a bank’s assets lose no value when
they are liquidated results in an understatement of the ex
pected loss of depositors and bank creditors in the various
cases. A study by the Federal Deposit Insurance Corpora
tion, Bovenzi and Murton (1988), reports that when the
FDIC liquidates the assets of failed banks, their liquidation
value averages about 70 percent of the book value of the
assets of failed banks.
13Boyd and Rolnick (1988) suggest 90 percent coverage of
deposits in their proposal for a form of co-insurance in
deposit insurance.

Table 2
A Model of Bank Profits
NET REVENUE
Revenue of the bank, net of the operating cost of servicing assets, is a random variable with a
discrete probability distribution. Let A be the assets of the bank. The probability distribution is as
follows.
Outcome
number

Net
revenue

1
2
3
4
5
6
7

0.4(1
0.3(1
0.2(1
0.1(1
0.0(1
-0 .1 (1
-0 .2 (1

A/10,000)A
A /10,000)A
A/10,000)A
A /10,000)A
A /10,000)A
A/10,000)A
A /10,000)A

Probability
0.01
0.04
0.10
0.70
0.10
0.04
0.01

COST
The cost of the bank is not a random variable. It is the same in each of the seven outcomes. Ex
pected profits are calculated for four cases, each involving a different assumption about the in
terest expense of the bank. In those outcomes in which the bank has a loss, the maximum loss
to the shareholders is their investment of $100.
Case A:

All Liabilities Fully Insured
The bank pays the risk-free rate of 8 percent on deposits, which equal A - $100.

Case B:

No Deposit Insurance
At each level of total assets of the bank, the interest rate on deposits is set at the
level that makes the expected return to holders of uninsured deposits equal to the
risk-free rate of 8 percent.

The interest rate on deposits in case B, for a given level of assets, can be derived by solving the
following equation for R, the interest rate on deposits. To economize on notation, let
A * = (1 - A/10,000)A. Then,
1.08(A - 100) =

0.01(1 +R)(A-100), or
0.01(1.4A*) if 0.4A* - R(A-100) < -1 0 0
+ 0.04(1 + R)(A - 100), or
0.04(1,3A*) if 0.3A* - R (A -1 0 0 ) < -1 0 0
+ 0.1(1 + R ) ( A - 100), or
0.1(1,2A*) if 0.2A* - R ( A - 100) < -1 0 0
+ 0.7(1 + R ) ( A - 100), or
0.7(1.1A*) if 0.1A* - R(A -1 0 0 ) < -1 0 0
+ 0.1(1 + R)(A - 100), or
0.1(A*) if - R(A - 100) < -1 0 0
+ 0.04(1 + R ) ( A - 100), or
0.04(0.9A*) if -0 .1 A * - R (A -1 0 0 ) < -1 0 0
+ 0.01(1 + R ) ( A - 100), or
0.01(0.8A*) if -0 .2 A * - R ( A - 100) < -1 0 0

JANUARY/FEBRUARY 1990

Table 2 continued
A Model of Bank Profits_________________________________
Case C:
Co-insurance
The calculation of the stated interest rate on deposits as in case B is modified by setting the
minimum return to depositors in each outcome at 90 percent of their principal plus stated
interest.
The equation for the interest rate on deposits is the same as that presented for case B, except
that in each of the seven possible outcomes, the return to the depositors can be no less than:
0.9(1+R)(A - 100).
Case D:

Subordinated Debt Requirement
The bank is required to have liabilities that are uninsured and subordinated to
deposits, equal to at least 10 percent of its total assets. Deposits are fully insured.
The interest rate on deposits is 8 percent. The interest rate on subordinated debt,
for a given level of assets, can be derived by solving the following equation for R.

1.08(0.1)(A) =
0.01(1 +R)0.1A, or
if 0.4A* - R(0.1A) - 0.08(0.9A -1 0 0 ) < -1 0 0 ,
0.01 (the greater of 1.4A* - 1.08 (0 .9 A - 100) or zero)
+ 0.04(1 + R)0.1A, or
if 0.3A* - R(0.1A) - 0.08(0.9A -1 0 0 ) < -1 0 0 ,
0.04 (the greater of 1.3A* -

1.08(0.9A-100) or zero)

+ 0.1(1 +R)0.1A, or
if 0.2A* - R(0.1A) - 0.08(0.9A - 100) < -1 0 0 ,
0.1 (the greater of 1.2A* - 1 .0 8 (0 .9 A -100) or zero)
+ 0.7(1 +R)0.1A, or
if 0.1A* - R(0.1A) - 0.08(0.9A - 100) < -1 0 0 ,
0.7 (the greater of 1.1A* - 1.08(0.9 A - 100) or zero)
+ 0.1(1 +R)0.1A, or
if - R(0.1A) - 0.08(0.9A - 100) < -1 0 0 ,
0.1 (the greater of A* -

1.08(0.9A - 100) or zero)

+ 0.04(1 +R)0.1 A, or
if -0 .1 A * -

R(0.1 A) - 0.08(0.9A-1 0 0 ) < -1 0 0 ,

0.04 (the greater of 0.9A* - 1.08(0.9A- 100) or zero)
+ 0.01(1 +R)0.1A, or
if - 0.2A* - R(0.1A) - 0 .0 8 (0 .9 A -100) < -1 0 0 ,
0.01 (the greater of 0.8A* -

Digitized forFEDERAL
FRASER RESERVE BANK OF ST. LOUIS

1.08(0.9A- 100) or zero)

subordinated debt liabilities equal to 10 percent
or m ore o f its total assets.14
In cases C and D, the interest rates on bank
liabilities also are set at levels that make ex
pected returns to bank creditors equal to the
risk-free rate. Equations fo r calculating the in
terest rates on bank liabilities are specified in
table 2.
Other reform proposals are o f interest but are
more difficult to incorporate into this simple
model. For instance, m ore detail w ould be ne
cessary to model the effects o f changing the
deposit insurance limit per account or limiting
FDIC coverage fo r each depositor to a given
amount at all insured institutions.

Case A: All Liabilities Fully Insured
The bank maximizes expected profits in this
case with total assets around $1800 (see figure
1). At this level, the bank fails (losses exceed the
$100 o f capital) in outcomes 5, 6 and 7. Thus,
the probability that the bank w ill fail is 15 per
cent, based on the probability distribution fo r
the return on assets in table 2. The FDIC’s ex
pected loss, $14.26, is about 0.84 percent o f in
sured deposits. The expected loss o f depositors,
o f course, is zero.

Case B: N o Deposit Insurance
Several reform proposals call fo r phasing out
deposit insurance (see table 1). W ith no deposit
insurance, depositors lose part o f their principal
plus interest if the bank fails (if losses exceed
the $100 o f capital). The interest rate on de
posits charged b y risk-neutral depositors is
positively related to the bank’s total assets (see
figure 2).

The FDIC’s expected loss in this case is zero.
The expected loss o f depositors w ith total assets
equal to $1000 is $4.21, which is about one-half
o f one percent o f deposits.

Case C: Co-insurance
Under the co-insurance option, federal deposit
insurance coverage w ould be limited to a frac
tion o f each deposit, w ith some low level o f
each account fully covered to protect small de
positors. Those w h o advocate co-insurance
argue that the depositors subject to fractional
coverage at the margin w ould monitor the risk
assumed by their banks and demand relatively
high interest rates on deposits at the banks that
assume relatively high risk.
T o simplify the illustration, all deposit ac
counts are subject to the same percentage o f in
surance coverage. In those outcomes in which
the bank fails, payments to depositors under the
co-insurance option w ould be the larger of:
(1) the liquidation value o f the bank’s assets,
or
(2) 90 percent o f the principal plus interest on
their deposits. Th e FDIC incurs a loss only
if the bank fails and the liquidation value
o f the bank is less than 90 percent o f the
principal plus interest on deposits.
As in case B, the market interest rate on
deposits is set at the level that makes the ex
pected return on deposits equal to 8 percent.
The difference in this case is that depositors
have the option o f receiving 90 percent o f their
principal plus interest from the FDIC if their
banks fail. Figure 2 indicates that fo r a given
level o f total assets, the market interest rate on
deposits is lo w er in case C than in case B,
because in case C the losses o f depositors are
limited by deposit insurance.

The bank maximizes its expected profits with
total assets equal to $1,000. It fails only in out
comes 6 and 7. Thus, the probability that the
bank w ill fail is only 5 percent, compared with
a 15 percent probability o f failure associated
w ith maximum profits in case A. Case B illus
trates how a bank that maximizes expected pro
fits can be induced to limit its probability o f
failure through market discipline imposed by its
creditors.

Under the assumptions o f case C, the bank
maximizes expected profits w ith total assets o f
$1100. The bank must pay 8.44 percent to at
tract the $1000 in deposits. W ith assets o f
$1100, the probability o f the bank failing is 5
percent. The FDIC’s expected loss is only 0.072
percent o f insured deposits, about 9 percent o f

14Case D involves a higher percentage of subordinated debt
to total assets than some of the proposals that call for
subordinated debt requirements. For instance, the recent
proposal of the Federal Reserve Bank of Chicago recom
mends that banks be required to maintain a 4 percent

ratio of subordinated debt to total assets. See Keehn
(1989). The 10 percent requirement in case D is chosen to
indicate that the degree of market discipline that can be
imposed through a co-insurance proposal can be matched
with a subordinated debt proposal.

JANUARY/FEBRUARY 1990

Figure 1
Expected Profits
Dollars

Dollars

Assets

Figure 2
Expected Loss to FDIC
Dollars

Dollars

Digitized forFEDERAL
FRASER RESERVE BANK OF ST. LOUIS

Assets

the loss rate fo r case A, w ith total assets at
$1800. The expected loss to depositors is $5.09,
which is about one-half o f one percent o f total
deposits.
From the FDIC’s perspective, there are tw o
advantages o f co-insurance (case C) over full
deposit insurance (case A). First, the bank
chooses a level o f assets associated w ith a low er
probability o f failure. Second, fo r a given level
o f total assets o f the bank, the FDIC’s expected
loss is lo w er under case C.15

Case D: Subordinated D ebt
Requirem ent
Some proposals fo r deposit insurance reform
w ould require banks to issue subordinated debt
liabilities that are not federally insured. The
term "subordinated” refers to the status o f
creditors o f a firm in bankruptcy. I f a failed
bank is liquidated, those w h o hold subordinated
debt w ould receive payments only if all deposi
tors are paid in full.
In case D, all deposits are fully insured by the
FDIC. The bank, how ever, must have uninsured
liabilities, which are subordinated to deposits,
that equal at least 10 percent o f its total assets.
The bank would choose to keep subordinated
debt liabilities at the 10 percent minimum since,
except at relatively low levels o f total assets, the
interest rate on subordinated debt exceeds the
risk-free rate paid on insured deposits.
As in cases B and C, those w h o invest in
subordinated debt are assumed to be riskneutral and know the probability distribution o f
the net return on assets. Figure 3 presents the
interest rate on subordinated debt as a function
o f the total assets o f the bank.16 For most levels
15Co-insurance, however, has one disadvantage. A change
from full coverage of insured deposits to co-insurance
creates an incentive for depositors to run on banks in
response to information (or rumors) about problems at
banks. Even with the FDIC insuring 90 percent of the prin
cipal and interest of deposit accounts, depositors have an
incentive to avoid the 10 percent loss by withdrawing their
deposits from a failing bank. Thus, in comparing cases A
and C, co-insurance reduces the significance of deposit in
surance in preventing runs on the banking system, placing
greater responsibility on the role of the Federal Reserve in
stabilizing the banking system in a financial crisis, as it
functions as the lender of last resort. If there is some
doubt that the Federal Reserve will execute its role as
lender of last resort, co-insurance may be less advan
tageous than full insurance of deposits.
16The humped pattern of the interest rate on subordinated
debt for case D in figure 3 reflects the particular discrete
probability distribution of returns on the assets used in this

o f total assets, the interest rates on subor
dinated debt is higher than the rates on
deposits in the cases analyzed earlier because
the expected loss is higher fo r those holding
subordinated debt. I f the bank’s losses exceed
the $100 investment o f the shareholders,
holders o f the subordinated debt receive some
payment only if the liquidation value o f the
bank exceeds total deposits.
The bank maximizes expected profits with
total assets equal to $1100. A t that level, the
bank must pay 12.12 percent on its subordi
nated debt liabilities. The FDIC incurs losses
only if the loss o f the bank exceeds the $100
capital o f the shareholders plus the subordi
nated debt. Th e bank has a 5 percent probabili
ty o f failure, and the expected loss o f the FDIC
w ith total assets equal to $1100 is 0.06 percent
o f insured deposits. Depositor losses are zero.

Comparison o f Cases B, C and D
A comparison o f risk in the operation o f the
banking system under various assumptions
depends on one’s assumption about the p ro
bability o f runs on the banking system. I f this
probability is assumed to be zero, the elimina
tion o f deposit insurance (case B) induces banks
to assume minimum risk. The FDIC’s expected
loss is zero in this case, and the bank is induced
by market forces to choose the lowest level o f
total assets. One advantage o f the subordinated
debt requirem ent over the other options is that,
w hile the bank is induced to choose a level o f
total assets below that in case A, the subor
dinated debt is not subject to runs. Thus, the
comparison o f risk betw een cases A and D does
not depend on assumptions about runs on the
banking system.
paper. With a continuous probability distribution, or a
discrete distribution with more possible outcomes, the plot
of the interest rate as a function of total assets would have
a less humped pattern.
The fact that the interest rate on subordinated debt is
higher at higher levels of total assets of the bank might in
dicate a way in which the management of the bank could
take advantage of those who invest in subordinated debt.
The bank could issue some subordinated debt at a low
level of total assets, at a relatively low interest rate, and
then increase total assets and issue more subordinated
debt at a higher rate. Investors in subordinated debt can
protect themselves from such actions by insisting on
covenants in the subordinated debt agreements that limit
additional debt. If management of the bank violates such a
covenant, the holders of the subordinated debt could go to
court to make their debt instruments payable on demand.
Restrictions on the issuance of additional debt are com
mon in bond covenants. See Smith and Warner (1979).

JANUARY/FEBRUARY 1990

Figure 3
Market Interest Rate on Bank Liabilities
Rate

Rate

Assets

Th e co-insurance option is not superior under
any combination o f assumptions. I f the possibili
ty o f runs on the banking system can be ruled
out, there is a subordinated debt requirem ent
that induces the same degree o f market disci
pline o f banking risk as co-insurance.

EM PIR ICAL STUDIES OF M AR KET
DISCIPLINE OF THE RISK
ASSUMED RY HANKS
Market forces w ill be effective in constraining
the risk assumed by banks only if investors can
assess the relative degrees o f risk assumed by
individual banks, and then set differential prices
on the stock and debt instruments issued by
17The studies described in this section include only those
based on data for individual banking organizations. Some
studies cited in the literature estimate indices of returns on
share prices or interest rates on bank liabilities for groups
of banks as functions of aggregate data on banking risk.
Such results are not relevant in determining whether par
ticipants in the equity and debt markets can distinguish
among the banking organizations, which would be
necessary if market discipline of bank risk were to be
effective.

FEDERAL RESERVE BANK OF ST. LOUIS

banks that reflect their inform ation about risk.
The results o f the studies described in table 3
are relevant in evaluating the effectiveness o f
market discipline. These studies estimate the in
fluence o f measures o f risk assumed by banks
on the stock prices o f banks and on the market
interest rates on uninsured deposits and the
subordinated debt o f banks.17 These studies do
not test the hypothesis that banks adjust their
risk in response to signals from the markets fo r
bank stocks and debt.18

The Market f o r Bank Equity
All but one o f these studies report evidence
that is consistent w ith the hypothesis that stock
prices are inversely related to the risk assumed
18Gendreau and Humphrey (1980) claim to have developed
a model in which there is feedback from adverse signals
in the bank equity market to bank leverage. It is difficult to
see a feedback relationship between the stock price and
leverage in this study, since the relationships among stock
prices, leverage and other variables are estimated using
contemporaneous observations. Estimating a feedback
relationship from market signals to variables under the
control of bank management would require dynamic
relationships.

