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Changes in the Size
Distribution of U.S. Banks:
1960–2005
Hubert P. Janicki and Edward Simpson Prescott
I
n 1960, there were nearly 13,000 independent banks. By 2005, the number
had dropped in half, to about 6,500. In 1960, the ten largest banks held
21 percent of the banking industry’s assets. By 2005, this share had
grown to almost 60 percent. A great deal of these changes started during the
deregulation of the 1980s and 1990s. (Figures 1 and 2 report the time paths
for these two measures.)
By any measure, these numbers represent a dramatic change in the bank
size distribution over the 1960–2005 period. This article documents the
extent of this change. It also documents the change in bank size dynamics,
that is, the entry and exit of banks and the movement of banks through the
size distribution. During this period, new banks formed, many more exited,
either because of failure or merger, and many others changed in their size.
For example, of the ten largest banks in 1960, only three were still among the
top ten largest in 2005.1
We document these facts because they are an important step in developing
a theory of bank size distribution. Although we do not provide one, such
a theory would be valuable because it could be used to answer important
questions such as: How costly were the pre-1980 limits on bank size? How
Hubert P. Janicki is a graduate student in economics at Arizona State University. The views
expressed in this article are those of the authors and not necessarily those of the Federal
Reserve Bank of Richmond or the Federal Reserve System. The authors would like to thank
Huberto Ennis, Yash Mehra, Andrea Waddle, and John Weinberg for helpful comments.
1 The ten largest banks in 1960 were Bank of America, Chase Manhattan Bank, First National
City Bank of New York, Chemical Bank New York Trust Company, Morgan Guaranty Trust Company, Manufacturer’s Hanover Trust Company, Bank of California, Security First National Bank,
Banker’s Trust Company, and First National Bank of Chicago. At the end of 2005, only three of
these banks still existed: Bank of America, Citigroup (formerly National City Bank of New York),
and JP Morgan Chase (an amalgamation of many of the banks listed above).
Federal Reserve Bank of Richmond Economic Quarterly Volume 92/4 Fall 2006
291
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Federal Reserve Bank of Richmond Economic Quarterly
Figure 1 Total Number of Independent Banks
13,000
12,000
Number of Banks
11,000
10,000
9,000
8,000
7,000
1960
1965
1970
1975
1980
1985
Year
1990
1995
2000
2005
Notes: All banks and bank holding companies that are under a higher level holding company are treated as a single independent bank. Section 3 gives a more precise definition
of an independent bank.
will the bank size distribution continue to evolve? And will there be more
concentration? If so, should policy do anything about it?
Our analysis of the size distribution emphasizes fitting the data to the
lognormal and Pareto distributions. These distributions are utilized because
they are commonly used to describe skewed distributions and frequently have
been used to describe firm size distribution. As we will demonstrate, the
lognormal poorly fits the upper right tail of the size distribution. The Pareto
distribution fits better with this part of the distribution, but the quality of the
fit is much better before deregulation.
We examine bank size dynamics along several dimensions. First, we
determine whether the data satisfies Gibrat’s Law, that is, whether growth
is independent of firm size. We find that Gibrat’s Law is a good description of the data during the 1960s and 1970s, before deregulation, but not a
good description afterward. After the 1970s, large banks grew faster than
small banks, though more so in the 1980s and 1990s than they did during the
H. P. Janicki and E. S. Prescott: Bank Size Distribution
293
Figure 2 Market Shares of Ten Largest Banks
Assets
Loans
Deposits
Employees
0.55
0.50
Fractions of Total
0.45
0.40
0.35
0.30
0.25
0.20
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
Year
Notes: The definition of a bank is given in Section 3.
starts in 1969.
Data on number of employees
2000–2005 period. Second, we document that entry into banking was remarkably stable over the entire period. Entry is cyclical but averages about
1.5 percent of total operating banks. Finally, we calculate transition matrices,
that is, the probability a bank will move from one size category to another,
over each of the decades. Following Adelman (1958) and Simon and Bonini
(1958), we use these transition probabilities and the entry data from 2000–2005
to forecast continued changes in the size distribution. The forecast predicts
a continued decline in the total number of banks, but at a much slower rate
than in the 1980s and 1990s, followed by a leveling off in the decline. It also
predicts that there will still be a large number of small banks as well as a sizable number of mid-size banks. If the present trends continue, the transition
in banking that began in the 1980s is slowing down and coming to an end.
1.
LITERATURE
In many industries, the distribution of firm size is highly skewed to the right,
that is, there are many small firms and a few large ones. One distribution that
has this characteristic and is frequently used to describe firm size distribution
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is the lognormal distribution. A random variable is lognormally distributed if
the logarithm of the random variable is normally distributed.2
Early studies of firm size distribution, namely Gibrat (1931), found that
the lognormal distribution fit the empirical data fairly well. Gibrat (1931)
also found evidence that firm growth was independent of firm size. This
latter finding, often called Gibrat’s Law or the Law of Proportionate Effect,
was important because a statistical process that satisfies it would generate a
lognormal distribution in the long run.
Later studies have found mixed support for Gibrat’s findings. In particular, studies in the 1980s found that the proportional rate of growth of a firm
conditional on survival decreases in size. Sutton (1997) is a good survey of
these results.
Another category of distributions used to fit the size distribution is based
on the power law. This category takes the form
f (x) = cx −α ,
where x > 0 and c > 0. In economics, the Pareto distribution is a power law
distribution often used to describe highly skewed data.3 It is similar to the
lognormal, but with a thicker right tail.
Power law distributions have been used in the sciences to fit data in a wide
variety of applications. Newman (2005) surveys various applications of the
power law, including studies of word frequency, magnitude of earthquakes,
diameter of moon craters, intensity of solar flares, and population of cities.
One property observed in many applications is that the size of the r-th largest
observation is inversely proportional to its rank.4 It is observed so frequently
that it is called Zipf’s Law. Like the lognormal distribution, Zipf’s Law can be
generated with appealing assumptions on the dynamics. For example, Simon
and Bonini (1958) study entry dynamics by assuming a constant probability
of entry and show that the distribution follows a power law in the upper tail.
See Gabaix (1999) for a detailed study of Zipf’s Law.
2 The probability density function of a lognormal distribution is
f (x) =
1 1 −(log(x)−μ)2 /(2σ 2 )
,
e
√
σ 2π x
where x > 0, μ and σ are the mean and standard deviation, respectively, of the natural log of x.
3 The probability density function of a Pareto distribution is
k x −(k+1)
f (x) = kxmin
where x > xmin , xmin > 0 and k > 0. The Pareto distribution is usually expressed as the
k x −k , which is
probability that random variable X is greater than x, that is, P rob(X ≥ x) = xmin
also a power law.
4 Formally, x = cr −α , where observation x has rank r, c is a constant, and α is close to
r
r
1. We solve for r, normalize the equation by dividing by N (where the N th ranked observation
is xmin ) and simplify to obtain the counter cumulative function: r/N = 1 − F (x) = (xmin /x)1/α .
This is the Pareto distribution with coefficient k = 1/α.
H. P. Janicki and E. S. Prescott: Bank Size Distribution
295
Whether Zipf’s Law fits U.S. firm size data is a matter of some debate.
Using 1997 U.S. census data, Axtell (2001) finds that it fits the firm size
distribution. Using a different data set, however, Rossi-Hansberg and Wright
(2006) find that it does not fit so well. They also find that establishment growth
and exit rates decline with size.
The banking literature has long been interested in the size distribution of
banks, partly because of the large degree to which laws and regulations limited
bank size. Recent studies include Berger, Kashyap, and Scalise (1995), Ennis
(2001), and Jones and Critchfield (2005).
While the banking literature has long noted that bank size distribution is
skewed, it has not typically tried to fit this using the previously mentioned
category of distributions. This absence has made it difficult to compare bank
size distribution with that of other industries.
There is part of this literature, however, that tests for Gibrat’s Law in
the banking industry but for smaller samples than those used in this study.
Alhadeff and Alhadeff (1964) analyze growth in assets of the 200 largest U.S.
banks between 1930–1960 and find the largest banks grew more slowly than
the banking system itself. They find, however, that the top banks that survived
throughout the sample period grew faster than the system as a whole and
attribute this to mergers among the largest banks. Rhoades and Yeats (1974)
analyze growth among U.S. banks by deposits for 1960–1971 and find that the
largest banks grew more slowly than the whole banking system. In a study
of the 100 largest international banks from 1969 to 1977 by assets, Tschoegl
(1983) finds that growth rates of banks are roughly independent of size, but that
growth rates exhibit positive serial correlation. Saunders and Walter (1994)
use international data for the 200 largest banks for the 1982–1987 period and
reject Gibrat’s Law, finding that the smaller banks grow faster than larger banks
in terms of assets. More recently, Goddard, McKillop, and Wilson (2002) find
evidence that during the 1990s in the United States, large credit unions grew
faster than their smaller counterparts.
As we discussed, the firm growth results are important because they ultimately determine the distribution of firm sizes. One interesting strand in
the literature calculates a Markov transition matrix of movement between size
categories and then calculates stationary size distributions. Early examples
include Hart and Prais (1956), Simon and Bonini (1958), and the study of the
steel industry by Adelman (1958). The only study that we are aware of that
applies this technique to banking is Robertson (2001).
More recently, the literature has attempted to generate firm size dynamics, and ultimately a size distribution, from models of maximizing behavior
of firms. Sutton (1997) surveys several such models. One paper in this literature is the learning model of Jovanovic (1982), in which new firms receive
productivity shocks, learn about them over time, and then decide whether to
continue or exit. Another prominent example is Hopenhayn (1992).
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Federal Reserve Bank of Richmond Economic Quarterly
LEGAL AND REGULATORY LIMITS ON BANK SIZE
Describing bank size distribution is particularly important because of the many
legal and regulatory limits on bank size that existed through the 1970s and
were removed during the 1980s and 1990s. As we will see, the removal of
these barriers coincide with dramatic changes in size dynamics and the size
distribution.
In 1960, banks could not branch across state lines and some states even
forbade branching within a state.5 The 1966 Douglas Amendment to the 1956
Bank Holding Company Act allowed interstate banking only with expressed
authorization by participating states. However, no state allowed interstate
banking at the time and the amendment was not even exercised until 1978
when Maine allowed out-of-state bank holding companies (BHCs) to operate
within the state.6 Over the next 12 years, many of the intrastate and interstate
restrictions were removed by the states.7
The remaining interstate banking restrictions were removed by the
Riegle-Neal Interstate Banking and Branching Efficiency Act of 1994. The
act permitted bank holding companies to acquire banks in any state and, beginning in 1997, allowed interstate bank mergers. (See Kane [1996] for a
summary of the act.) More recently, the Gramm-Leach-Bliley Act of 1999
allowed banks to engage in nonbanking financial activities such as insurance.
3. THE DATA
Many banks operate under a bank holding company structure. A bank holding
company is a legal entity that i) directly or indirectly owns at least 25 percent
of the bank’s stock, ii) controls the election of a majority of a bank’s directors,
or iii) is deemed to exert controlling influence of bank policy by the Federal
Reserve (Spong 2000). Many bank holding companies have multiple banks
and even other holding companies under their control. Historically, this legal
organization was used to avoid some of the restrictions on branching (Mengle
1990). In many cases, a bank holding company would operate many activities jointly. For this reason, we follow Berger, Kashyap, and Scalise (1995)
and treat all banks and bank holding companies under a higher level holding
5 The 1927 McFadden Act forbade interstate branching by federally chartered banks. Later,
the Federal Reserve extended the ruling to include all state-chartered banks that are regulated by
the Federal Reserve.
6 There were some means around these restrictions. For example, the 1956 Bank Holding
Company Act did not limit the location of nonbank subsidiaries of bank holding companies, so
some banks had a cross-state network of nonbranch offices that would specialize in activities like
lending. Also, some exemptions were allowed for the acquisition of insolvent banks from government deposit insurance funds (Kane 1996).
7 Jayaratne and Strahan (1997) list when states removed restrictions to interstate banking and
intrastate branching.
H. P. Janicki and E. S. Prescott: Bank Size Distribution
297
company as a single independent banking enterprise. For convenience, we
will typically refer to each of these entities as a bank.
Data on banks are taken from the Reports on Condition and Income (the
“Call Report”) collected by federal bank regulators. We use fourth quarter
data on all commercial banks in the United States. We look specifically at
commercial banks and exclude savings banks, savings and loan associations,
credit unions, investment banks, mutual funds, and credit card banks. Individual commercial banks that belong to a holding company are then grouped
according to a unique bank holding company regulatory number, and their
assets, deposits, loans, and employees are summed and replaced by one entry
in our data set. Our data set, therefore, tracks bank holding companies and
independent commercial banks not affiliated with a holding company. It does
not distinguish between a merger and a failure. Both events are treated as an
exit.
We use four measures of bank size. The first is commercial bank assets.
For this variable, we have data from 1960–2005. Prior to 1969, we only have
domestic holdings, but after 1969, we have foreign and domestic holdings.
The next two size measures are domestic holdings of deposits and loans. For
both of these variables, we have data from 1960–2005. The final size measure
is the number of domestic employees.8 For this last measure, we only have
data for 1969–2005. Assets and loans are adjusted to include off-balance-sheet
items starting in 1990. (See the Appendix for details.) All the variables are
adjusted for the total aggregate size of that variable in each year. In particular,
firm size data are converted into market share numbers for that year and then
multiplied by the total quantity in the banking industry of that variable in
2004. The market share adjustment facilitates comparison across years, while
the scaling by 2004 aggregate quantities gives a sense of the size in terms of
recent quantities.