Table 3
Implications of Empirical Studies for the Effectiveness of Market Discipline
of Bank Risk

Authors

Relationships estimated

Results consistent
with the
effectiveness
of market
discipline

Results
MARKET FOR BANK EQUITY

Beighley, Boyd
and Jacobs
(1975)

Share prices of bank stocks estimated
as a function of (1) capital ratios,
(2) earnings and growth of earnings,
(3) asset size, and (4) loss rates.

Holding constant the influence of earn
ings banks with higher capital ratios and
lower loss rates tend to have higher
share prices.

Yes

Pettway (1976)

Betas for individual banks (a measure
of risk derived from stock prices) esti
mated as a function of the capital ratios
of individual banks.

The coefficient on the capital ratio is
negative for one year but insignificant for
other years. The negative coefficient on
the capital ratio indicates that investors
consider banks with higher capital ratios
to be less risky.

Yes

Pettway (1980)

For several large banks that failed,
returns to shareholders are simulated
for several years prior to their failure.
Simulations are based on returns from
holding stocks of large banks that did
not fail.

On average, returns on the stocks of
banks that failed declined relative to
simulated returns two years before
failure.

Yes

Brewer and Lee
(1986)

Betas for individual banks are estimated
as functions of ratios from balance
sheets and income statements used
by bank supervisors to reflect risk.

Some of the measures chosen to reflect
risk have positive, significant regres
sion coefficients.

Yes

Cornell and
Shapiro (1986)

Returns to shareholders of 43 large
banks are estimated as functions
of the composition of their assets and
liabilities in the years 1982-83.

The percentage that Latin American
loans was of total assets had a signi
ficant, negative impact on returns in
1982. Energy loans had a negative
impact in 1982-83. Loans purchased
from Penn Square Bank had a negative
impact on returns in the month in
which that bank failed.

Yes

Shome, Smith
and Heggestad
(1986)

Prices of bank stocks are estimated as a
function of its earnings and capital
ratios.

The coefficient on the capital ratio is
positive and significant for some years,
insignificant for other years.

Yes

Smirlock and
Kaufold (1987)

Changes in stock prices of large
banks at the time of the announce
ment by Mexico in 1982 of its mora
torium on debt payments as a function
of the ratio of Mexican debt to equity
capital at individual banks.

Coefficient on the ratio of Mexican
debt to equity capital is negative and sig
nificant. Banks were not required to
disclose their Mexican debt at the time
of the 1982 moratorium.

Yes

JANUARY/FEBRUARY 1990

Table 3 continued
Implications of Empirical Studies for the Effectiveness of Market Discipline
of Bank Risk

Authors

Relationships estimated

Results

Results consistent
with the
effectiveness
of market
discipline

MARKET FOR BANK EQUITY continued
James (1989)
and Cargill
(1989)

Returns on holding the stock of BHCs
estimated as a function of the change
in the market value of the BHCs’ loans
to less-developed countries and dum
my variables for individual banks and
individual time periods.

The change in the market value of
loans to less-developed countries has a
positive, significant coefficient which is
not significantly different from unity.

Yes

Randall (1989)

This is a case study of 40 BHCs that
reported relatively large losses in the
1980s. For each BHC, a time period
is designated when it began assuming
relatively high risk and a time period
when problems became public know
ledge. Stock prices are compared to
market averages before and after the
problems became public knowledge.

Stocks prices of the BHCs that re
ported relatively large losses declined
relative to market average stock prices
only after the problems became public
knowledge, not during the periods
which the banks began assuming
relatively high risk.

MARKET FOR UNINSURED DEPOSITS
The interest rate on large denomination certificates of deposit is the dependent
variable in each study.
Crane (1976)

Identifies the determinants of the CD
rate using factor analysis.

The factor that reflects profit rates and
capital ratios is not a significant vari
able in explaining the CD rate.

Herzig-Marx and
Weaver (1979)

Estimates CD rates as a function of
variables used by bank supervisors to
reflect risk.

Of bank risk variables, only the liquidi
ty measure has a significant coefficient.
Capital and loss ratios have insignifi
cant coefficients.

Baer and Brewer
(1986)

CD rate estimated as a function of
variables used by bank supervisors to
reflect risk, and separately, as func
tions of level and variability of the
prices of bank stocks.

Coefficients on risk measures used by
bank supervisors are not significant.
Measures of the level and variability of
stock prices help explain CD rates.

James (1987)

The average interest rates paid by 58
large banks on their large denomina
tion deposits are estimated as func
tions of leverage, loan loss provision
divided by total loans and the variance
of stock returns.

Each of these measures of risk have
positive, significant coefficients.

Yes

These three variables have significant
coefficients. CD rates tend to be higher
at banks with more variable income and
lower capital ratios, holding constant
the influence of total assets.

Yes

Hannan and
CD rate is estimated as a function of (1)
Hanweck (1988) the variability of the ratio of income to
assets, (2) the capital ratio and (3) bank
assets.

Digitized forFEDERAL
FRASER RESERVE BANK OF ST. LOUIS

Table 3 continued
Implications of Empirical Studies for the Effectiveness of Market Discipline
of Bank Risk

Authors

Relationships estimated

Results consistent
with the
effectiveness
of market
discipline

Results

MARKET FOR UNINSURED DEPOSITS continued

James (1989)

Interest cost on large CDs estimated
as a function of risk measures: domestic
loans/capital, foreign loans/capital and
the loan loss provision/total loans.

Interest cost positively related to the
ratio of domestic loans to capital and
the loan loss provision. The negative
relation between interest cost and the
ratio of foreign loans to capital is inter
preted as evidence of an implicit
government guarantee of foreign loans.

Yes

MARKET FOR SUBORDINATED DEBT:
In each study the measure of the interest rate on the subordinated debt of banks is
the rate on the subordinated debt minus the rate on long-term U.S. Treasury
securities, called the rate premium.
Pettway (1976)

The rate premium is estimated as a
function of the capital ratio of banks
and other independent variables.

The coefficient on the capital ratio is
not significant.

Beighley (1977)

The rate premium is estimated as a
function of several measures of risk,
including a loss ratio and a leverage
ratio.

The coefficients on the loss and lever
age ratios are positive and significant.

Yes

Fraser and
McCormack
(1978)

The rate premium is estimated as a
function of the capital ratio and the
variability of profits divided by total
assets.

Neither independent variable has a
significant coefficient.

Herzig-Marx
(1979)

The rate premium is estimated as a
function of several measures of risk
assumed by banks.

None of the risk measures have signi
ficant coefficients.

Avery, Belton
and Goldberg
(1988)

The rate premium is estimated as a
function of risk measures derived from
balance sheets and income statements
and of the asset size of banks.

Coefficients on the risk measures de
rived from balance sheets and income
statements are not significant.

Gorton and
Santomero
(1988)

Use data in Avery, Belton and Goldberg
(1988) to derive a measure of the vari
ance of assets of banks implied by a
contingent claims valuation model. The
measure of the variance of assets is
estimated as a function of the risk
measures derived from balance sheets
and income statements.

Some of the risk measures derived
from the balance sheets and income
statements have significant coefficients.

Yes

JANUARY/FEBRUARY 1990

by banks, holding constant other determinants
o f stock prices. Th e one study that concludes
that stock prices do not reflect the risk assumed
by banks, by Randall (1989), examines m ove
ments in the stock prices o f bank holding com
panies that reported relatively large losses in
the 1980s. Randall concludes that these stock
prices fell relative to the stock prices at other
banks after their problems became common
knowledge; how ever, they did not decline dur
ing the periods w hen the banks w e re assuming
the relatively high risk that led to losses. Ran
dall concludes that the stock market does not
discipline the risk assumed by banks, since the
relative declines in bank stock prices did not
precede public information on the consequences
o f risk assumed by these banks.
Randall’s study, how ever, has several w eak
nesses. It is a case study, not a statistical study
o f the determinants o f stock prices. The dating
o f points at which problems became common
knowledge is arbitrary; the choice o f such
dates, how ever, determines the results. About
half o f the cases involve banks in the Southwest
W e w ould not expect relative declines in the
stock prices o f these banks before the large
decline in oil prices. W e cannot expect the par
ticipants in the market fo r bank stocks to have
greater foresight in predicting the decline in the
price o f oil than the participants in the market
fo r oil.
T w o studies are particularly interesting in
terms o f investors’ ability to differentiate among
banks on the basis o f risk. Pettway (1980) com
pares stock prices o f large banks that failed
w ith simulated stock prices that w ere based on
data from banks o f comparable size that did not
fail. Returns to stockholders o f the failed banks
declined relative to their simulated returns
about tw o years b efore the banks failed. Rela
tive returns o f the failed banks also declined
before the bank supervisors put them on the
problem bank list. Smirlock and Kaufold (1987)
find that, w hen Mexico announced the m orato
rium on its debt payments in 1982, the declines
in the stock prices w ere proportional to the
Mexican debt held by banks relative to the book

19This contrast can be illustrated using some recent studies
and bank failure cases. Avery, Belton and Goldberg (1988)
use observations for the 100 largest BHCs, which had total
assets above $3 billion in 1985 and 1986. The total assets
of the banks in the sample used by Hannan and Hanweck
(1988) average $4 billion. In 1985 and 1986, 69 failed
banks did not have their liabilities assumed by surviving

FEDERAL RESERVE BANK OF ST. LOUIS

value o f their equity capital. At the time o f the
moratorium, banks w ere not required to dis
close their loans to other nations. Nevertheless,
investors appeared to have sufficient inform a
tion, without such requirements, to make the
appropriate adjustments to the prices o f bank
stocks.

The Markets f o r Uninsured
Deposits and Subordinated D ebt
The findings about the relationship betw een
risk and interest rates on uninsured deposits
and on subordinated debt are m ore mixed.
Th ree o f the six studies o f bank CD rates report
no evidence that higher CD rates are paid by
banks that assume m ore risk. Four o f the six
studies o f the determinants o f rates on the sub
ordinated debt o f banks find no significant e f
fects o f risk measures on interest rates.

Implications f o r the Effectiveness
o f Market Discipline
In evaluating these results, it is important to
note that, under the procedures follow ed by
federal bank regulators in recent years, risk has
a m ore certain implication fo r bank profits than
fo r the returns to the holders o f uninsured de
posits or subordinated debt. Losses on bank
assets reduce profits, and if losses fo rce a bank
to fail, the bank shareholders are likely to
receive nothing after the liquidation or sale o f
the bank.
Uninsured depositors and holders o f subor
dinated debt, in contrast, receive less than the
principal plus contracted interest only if a bank
fails. In most cases, the failed bank is m erged
w ith another bank, and the surviving banks
assume all liabilities o f the failed banks, in
cluding those in the form o f uninsured deposits
and subordinated debt. Most o f the cases in
which uninsured depositors and holders o f
subordinated debt absorb some losses involve
banks smaller than those included in the studies
described in this paper.19 These observations
are consistent w ith the conclusion that interest
rates on bank liabilities w ould be m ore sensitive

banks. Of these 69 failed banks, 66 had total assets less
than $100 million, while the remaining three had total
assets less than $200 million. The failure of some large
banking organizations in the Southwest, in which the
BHC’s bondholders absorbed losses, occurred after the
periods covered by these studies.

to the risk assumed by banks if bank creditors
lost at least part o f their principal plus interest
in each bank failure.
The empirical results cannot be used to indicate
the degree o f risk that banks would assume if
bank supervisors eliminated various forms of
supervision and regulation, relying instead on
market forces to limit bank risk. To illustrate
such a change in policies, suppose bank super
visors eliminate capital requirements and restric
tions on the types o f assets that banks may ac
quire, substituting a requirement that banks issue
subordinated debt. The empirical results do not
tell us whether the probability o f bank failures
would increase or decrease under such a change
in policies. The only useful information from the
empirical studies is that investors in bank stocks,
w ho have the strongest incentives to be sensitive
to the risk assumed by banks, are able to dif
ferentiate among banks on the basis o f risk.

CONCLUSIONS
This theoretical exercise illustrates how
market forces could limit the incentives fo r
banks to assume risk. The incentives fo r banks
to assume relatively high risk are reduced if the
insurance coverage o f bank creditors drops
from full to partial coverage. One o f the im por
tant differences among the various approaches
to prom oting market discipline o f banking risk
involves the vulnerability o f banks to runs.
Banks are m ore vulnerable to runs if depositors
are at risk than if the risks are borne by those
holding long-term bank debt that is subor
dinated to deposits.
Empirical studies o f the effectiveness o f
market discipline report mixed results. The
most consistent result is that the stock prices o f
individual banks reflect the risk assumed by
banks. Market discipline o f such risk w ould
tend to be m ore effective if bank creditors w ere
forced to absorb losses in a m ore consistent
fashion in bank failure cases.
The empirical studies do not indicate the
degree o f risk that banks w ould assume if
deposit insurance w ere reform ed to enhance
the effectiveness o f market discipline. Thus, the
empirical studies do not perm it us to determine
w hether the probability o f bank failures would
rise or fall if the current form s o f bank regula
tion w ere eliminated in favor o f market disci
pline by bank shareholders and creditors.

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the Currency (Vol. 7, No. 3, 1988), pp. 9-20.
Pettway, Richard H. “ Market Tests of Capital Adequacy of
Large Commercial Banks,” Journal of Finance (June 1976),
pp. 865-75.
________“ Potential Insolvency, Market Efficiency, and the
Bank Regulation of Large Commercial Banks,” Journal of
Financial and Quantitative Analysis (March 1980), pp.
219-36.
Randall, Richard E. “ Can the Market Evaluate Asset Quality
Exposure in Banks?” New England Economic Review (Ju
ly/August 1989), pp. 3-24.
Schwartz, Anna J. “ The Effects of Regulation on Systemic
Risk,” Proceedings of a Conference on Bank Structure and
Competition, Federal Reserve Bank of Chicago (May 1988),
pp. 28-34.
Shome, D.K., S.D. Smith, and A.A. Heggestad. “ Capital Ade
quacy and the Valuation of Large Commercial Banking
Organizations,” Journal of Financial Research (Winter 1986),
pp. 331-41.
Short, Eugenie D., and Gerald P. O ’Driscoll. “ Deregulation
and Deposit Insurance,” Federal Reserve Bank of Dallas
Economic Review (September 1983), pp. 11-22.
Smirlock, Michael, and Howard Kaufold. “ Bank Foreign
Lending, Mandatory Disclosure Rules, and the Reaction of
Bank Stock Prices to the Mexican Debt Crisis,” Journal of
Business (July 1987), pp. 347-64.
Smith, Clifford W., and Jerold B. Warner. “ On Financial Con
tracting: An Analysis of Bond Covenants,” Journal of Finan
cial Economics (June 1979), pp. 117-61.
Smith, Fred. “ Cap the Financial Black Holes,” Wall Street
Journal, September 29, 1988.
Wall, Larry D. “A Plan for Reducing Future Deposit In
surance Losses: Puttable Subordinated Debt,” Federal
Reserve Bank of Atlanta Economic Review (July/August
1989), pp. 2-17.

Mark D. Flood
Mark D. Flood is a visiting scholar at the Federal Reserve
Bank of St. Louis. David H. Kelly provided research assistance.