4. THE SIZE DISTRIBUTION
The size distribution of banks has always been skewed, but it has become more
so since the 1960s. Figure 3 shows the distribution of assets for 1960, 1980,
and 2005. Each year is normalized by the total assets in that year relative
to 2004 so that the distributions are comparable over time. The distribution
is plotted on a log scale. Because the size distribution is so skewed to the
right, the log scale—or something similar—is needed to fit all the banks on
the graph.
Figure 3 demonstrates that there are a large number of small banks and
a few large banks. As is evident, there is a shift in the distribution to the left
8 The Call Report counts employees in terms of full-time equivalents.
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Federal Reserve Bank of Richmond Economic Quarterly
Figure 3 Change in Bank Size Distribution Over Time
1960
1980
2005
0.06
Probability
0.05
0.04
0.03
0.02
0.01
0.00
1
10
2
10
3
4
10
10
Assets (Millions USD, 2004) Log Scale
10
5
6
10
Notes: Each line is a probability distribution of bank size as measured by assets for a
given year.
over time. Since we have scaled assets in each year to be the same scale, this
change means that a higher fraction of the assets are being held by the small
number of large banks, as indicated earlier in Figure 2.
Visually, the graphs suggest that the distribution might be accurately represented by a lognormal distribution. Figure 4 reports the actual distribution
and an estimated lognormal distribution for assets in 2005, where the lognormal has parameters μ and σ 2 obtained from the 2005 data set. However,
the distribution fails the Kolmogorov-Smirnov test for goodness-of-fit. This
is true for almost all the years in the data set, as well as for the other size
measures.
The estimated lognormal distribution does a particularly poor job of fitting
the right tail of the distribution. This is hard to see in Figure 4 because of the
small number of large banks. The fit at the right tail is better seen if we use
a rank-frequency, or Zipf plot. For a power law distribution plotted on a log
scale, this type of graph has the valuable property that the slope will be linear.
For example, let xr = cr −α , where r is the rank of a variable. Taking the
H. P. Janicki and E. S. Prescott: Bank Size Distribution
299
Figure 4 Size Distribution of Banks in 2005
0.06
Data
2
Lognormal, ( μ =11.7234, σ =1.8727)
0.05
Probability
0.04
0.03
0.02
0.01
0.00
1
10
2
10
3
4
10
10
Assets (Millions USD, 2004) Log Scale
5
10
6
10
Notes: The parameters μ and σ are the mean and standard deviation, respectively, of the
natural log of assets. The lognormal distribution is calculated using these parameters.
logarithm of both sides, we obtain the equation,
ln(xr ) = ln(c) − αln(r).
If α = 1, then Zipf’s Law holds.
Figure 5 is a Zipf plot for the bank size distribution in 2005.9 It plots the
data and the lognormal distribution using the sample mean and variance.
Figure 5 demonstrates that the lognormal distribution underestimates the
density of the right tail (which is the left side in the Zipf plot). Indeed,
if the plot is linear, we know that the right tail of the distribution of bank
holding companies can be better approximated by the Pareto distribution.
Note, however, that only the right tail of the distribution—which corresponds
to the left side of the figure—appears to fit Zipf’s Law. That is, the distribution
of bank holding companies seems to be lognormally distributed with a Paretodistributed tail.
9 Zipf plots for different size measures look very similar.
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Federal Reserve Bank of Richmond Economic Quarterly
Assets (Millions USD, 2004) Ln Scale
Figure 5 Zipf Plot of Banks in 2005
10
6
10
5
10
4
10
3
10
2
10
1
10
0
Data
2
Lognormal, μ=11.7234, σ =1.8727)
Pareto, k=0.880
0
10
1
2
10
10
Rank (Ln Scale)
3
10
Notes: The parameters μ and σ are the mean and standard deviation, respectively, of
the natural log of assets. The lognormal distribution is calculated using these parameters.
The estimate for the Pareto distribution is only for the 3,000 largest banks.
We can formally test whether the right tail of the distribution satisfies
Zipf’s Law. One common method is to estimate the slope by fitting a linear
regression of the form,
ln(xi ) = α 0 + α 1 ln(ri ) + i ,
(1)
where r is the rank of the bank. The coefficient α 1 is a power exponent in the
Pareto distribution and if it is equal to –1, then Zipf’s Law holds.
Ordinary least squares estimates of (1) will underestimate α 1 (see Gabaix
and Ioannides 2004). For this reason, we use the maximum likelihood estimator in Newman (2005) to estimate the power coefficient in the Pareto
distribution, or equivalently the slope α 1 :
α 1 = −n−1 [
n
(ln(xi ) − ln(xmin ))].
i=1
H. P. Janicki and E. S. Prescott: Bank Size Distribution
301
Table 1 Zipf’s Law: Maximum Likelihood Estimates
Assets
Deposits
Loans
Employees
1960
–α 1
SE
1.001 0.001
0.998 0.001
1.028 0.001
–
–
1970
–α 1
SE
1.030 0.001
1.016 0.001
1.051 0.001
1.022 0.065
1980
–α 1
SE
1.018 0.001
0.983 0.001
1.007 0.001
1.031 0.073
1990
–α 1
SE
1.017 0.001
0.982 0.001
1.082 0.001
1.005 0.084
2005
–α 1
SE
1.136 0.001
1.092 0.001
1.187 0.001
1.052 0.096
Notes: The estimates are reported as −α 1 .
The maximum likelihood estimates for several different years are reported in
Table 1.10 We restrict the estimates to the tail by limiting the sample to the
largest 3,000 banks for each year. Although they are not identical across years,
they broadly support Zipf’s Law in the upper tail, but the results are sensitive
to the cutoff.
The worst fit in Table 1 is in 2005. This can be seen in Figure 5. The straight
line in Figure 5 graphs the power distribution for the maximum likelihood
estimate based on the 3,000 largest banks. The distribution is a straight line
with slope –1.136. The slope is too high for Zipf’s Law. Furthermore, the
slope is high because the larger banks are larger than predicted by Zipf’s Law,
though this is less true for the largest.
In contrast, the fit for assets in 1960 is excellent. Figure 6 is a Zipf plot for
assets in 1960. The distribution is a straight line with slope –0.999, practically
the same as in Zipf’s Law.
The estimates for assets in 1970, 1980, and 1990 are similar, but this hides
an important difference. The year 1970 is similar to 1960 in that many of the
largest banks are smaller in size than predicted by the estimate. In 1980, this
pattern changes to one where the largest banks are larger than predicted. The
differences in the predictions for the largest banks are even larger in 1990 and
look similar to that of 2005 (see Figure 5).
To summarize, only the right tail can reasonably be considered to be fitted
by Zipf’s Law, and the fit depends on the year. It does well in 1960, but starting
in 1980 Zipf’s Law predicts that the largest banks will be smaller than in the
data. The lognormal distribution poorly fits the size of the largest banks but
better fits the small banks.
10 We calculate standard errors using the method outlined in Gabaix and Ioannides (2004).
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Federal Reserve Bank of Richmond Economic Quarterly
Figure 6 Zipf Plot of Banks in 1960
10
6
Assets (Millions USD, 2004) Ln Scale
Data
2
Lognormal, μ=12.1774, σ =1.4948)
Pareto, k=0.999
10
5
10
4
10
3
10
2
10
1
0
10
1
10
2
10
Rank (Ln Scale)
3
10
4
10
Notes: The parameters μ and σ are the mean and standard deviation, respectively, of
the natural log of assets. The lognormal distribution is calculated using these parameters.
The estimate for the Pareto distribution is only for the 3,000 largest banks.
5.
DOES GIBRAT’S LAW HOLD FOR BANKS?
Gibrat’s Law states that firm size growth is independent of firm size. For our
first test of this law, we fit the following linear equation,
ln(xit+1 ) = β 0 + β 1 ln(xit ) + it ,
(2)
where xit is the size measure (assets, employees, etc.) of bank i at time t. A
coefficient value of β 1 = 1 means that growth is independent of size.
We estimated (2) over each decade and over 2000–2005 using ordinary
least squares. We only considered banks in the sample that were around at the
beginning and end of the estimation period. Those that exited were dropped
from the sample.11 The estimates are reported in Table 2.
11 Sometimes the literature includes exiting banks and sometimes it does not. See Sutton
(1997) for more information. To reduce survivorship bias, we also estimated equation (2) over
each year and found similar results to those reported in Table 2.
H. P. Janicki and E. S. Prescott: Bank Size Distribution
303
Table 2 Gibrat’s Law: Estimates
1960–1969
1970–1979
1980–1989
1990–1999
2000–2005
Assets
β 0 0.143 (0.031)
β 1 0.987 (0.003)
R 2 0.929
0.574
0.953
0.915
(0.034) 0.043 (0.057) –0.187 (0.074) 0.015 (0.061)
(0.003) 1.011 (0.005) 1.024 (0.006) 0.998 (0.005)
0.847
0.823
0.869
Deposits
β 0 0.435 (0.030)
β 1 0.971 (0.003)
R 2 0.923
1.004
0.929
0.902
(0.033) 0.131 (0.058)
(0.003) 0.999 (0.005)
0.826
0.654
0.972
0.787
(0.075) 0.379 (0.063)
(0.007) 0.970 (0.005)
0.847
Loans
β 0 0.545 (0.037)
β 1 0.960 (0.003)
R 2 0.889
1.368
0.901
0.870
(0.039) 0.000 (0.071)
(0.003) 0.993 (0.006)
0.767
0.635
0.964
0.782
(0.074) 0.444 (0.069)
(0.007) 0.962 (0.006)
0.812
Employees
β0
–
β1
–
–
R2
–0.021 (0.012) 0.240 (0.016)
1.006 (0.003) 1.001 (0.004)
0.897
0.862
0.284
1.008
0.829
(0.022) 0.101 (0.017)
(0.006) 0.994 (0.004)
0.900
–
–
Notes: The table provides ordinary least squares estimates of equation (2) for each sample
period. Standard errors are in parentheses. Numbers in bold are the slope estimates of
the effect of firm size on growth. An estimate close to one is consistent with Gibrat’s
Law.
The estimates in Table 2 are close to one for all the decades and variables.
While broadly supportive of Gibrat’s Law, these estimates put a great deal of
emphasis on small banks because they comprise most of the sample. For this
reason, we broke the sample into different size categories and then calculated
the annualized growth rates over the same periods for banks in each category.
As before, we only considered banks that survived. Figure 7 reports the growth
rates for assets, deposits, and loans.12
Just as in Table 2, growth measures do not vary much with size in the
1960s and 1970s.13 In the 1980s and 1990s, however, the numbers in Figure
7 present a different picture than the estimates in Table 2. As demonstrated
by the figure, the largest banks clearly grew faster than the small banks. This
is also true for the 2000–2005 period, but the effect is less pronounced.
12 A figure showing employee growth rates is excluded because the growth rates jump around
significantly and do not show any clear patterns.
13 We have also calculated annual growth rates over each period and then averaged them to
get a similar figure. (Over the 1960–1969 period, for example, this meant calculating the 1960–
1961, 1961–1962, etc., growth rates and then averaging them.) The results are similar.
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Federal Reserve Bank of Richmond Economic Quarterly
Figure 7 Annualized Growth Rates by Size Categories
10
1960-1969
1970-1979
1980-1989
1990-1999
2000-2005
Annualized Growth (%)
8
6
4
2
0
-2
10 2
3
4
10
5
10
10
Assets in Millions (Log Scale)
10
1960-1969
1970-1979
1980-1989
1990-1999
2000-2005
Annualized Growth (%)
8
6
4
2
0
-2
2
3
10
4
10
10
Deposits in Millions (Log Scale)
10
1960-1969
1970-1979
1980-1989
1990-1999
2000-2005
Annualized Growth (%)
8
6
4
2
0
-2
2
10
3
10
4
10
5
10
Loans in Millions (Log Scale)
Notes: The size measure is broken into seven size categories and then average annualized
growth rates are reported for each category.
H. P. Janicki and E. S. Prescott: Bank Size Distribution
305
Gibrat’s model that yields the law of proportionate effect assumes that
growth rates are not persistent over time. To check the validity of this assumption, we calculate the correlation of growth rates for surviving banks in
each decade. We find that correlations of growth rates for all variables are
very low. In the 1960s, it is 0.0610; in the 1970s, it is 0.0331; in the 1980s, it
is 0.0691; in the 1990s, it is 0.0722; and after 2000, it is 0.1617. Correlation
coefficients of growth in the remaining variables are of similar magnitude.
To conclude, growth rates appear to be independent of size in the 1960s
and 1970s, but they are positively related to size in the 1980s and 1990s. In
the 2000–2005 period, the growth rates are also higher for the largest banks,
but less so than in the previous two decades. It appears that the 1960s and
1970s were relatively stationary periods, but the 1980s and 1990s were a
long transition period, no doubt due to the legal, regulatory, and technological
changes of the period. Finally, the 2000–2005 period appears to be the end of
the transition, as the size dynamics seem to be returning slowly to the numbers
of the 1960s and 1970s. Of course, this conclusion is tentative because trends
calculated from five years of data can easily be transitory.
6.
ENTRY AND EXIT
Despite the large number of banks that have exited the industry over the last
45 years, there has been a consistent flow of new bank entries. The number
of entries and exits (including mergers) expressed as a fraction of the banking
population is reported in Figure 8.