On the Use of Option Pricing
Models to Analyze Deposit
Insurance

T„e

FAILURE rate o f banks and thrifts has
exploded over the past decade, making reform
o f the deposit insurance system a topic o f con
siderable interest to regulators, bankers, and
economists. As illustrated by figure 1, which
shows the total number o f failed commercial
banks (excluding thrifts) fo r each year since the
chartering o f the Federal Deposit Insurance Cor
poration (FDIC), the annual number o f com m er
cial bank failures in each o f the last several
years has exceeded its previous peak, attained
during the Great Depression. The status o f the
thrift industry is even m ore grim, with losses to
the Federal Savings and Loan Insurance Cor
poration (FSLIC) estimated at $160 billion or
more. The prim ary consequence o f these fail
ures fo r public policy is the enormous losses,
especially to the FSLIC, as depositors in these
failed institutions are reimbursed.
This article considers a particular set o f eco
nomic tools used to evaluate deposit insurance.1
Option pricing models are among the techniques
available fo r analyzing the deposit insurance
system. These models can be used to assign
1This article does not address the issue of why we should
have deposit insurance. The rationale for the current
system of bank regulation and for deposit insurance in
particular is based on two related principles: protection of
the depositor and the mitigation of contagious bank runs.
See FDIC (1984) and U. S. Treasury (1985). Benston and

specific values to the claims o f each o f the in
terested parties involved in the deposit insur
ance system — the insurer, financial institutions,
and depositors. Such valuations can then be
used, fo r example, to estimate the net value o f
the governm ent’s insurance fund or to deter
mine a fair price that a bank should pay fo r its
insurance. M ore generally, by comparing in
surance valuations with different m odel parame
ters, one can investigate the system o f incen
tives under a given regulatory scheme, such as
the risk incentives fo r bank shareholders and
depositors under the present system. Finally,
comparisons o f insurance values and incentives
can be made across various proposed regulatory
schemes. These applications are illustrated be
low w ith some examples.
The usefulness o f option pricing models fo r
evaluating deposit insurance is o f special in
terest fo r tw o reasons. First, the consensus
among the interested parties is that the present
deposit insurance system has contributed to the
current crisis. Second, in the context o f this
debate, a number o f economists have used the
Kaufman (1988) identify three reasons for bank regulation
in addition to the two traditional rationales, namely disrup
tion to communities from localized bank failures, moral
hazard induced by deposit insurance and restrictions on
competition.

JANUARY/FEBRUARY 1990

Figure 1
Bank Failures (Insured and
Uninsured)
Number of Failed Banks
200

Number of Failed Banks

200

0 1-------------- ii a a s r .
1934

— r ........ —..........J
54

1988

Source: FDIC (1989)

m odern theory o f option pricing to explain the
incentives and measure the costs both o f the
current system o f deposit insurance and of
some suggested alternatives. This paper p re
sents the basic theory o f option pricing, explains
how it can be applied to deposit insurance, and
analyzes some o f the issues involved in its use.

A PR IM ER ON O P T IO N PR ICING
This paper presumes no knowledge o f options
or o f the various economic models that have
been used in the academic literature to assign
values to options. Thus, it begins with a brief
description o f options and some o f the major
contributions to the theory o f pricing options
that have been made in the past tw o decades.

Contingent Claims, or Options
A call option is a legal contract that gives its
ow n er the right to buy a specified asset at a fix
ed price on a specified date.2 Similarly, a put op
tion gives its ow n er the right to sell a specified
asset at a fixed price on a specified date. Option
contracts are usually sold b y one party to an
other.3 The person w ho owns an option con
tract is called the holder o f the option. The per

2This definition is a paraphrase of the definition given by
Cox and Rubinstein (1985), p. 1. It describes a “ Euro
pean” option, which is distinguished from an “ American”
option. An American option gives its owner the right to buy
at any time on or before the specified date.
3They are sold, because options have a non-negative value;
because they are a right to buy (or sell) the asset, they do

Digitized forFEDERAL
FRASER RESERVE BANK OF ST. LOUIS

son w ho sells an option contract — that is, the
person w ho w ill be compelled to perform if the
option holder invokes her right as specified in
the contract — is called the writer o f the option.
The act o f invoking the contract is called exer
cising the option. The fixed price identified by
the option contract is called the striking price.
The date at which the option can be exercised
is called the expiration date o f the option.
These legal contracts are probably best known
by the stock options that are bought and sold
by brokers in the trading pits o f organized op
tions exchanges in Chicago, N ew York and
elsewhere. In addition to options on common
stock, there are active markets fo r options on
agricultural comm odity futures, foreign curren
cies, stock index portfolios, and governm ent
securities, to name only a few . The definition o f
an option, how ever, does not limit the term to
those contracts actively traded on the floors o f
organized financial exchanges. By definition, an
option is any appropriately constructed legal
contract betw een the w riter and the holder,
regardless o f w hether it is ever traded.

Expiration-date Values o f Options
Consider now the value to the holder o f an
expiring put option, as illustrated in figure 2.
The value o f the underlying asset specified by
the contract is given on the horizontal axis,
w hile the value o f the option itself is given on
the vertical axis. The point K on the horizontal
axis is the specified striking price fo r the asset.
If the value o f the underlying asset is above the
striking price on the expiration date, then the
put option w ill not be exercised; anyone w ho
truly wanted to sell the asset w ould do so
outright at the going price, rather than using
the option and receiving the striking price. In
this case, the option expires worthless, and the
option holder experiences no gain or loss on the
expiration date.
On the other hand, if the value o f the asset is
below the striking price, then the holder w ill
exercise her option and receive the striking

not compel the owner of the contract to do anything.
Although they are valuable, nothing in the definition of an
option requires that they be offered for sale; that is, their
value does not depend on how they were obtained.

Figure 2
Value of Put Option to Holder
Expiration Value of Option

Figure 3
Value of Put Option to Writer
Expiration Value of Option

(1)

State

Action

Option value

AT < K
At > K

Exercise
No exercise

P = K - AT
P = 0.

For this reason, options are also referred to as
"contingent claims” on the underlying assets.
The corresponding net payoffs to the w riter
o f the put option are given in figure 3. Notice
that his payoffs are exactly the inverse o f those
fo r the option holder. Also note that the payoff
at expiration to the w riter o f an option is never
positive; at best it is zero. It is fo r this reason
that options are sold to the holder, rather than
being given away fre e o f charge. The price in
itially paid fo r the option — the option price or
option premium — could be incorporated into
the figures by simply shifting the holder's
payoffs dow n and the w riter’s payoffs up by
the appropriate amount.
The payoffs at expiration to the holder and
w riter o f a call option are given in figures 4
and 5, respectively. Th e corresponding analysis
fo r call options is precisely analogous to the
analysis just given fo r put options.

The Black-Scholes Option Valua
tion M od el

----------------------------- |-----------------0

K
Value of Underlying Asset at Expiration

price fo r the asset. In this case, her net gain on
the expiration date w ill be (K - A t ), the d iffe r
ence betw een the striking price and the current
price, since she can turn around and replace
the asset immediately, if she wants to. Thus, the
expiration value o f the option and the decision
about w hether to exercise are contingent upon
the value o f the underlying asset at that time:

4For an exposition of the arbitrage bounds on option prices,
see Merton (1973), or Cox and Rubinstein, ch. 4.
5Almost all derivations of option pricing models, including
that of Black and Scholes, are stated in terms of call
rather than put options. As it happens, this distinction is
largely irrelevant, because call option valuations are readi
ly converted to put option valuations, and vice-versa, via
an arbitrage relationship known as “ put-call parity” . Put-

Having described the value o f an option at ex
piration leaves the question o f its value prior to
expiration unanswered. Instead o f being a sim
ple function o f A t and K, the value o f an option
before maturity depends on several additional
factors. Although a num ber o f bounds had been
placed on the value o f an unexpired option by
using relatively simple arbitrage arguments, an
important advance in the valuation o f unexpired
options was made by Black and Scholes (1973).4
They obtained an exact equation fo r the value
o f a put option under an unrestrictive set o f
assumptions.5 Th eir result has since been elab
orated and generalized by others.6
In their model, the value o f an unexpired op
tion depends on five things:

call parity is an exact relationship for European options
and an approximate one for American options (see Cox
and Rubinstein, pp. 150-52); throughout this paper it is
treated as exact. Put-call parity is first presented by Stoll
(1969).
6One such generalization is found in the shaded insert. For
a partial survey of option pricing models, see Cox and
Rubinstein, ch. 7.

JANUARY/FEBRUARY 1990

K
A
T
a
R

=
=
=
=
=

the
the
the
the
the

striking price
current asset price
time remaining to expiration
volatility o f the asset price
risk-free interest rate.

Figure 4
Value of Call Option to Holder
Expiration Value of Option

Almost as notable is what the option’s value
does not depend on: any characteristic o f the
holder or the w riter.7 Under their assumptions,
Black and Scholes are able to include an option
in a riskless portfolio. Such a portfolio must
earn the risk-free interest rate, and they are
able to use this result, along with an assumption
about the probability distribution o f the asset
price, to identify an exact value, P, fo r a put
option:
P = (K*e - RT)*N(X + o\/t ) - A-N(X),
where:
X
-

1 - [ln (K -e -RT)- ln (A )]

- VzcrJ T

<rJT
N (»)

= the Vstandard normal cumulative
probability function

Figure 5
Value of Call Option to Writer
Expiration Value of Option

ln («) = the natural logarithm function
e H the base o f the natural logarithm
Although this form ula may at first appear
complicated, a rough intuition can be provided
relatively painlessly.8 First, e ~ RT is just the p re
sent value discount factor fo r T periods at in
terest rate R w ith continuous compounding, so
that K-e_RT is the present value o f the striking
price. Keeping in mind that N(X + oV t ) is a pro
bability, the first term is the expected present
value o f the striking price at expiration, given
that A t < K. Similarly, the second term, A*N(X),
is the expected present value o f the expirationday asset price, again given that A t < K.9 Thus,
the value o f the option is the expected present
value o f its value at expiration, given by condi
tion 1 above.
Unfortunately, no easy, correct interpretation
can be attached to the specific probabilities,
N(X + oV t ) and N(X), in the tw o terms. These

7For example, one might suspect that the holder’s attitudes
toward risk or her beliefs about the asset price at expira
tion should influence the option’s value to her. This is not
the case, however. Also note that four of the five factors,
at least theoretically, are well-defined and directly obser
vable at the time of valuation. The exception is asset
volatility, which must be estimated from observable fac
tors; see Cox and Rubinstein, pp. 280-87, for an example
of an estimation technique.
8A full derivation of the formula is fairly involved and will
not be presented here. Interested readers are referred to

FEDERAL RESERVE BANK OF ST. LOUIS

probabilities are closely related to the probabili
ty that A t < K, but they are not quite the
same, because the present value o f the striking
price is known w ith certainty, w hereas the p re
sent value o f the asset’s price on the expiration
day, A T*e~RT, is not; the current asset price, A,
appears instead.

Malliaris (1983) for a mathematically advanced approach
or to Cox and Rubinstein, ch. 5, for a longer but less
technical derivation.
9The corresponding expected present values for the case
when A t is greater than K are both zero, because then the
expiring option is worthless and will not be exercised;
hence, this possibility adds nothing to the current value of
the option.

In spite o f its complexity, the option pricing
equation is still a useful tool. In one sense, the
formula can be treated as a black box in which
the five parameters (K, A, T, o and R) enter at
one end, and the value o f the put option, P,
comes out at the other; a computer spreadsheet
or calculator can be program m ed to perform
the intervening calculations defined by the fo r
mula. For example, if the current asset value is
A = $985, the standard deviation o f asset
returns is o = 0.3 percent, the striking price o f
the option is K = $1000, the time to expiration
is one year, and the riskless interest rate is 8
percent per year, then the Black-Scholes equa
tion tells us that the put option is w orth $85.45.
Figure 6 graphs the Black-Scholes value o f a put
option fo r a range o f current asset values from
zero to $1500, w h ere the values o f the other
four parameters are the ones just given.

The Brownian M otion Assumption

Figure 6
Value of Put Option to Holder
Option Value Prior to Maturity
(K =1000, T=1, o = .3 , Rf = .08)

1000
932.12

o
Asset Value

Figure 7
Value of Net Asset Position
Net Position Value

Not surprisingly, the distribution o f asset
prices is a crucial factor in determining the ex
act form o f the option pricing equation. In their
derivation, Black and Scholes assumed that the
price o f the underlying asset progressed ran
domly through time according to geom etric
Brownian motion. This is the assumption that
leads to the specific normal probability func
tions in their pricing equation.
Brownian motion was first used to describe
the random progress o f a single molecule
through a gas from a given starting point.10 It is
a mathematical model o f motion that identifies
the w ay the particle can move. Three restric
tions are implied by Brownian motion:
1)The path follow ed must be continuous;11
2)A11 future movements are independent o f
all past m ovem ents;12
3)The change in position betw een time s and
time t is normally distributed w ith mean
equal to zero and a standard deviation
equal to o V (t-s ).
Note that standard deviation is directly p ro
portional to the amount o f time that has passed.
10We are here concerned with only a single dimension of
motion, for example, the East-West coordinate of the
molecule or the price of an asset.
"A lth ou gh this may be true of molecules, it need not be the
case for asset prices, as is considered in the shaded in
sert on Merton’s jump-diffusion model.
12This implies, for example, that the molecule cannot build
up momentum or that prices do not have a predictable

Thus, the longer one waits, the less certain one
is about the location o f the molecule, or, in our
case, the price o f the asset.
Simple Brownian motion is not completely
satisfactory fo r describing asset prices, h ow
ever. W hile a norm ally distributed random vari
able can take on negative values, an asset price
cannot. Th erefore, geom etric Brownian motion,
a variant, is assumed fo r the Black-Scholes
model.13 Under geom etric Brownian motion, the
third restriction is m odified, so that the loga
rithm o f the change in position, rather than the
trend. It does not mean that the future location is indepen
dent of the past location.
13By way of terminology, simple Brownian motion (also
known as arithmetic Brownian motion) and geometric
Brownian motion are examples of the Wiener process (also
known as the Gauss-Wiener process), which, in turn, is a
special case of the Ito process.

JANUARY/FEBRUARY 1990

change in position itself, is norm ally distributed
w ith mean zero and standard deviation o V (t-s ).
This distributional assumption gives us the spe
cific functional form which appears in the BlackScholes equation. Thus, this assumption is im
portant: a different distribution w ould generally
yield a different pricing equation, as illustrated
by M erton’s (1976) jump-diffusion option pricing
model, which is presented in the shaded insert
on the opposite page.

Risk and Hedging in Options
The option pricing equation has the paradox
ical property that, although risk (as measured
by volatility in the asset price) is itself a factor
in the option’s value, the attitudes tow ard risk
o f the holder and the w riter (and anyone else)
are not. The option’s value is a function o f five
variables, none o f which depends on the charac
teristics o f the individuals involved. Black and
Scholes achieved this by showing that the op
tion can be made part o f a com pletely hedged
(that is, riskless) portfolio. Any option w riter
w ho offered a risk discount w hen selling an op
tion w ould find him self selling many option con
tracts to investors who, in turn, could hedge
the risk com pletely and pocket the risk discount
as an arbitrage profit. It is fo r this reason that
the discount rate which appears in the pricing
equation is the risk-free rate o f interest, and at
titudes tow ard risk are irrelevant to the value
o f the option.
T o see how the hedged portfolio works, con
sider the value o f a put option to the holder
before expiration, depicted in figure 6, and the
value o f the underlying asset purchased fo r the
amount A, depicted in figure 7. The value o f
the net asset investment increases one fo r one
as the price o f the asset increases, and the val
ue o f the option decreases, although not in a
constant proportion.
The key to the hedged portfolio is to buy put
options and underlying assets in the appropriate
ratio, so that, w hen the asset price increases,
the increase in value o f the net asset investment
w ill be precisely offset by the decrease in value
o f the option position, and vice-versa. This im
plies a riskless total portfolio. O f course, the ap
propriate ratio (called the "hedge ratio” or "op
tion delta”) also changes as the asset price
changes, because the value o f a put option does
14This connection was first made by Black and Scholes and
first applied to deposit insurance by Merton (1977).