Visually, it is apparent that the flow of new banks is a relatively constant
fraction of the banking population. To check this, we estimated a linear time
trend of the number of entries as a fraction of the number of banks operating.
Specifically we estimated the equation,
yt = γ 0 + γ 1 t + t,
where yt is the fraction of entries per year and t is the time trend. The ordinary
least squares estimates are γ 0 = 0.0145 (0.0018), γ 1 = 0.0001 (0.0001) with
R 2 =0.0612. Standard errors are in parentheses. There is no time trend in the
flow of new banks, though there is a significant amount of cyclical variation.
The fraction of banks that exit varied a great deal over time. There was
a significant increase starting in the 1980s and, except for a short dip in the
early 1990s, the high level continued through the late 1990s. No doubt much
of this exit was due to mergers, particularly those that occurred in the 1990s,
but our data does not allow us to distinguish between these two sources of
exit. It is only in the last five years that the rate of exit seems to slow down.
It is striking that despite the huge number of bank exits starting in the
1980s, entry remained strong throughout the entire period. Interestingly, it
is virtually uncorrelated with exit. For example, the correlation between exit
306
Federal Reserve Bank of Richmond Economic Quarterly
Figure 8 Fraction of Banks that Enter and Exit by Year
0.02
Fraction of Entries and Exits
0.01
0.00
-0.01
-0.02
-0.03
-0.04
-0.05
-0.06
Entries
Exits/Mergers
1965
1970
1975
1980
1985
1990
1995
2000
2005
Year
Notes: The chart reports the gross flow of banks that enter and exit expressed as a
fraction of banks in each year.
and entry for the 1985–2005 period is only –0.07. This evidence means that a
theory of the bank size distribution needs to be able to generate robust entry,
even when the total number of banks is declining.
7. THE DYNAMICS OF THE SIZE DISTRIBUTION
In this section, we use the information previously documented—the size distribution at a point of time, entry and exit rates, and bank dynamics—to make
forecasts of what will happen to the size distribution. We also perform counterfactual experiments such as what would have happened to the bank size
distribution if the dynamics had not changed. The purpose of these exercises
is to develop a sense of how changes in size dynamics have mattered for the
size distribution.
To make these calculations, we do not use growth rates at the individual
bank level. Instead, we do something similar by constructing a simple Markov
chain model following Adelman (1958). A Markov chain model splits the
H. P. Janicki and E. S. Prescott: Bank Size Distribution
307
Figure 9 Projected Number of Banks
13,000
12,000
Number of Banks
11,000
10,000
9,000
8,000
7,000
Data
Transition Matrix 1960-1969
Transition Matrix 1970-1979
Transition Matrix 1980-1989
Transition Matrix 1990-1999
Transition Matrix 2000-2005
6,000
5,000
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
2010
Year
Notes: Each projection is made by taking the transition matrix estimated over a given
period and then calculating the total number of banks that would operate after the last
year in that given period.
banks into a finite number of size categories. Probabilities of moving between
size categories are summarized with a Markov, or transition, matrix. A Markov
matrix is a square matrix, P , where element Pij specifies the probability a bank
that starts in size category i will move into size category j in the next period.
A Markov model allows for straightforward predictions about changes to
a size distribution. For example, if the size distribution at time t is st , then the
size distribution at time t + n is
st+n = P n st .
If a Markov model has the property that a bank starting in any category
has a positive probability of moving to any other size category in a finite
number of periods, then several useful theorems apply. First, there exists a
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Federal Reserve Bank of Richmond Economic Quarterly
Table 3 Transition Matrix Size Categories (Scaled Dollars)
Size Categories
1
2
3
4
5
6
7
Assets
<100m
100m-500m
500m-1b
1b-5b
5b-10b
10b-50b
50b<
Notes: A list of size categories used in the Markov model. In all years, data are converted
into market share numbers and then multiplied by the total quantity of assets in 2004.
stationary distribution, that is, there exists s such that s = P s.14 Second,
the stationary distribution is unique and independent of the initial distribution
of banks. Therefore, regardless of the initial distribution, if the transition
matrix, P , is repeatedly applied to the distribution, then the size distribution
will approach the unique stationary distribution.
We construct seven different size categories. These are listed in Table 3.
All the data are scaled as we discussed earlier to make it possible to compare
across years. We also include an eighth category that represents banks that are
inactive. New banks come from this category and exiting banks move into it.
We calculate the number of banks in each of the seven active categories. For
the inactive banks, we assume that there is a large pool of 100,000 potential
entries.
Our transition matrix is calculated by counting the fraction of banks that
move from size category i to j each year over the specified time period. Entry
rates are calculated so that a constant fraction of potential banks enter.
For our first exercise, we use the transition matrices estimated over several
ranges of time periods to forecast the aggregate number of banks in the industry. We estimate several transition matrices and make predictions to 2013.
The results are illustrated in Figure 9. For each period, we calculate the transition matrix and then forecast the change in the total number of banks as if
the transition probabilities had not changed from that time period forward.
This exercise is similar to one in Jones and Critchfield (2005), although our
methodology is very different. An advantage of our Markov chain model is
that we have information on the entire distribution at each point in Figure 9.
As is clear from the earlier analysis, as well as from Figure 9, the size
dynamics changed significantly over this period. Nevertheless, the exercise
14 Let s j be the fraction of banks in size category j . The stationary distribution s is the
solution to the set of equations s = P s and
j sj = 1.
H. P. Janicki and E. S. Prescott: Bank Size Distribution
309
Table 4 Stationary Distribution of Banks by Assets
Size
Categories
1
2
3
4
5
6
7
1960–1969
1969
SS
0.333 0.357
0.504 0.516
0.085 0.076
0.060 0.042
0.008 0.004
0.009 0.005
0.002 0.000
1970–1979
1979
SS
0.379 0.439
0.492 0.485
0.064 0.044
0.046 0.021
0.007 0.003
0.010 0.005
0.002 0.003
1980–1989
1989
SS
0.379 0.471
0.488 0.435
0.067 0.049
0.043 0.027
0.008 0.005
0.010 0.007
0.005 0.007
1990–1999
1999
SS
0.440 0.523
0.436 0.377
0.061 0.051
0.044 0.037
0.006 0.004
0.007 0.005
0.005 0.002
2000–2005
2005
SS
0.480 0.540
0.399 0.363
0.060 0.050
0.043 0.033
0.008 0.006
0.006 0.005
0.005 0.003
Notes: Columns under a year list the actual size distribution in that year. Columns under
“SS” list the stationary distribution as calculated from the estimated transition probabilities
for that period.
is interesting because it illustrates how the dynamics matter for the size distribution. For example, the 1960s and 1970s look relatively stable. If these
dynamics had not changed, there would not have been much change in the total
number of banks. In the 1980s and 1990s, the dynamics predicted continued
substantial declines in the number of banks. However, the transition probabilities changed significantly in the 2000–2005 period and as a consequence,
the decline in the number of banks leveled off.