FEDERAL RESERVE BANK OF ST. LOUIS

not decrease as a constant proportion o f the
asset value (the put option’s value is represented
by a curved line). This implies that the holder
o f a com pletely hedged portfolio must con
tinuously adjust the relative proportions o f op
tions to assets if the hedged portfolio is to re
main riskless. Black and Scholes presume that at
least some investors are large and sophisticated
enough to do this.
Because the risk o f an option can be com
pletely diversified, the risk-free rate is the ap
propriate interest rate to use fo r discounting
the option’s uncertain p a yoff at expiration.
Nevertheless, the risk (defined as price volatility)
o f the underlying asset is a factor in the op
tion’s value, because asset risk affects the ex
pected value o f the option's payoff. This is due
to the limited liability nature o f the option.
Although increasing the volatility o f the asset
price increases both the chance o f getting a
very high expiration-day asset price and the
chance o f getting a ve ry low expiration-day
asset price, the bad (high price) outcomes all
have a w eight o f zero in the put option valua
tion, w hile the good (low price) outcomes have
a w eight o f (K - A t ). The volatility o f the op
tion’s value also increases w ith that o f the asset
price, but the volatility o f the option’s value is
irrelevant, because it can be com pletely hedged.

D EPO SIT INSURANCE AN D
O PT IO N S
The analysis o f deposit insurance is a natural,
albeit not obvious, extension o f option pricing
models. The connection betw een the tw o comes
through the limited liability property common
to both options and common stock.14 This p ro
perty implies an "expiration-day” p a yoff fo r
deposit insurance that can be m odeled as an or
dinary put option. Similarly, other claims on a
financial interm ediary’s assets can be modeled
as options or combinations o f options. The
benefit is that, given such a model, option pric
ing theory allows us to assign values to each o f
the claims. These values are the key to option
pricing’s usefulness in this context, because they
allow tw o sorts o f comparisons to be made.
First, variations in the parameters o f the op
tion pricing equation can be considered.15 Such
variations are o f special interest, because, in the
15For example, in Black and Scholes’ model, the five
parameters: K, A, T, o and R would be varied.

Merton's Jum p-Diffusion Model
The Black and Scholes (1973) derivation o f
an option’s value was based, in part, on a
particular assumption about the random be
havior o f the price o f the underlying asset.
Th eir assumption o f geom etric Brownian m o
tion as a description o f the movements o f
asset prices is by no means the only possibili
ty. In general, different assumptions about
the statistical properties o f asset price behav
ior produce different valuation equations fo r
an option on that asset. M erton’s jumpdiffusion model is one o f several alternative
formulations that have been developed.1
Just as arithmetic Brownian motion was not
an apt model o f asset price movements, em
pirical research suggests that geom etric
Brownian motion, at least fo r some assets, is
similarly inappropriate. An alternative, p ro
posed by M erton (1976), is a combination o f
geom etric Brownian motion with random,
discontinuous jumps in the asset price, such
as might occur at the announcement o f some
news event.2 This arrangem ent violates the
first condition fo r Brownian motion and
modifies the third condition again.3 W hen a
jump occurs, the asset price is abruptly
shifted by a random amount; the logarithm
o f this shift is normally distributed, analo
gously to geom etric Brownian motion.
This new process has tw o important im
plications fo r the option pricing model. First,
the option pricing equation is different from
the Black and Scholes formula, since it must
account fo r the new jumps. I f w e relabel the
Black-Scholes value as Pbs = Pbs(K,A,T,o,R),
'Some others are McCulloch’s (1981, 1985) Paretianstable process, and Cox and Ross’s (1976) alternative
jump processes and constant elasticity of variance
(CEV) Ito process.
2Merton’s derivation is only one of several alternatives
to the Black and Scholes model. It was chosen to il
lustrate some of the issues involved in selecting an ap
propriate pricing model, not because it outperforms the
others in some sense. See Rubinstein (1985) for a per
formance comparison of several models.

context o f deposit insurance, some o f the pa
rameters can be controlled or influenced by the
parties to the option contract. Thus, each party
has a clear interest in influencing the parame
ters to his ow n benefit and th erefore to the

then w e can w rite the M erton formula as a
function o f the Black-Scholes value:
oo

Pm = . I J f t - P bl(K ,A,T,0I,R)
i = 0*+ (1 -/?,) * (K *e-RT - A )],
where:

pi - £ ^ 1M U
i!

©i

- V (o2+ hzi/t)

- the Poisson frequency o f jumps

h2 ■ the variance o f the shift distribution.
Not surprisingly, the M erton equation is
m ore complicated than the Black and Scholes
equation. Nevertheless, it is still a function
o f variables that are at least hypothetically
observable.4
Second, the presence o f the jumps com
plicates the diversification problem. The sim
ple hedging portfolio used fo r the BlackScholes model w ill not do, because even con
tinuous rebalancing cannot insure the port
folio’s value at the jump points. If, how ever,
the jumps are idiosyncratic (that is, firmspecific), then their risk can still be elimi
nated by holding a well-diversified portfolio
that includes the assets o f many firms in
many industries. If the jump disturbances are
idiosyncratic, then the risk-free interest rate
is still the appropriate discount rate fo r the
expected option payoff, and individual at
titudes tow ard risk are not a factor in the op
tion’s value before expiration.
3The discontinuous jumps arrive according to a Poisson
process, which conforms to the second condition.
4ln practice, the statistical parameters: o, h and g would
have to be estimated, either from previous observations
or via some other technique. The equation given here
is a special case of a more general formulation given
by Merton (1976). His derivation is of a call option
price, which has been re-arranged here using put-call
parity.

detriment o f the other. The risk-incentive pro
blem presented below exemplifies this sort o f
application. Comparisons based on option pric
ing models indicate not only the direction, but
also the magnitude, o f such incentives.

JANUARY/FEBRUARY 1990

Second, various deposit insurance structures
can be compared. The structure o f deposit in
surance is defined here by the number and
type o f options pertaining to each o f the in
terested parties. Changes in deposit insurance
structure are different from the parameter
changes w ithin an insurance structure, con
sidered in the preceding paragraph. Thus, fo r
example, the FDIC could use option models to
estimate the net increase or decrease in the p re
sent value o f the insurance fund caused by a
switch from one structure to another; or it
could examine the change in risk incentives oc
casioned by the same switch. Th ree different
structures, illustrating some o f the issues involv
ed, are presented in the follow ing examples.16

100 Percent Deposit Insurance
Coverage
T o see how deposit insurance and options are
related, consider the follow ing simplified bank
ing scenario. A single banker both owns and
runs a bank, a single large depositor provides
the entire liability portfolio o f the bank, and a
single insurer, the FDIC, insures deposits and
w ill liquidate the bank in the event o f insolven
cy.17 The liability portfolio consists o f a single
deposit due at year-end. Also at year-end, the
FDIC examines the bank to determine the value
o f assets, which will, in turn, determine w heth
er liquidation occurs. If the bank is econom i
cally insolvent, it is closed by the FDIC, which
liquidates the assets at market values and pays
o ff the depositor in full.18 If the bank is eco
nomically solvent, control remains with the
banker, w h o can either renegotiate the deposit
or liquidate the bank.

16Full coverage is considered first, because it is the simplest
insurance structure possible, and because it approximates
the current system of extensive coverage combined with
the FDIC’s tendency to arrange purchase and assumption
transactions, rather than deposit payouts, for failed banks.
17The assumptions of a single owner-manager for the bank
and a lone depositor are clearly broad abstractions from
reality. The owner-manager assumption allows us to ignore
principal-agent incentives; see Barnea, Haugen and
Senbet (1985). Similarly, the assumption of a single
depositor effectively precludes the ability of depositors to
withdraw their funds individually without forcing an im
mediate closure of the bank. Although these are both im
portant issues, the purpose of the present analysis is to il
lustrate the general principles involved in the application of
option models to deposit insurance, rather than to model a
bank in its full complexity.
18The bank is defined as economically insolvent when the
market value of its assets is less than the present value of
its liability to the depositor. This terminology is meant to

FEDERAL RESERVE BANK OF ST. LOUIS

N ow consider the payoffs to the three in
terested parties — banker, FDIC and depositor
— when the year-end audit is perform ed. These
payoffs are illustrated in figures 8-10. Each par
ty’s year-end payoff is plotted as a function o f
the year-end value o f the bank’s assets. Note
that the sum o f the payoffs to all o f the parties
(obtained by adding the graphs vertically) equals
the value o f the bank's assets. These functions
show how the bank's assets w ill be distributed
after the audit is perform ed. Also note the
shape o f the payoff functions fo r the banker
and the FDIC; in effect, the banker's portfolio
consists o f the bank’s assets, whose value is
uncertain before the audit but known after
ward, the bank’s deposits, w hose value is known
to be L, and a put option w ith striking price L
w ritten by the FDIC.19 The FDIC, on the other
hand, has effectively w ritten the put option on
the assets o f the bank and sold that option to
the banker fo r the price o f the deposit insur
ance premium.20 The depositor has issued the
bank a risk-free loan, which pays o ff the amount
L, including accrued interest.
W ith this in mind, the usefulness o f an option
pricing model to evaluate deposit insurance b e
comes m ore apparent. An option pricing model
provides an estimate o f the actuarial dollar
value o f deposit insurance, as w ell as a tool
w ith which to analyze the economic incentives
that deposit insurance creates. The depositor,
fo r example, has a portfolio, D, that is worth, at
the beginning o f the year, simply the present
value o f the deposit liability discounted at the
riskless rate, L*e_RT; if his year-end payoff, L, is
$1000, and the riskless rate, R, is 8 percent,
then the value o f this portfolio at the beginning

contrast with an illiquidity, or legal insolvency, which is
brought on by an inability to meet maturing short-term
liabilities with liquid assets. The legal profession has a
separate terminology for these two concepts: the “ balance
sheet test” is used to determine economic insolvency, and
the “ equity test” is used to determine legal insolvency.
See Symons and White (1984), pp. 603-16, for an exposi
tion. Since there are no short-term liabilities in the
simplified world here, legal insolvency is not germane.
19The bank’s deposits represent a “ short” position, or bor
rowing, for the banker. The net payoff shown in figure 8
can be gotten by drawing the individual payoff graphs for
the components of the portfolio and adding these together
vertically as before. This portfolio is also equivalent, via
put-call parity, to a simple call option on the assets of the
bank.
“ Compare figure 9 with figure 3. The insurance premium is
considered a sunk cost at the time of the audit and hence
is not included in the graphs.

o f the year is $1000e 08 = $923.12. The FDIC,
on the other hand, has w ritten a put option
with striking price L = $1000; if, fo r example,
the standard deviation o f the bank’s asset re
turns is o = 0.3 percent, and the current value
o f the bank’s assets is $985, then the value o f
the FDIC’s portfolio is given by the BlackScholes equation as - P = -$85.45.

Figures 8-10
Payoff to
Banker

.. ..............

f ----------------------------L

Asset Value

Payoff to
Depositor

0 --------------------- 1-----------------------------L

Asset Value

21Recall that the value of an option to the holder increases
with the volatility of the underlying asset. For the Black
and Scholes model, risk is defined as the standard devia
tion of the logarithm of the asset’s value.
22This is the function of bond rating services, such as
Moody’s and Standard and Poor’s. See Barnea, Haugen,
and Senbet, especially pp. 33-35, for an exposition of the
risk-incentive problem.
23For this reason, risk-taking is restricted by extensive
regulation of commercial bank activities. In practice, the
banker’s incentive may also be mitigated by the potential
loss of a valuable bank charter or by nonpecuniary factors,
for example, the potential loss of a bank manager’s pro
fessional reputation in the event of a failure. These factors
are beyond the scope of the option model.

O f particular interest are the incentives cre
ated by deposit insurance. Under 100 percent
insurance, the depositor does not care about the
value o f the bank's assets, since he receives his
deposit back with interest, regardless o f the
bank’s condition. The banker, how ever, receives
the positive equity capital, if the bank is solvent;
if the bank is insolvent, the loss is charged to
the FDIC. This "heads I win, tails you lose" ar
rangement is certainly not peculiar to banks; it
applies to any corporate entity w ith limited
stockholder liability. In the absence o f other in
centives, the banker will make the corporation
as risky as possible.21
W hat is peculiar to banks under 100 percent
flat-rate deposit insurance is the absence o f
such other incentives fo r the depositor and the
banker to limit risk. Normally, creditors impose
a risk premium on corporations, based on the
riskiness o f the firm ’s assets.22 By definition,
flat-rate insurance implies that the FDIC charges
no risk premium. Similarly, the depositors
charge no risk premium, because they are fully
insured. The result, in our simplified model, is
that the banker has an unmitigated incentive to
increase the riskiness o f the bank's assets, while
the FDIC has the inverse incentive to reduce the
bank’s risk-taking.23 The risk incentive implied
by extensive, flat-rate deposit insurance is the
impetus fo r most o f the current proposals fo r
deposit insurance reform .24 In analyzing both
the current system and proposed reforms, many
authors have used option pricing models.25
24Reform proposals include: risk-based insurance premia,
risk-based capital requirements, larger capital re
quirements, reduced insurance coverage, depositor coinsurance, subordinated debt requirements, increased
supervision and more stringent asset regulation. See
White (1989) for a survey of current proposals.
25See, for example, Merton (1977, 1978), McCulloch (1981,
1985), Sinkey and Miles (1982), Pyle (1983, 1984, 1986),
Brumbaugh and Hemel (1984), Marcus (1984), Marcus and
Shaked (1984), Ronn and Verma (1986, 1987), Thomson
(1987), Furlong and Keeley (1987), Pennacchi (1987a,
1987b), Osterberg and Thomson (1988), Flannery (1989a,
1989b), and Allen and Saunders (1990).

JANUARY/FEBRUARY 1990

A P P L IC A T IO N S TO OTHER
D EPO SIT INSURANCE
ARRANGEM ENTS
W e can now extend the options model to
other arrangements fo r deposit insurance, to
evaluate their relative impacts. T w o illustrative
cases will be presented: a coverage ceiling clause
and a deductible clause. A significant character
istic o f both these cases is that they impose a
portion o f the bank's asset risk on the depos
itor. The FDIC benefits directly from such pro
visions, because they shift some potential losses
directly to the depositors. In addition, imposi
tion o f a possible loss on the depositor mitigates
the risk-incentive problem that exists under 100
percent, flat-rate deposit insurance. In general,
the depositor w ill m onitor the bank m ore close
ly and w ill require a higher interest rate to
compensate fo r the possibility o f default.

Maximum Insurance Coverage
Limit
Although 100 percent coverage is often treated
as the status quo de fa cto o f federal deposit in
surance, coverage extends legally only to the
first $100,000 per depositor per institution.26 A
maximum coverage limit is a form o f co-insur
ance, a technique used by insurers to reduce
the moral hazard problem (the tendency o f in
surance to alter the behavior o f the insured).
Other basic forms o f co-insurance are the deduct
ible and fixed proportional sharing o f losses.27

Under a deposit payout closure, the FDIC
itself takes all o f the bank’s assets and liabilities
into receivership. It then sells the assets and
pays the depositors up to the maximum cover
age limit plus any excess o f asset sales over in
surance claims, distributed on a pro rata basis.
As a result, this method is best modelled by the
deductible considered below. The upshot is that
the payoffs under the FDIC’s maximum cover
age limit do not conform to the familiar (from,
say, automobile or health insurance) maximum
coverage arrangement illustrated here.
Consider a maximum coverage limit o f M
dollars fo r the depositor (w here M < L), illus
trated in figures 11— 13. Under this arrange
ment, the depositor receives the full deposit
amount, L, in the event o f any insolvency or
shortfall, up to the amount, M, o f the coverage
limit. Thus, the depositor’s portfolio contains the
deposit amount, L, and he has w ritten a put op
tion on the bank's assets with striking price (L
- M). This put option is the result o f the max
imum coverage limit. The FDIC holds the put
option w ith striking price (L - M), but has w rit
ten a second put option on the bank’s assets
with striking price L. As before, this put option
(with striking price L) is held by the banker,
w ho also owns the assets and ow es the amount
L to the depositor.