Table 4 reports stationary size distributions calculated from the estimated
transition matrices for several periods in the data. We compare them with
the actual distribution at the end of each period in order to illustrate whether
the existing distribution was close to the stationary distribution. Interestingly,
for every period, we find that the fraction of banks that are in the smallest
category is higher in the stationary distribution than in the final year of the
period. We also find that for every other period, the fraction of banks in each
of the other size categories is less in the stationary distribution than in the final
year (the only exceptions are during Category 2 [1960–1969] and Category 7
[1970–1979] and [1980–1989]).
Table 4 has several implications. First, even in the relatively stable decades
of the 1960s and 1970s, the size distribution was not at a stationary point
and under the estimated transition probabilities, the distribution would have
continued to change. Second, there will continue to be large numbers of small
banks, even if the fraction of assets they hold is not large. Third, we can see that
the dynamics for the 1990s imply even more concentration than the dynamics
from the 1980s. This was also suggested by Figure 9. Finally, there will
continue to be a large number of mid-size banks. The most recent merger wave
led some commentators to speculate that the bank size distribution would take a
“barbell” shape with only small banks and large banks. The small banks would
310
Federal Reserve Bank of Richmond Economic Quarterly
Table 5 Distribution of Bank Assets
2005
Size
Categories
Fraction
of Banks
Fraction
of Assets
1
2
3
4
5
6
7
0.480
0.399
0.060
0.043
0.008
0.006
0.005
0.014
0.050
0.024
0.047
0.034
0.068
0.763
Stationary
Fraction
Fraction
of Banks
of Assets
0.540
0.363
0.050
0.033
0.006
0.005
0.003
0.022
0.062
0.027
0.050
0.035
0.083
0.721
Notes: Fraction of assets for the stationary distribution was calculated by assuming that
the mean assets in each size category are the same as in 2005.
survive because of their comparative advantage at small business lending and
the big banks would take advantage of their scale economies. Based on the
stationary distribution calculated from the 2000–2005 transition probabilities,
there will continue to be more mid-size banks than large banks. Indeed, the
2000–2005 stationary distribution has a higher fraction of Category 5 banks
than the stationary distributions of the 1960s and 1970s.
Table 5 reports the fraction of assets held by each size category in 2005
and in the stationary distribution based on the 2000–2005 data. Interestingly,
in the stationary distribution, only the largest size category holds a smaller
percentage of assets than it does in 2005. This striking finding demonstrates the
importance of the transition probabilities for determining the size distribution.
Finally, we report in Table 6 the estimated transition matrix for the 2000–
2005 period. The first row lists the probability of a new bank forming and
starting in each size category. The first column lists the probability a bank
of each size category exits (and for an inactive bank, stays inactive). For the
given time period, all entering banks enter into the smallest size category. The
first column shows the probability of exit, either by failure or merger, from the
industry. Exit is most common in the largest category and represents mergers.
We see that banks in any size category are most likely to remain in the same
bin as denoted by the diagonal entries. Finally, the matrix shows that banks
gradually change in size. Except for exits, banks almost always stay within
one size category of the previous year.
8.
CONCLUSION
In this paper, we documented the large changes in the size distribution of banks
that occurred starting in the 1980s. We found that the lognormal distribution
poorly fits the right tail of the size distribution. Zipf’s Law fits better, but for
H. P. Janicki and E. S. Prescott: Bank Size Distribution
311
some size measures in some periods, this distribution does not fit the largest
banks that well.
We also documented some differences and similarities in the size dynamics
over time. First, we found that new banks are a constant fraction of the
total number of banks. Second, we also found that Gibrat’s Law is a good
approximation for the 1960s and 1970s, before deregulation, but does not
describe the 1980s and 1990s. In these decades, the large banks grow the
fastest. The last five years of data suggest that the dynamics are returning to
the earlier, more stable period. Of course, five years of data are not enough to
make a strong prediction.
We also performed a simple forecasting exercise, using the transition probabilities taken from different time periods. Again, the relative constancy of
the number of banks in the 1960s and 1970s suggests that this period was
relatively stable. The projected rapid decline in the number of banks using
1980s and 1990s transition probabilities is evidence of the rapid changes that
occurred in the banking industry during that time. Finally, the 2000–2005
transition probabilities predict a leveling off in the number of banks.15 If that
trend continues, then we will be returning to a relatively stable period in banking, at least as measured by the number of banks. The size dynamics imply
that the U.S. banking structure will continue to have large numbers of small
banks and a decent number of mid-size banks.
As illustrated by the transition probability analysis, the size distribution depends, ultimately, on the size dynamics. Therefore, a theory of the changes in
bank size distribution needs an explanation of why the size dynamics changed
and by how much. The data demonstrate that these changes started in the
1980s as deregulation proceeded, so the natural place to start is with an understanding of how removals to growth and size limits change the growth rates of
different size banks. A successful theory would also need to account for the
robust entry over this period, despite the large number of banks that exited.
15 Jones and Critchfield (2005) make a similar prediction, using a different forecasting method.
312
Federal Reserve Bank of Richmond Economic Quarterly
Table 6 Transition Probability Matrix for Bank Assets: 2000–2005
Size Categories
Inactive
1
2
3
4
5
6
7
Inactive
1
2
3
4
5
6
7
0.999
0.024
0.030
0.040
0.050
0.028
0.053
0.061
0.001
0.927
0.043
0.000
0.000
0.000
0.000
0.000
0.000
0.049
0.908
0.076
0.003
0.000
0.000
0.000
0.000
0.000
0.018
0.827
0.048
0.000
0.000
0.000
0.000
0.000
0.000
0.058
0.879
0.060
0.000
0.000
0.000
0.000
0.000
0.000
0.020
0.827
0.068
0.000
0.000
0.000
0.000
0.000
0.001
0.086
0.849
0.020
0.000
0.000
0.000
0.000
0.000
0.000
0.030
0.920
Notes: This matrix has the property that a bank starting in any size category will reach
with positive probability any other size category in a finite number of periods. Probabilities that are significantly different than zero are highlighted in bold.
APPENDIX:
OFF-BALANCE-SHEET ITEMS
Banks can make commitments that are not directly measured by a traditional
balance sheet. For example, a loan commitment is a promise to make a loan
under certain conditions. Traditionally, this kind of promise was not measured
as an asset on a balance sheet. As documented by Boyd and Gertler (1994),
providing this and other off-balance-sheet items have become an important
service provided by banks, which means that traditional balance sheet numbers
did not accurately report some of the implicit assets and liabilities of a bank.
We account for loan commitment and other off-balance-sheet items such
as derivatives by converting them into credit equivalents and then adding
them to on-balance-sheet assets and loans. This method is similar to the
“Basel Credit Equivalents” series found in Boyd and Gertler (1994). Offbalance-sheet items are weighted by a credit conversion factor to create credit
equivalents. We make these adjustments starting in 1989 because it is only
from this year that we have the complete data to make them. Both panels of
Figure 10 demonstrate the importance of the adjustment by plotting aggregate
assets and loans with and without the adjustment, as well as by plotting credit
equivalents as a share of total loans. A detailed list of off-balance-sheet items
and credit equivalent weights is found in Table 7. These weights are used
by federal regulators to determine credit equivalents for regulatory capital
purposes.
H. P. Janicki and E. S. Prescott: Bank Size Distribution
313
Figure 10 Total Credit Equivalents in Assets and Loans
Adjusted Assets (Includes Credit Equivalent)
Unadjusted Assets
Adjusted Loans (Includes Credit Equivalent)
Unadjusted Loans
11,000
10,000
Billions (USD, 2004)
9,000
8,000
7,000
6,000
5,000
4,000
3,000
2,000
1,000
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
Year
Credit Equivalent as Fraction of Total Loans
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
1990
1995
2000
2005
Year
Notes: Top panel reports assets and loans both unadjusted and adjusted for off-balancesheet items. Bottom panel reports the fraction of adjusted loans that are due to the credit
equivalent adjustment. Credit equivalents are based on the weights used by regulators to
determine regulatory capital requirements.
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Federal Reserve Bank of Richmond Economic Quarterly
Table 7 Off-Balance-Sheet Items and Credit Equivalents
Item
Financial Standby Letters of Credit
Performance and Standby Letters of Credit
Commercial Standby Letters of Credit
Risk Participations in Bankers’ Acceptances
Securities Lent
Retained Recourse on Small Business Obligations
Recourse and Direct Credit Substitutes
Other Financial Assets Sold with Recourse
Other Off-Balance-Sheet Liabilities
Unused Loan Commitments (maturity >1 year)
Derivatives
Conversion Factor
1.00
0.50
0.20
1.00
1.00
1.00
1.00
1.00
1.00
0.50
–
Notes: Conversion factors used by regulators for determining credit equivalents of
off-balance-sheet items. The source is FFIEC 041 Schedule RC-R retrieved from:
www.ffiec.gov/forms041.htm (accessed on November 10, 2005).
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Bond Price Premiums
Alexander L. Wolman
T
his article provides a detailed introduction to consumption-based bond
pricing theory, a special case of the consumption-based asset
pricing theory associated with Robert Lucas (1978). To help make
the theory more accessible to novices, we organize the article around the two
famous interest rate decompositions associated with Irving Fisher. These complementary decompositions relate real or nominal long-term interest rates to
expected future short-term interest rates (the expectations theory of the term
structure), and relate short- or long-term nominal interest rates to the ex ante
real interest rate and the expected inflation rate (the Fisher equation). According to consumption-based theory, the Fisherian relationships hold exactly only
under certain restrictive conditions. We show what those conditions are, and
we show that generalizations of the Fisherian relationships hold quite broadly
in the consumption-based model.
The pure Fisherian relationships are shown to hold only as special cases of
the relationship between individual preferences, future economic activity, and
the returns on assets. Notable sufficient conditions for the pure expectations
hypothesis are that households be neutral to risk and the price level behave like
a random walk; the pure Fisher equation requires only risk neutrality.1 In turn,
long-term nominal bond prices may lie above or below the values dictated by
the pure expectations hypothesis and the pure Fisher relationship—forward
premiums and inflation-risk premiums may be positive or negative.
Interpreting bond prices of various maturities is an important challenge
for the Federal Reserve. Nominal bond prices contain information about the
public’s expectations of inflation and of future short-term rates. And they
contain information about the levels of short-term and long-term real interest
rates. All these variables can be valuable signals to the Federal Reserve of the
For helpful comments, we would like to thank Kartik Athreya, Brian Minton, Ned Prescott,
and Roy Webb. This article does not necessarily represent the views of the Federal Reserve
Bank of Richmond or the Federal Reserve System.
1 Risk aversion lies at the heart of much of asset pricing theory. For example, it is what
leads us to assume that riskier assets will have a higher average return than safer assets.
Federal Reserve Bank of Richmond Economic Quarterly Volume 92/4 Fall 2006
317
318
Federal Reserve Bank of Richmond Economic Quarterly
appropriateness of its policy.2 However, extracting these signals requires an
understanding of the potential limitations of the pure expectations hypothesis
and the pure Fisher relationship.3
The article proceeds as follows. In Section 1 we provide a brief historical
overview of the two interest rate decompositions. Section 2 lays out a modeling
framework for thinking about bond price determination, and derives the basic
bond pricing equations from which all else will follow. Section 3 derives the
generalized expectations theory of the term structure and Section 4 derives the
generalized Fisher equation. Section 5 combines the results of the previous two
sections for a general discussion of the yield differential between short- and
long-term bonds. Sections 2–5 provide a textbook treatment of bond pricing
relationships.4 Section 6 provides a selective review of applied research based
on bond pricing theory. Section 7 concludes the article.
Although the usual statements of the expectations hypothesis and the
Fisher equation are made in terms of interest rates, most of our derivations use
zero-coupon bond prices. This is for analytical simplicity; working with bond
prices is slightly easier, especially when the bonds are zero-coupon bonds.
And given an expression for the price of a bond, one can always work out the
corresponding interest rate.
1.
BRIEF HISTORY OF INTEREST RATE DECOMPOSITIONS
The expectations hypothesis of the term structure and the Fisher equation both
made early appearances in Irving Fisher’s Appreciation and Interest (1896).5
Chapter 2 of that work is devoted to a discussion of the equation, or “effect,”
that would later bear the author’s name. The Fisher equation is typically
thought of as relating “real” and “nominal” interest to the expected rate of
inflation, but Fisher’s analysis in Appreciation and Interest is more general. He
relates the interest rates between two standards (for example real vs. nominal,
or dollars vs. yen) to the relative rate of appreciation of the standards, as
1 + j = (1 + a)(1 + i),
(1)
2 See Bernanke and Woodford (1997), however, on the risks involved in the Federal Reserve
basing its policy actions solely on such data.
3 The empirical limitations of the pure expectations hypothesis have been well documented,
for example, by Campbell and Shiller (1991). There has been less emphasis on violations of the
Fisher relationship. Sarte (1998) presents evidence that the violations are small using a standard
form of preferences. Kim and Wright’s (2005) results suggest larger violations, using a different
approach outlined in Section 6.
4 Textbook treatments are available, see for example, Sargent (1987) and Cochrane (2001).
However, they tend to provide fewer details, concentrating instead on the method by which one
can price any asset in the consumption-based framework.