The applicability o f the maximum coverage
limit considered here is complicated by the
FDIC’s current closure protocol. Bank closures
by the FDIC can take one o f tw o forms: pur
chase and assumption or deposit payout. Under
a purchase and assumption closure, healthy
assets and all deposits are transferred to an
other healthy bank, with the FDIC absorbing
the problem assets and any net loss.28 This sort
o f transaction is best modelled by the 100 per
cent coverage considered above.

Because o f the put option w ritten by the
depositor and held by the FDIC, the depositor
now shares in the risk o f the bank’s assets. His
deposit is now w orth less, and he w ill discount
the promised p a yoff m ore steeply. Extending
the example given above fo r the case o f 100
percent coverage, the depositor’s portfolio, which
contained only the riskless deposit, worth $932.12
w hen discounted, is now augmented by the put
option w ritten w ith a striking price o f (L - M).
If, fo r example, the coverage limit is set at M =
$100, so that the striking price is $900, then
with o = 0.3 percent, R = 8 percent and A =
$985 as before, the Black-Scholes value o f this
put option to the depositor is - P = -$47.96,
and the total value o f his portfolio is D =

26The original limit was $2500 under the Banking Act of
1933; see FDIC (1984), pp. 44, 69. The impact of the
coverage ceiling is limited by the availability of brokered
deposits and by the tendency of the insurer to arrange
purchase and assumption solutions to bank failures.

in the United States was to have sharing in staggered pro
portions [see FDIC (1984), p. 44]. Analyses of co-insurance
tend to focus on proportional sharing arrangements. See
Boyd and Rolnick (1989), and Benston and Kaufman
(1988), ch. 3.

27There are other possibilities. For example, deposit in
surance in the United Kingdom involves fixed proportional
sharing combined with a coverage ceiling [see Llewellyn
(1986), p. 20], and a temporary deposit insurance program

FEDERAL RESERVE BANK OF ST. LOUIS

28Defining a "healthy” asset is a difficult chore. The task is
generally accomplished by individual evaluation of assets,
rather than the application of a generic rule.

Figures 11-13

the risk premium that the depositor w ould
charge under this risk-sharing arrangement. On
the one hand, w e can think o f the deposit as a
riskless deposit combined w ith a put option. On
the other hand, w e can think o f it as a single
risky promise o f repaym ent from the banker, to
be discounted at some risk-adjusted interest
rate, r, so that the value o f the deposit is:
D = L*e_rT. W e can equate these tw o inter
pretations thus:
D = L - e 'RT - P(L - M,A,T,o,R)
= L-e_rT,
w h ere P (») is the value o f the put option before
expiration, as defined, fo r example, by the
Black-Scholes model, and r is the risk-adjusted
discount rate implied by the presence o f the
coverage limit. Given the other variables, w e
can rearrange this to find the risk premium:
(2)

(r - R) = - - i ln [ e ~ RT -

P(*) ] - R.

Applying this to the numerical example, the
stated risk-adjusted interest yield on the deposit is:
r = - ln fe" 08(,) - J ? l 796 1
L
$ 1 0 00 .0 0 J
= - ln(0.87515) = 13.3%,
implying a risk premium, r - R, o f 5.3 percent.
In practical terms, such an estimate o f the
magnitude o f the risk premium implied by a
given coverage ceiling might be useful in calibra
ting the degree o f market discipline in a reform
o f the insurance system. If bank risk-taking is to
be curbed by limiting deposit insurance, forcing
riskier banks to pay higher risk premia as some
have suggested, then the insurance limitations
must be such that the risk premium implied by
equation 2 is large enough to make bankers alter
their behavior.29
$932.12 - $47.96 = $884.16. In other words,
given deposit insurance w ith a $100 coverage
limit, $884.16 is the amount deposited at the
beginning o f the year in exchange fo r a p rom
ised year-end payoff o f $1000.
In general, w e can use an option pricing
m odel to solve algebraically fo r a measure o f

29For some examples of market discipline proposals, see
Boyd and Rolnick (1989), Gilbert (1990), Gorton and Santomero (1988, 1989), or Thomson (1987).

The magnitude o f the risk premium might also
serve as a readily observable vital sign, registering
the financial health o f the bank’s assets, and
aiding the regulator in scheduling audits. This
presumes that depositors have some advantage
over regulators in assessing the bank’s risk bet
w een audits.30 Such applications, how-

the amount lent to a bank based on that bank’s risk, in
addition to pricing that risk,

30Note that uninsured or co-insured depositors might find it
more practical to engage in capital rationing, i. e., limiting

JANUARY/FEBRUARY 1990

ever, are subject to some limitations which are
illustrated by the next example.

Figures 14-16

Deductible
Another form o f co-insurance is a deductible.
The case o f a deductible on insurance coverage
introduces a twist to the problem. Now the
depositor’s portfolio effectively consists o f tw o
put options, one w ritten and one held, in addi
tion to the promised repayment o f the deposit
with interest. This case is o f special interest,
because it applies to a deposit payout closure as
considered above and because it can also be ap
plied to subordinated debt, which is the object
o f a recent debate on sources o f market disci
pline o f bank risk-taking. In both cases, the
payoffs to one o f the bank's creditors can be
modeled as a pair o f put options with different
striking prices.31
In this example, the depositor is promised the
return o f his deposit amount, w ith accrued in
terest, fo r a total o f L dollars. Because o f the
deductible provision, how ever, this promised
repayment is not certain; in the event o f the
bank’s insolvency, the depositor w ill be the first
to share in the shortfall. For year-end asset
levels below L, the shortfall is deducted from the
depositor's payoff until the deductible amount, U,
is exhausted. Any shortfall beyond that is ab
sorbed by the FDIC. Thus, the depositor effe c
tively holds the deposit amount L and has w rit
ten a put option w ith striking price L, that is
held by the banker; in addition, he holds a put
option w ith striking price (L - U), which is
w ritten by the FDIC. These payoffs are illus
trated in figures 14-16.
The deductible provides a cushion fo r the
FDIC, which, in the preceding examples, had
written a put option w ith striking price L
rather than (L - U). The year-end payoffs fo r
the banker are the same as before. The real dif
ference applies to the depositor's incentives and
the resulting impact on the price he charges the
banker fo r the deposit. Although he always
prefers a higher asset value, as before, his at
titude tow ard the riskiness o f the bank’s assets
is now ambiguous, because he is long one put
option and short another, with tw o different
striking prices. Volatility in the bank’s asset

31The relevant creditors in each case are the depositor and
the subordinated debt-holder, respectively. For an analysis
of option models in the case of subordinated debt, see
Black and Cox (1976) and Gorton and Santomero (1988,
1989). For an analysis of subordinated debt and bank

FEDERAL RESERVE BANK OF ST. LOUIS

Payoff to
Depositor

L-U

Asset Value

returns increases the value o f the long position
and decreases the value o f the short position.
As Black and Cox (1976) point out, the net im
pact o f these countervailing forces w ill depend
on the current asset value relative to the strik
ing prices. Specifically, there is an inflection

regulation, see Gilbert (1990). The approach here is at
odds with that of Ronn and Verma (1986), who calculate
the value for a single put on total debt and then scale that
value down by the proportion of insured to total liabilities.

point equal to the discounted geom etric mean
o f the tw o striking prices.32 For an asset value
above the inflection point, w hich includes all
cases in which the bank is solvent (i. e., A >
L ’e _RT), the effect o f the short position out
weighs that o f the long position, and the de
positor w ill p re fer less risk. Conversely, w hen
the current market value o f assets falls below
the inflection point, the long position outweighs
the short, and the depositor w ould p refer a
riskier asset portfolio, given the low asset value.
Thus, a decrease in the "risk prem ium ” charged
by the depositor no longer necessarily implies
that the bank's assets are less risky; fo r exam
ple, such a decrease could instead be the result
o f an increase in the current asset value and an
increase in the volatility o f those assets.
Under such circumstances, it is a reasonable
taxonomic question w hether the interest rate
markup over the riskless rate should be called a
risk premium at all. The current value o f the
depositor’s claim and the implicit risk premium
can be calculated as before:

value o f the deposit resembles m ore and m ore
the staggered year-end p a yoff function o f figure
16. In fact, if the year-end p a y o ff o f figure 16 is
scaled dow n by the risk-free present value dis
count factor, e -RT, it becomes identical to the
extreme case o f figure 17 w here o = 0. Figure
17 also illustrates graphically the Black and Cox
argument that fo r some (higher) asset levels, de
positors will charge a risk premium, while for
other (lower) levels, they will offer a risk discount.
It is also clear from the picture, however, that
the asset level has a much more significant effect
on the value o f the claim than does the volatili
ty.34 All o f this suggests that bankers, depositors
and policymakers should give considerable care to
an appropriate definition o f risk in this context,
and that similar care should be given to designing
a practical measure o f that risk. Risk defined as
volatility in bank asset returns and measured by
the risk premium charged on equity, subor
dinated debt or uninsured deposits may not be
apt for the tasks to which it has been applied.

D = L-e“ RT- P(L,A,T,o,R) + P (L-U ,A ,T ,o,R )
= L-e_rT

SOME CAVEATS

(r - R) =
- -1 lnj^e“ RT- -1 [P (L ,«) - P(L - U ,«)] j

- R,

but the risk premium so defined is a measure
o f the expected difference betw een cash pro
mised, L, and cash ultimately received. It is a
poor measure o f the volatility o f the returns on
the bank’s assets, because the expected dif
ference betw een cash promised and cash re
ceived depends on several factors and is no
longer a simple direct relation o f the volatility
o f assets.
Figure 17 graphs the value o f the depositor’s
claim fo r a range o f asset levels and volatili
ties.33 In interpreting this graph, note the con
nection between it and figure 16. In particular,
as the volatility, o, goes to zero in figure 17, the
32The discount rate used to calculate this inflection point in
cludes a risk premium. Since all other discounting in this
context is at the riskless rate, this means that the inflec
tion point can fall below the present value of the lower dis
count rate, (L - u)-e~RT, if the risk premium, o2/2, is large
enough. For the same reason, the inflection point is
always smaller than the solvency point, L-e
.
33The riskless rate was set at 8 percent, time to maturity
was one year, promised repayment was $1000, and the
deductible amount was $200. Similar graphs with other
maturities and deductible amounts reveal no surprises.

The preceding analysis has illustrated some uses
o f option pricing models in evaluating deposit in
surance. There are some limitations, however, on
the use o f options models in this context. Most of
these limitations derive from the assumptions that
form the basis for the option pricing equation,
and the extent to which these assumptions are
valid for the case at hand.
Perhaps the most basic problem is the ques
tion, w ho truly holds the option.35 Until now, it
has been presumed, based on the end-of-period
payoffs, that deposit insurance represents a put
option w ritten by the FDIC and held by the
banker. In fact, how ever, the FDIC decides
w hether a bank is insolvent, and, m ore im por
tantly, w hether to close a bank that is already
insolvent (or one that is not quite insolvent).36 In
the face o f a large-scale bank failure or run,
34This fact is noted by Pyle (1983), p. 13.
35This issue is addressed by Brumbaugh and Hemel (1984)
and Allen and Saunders (1990).
36Recall that the definition of insolvency used in this paper
ignores the possibility that the bank might be deemed in
solvent on the basis of its current ratio — i. e., its inability
to meet maturing liabilities with liquid assets. This problem
relates to the maturity structure of the bank’s portfolio,
which will be considered briefly below.

JANUARY/FEBRUARY 1990

Figure 17
Value of Depositor’s Claim
z

RT =923.12

(L-U ) e

07 =738.49

1200
923.12
738.49

short-term political considerations may over
whelm any prior prescriptions on closure poli
cy. Th e FSLIC’s actions in the thrift crisis in
dicate that this is not idle speculation; numerous
thrifts w ere left open long after their insolvency
had been discovered. Conversely, as Benston
and Kaufman (1988) suggest, the FDIC could
close solvent banks, if they have come close
enough to insolvency. If the insurer w ere to
follow scrupulously a well-defined rule one w ay
or the other, there w ould be little at issue, since
the striking price and year-end payoffs could
then be easily adjusted within the context o f the
current model. As matters stand, how ever,
bankers and depositors effectively face a ran
dom striking price, because the insurer decides
if the option w ill be exercised. As a practical
37A rule is well-defined if it leaves no doubt about the circumstances which imply closure, with no room for FDIC
discretion. Note that defining a closure rule, in general, is

FEDERAL RESERVE BANK OF ST. LOUIS

matter, it is difficult to envision how such a
well-defined closure rule might be im
plem ented.37
A related issue is the measurement o f bank
asset values. The option pricing m odel p re
sented above presumes that the current asset
price can be readily observed. For stock op
tions, this is an uncontentious assumption,
because stock prices can be observed on the
floor o f the exchange or in the over-the-counter
market. For an option on the assets o f the bank,
how ever, the relevant price is not readily obser
vable. Indeed, one o f the prim ary functions o f
bank credit analysis is to assign values to assets
fo r which there is no active market. Similarly,
not sufficient for our purposes; the closure rule must be
defined so that the values of the resultant claims conform
to the values given by the option pricing equation.

the insurer must invest significant effort, in the
form o f an audit, to determine the year-end
asset value.

that this choice is a salient factor in the option’s
value, because it determines the probability o f
each o f the possible year-end payoffs.

The inherent inaccessibility o f bank asset values
has tw o implications fo r option models. First, it
is no longer possible fo r an option holder to
construct the appropriately hedged portfolio
described in the Black and Scholes derivation,
because the hedge ratio depends on the value o f
the underlying asset. This casts doubt upon the
appropriateness o f the riskless rate in discoun
ting the expected end-of-period payoffs.38 Se
cond, the current asset value is important, be
cause it partly determines the probabilities fo r
the various end-of-period payoffs. Ignorance o f
the current asset value adds another layer o f
uncertainty, and this additional uncertainty
significantly affects the value o f the option.39
A closely related issue is the measurement o f
asset risk. The option pricing models presented
here use the variance o f the asset’s returns as a
measure o f risk.40 Producing an accurate assess
ment o f the variance is problematic, even fo r
stock options, because the volatility that matters
is the variance o f the process over the fu tu re
life o f the option. For bank assets, the measure
ment problem is compounded, because even
past values are generally unavailable. Pyle (1983)
and Flannery (1989b) consider some o f the im
plications o f this problem in using the option
models to price deposit insurance.

The empirical evidence to date testifies to the
sensitivity o f the results to the specification
employed. Marcus and Shaked (1984) use the
basic Black-Scholes model, adjusted fo r divi
dends, and find that federal deposit insurance
is currently substantially overpriced relative to
the "actuarially fair” estimates provided by their
option model.41 They note, how ever, that
"McCulloch’s (1981, 1983) estimates o f insurance
values derived from the Paretian-stable distribu
tion greatly exceed” their ow n.42 Pennacchi
(1987b) uses a m ore complicated model which
includes the degree o f regulatory control w ield
ed by the insurer. He finds that deposit insur
ance may be either overpriced or underpriced,
depending on the level o f regulation assumed.
McCulloch’s (1985) study assumes non-normal,
Paretian-stable asset returns and non-stationary
random interest rates. He finds that insurance
values are highly sensitive to the level and
volatility o f interest rates. Ronn and Verma
(1986), how ever, using a variant o f M erton’s
(1977) model, conclude that neither random in
terest rates nor non-stationary equity returns
significantly affect the insurance valuations. In
brief, the empirical evidence suggests that a
w ide range o f insurance valuations can be
reached by varying the returns process em
ployed in the model.

Just as asset values and the volatility o f re
turns are not observable directly, there is the
m ore general moot question o f which stochastic
returns-generating process should be incor
porated in the option pricing model. As w e ’ve
seen above, the difference in the assumed re
turns process betw een Black and Scholes’s
model and M erton’s model resulted in a
substantially different pricing equation.
Although the choice o f an appropriate returns
process fo r modeling a bank’s assets is beyond
the scope o f this paper, it is sufficient to note

Finally, it has been assumed in the preceding
examples that there is a single deposit that does
not mature until the end o f the year. In fact, o f
course, banks maintain many deposit accounts
w ith a w ide range o f maturities, starting with
the instant maturity o f demand deposits. This is
significant, because it gives many depositors
another type o f insurance. A depositor w ho can
w ithdraw his funds from a failing bank sooner
than the FDIC can close it has 100 percent in
surance, regardless o f the balance in the ac
count or the insurance scheme in effect.43 The

38One might resort to the argument that the riskless rate is
appropriate if the asset has no idiosyncratic (that is, firmspecific) risk, as noted in the context of Merton’s (1976)
pricing model (see the shaded insert), but such an
assumption is not particularly credible for bank asset port
folios, prima facie.
39See Pyle (1983) for an analysis of asset value uncertainty.
Figlewski (1989) and Babbel (1989) consider the im
possibility of hedging, along with many other difficulties in
the application of stock option pricing models.

41For an exposition of the dividend adjustment to the BlackScholes model, see Merton (1973).
42Marcus and Shaked (1984), p. 449. Their reference to
McCulloch (1983) was a working paper, later published as
McCulloch (1985).
43lt need only be the case that the bank funds the deposit
outflow somehow, for example, through a fire sale of its
assets. Such a run on Continental Illinois by institutional
depositors prompted the FDIC to extend 100 percent in
surance to uninsured depositors.

40Merton’s (1976) approach uses that variance together with
the parameters of the jump distributions.

JANUARY/FEBRUARY 1990

result is that the simple application o f an option
pricing model does not provide an accurate
evaluation o f such deposits, because it ignores
certain relevant strategies.

CONCLUSIONS
The usefulness o f option models in the study
o f deposit insurance results from tw o important
characteristics. First, these models distill the
host o f economic factors involved down to a
handful o f relevant parameters whose interac
tion is well-defined by the option-pricing equa
tion. Second, they are able to evaluate deposit
insurance claims under a w ide variety o f in
surance structures. Although only three such
structures w ere elaborated here, they can be
generalized to other applications. Thus, option
models provide a unified context fo r analyzing
incentives w ithin an insurance structure, as w ell
as fo r comparing alternative insurance schemes.
Unfortunately, option pricing models, like
most economic models, are an im perfect tool
when directly applied to the complexities o f the
real w orld. Beyond certain fundamental qualita
tive results, there are theoretical and empirical
reasons to believe that the insurance valuations
given by any particular option pricing model
w ill be incorrect, highly sensitive to changes in
their specification, or both. As a result, the ab
solute dollar magnitudes provided by options
models o f the value o f deposit insurance are
suspect. The contradictory empirical evidence
on fair pricing is indicative o f this problem.
In defense o f these models, how ever, there is
no reason to believe that option models are any
w orse in this regard than any alternative
economic model. Indeed, there is some reason
to believe that, although the absolute magnitude
o f the valuations provided by option models
may be unstable, the rankings they provide fo r
a sample o f banks are not.44 Similarly, inac
curacies in determining the scale o f insurance
values do not deny the ability o f option models
to identify the direction o f incentives or the im
pact o f marginal changes in the structure o f
deposit insurance. Th erefore, used judiciously,
option pricing models can be an effective ana
lytical tool in the study o f deposit insurance.

"T h e rank-order correlation of risk measures for a sample
of banks over time [Marcus and Shaked, (1984), p. 455]
and over alternative risk metrics [Ronn and Verma, (1987),

FEDERAL RESERVE BANK OF ST. LOUIS

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Boyd, John H., and Arthur J. Rolnick. “A Case for Reforming
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Cox, John C., and Stephen A. Ross. “ The Valuation of Op
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Cox, John C., and Mark Rubinstein. Options Markets
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a Changing Environment, (FDIC, Washington, D.C., 1983).
_______ . Federal Deposit Insurance Corporation: The First
Fifty Years, (FDIC, Washington, D.C., 1984).
_______ . 1988 Annual Report, (FDIC, Washington, D.C.,
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Us About Option Prices?” Financial Analysts Journal
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Flannery, Mark J. “ Capital Regulation and Insured Banks’
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_______ . “ Pricing Deposit Insurance When the Insurer
Measures Bank Risk with Error,” Working Paper, School of
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Furlong, Frederick T., and Michael C. Keeley. “ Does Capital
Regulation Affect Bank Risk-taking,” Working Paper No.
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Gilbert, R. Alton. “ Market Discipline of Bank Risk: Theory
and Evidence,” this Review (January/February 1990), pp. 3-18.

p. 511] indicate that the stability of the rankings is signifi
cant but imperfect. Further research along these lines
would be welcome.

Gorton, Gary, and Anthony M. Santomero. “ The Market’s
Evaluation of Bank Risk: A Methodological Approach,” in:
Proceedings of a Conference on Bank Structure and Com
petition, (Federal Reserve Bank of Chicago, May 1988), pp.
202-18.
_______ . “ Market Discipline and Bank Subordinated Debt,”
Working Paper, Wharton School, University of Penn
sylvania, August 1989.
Huertas, Thomas F., and Rachel Strauber. “An Analysis of
Alternative Proposals for Deposit Insurance Reform,” Ap
pendix E to Hans H. Angermueller’s testimony in: Structure
and Regulation of Financial Firms and Holding Companies
(Part 3): Hearings before a Subcommittee of the Committee
on Government Operations, U.S. House of Representatives,
99th Cong., 2 Sess., December 17-18, 1986 (GPO, 1987),
pp. 390-463.
LLewellyn, D. T. The Regulation and Supervision of Financial
Institutions, Gilbart Lectures on Banking, (Institute of
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McCulloch, J. Huston. “ Interest Rate Risk and Capital Ade
quacy for Traditional Banks and Financial Intermediaries,”
in S. J. Maisel, ed., Risk and Capital Adequacy in Commer
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pp. 223-48.
_______ . “ Interest-risk Sensitive Deposit Insurance Premia:
Stable ACH estimates,” Journal of Banking and Finance
(March 1985), pp. 137-56.
Malliaris, A. G. “ Ito’s Calculus in Financial Decision Mak
ing,” SIAM Review (October 1983), pp. 481-96.
Marcus, Alan J. “ Deregulation and Bank Financial Policy,”
Journal of Banking and Finance (December 1984), pp.
557-65.
Marcus, Alan J., and Israel Shaked. "The Valuation of FDIC
Deposit Insurance Using Option-pricing Estimates,” Journal
of Money, Credit and Banking (November 1984), pp. 446-60.
Merton, Robert C. “ Theory of Rational Option Pricing,” Bell
Journal of Economics and Management Science (Spring,
1973), pp. 141-83.

Osterberg, William P., and James B. Thomson. “ Capital Re
quirements and Optimal Bank Portfolios: A Reexamina
tion,” Working Paper No. 8806, Federal Reserve Bank of
Cleveland, 1988.
Pennacchi, George G. “Alternative Forms of Deposit In
surance,” Journal of Banking and Finance (June 1987a), pp.
291-312.
_______ .“A Reexamination of the Over- (or Under-) Pricing
of Deposit Insurance,” Journal of Money, Credit and Bank
ing (August 1987b), pp. 340-60.
Pyle, David H. “ Pricing Deposit Insurance: The Effects of
Mismeasurement,” Working Paper No. 83-05, Federal
Reserve Bank of San Francisco, October 1983.
_______ . “ Deregulation and Deposit Insurance Reform,”
Economic Review, Federal Reserve Bank of San Francisco
(Spring 1984), pp. 5-15.
_______ . “ Capital Regulation and Deposit Insurance,” Jour
nal of Banking and Finance (June 1986), pp. 189-201.
Ronn, Ehud I., and Avinash K. Verma. “ Pricing RiskAdjusted Deposit Insurance: An Option-Based Model,”
Journal of Finance (September 1986), pp. 871-95.
_ _ _ _ _ . “A Multi-Attribute Comparative Evaluation of
Relative Risk for a Sample of Banks,” Journal of Banking
and Finance (September 1987), pp. 499-523.
Rubinstein, Mark. “ Nonparametric Tests of Alternative Option
Pricing Models Using All Reported Trades and Quotes on
the 30 Most Active CBOE Option Classes from August 23,
1976 Through August 31, 1978,” Journal of Finance (June
1985), pp. 455-80.
Sinkey, Joseph F., Jr., and James A. Miles. “ The Use of
Warrants in the Bail Out of First Pennsylvania Bank: An
Application of Option Pricing,” Financial Management
(Autumn 1982), pp. 27-32.
Stoll, Hans R. “ The Relationship Between Put and Call Op
tion Prices,” Journal of Finance (December 1969), pp.
802-24.
Symons, Edward L., Jr., and James J. White. Banking Law
(West Publishing, 1984).

_______ . “ Option Pricing When Underlying Stock Returns
Are Discontinuous,” Journal of Financial Economics
(January/March 1976), pp. 125-44.

Thomson, James B. “ The Use of Market Information in Pric
ing Deposit Insurance,” Journal of Money, Credit and Bank
ing (November 1987), pp. 528-37.

_______ . “An Analytic Derivation of the Cost of Deposit In
surance and Loan Guarantees: An Application of Modern
Option Pricing Theory,” Journal of Banking and Finance
(June 1977), pp. 3-11.

U.S. Treasury Department, The Working Group of the
Cabinet Council on Economic Affairs. “ Recommendations
for Change in the Federal Deposit Insurance System”
(GPO, 1985).

_______ . “ On the Cost of Deposit Insurance When There
Are Surveillance Costs,” Journal of Business (July 1978),
pp. 439-52.

White, Lawrence J. “ The Reform of Federal Deposit In
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11-29.

JANUARY/FEBRUARY 1990

Cletus C. Coughlin and Kees Koedijk
Cletus C. Coughlin is a research officer at the Federal Reserve
Bank of St. Louis. Kees Koedijk is a professor of economics at
Erasmus University, Rotterdam. Thomas A. Pollmann provided
research assistance.

What Do We K now About the
Long-Run Real Exchange
Rate?

REAL EXCHANGE rate is defined as the
foreign currency price o f a unit o f domestic
currency (that is, the nominal exchange rate)
multiplied by the ratio o f the domestic to the
foreign price level. The real exchange rate has
been at the center o f economic policy discus
sions in the 1980s fo r at least tw o reasons.
First, this relative price has been m ore variable
in the floating-rate period than in the preceding
era o f fixed (nominal) exchange rates.1 Second,
this price is related to international trade pat
terns because the competitive position o f an in
dividual exporting (import-competing) firm in a

'See Frankel and Meese (1987) and Dornbusch (1989) for
surveys of this literature. As noted by Dornbusch, the in
creased variability and lack of knowledge have contributed
to divergent policy recommendations, which include a
return to some form of a managed exchange-rate system,
taxes on foreign exchange transactions as well as doing
nothing.
2The U.S. dollar has been at the center of the controversy,
with the dollar allegedly being undervalued in the late
1970s/early 1980s and overvalued in the mid-1980s. Dur
ing the period of undervaluation, U.S. tradeable goods in
dustries were stimulated and induced to overexpand. The
costs of this alleged overexpansion were exacerbated by
the subsequent overvaluation, which resulted in layoffs,
plant closings and bankruptcies in these same industries.
3This elementary principle is ignored when the exchange
rate in macroeconomic settings is treated as an ex

FEDERAL RESERVE BANK OF ST. LOUIS

country is affected adversely by an appreciating
(depreciating) real exchange rate.2
Despite much research, h ow ever, there is no
consensus on w hich variables cause changes in
the real exchange rate. Like any asset price,
real exchange rates are related to the determ i
nants o f the relevant supply and demand curves
now and in the future.3 W ith real exchange
rates, the relevant determinants are those affec
ting the relative supplies and demands fo r the
currencies o f tw o countries. Claims have been
made, how ever, that the real exchange rates

ogenous rather than endogenous variable. For example, a
standard assertion is that a depreciating dollar boosts U.S.
manufacturing output. A declining dollar is expected to
raise the dollar prices of U.S. imports and lower the
foreign currency prices of U.S. exports. Consequently,
consumption and production of U.S. exports and importcompeting goods would rise. This analysis is faulty
because changes in the value of the dollar are not in
dependent of U.S. industrial developments and, in fact,
can be the direct result of industrial developments. For ex
ample, an economic policy that boosts productive capacity
can generate a positive relationship between the value of
the dollar and U.S. manufacturing output. Details on this
argument can be found in Tatom (1988).

often d iffer substantially from levels consistent
w ith the underlying econom ic fundamentals and
that these differences persist fo r long periods.
A prim ary goal o f our research is to provide
an elem entary understanding o f the major theo
retical approaches to the determination o f longrun real exchange rates. These approaches iden
tify numerous variables that have been tested
fo r their relationships to the changing values o f
the real exchange rate. Empirically, w e examine
the six bilateral real exchange rates among the
United States, W est Germany, Japan and the
United Kingdom.4 Using a data set covering ap
proxim ately the same time period, w e make a
straightforw ard comparison o f the three p ri
mary approaches and present a clear picture o f
what can be said about the determinants o f real
exchange rates.
Research to explain movements in the longrun real exchange rate is unnecessary if pur
chasing p ow er parity (PPP) holds in the long
run. Thus, w e begin by review in g the literature
on PPP in the long run. This provides a natural
starting point from which to examine the dif
ferent theoretical approaches to real exchange
rate determination and the major empirical find
ings. Next, w e undertake unit root and cointe
gration tests to examine w hether long-run rela
tionships exist betw een the real exchange rate
and some o f its potential determinants.

IS THE R EAL EXCHANGE RATE A
R A N D O M W ALK?
As a point o f departure, it is useful to define
the real exchange as it is used throughout this
paper. A standard representation expresses all
variables in logarithms, so that a real exchange

4Our selection of countries is based on research by Koedijk
and Schotman (1989), which indicates that the movements
of real exchange rates for 15 industrial countries can be
partitioned into four groups led by the United States, West
Germany, Japan and the United Kingdom.
5Wholesale price indexes are also frequently used in the
calculation of real exchange rates. The use of wholesale
rather than consumer prices can generate different results.
For an example, see McNown and Wallace (1989). For a
brief discussion of why a broad-based measure of prices
such as consumer prices is more appropriate than one of
wholesale prices in calculating real exchange rates, see
Cox (1987).

rate, q, is defined as follows:
(1) q = e + p - p * ,
w h ere e is the foreign currency price o f a unit
o f domestic currency, p is the domestic price
level as measured by the consumer price index
and p* is the similarly measured foreign price
level.5
Since the advent o f flexible exchange rates in
1973, real exchange rates have been m ore vari
able than they w ere previously. This point is il
lustrated in figure 1 over 1957 to 1988 fo r the
pound/dollar, mark/dollar and yen/dollar real ex
change rates. The increased variability has in
duced many researchers to focus on the fun
damental relationships that determine real ex
change rates.
The concept o f purchasing p ow er parity has
been one o f the most important building blocks
fo r nominal, as w ell as real, exchange rate
modeling during the 1970s and 1980s. In its
absolute version, PPP states that the equilibrium
value o f the nominal exchange rate betw een
the currencies o f tw o countries w ill equal the
ratio o f the countries’ price levels.6 Thus, a
deviation o f the nominal exchange rate from
PPP has been view ed as a measure o f a curren
cy’s over/undervaluation. In its relative version,
PPP states that the equilibrium value o f the
nominal exchange rate w ill change according to
the relative change o f the countries’ price levels.
A notew orthy implication o f both versions o f
PPP is that the real exchange rate w ill remain
constant over time.
Economists have debated w hether PPP applies
in the short run, long run or neither. By the
end o f the 1970s, PPP, at least in the short run,
was rejected convincingly by the data.7 W hether
PPP in the long run can be rejected is less clear.
A standard theoretical argument in support o f

rate (domestic currency value per unit of foreign currency),
P is an index of domestic prices, P* is an index of foreign
prices and K is a scalar. In this view, the PPP hypothesis
is a homogeneity postulate of monetary theory rather than
an arbitrage condition. Thus, a monetary disturbance
causes an equiproportionate change in money, commodity
prices and the price of foreign exchange, while relative
prices are unchanged. The influence of real factors on the
relationship between exchange rates and national price
levels is captured by K, which is a function of structural
factors that can alter the relative prices of goods.
7See Adler and Lehmann (1983) for the references underly
ing this consensus.

6More generally, PPP has been stated in Edison and
Klovland (1987) as E = K(P/P*), where E is the exchange

JANUARY/FEBRUARY 1990

Figure 1
Real Bilateral Exchange Rates in the Fixed and Floating
Rate Periods

1957

FEDERAL RESERVE BANK OF ST. LOUIS

1989

PPP is that deviations from parity, assuming
zero transportation costs and no trade barriers,
indicate profitable opportunities fo r commodity
arbitrage. Deviations from PPP imply that the
same good, after adjusting fo r the exchange
rate, w ill sell at different prices in tw o loca
tions. Simultaneously buying the good in the
low-price country and selling the good in the
high-price country w ill force the nominal ex
change rate to PPP and the real exchange rate
to some constant value.
Th e chief issue is w hether the real exchange
rate returns over time to a fixed value, the
long-run equilibrium real exchange rate. On the
other hand, it is possible that the equilibrium
value o f the real exchange rate is not constant
over time, but instead changes in response to
changes in some fundamental economic vari
ables. For example, an increase in a country’s
real interest rate, ceteris paribus, could cause an
appreciation o f the country’s real exchange rate.
One conclusion, how ever, is clear: if the real
exchange rate follow s a random walk, long-run
PPP does not hold. A variable is said to follow a
random walk if its value in the next period
equals its value in the current period plus a
random error that cannot be forecast using
available information. I f the real exchange rate
follows a random walk, then it w ill not return
to some average value associated with PPP over
time. In fact, its deviation from the PPP value
becomes unbounded in the long run.
The unit root test is a common procedure to
use in determining w hether a variable follow s a
random walk.8 I f the existence o f a unit root

eThe issue is whether the real exchange rate is stationary.
If the real exchange rate is stationary, then random distur
bances have no permanent effects on this rate. If the real
exchange rate is nonstationary, then there is no tendency
for this rate to return to an “ average” value over time. To
determine whether the real exchange rate is stationary, a
standard procedure is to use the Dickey-Fuller test for unit
roots. This procedure is described later in the text.
9Examples include Cumby and Obstfeld (1984) and Frankel
(1986). Using monthly data between September 1975 and
May 1981, Cumby and Obstfeld rejected the random-walk
hypothesis for the real exchange rate between the United
States and Canada. On the other hand, they were unable
to reject the random-walk hypothesis for the real exchange
rate when the United States was paired with each of the
following countries— United Kingdom, West Germany,
Switzerland and Japan. Frankel (1986) rejected the
random-walk hypothesis for the real U.S. dollar/British
pound exchange rate using annual data between 1869 and
1984, but was not able to reject the hypothesis using data
for 1945-1984.

cannot be rejected, then the variable is said to
follow a random walk. Using data from various
developed countries, recent studies by Darby
(1983), A dler and Lehmann (1983), Huizinga
(1987), Baillie and Selover (1987) and Taylor
(1988) could not reject the unit root hypothesis
fo r the real exchange rate in the current float
ing rate period and, hence, rejected the notion
o f long-run PPP.
The issue, nevertheless, remains controversial.
One reason is that some researchers have found
evidence to reject the random-walk hypothesis
in some cases.9 In addition, doubts about the
p o w er o f standard tests to discriminate betw een
true random walks and near random walks
have been raised. For example, Hakkio (1986)
demonstrated that, w hen the real exchange rate
differs modestly from a random walk, the re
sults o f standard tests are biased in favor o f the
random-walk hypothesis. In other words, there
is a high probability o f failing to reject the ran
dom-walk hypothesis even if it is false.10
Another possibility is that the current floatingrate period is too b rie f to assess accurately the
validity o f PPP. Lothian (1989), using unit root
tests and annual data fo r over 100 years for
Japan, the United States, the United Kingdom
and France, found that real exchange rates
tended to return to their long-run equilibrium
values, but that the period o f adjustment was
quite long. For example, adjustment periods
ranging from three to five years w ere found.
Consequently, the current floating-rate period
might not be long enough to identify the longrun tendency o f the real exchange rate to re
turn to an equilibrium.

10The preceding problem motivated Sims (1988) to develop
a new test for discriminating between true and near ran
dom walks. Applying this new test, Whitt (1989) was able
to reject the hypothesis that the real exchange rate was a
random walk. A forthcoming issue of the Journal of
Econometrics, however, concludes that the ap
propriateness of Bayesian approaches in detecting unit
roots remains in doubt because many questions, some re
quiring highly technical responses, have not been
answered. Consequently, we did not use this technique in
our analysis.

JANUARY/FEBRUARY 1990

APPR O A C H E S T O R EAL EX
CHANGE R ATE D ETER M IN ATIO N
IN THE LONG RUN
Our goal is not to resolve the preceding con
troversy about PPP in the long run. Rather, it is
to examine, as w ell as extend empirically, the
research efforts o f those w ho have provided
models that allow fo r the long-run real ex
change rate to vary over time. In other words,
our goal is to examine the attempts b y resear
chers skeptical about PPP in the long run to ex
plain movements in the long-run real exchange
rate. T w o real approaches and a m onetary ap
proach to exchange-rate determination have
been used to explain movements in the equilib
rium real exchange rate. The first real approach
is concerned w ith movements in the real ex
change rate that arise from incorporating the
difference betw een tradeable and non-tradeable
goods prices. Th e other real approach deals
w ith the implications o f incorporating a balance
o f payments constraint. The m onetary ap
proach, in contrast, focuses on the relationship
betw een real exchange rates and real interest
rates.

Tradeables and Non-Tradeables
Absolute PPP implies that the equilibrium
value o f the nominal exchange rate betw een the
currencies o f tw o countries w ould equal the
ratio o f the countries’ price levels, w hich is
commonly measured by the respective consum
er price indexes. Ignoring transportation costs,
free international trade eliminates the price dif
ference betw een the same good in tw o coun
tries; how ever, price differences across coun
tries fo r non-traded goods may persist and may
change substantially over time. Frequently, this
possibility is referred to as PPP holding only fo r
internationally traded goods; how ever, one
could view this possibility as a substantial m odi
fication o f PPP. T o prevent confusion, w e do
not call this PPP, but rather characterize it as
the law o f one price fo r traded goods.
A simple model illustrating this approach is
presented below. Let p be the logarithm (log) o f

"T h e se indexes suggest that the price level is constructed
as follows: P = P‘ ‘ ‘ *| p *T, where the upper-case Ps repre
sent levels. Price indexes are not really constructed this
way; however, following Hsieh (1982), this construction was
chosen to simplify the derivation. Hsieh has argued that
his empirical results were not distorted by this assumed
construction because he used highly aggregated data.

FED ERAL RESERVE BANK OF ST. LOUIS

the overall price level, and pT and pNT be the
logs o f the price levels o f traded and non-traded
goods; an asterisk denotes the foreign country.
The overall price level is related to the prices o f
tradeable and non-tradeable goods by
(2) p = (1 - a)pT + apNT
and
(3) P * = ( i -

p)P;

/?P * T,

w h ere a and ft denote the shares o f the nontradeable goods sectors in the economies.11
Assuming the law o f one price fo r tradeable
goods,
(4) e + pT - p* = 0,
w h ere e is the log o f the nominal exchange
rate, measured as the foreign currency price o f
a unit o f domestic currency.12
By substituting equations 2, 3 and 4 into equa
tion 1, the real exchange rate, q, can be w ritten
as
(5) q = - a ( p T - pNT) + P (p ; - p *T).
Thus, the real exchange rate depends on rela
tive prices betw een tradeable and non-tradeable
goods as w ell as the sizes o f the non-tradeable
goods sectors in the tw o countries. Our focus is
restricted to the possibility that persistent d if
ferences betw een the price changes o f tradeable
and non-tradeable goods across the tw o
economies can cause real exchange rate
movements.
T w o main proxies, one using relative prices,
the other using output measures, have been us
ed to measure the tradeables/non-tradeables
distinction. As W o lff (1987) has noted, a stan
dard empirical proxy in analyzing relative prices
in a w orld with internationally traded and nontraded goods is the ratio o f wholesale prices to
consumer prices. The reasoning is straightfor
ward. Wholesale price indexes generally pertain
to baskets o f goods that contain larger shares o f
traded goods than consumer price indexes do.
Consumer price indexes tend to contain relative
ly larger shares o f non-traded consumer ser
vices. T o date, how ever, empirical evidence on

12As a check, using wholesale prices for the prices of traded
goods, we found that e + pT- p* was not stationary. Thus,
one of the building blocks for this approach does not hold
for our data. In addition, even if one were to define the
real exchange rate using wholesale rather than consumer
prices, PPP would not appear to hold in the long run.

the importance o f relative prices in explaining
real exchange rate movements is lacking.

The Real Approach Using the
Balance-of-Paym ents Equilibrium

The other proxy fo r the tradeables/nontradeables distinction was highlighted by Balassa
(1964). Balassa assumed that the law o f one
price held fo r traded goods, that wages in the
tradeable goods sector are linked to productivity
and that wages across industries are equal.
These assumptions cause the price o f non-tradeable goods relative to tradeable goods to in
crease m ore over time in a country w ith high
productivity grow th in the tradeable goods sec
tor than in a country with low productivity
growth. Such a productivity differential, in con
junction with a general price index that covers
both traded and non-traded goods, w ill result in
a real exchange rate appreciation fo r fastgrowing countries even with the prices o f traded
goods equalized across countries.

An alternative real approach to analyze m ove
ments in the real exchange rate is to include a
balance-of-payments constraint.13 This approach
focuses on the theoretical relationship between
changes in the equilibrium real exchange rate
and changes in the current account. The longrun equilibrium real exchange rate is the rate
that equilibrates the current account in the long
run. Recall that balance-of-payments accounting
ensures that the current account is identical to
the negative o f the capital account, which is
simply the rate o f change o f net foreign hold
ings. Thus, the current account equilibrium in
the long run is determined by the rate at which
foreign and domestic residents wish to change
their net foreign asset positions in the long run.

For the empirical application o f the productivi
ty approach, Balassa suggested that there should
be a positive link betw een the real exchange
rate and real per capita gross national product,
which assumes that inter-country productivity
differences are reflected in per capita income
levels. The effect o f shifts in sectoral productivi
ty have been investigated by Hsieh (1982) and
Edison and Klovland (1987). Hsieh found that
real exchange-rate changes fo r W est Germany
and Japan could be explained by differences in
the relative grow th rates o f labor productivity
betw een traded and non-traded sectors fo r these
countries and their major trading partners.
Similarly, Edison and Klovland, using annual
data, found a long-run equilibrium relation be
tw een the pound/Norwegian krone real ex
change rate and the real output differential and
betw een the real exchange rate and the com
modity/service productivity ratio differential.
The results o f Edison and Klovland raise a
number o f interesting questions because the
data cover a period that is both long, 1874-1971,
and does not encompass the current floatingrate period. Consequently, one is left w ondering
w hether 15 years o f data, which require the
use o f data m ore frequent than annual observa
tions, is sufficient to reach strong conclusions
about the current period and w hether Edison
and Klovland’s results w ould be altered by data
from the current period.

Any fundamental econom ic factor that in
fluences the current account affects the real ex
change rate. Consequently, the long-run equilib
rium real exchange rate depends on real fac
tors—whose changes can either be anticipated
or unanticipated—that cause shifts in the de
mand fo r and supply o f domestic and foreign
goods. The most notable example is the relative
output differential. Relatively faster output
grow th domestically w ill induce an appreciation
o f the long-run equilibrium real exchange rate.
A key aspect o f this approach focuses on the
possibility that unanticipated changes in the cur
rent account affect the long-run real exchange
rate. Unexpected changes in the current ac
count are assumed to reflect changes in under
lying determinants that, in turn, require offset
ting changes in the real exchange rate to ensure
current account equilibrium in the long run. A
long-run balance-of-payments constraint sug
gests that any revisions in expectations about
the long-run values o f variables that affect the
balance o f payments affect the expected value
o f the long-run real exchange rate. As Isard
(1983) notes, the substantial changes in the rela
tive price o f oil during the 1970s are excellent ex
amples o f how unexpected changes in a determi
nant o f the current account caused revised expec
tations about the long-run real exchange rate.
An illustration highlighting the importance o f
unanticipated current account changes is pre-

13Examples may be found in Isard (1983) and Frenkel and
Mussa (1985).

JANUARY/FEBRUARY 1990

sented by Dornbusch and Fischer (1980). In
their model, a current account surplus causes a
rise in wealth through the net in flow o f foreign
assets. Assuming the rise in wealth is unantici
pated, excess demand in the domestic goods
market occurs. In turn, an increase in the real
exchange rate is required fo r the new goods
market equilibrium. This increase induces the
necessary shift from domestic to foreign goods
by domestic and foreign consumers to eliminate
the excess demand.
H ooper and M orton (1982) use this fram ew ork
to relate changes in the real exchange rate to
economic fundamentals. They use the cumu
lated current account as a determinant o f the
long-run equilibrium real exchange rate. In their
model, unanticipated changes in the current ac
count are assumed to provide inform ation about
shifts in the underlying determinants that ne
cessitate offsetting shifts in the real exchange
rate to maintain current account equilibrium in
the long run. Consistent with this balance-ofpayments approach, their results indicate that,
betw een 1973 and 1978, movements in the cur
rent account have been a significant determ i
nant o f movements in the real exchange rate
fo r the U.S. dollar, predominantly through
changes in expectations.

The M onetary Approach
As m entioned previously, the m onetary ap
proach focuses on the relationship betw een
real exchange rates and real interest rates. A
straightforw ard exposition o f this approach,
which can be found in Meese and R ogoff (1988),
is based on models developed by Dornbusch
(1976), Frankel (1979) and Hooper and M orton
(1982). These models are "sticky-price” versions
o f the m onetary m odel o f exchange rates; they
assume that prices o f all goods adjust slowly in
response to disturbances. Thus, tem porary devi
ations in the real exchange rate from its longrun equilibrium value (that is, purchasing
p ow er parity) are possible.
These tem porary deviations necessitate an
exchange-rate adjustment mechanism to restore
the long-run equilibrium value. A standard as

14For example, a comparison of 0= .6 with 0 = .4 after two
periods reveals that, in the former case, the expected dif
ference between the actual and long-run equilibrium is .36
of the difference in the current period, while in the latter
case the expected difference is .16 of the difference in the
current period.

FEDERAL RESERVE BANK OF ST. LOUIS

sumption is that the deviations are eliminated at
a constant rate. The adjustment process can be
represented as follows:
(6) E,(qt+k - q t+k) = 0k(q, - qt), O < 0 < 1 ,
w h ere E is an expectations operator, the sub
scripts designate the time period, q is the loga
rithm o f the real exchange rate, the bars in
dicate values that w ould prevail if all prices
w ere fully flexible instantaneously and 0 is the
speed-of-adjustment parameter. Consequently,
there is a monotonic adjustment o f the real ex
change rate to the long-run equilibrium, q „ over
time with low er values o f 0 indicating a quicker
adjustment process.14
Th e long-run equilibrium value changes with
random real shocks; how ever, assuming all real
shocks fo llow random-walk processes, these
shocks do not affect the expected long-run equi
librium exchange rate. Consequently,
(7) Etqt+k = qt.
Substituting equation 7 into equation 6 yields
(8) q, = d(E,qt+k - qt) + q„
w h ere 6 = l/(0k - 1) < - 1 . The observed real
exchange rate is its tem porary deviation from
its long-run equilibrium value plus its long-run
equilibrium value.
T o complete the model, uncovered interest
parity is assumed.15 This assumption is express
ed as follows:
(9) Etet+k - e, = kr* - kr„
w h ere e is the logarithm o f the nominal ex
change rate (foreign currency per domestic cur
rency unit), kr t is the k-period nominal interest
rate at time t and the asterisk denotes a foreign
value. In other words, changes in the nominal
exchange rate are directly related to nominal in
terest rate differentials. As domestic nominal in
terest rates rise relative to foreign rates, the
nominal exchange rate o f the domestic country
is expected to depreciate.
Equation 9 implies that the expected change
in the real exchange rate reflects the expected
real interest rate differential. In symbols,

15The appropriateness of this assumption can be ques
tioned. The uncovered interest parity assumption requires
that the forward rate be an unbiased and efficient predic
tor of the future spot rate; however, the empirical results
summarized by Baillie and McMahon (1989) suggest
otherwise.

(10) E,(qt+k - q.) = kRt* - kR„
w here the k-period interest rate, kR„ is the dif
ference betw een the nominal interest rate less
the expected rate o f change in prices. Substi
tuting equation 10 into equation 8 yields
(11) q, = d(kR* - kRt) + qt.
Th erefore, the essence o f the monetary ap
proach is that changes in the real exchange rate
are directly related to changes in the real in
terest differential. As expected real domestic in
terest rates rise relative to foreign rates; the
real exchange rate o f the domestic country rises
as well.
Equation 11 provided the foundation fo r vari
ous statistical tests by Meese and R ogoff (1988).
As noted in the appendix, the measurement o f
expected real interest rates is problematic. W hile
the sign o f the relationship betw een the long
term real interest rate differential and the real
exchange rate was consistent with theory, the
relationship was not statistically significant.

A N EM PIR ICAL AN ALY SIS OF
REAL EXCHANGE RATES AN D
PO T E N T IA L DETERM INANTS
Our empirical analysis proceeds in tw o steps.
First, using unit root tests, w e test fo r the stationarity o f the six real exchange rates that
result from pairwise combinations o f the fo r
eign exchange rates o f the United States, Japan,
W est Germany and the United Kingdom. The
stationarity o f five potential determinants fo r
these exchange rates is examined as well. D e
tails on the construction o f these variables are
presented in the appendix. Unless noted other
wise, w e used monthly data from June 1973 to
June 1988 fo r all variables. Thus, in the first
step, w e provide additional evidence on the ex
istence o f PPP in the long run. Second, w e test
fo r cointegration betw een the real exchange
rates and each o f the potential determinants.
The goal is to identify which variables, if any,
from the models that w e have review ed explain
variations in the real exchange rate over time.

Testing f o r Unit R oots
Potential Determinants
In summary, the existing literature points to
five potential determinants o f long-run real ex
change rates that w e use. The real approach
identifies three possibilities, tw o based on the
tradeables/non-tradeables distinction and one
based on the balance-of-payments equilibrium.
The proxies to measure the tradeables/nontradeables distinction have used the ratio o f
wholesale to consumer prices and real per cap
ita gross national product differences, w hile the
cumulated current account difference is used
fo r the balance o f payments. The other major
approach, the m onetary approach, highlights
the role o f interest rate differentials. Both short
term and long-term interest rate differentials
across countries have been used.

16See Trehan (1988) for a basic introduction to the intuition
underlying unit roots and cointegration, as well as a practical illustration.

W e used the test developed by Dickey and
Fuller (1979) fo r testing fo r unit roots.16 In the
present case, the test consists o f regressing the
first difference o f the variable under considera
tion on its ow n lagged level, a constant and, to
control fo r autocorrelation, an appropriate num
ber o f lagged first differences.17 The coefficient
estimate on the lagged level is crucial, because
the null hypothesis o f a unit root implies that it
is zero. The test-statistic is simply the estimate
o f the coefficient divided by its standard error.
This test-statistic, which does not have the usual
t-distribution, is then compared w ith critical
values tabulated in Fuller (1976).
The results listed in table 1 show that w e can
not reject the null hypothesis o f a unit root fo r
any o f the bilateral real exchange rates.18 The

percent. Given the nature of the cointegration tests, this
caveat applies to these results as well,

17lf lagged first differences are needed, then the test is an
“ augmented” Dickey-Fuller test; otherwise, the test is
simply a Dickey-Fuller test. The chosen lag length is the
smallest lag length for which there is no autocorrelation.
18An important caveat concerning the interpretation of unit
root tests is the extremely low power of these tests. Given
a sample size of approximately 100 observations, the prob
ability of accepting a coefficient of 1.0 on the lagged
dependent variable when it is actually 0.95 is roughly 80

JANUARY/FEBRUARY 1990

Table 1
Unit Root Tests For Real Exchange Rates and Potential
Determinants
Countries

PW /PC -PW '/PC*

GNP-GNP*

TB-TB*

RS-RS*

RL-RL*

UK/US

1.63

0.31

2.20

1.86

2.96’

2.77

WG/US

1.27

0.13

1.51

2.72

3.151

1.52

JP/US

0.88

1.44

-0 .4 7

0.10

3.66'

1.48

UK/WG

1.56

1.15

2.48

2.08

2.04

2.03

JP/WG

0.84

0.43

0.70

2.56

2.921

2.68

UK/JP

1.27

0.68

0.32

1.91

4.40'

2.73

1 Statistically significant at the 0.05 level.
NOTE: The test-statistic reported is minus the regression t-statistic on a in a regression of the follow
ing general form:
n

Ax, * c - ax_, + 1/3, to ., + £,.
i« 1
The lag length n is chosen as the smallest value for which no autocorrelation exists. For a
sample of 100 observations, the critical value for a significance level of 5 percent is 2.89.

real exchange rate measures are nonstationary
since there is no tendency fo r the real exchange
rates to return to an average value over time.
Thus, consistent w ith studies cited previously,
w e reject long-run PPP.
Table 1 also contains the results o f unit root
tests fo r the potential determinants o f real ex
change rates. Both proxies used to measure the
tradeables/non-tradeables distinction, the dif
ference betw een countries o f their ratios o f
wholesale to consumer prices (PW/PC-PW*/PC*)
and real per capita gross national products
(GNP-GNP*), are nonstationary. An identical con
clusion is reached fo r the cumulated current ac
count difference (TB-TB*), the proxy based on
the balance-of-payments approach. In every
case, w e cannot reject the null hypothesis o f a
unit root.
Th e results fo r the tw o proxies based on the
m onetary approach are mixed. The difference
betw een the short-term interest rates (RS-RS*)
appears to be stationary in most cases. In only
one case, United Kingdom/West Germany, the
null hypothesis o f a unit root is accepted. On
the other hand, the difference betw een the
long-term interest rates (RL-RL*) appears to be
nonstationary because the null hypothesis o f a
unit root is accepted in each case.

FEDERAL RESERVE BANK OF ST. LOUIS

Testing f o r Cointegration
Even though the real exchange rate has a unit
root, it is possible that there is a long-run rela
tionship betw een it and other variables that also
contain unit roots. For an equilibrium relation
ship to exist betw een these variables, the distur
bances that cause nonstationary behavior in one
variable must also cause nonstationary behavior
in the other variable.
To test w hether there is a long-run relation
ship betw een variables that contain unit roots,
the residuals from an ordinary-least-squares
regression betw een the variables can be exam
ined fo r stationarity. In other w ords, a DickeyFuller test is perform ed on the residuals resul
ting from regressing one variable—the real ex
change rate in our case—on a potential determ i
nant. Th e first difference o f the residual series
is regressed on its lagged level, a constant and
an appropriate num ber o f lagged first d iffe r
ences. A hypothesis test is perform ed using the
coefficient estimate on the lagged level. I f the
null hypothesis o f a coefficient o f zero can be
rejected, then the residuals are stationary. I f the
residuals are stationary, then the variables w ill
not drift away from each other. Such variables
are said to be cointegrated. Since cointegration

Table 2
Cointegration Tests of Real Exchange Rates and Potential
Determinants
Countries

PW /PC-PW '/PC*

GNP-GNP*

TB-TB*

RS-RS*

RL-RL*

UK/US

1.62

1.64

1.57

1.83

1.71

WG/US

1.49

1.53

1.64

2.45

3.12'

JP/US

1.90

1.07

1.21

1.23

UK/WG

2.76

2.11

2.29

2.11

2.37

JP/WG

2.63

3.501

2.06

1.39

1.33

UK/JP

2.41

1.49

1.32

1.51

1.80

1 Statistically significant at the 0.05 level.

tests are oriented tow ard rejecting any long-run
relationship, finding cointegration suggests the
existence o f a significant statistical link betw een
the variables.
Table 2 shows the results o f such cointegra
tion tests. Overall, there is little evidence to in
dicate that the examined variables explain varia
tions in the real exchange rate over time. Real
exchange rates and the differences betw een
ratios o f wholesale to consumer prices across
countries do not appear to be cointegrated.
Despite augmented Dickey-Fuller statistics that
approach the critical value in tw o cases, United
Kingdom/West Germany and West Germany/Japan,
the residuals are nonstationary in each case.
The real exchange rate and the difference be
tw een the real per capita gross national pro
ducts are cointegrated only fo r W est Germany
and Japan. The cointegration tests also reveal
that the residuals are nonstationary fo r both the
cumulated trade balance and short-term interest
rate differences.
One interesting result is the significant rela
tionship betw een the real exchange rate and the
long-term real interest differential fo r the
United States and W est Germany. This result

contrasts w ith Campbell and Clarida (1987) and
Meese and R o go ff (1988) w ho fail to find cointe
gration betw een these variables. Since our p ro
xy fo r the long-term real interest differential is
exactly the same as that used b y Meese and
Rogoff, the additional 18 m ore recent months o f
data seem to account fo r the different result.19
In figure 2, w e have plotted the long-term real
interest differential betw een the United States
and Germany and the real mark/dollar exchange
rate. As is apparent from the figure, there is a
strong link betw een these variables: a higher
long-term real interest rate in the United States
relative to Germany means a stronger dollar.

SUM M ARY
Unfortunately, as our review o f a host o f
studies reveals, little is known about the deter
minants o f real exchange rates in the long run.
Our systematic survey o f five potential explana
tory variables suggests that no approach to this
issue is satisfactory. For example, fo r certain ex
change rates, the real approach based on the
tradeables/non-tradeables distinction yields
evidence o f an equilibrium relationship between

19The sample used by Meese and Rogoff (1988), as stated
in their table V, terminated in December 1985, while our
sample using long-term interest differentials between West
Germany and the United States terminated in June 1987.
When we delete these 18 months of observations, the
regression t-statistic is 2.77 rather than the 3.12 reported
in our table 2. Like Meese and Rogoff, we would fail to
find cointegration.

JANUARY/FEBRUARY 1990

Figure 2
Cointegration of Real Mark/Dollar Exchange Rate
and Real Long Interest Rate Differential

1973

the real exchange rate and differences in real
per capita gross national product across coun
tries. On the other hand, the m onetary ap
proach seems to have some value in the W est
German/United States case. An equilibrium rela
tionship appears to exist betw een the real ex
change rate and the difference in long-term in
terest rates. In view o f the low p ow er o f unit
root tests, this finding is especially noteworthy.
Th e fact remains that our knowledge o f the
determination o f real exchange rates is meager,
at best. The logical question is to ask w h ere
research might be directed to expand our
knowledge in this area. Numerous explanations,
in addition to measurement problems, can be
offered fo r the fact that fundamental variables
have not yielded good explanations o f real ex
change rate movements. One possibility is that
the recent period o f floating exchange rates is

FEDERAL RESERVE BANK OF ST. LOUIS

too brief, especially in view o f our statistical
tools, to draw conclusions about the long-run
behavior o f real exchange rates. Assuming no
change in exchange-rate regime, the passage o f
time w ill ultimately rectify this problem. Until
sufficient time passes, how ever, it w ould be p re
m ature to discard the theoretical approaches
that have been proposed.
A second possibility is that the existing models
are deficient. Our review identified instances in
which the assumptions underlying the models did
not hold. In addition, since real exchange rates are
asset prices, the role o f expectations is an aspect
o f the modeling process that deserves additional
scrutiny. The failure o f existing models might re
sult from the fact that expectations are formed
differently than our models suggest. Consequently,
the development o f alternative expectations for
mation mechanisms might prove productive.

A final possibility is that random shocks of
various origins, such as oil price shocks, have
moved the real exchange rate. While identification
o f the real factors that might have affected ex
change rates is difficult, Meese and Rogoff (1988)
suggest that further attention should be focused
on the role o f real shocks. Thus, models utilizing
modern real business-cycle research might
generate some insights. All o f these “mights” serve
as a final rem inder o f how little w e know.

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JANUARY/FEBRUARY 1990

Appendix
Data Sources and the Construction of Variables
For all variables except interest rates, w e use
monthly data from International Financial Statis
tics fo r June 1973 to June 1988. Recall that q is
the log o f the real exchange rate with the con
sumer price index o f the respective countries as
the deflator. GNP is the log o f real gross na
tional product, w h ere nominal gross national
product is deflated w ith the gross national p ro
duct price deflator. Since gross national product
data are not available monthly, w e interpolated
from quarterly to monthly observations using
industrial production. Note that all the potential
determinants are defined as differences, w ith an
asterisk indicating the respective value in the
other country. PW/PC is the log o f the w h ole
sale price index divided by the consumer price
index. TB is the cumulated trade balance (that
is, the sum from January 1972 to time period t
o f the difference betw een exports and imports)
divided by gross national product.
For the interest rates not involving Japan, the
time period is June 1973 to June 1987, while
fo r those involving Japan, the time period is
July 1977 to June 1987. The nominal short-term
interest rates and their sources are as follows:
Japan—one-month Gensaki rate from the Bank
o f Japan; United Kingdom —one-month interbank
deposit rate from the Financial Times', W est
Germany—one-month interbank rate from the
Frankfurter Allgemeine Zeitung; and United
States—the yield on one-month Treasury bills
until A pril 1984 and, afterward, the interest

FEDERAL RESERVE BANK OF ST. LOUIS

rate on three-month Treasury bills from the
Federal Reserve Board.
The nominal long-term interest rates and their
sources are as follows: Japan—average yield to
maturity on governm ent bonds w ith constant
remaining maturity o f nine years from the Bank
o f Japan; United Kingdom—average yield to
maturity on governm ent bonds w ith remaining
maturity betw een eight and 12 years from the
Financial Times; W est Germany—average yield
to maturity on governm ent bonds w ith remain
ing m aturity over eight years from the Frank
fu rte r Allgemeine Zeitung; and United States—
yield to m aturity o f governm ent bonds w ith re
maining maturity o f 10 years from the Federal
Reserve Board. W e calculated the real short
term and long-term interest rates, RS and RL, in
the same manner as Meese and R o go ff (1988).
Thus, actual inflation rates based on the preced
ing 12 months as measured by the consumer
price index are subtracted from the nominal in
terest rates to generate the real rates. As a re
sult, the inflation measure does not correspond
to the term o f the interest rates. A problem
w ith this construction, as w ell as many others,
is that negative real interest rates are computed.
For example, the long-term (short-term) real in
terest rate is negative fo r 25 (42) percent o f the
U.S. observations, 38 (30) percent o f the British
observations, 0 (9) percent o f the W est German
observations and 6 (7) percent o f the Japanese
observations.

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St. Louis, Missouri 63166

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