5 See Humphrey (1983) for the intellectual history of the Fisher equation before Fisher.
Humphrey shows that the relationship was well understood before Fisher.
A. L. Wolman: Bond Prices
319
where i is the rate of interest in the appreciating standard, a is the rate of
appreciation, and j is the rate of interest in the depreciating standard. In
Fisher’s words,
The rate of interest in the (relatively) depreciating standard is equal to
the sum of three terms, viz., the rate of interest in the appreciating
standard, the rate of appreciation itself, and the product of these two
elements. (p. 9)
In our context, j is the nominal rate, i is the real rate, and a is the expected
inflation rate.
In Chapter 5 and to some extent in Chapters 3 and 4, one can find the
essence of the expectations hypothesis of the term structure. Most notably
perhaps, on pages 28 and 29, Fisher writes,
A government bond, for instance, is a promise to pay a specific series
of future sums, the price of the bond is the present value of this series
and the “interest realized by the investor” as computed by actuaries is
nothing more or less than the “average” rate of interest in the sense above
defined.
By “ ‘average’ rate of interest in the sense above defined,” Fisher means what
we now understand to be the expected future path of short-term rates.
John Hicks (1939) and F. A. Lutz (1940) elaborated on Fisher’s version of
the expectations hypothesis in the 1930s and 1940s. Their versions of these
interest rate decompositions continued to be based on reasoning regarding
how returns among different assets should be related. Later, the development
of consumption-based asset pricing theory (Lucas 1978) gave a formal foundation to Fisher’s reasoning, while making clear that restrictive assumptions
were needed for the Fisherian relationships to hold exactly. The discipline
provided by consumption-based theory and the rise of rational expectations
and dynamic equilibrium modeling in macroeconomics also led economists
and finance theorists to de-emphasize certain elements of Fisher’s theories
regarding interest rates. For example, with respect to the expectations hypothesis, early versions were “usually understood to imply . . . that interest rates on
long-term securities will move less, on the average, than rates on short-term
securities” (Wood 1964). It is now well understood that whether this will
be true depends on the behavior of monetary policy and on the real shocks
hitting the economy (Watson 1999). And with respect to the Fisher equation,
prior to the consumption-based theory, researchers often emphasized not just
the decomposition into real rates and expected inflation, but also the extent
to which the real rate was invariant to changes in expected inflation (Mundell
1963). It is now understood that one cannot make general statements about this
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Federal Reserve Bank of Richmond Economic Quarterly
invariance, even though a version of the Fisher equation holds under general
conditions.
2.
MODELING FRAMEWORK
We use the modern theory of consumption-based asset pricing, first developed
by Robert Lucas (1978), to study bond prices. For our purposes, the crucial
elements in this theory are as follows. There is a representative consumer who
has an infinite planning horizon, has a standard utility function (exhibiting risk
aversion) over consumption each period, and discounts the utility from future
consumption at a constant rate.6 The consumer has a budget constraint which
states that the sum of income from sales of real and financial assets (including
income from maturing bonds), and income from other sources must not be
exceeded by the sum of spending on current consumption, on purchases of real
and financial assets, and on any other uses. With this framework, it is possible
to price any asset. To do this, we use conditions describing individuals’optimal
behavior.
Preferences and Budget Constraint
The consumer’s preferences in period t are given by
v t = Et
∞
β j u ct+j ,
(2)
j =0
where u ct+j is a utility function that is increasing and strictly concave in
consumption (c), and the discount factor β ∈ (0, 1). Before specifying the
budget constraint, it will be helpful to provide more detail about the set of
financial assets that play a role in our analysis. They are
1. n-period real discount bonds; if issued in period t, they pay off one unit
of consumption with certainty in period t + n. Their price in period t
in terms of goods is qt(n) , and the quantity that the consumer purchases
in period t is bt(n) .
2. n-period nominal discount bonds; if issued in period t, they pay off
a dollar with certainty in period t + n. Their dollar price in period
t is denoted Q(n)
t , and the quantity that the consumer purchases in
period t is denoted Bt(n) . Notice that if the dollar-denominated price of
consumption at date t + n is very high, a nominal bond provides little
6 The representative consumer idea can be taken literally or can be viewed as a shortcut for
the assumption that whatever individual-level heterogeneity does exist has been insured away by
the existence of a complete set of Arrow-Debreu securities (state-contingent claims).
A. L. Wolman: Bond Prices
321
consumption. Therefore, an asset that yields a dollar with certainty is
still a “risky” asset.
3. One-period nominal forward contracts, s periods ahead; these contracts
represent a commitment in period t to purchase a one-period nominal
f
discount bond in period t + s at the pre-specified dollar price Qt,s . The
quantity that the consumer commits in period t to purchase in period
f
t + s is Bt,s .
4. One-period real forward contracts, s periods ahead; these contracts represent a commitment in period t to purchase a one-period real discount
f
bond in period t + s at the pre-specified price in terms of goods qt,s .
The quantity that the consumer agrees in period t to purchase in period
f
t + s is bt,s .
With this set of assets, the consumer’s flow budget constraint in period t
is
∞
f
(n)
Q(n−1)
Bt−1
+ t−1
Bt−s−1,s +
t
n=1
∞ (n−1) (n)
s=1
f
Pt n=1 qt
bt−1 + Pt t−1
s=1 bt−s−1,s + Wt
∞
f
f
= Pt ct + n=1 Q(n)
B (n) + ts=1 Qt−s,s Bt−s,s +
∞ (n) (n) t t t
f
f
Pt n=1 qt bt + Pt s=1 qt−s,s bt−s,s + Zt ,
(3)
where Pt is the price level—the dollar price of the consumption good, Zt is
purchases of other assets, and Wt is labor income and income from all other
sources. The left-hand side of (3) represents income and the right-hand side
represents spending.
There are several things to note with regard to the budget constraint. First,
in period t, for any j > k > 0, a j -period bond issued in period (t − k) is
identical to a (j − k)-period bond issued in period t, because they have the
same maturity and the same payoff at maturity.7 Thus, in the budget constraint
we include only the latter on the right-hand side. Second, a discount bond
that matures in period t can be thought of as having a price of one dollar (for
a nominal bond) or one good (for a real bond) in period t. Thus, on the left(0)
hand side of the budget constraint we have imposed Q(0)
= 1. Third,
t = qt
the prices of nominal bonds are written in terms of dollars and the prices of
real bonds are written in terms of goods; with the budget constraint written
in nominal terms, this means that prices of real bonds must be multiplied by
Pt . Finally, it is important to be clear about the forward contracts that appear
f
in the budget constraint. On the left-hand side, the terms t−1
s=1 Bt−s−1,s and
7 For example, set j = 10, k = 3 and t = 20. In period 20, a ten-period bond issued in
period 17 is identical to a seven-period bond issued in period 20.
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Federal Reserve Bank of Richmond Economic Quarterly
f
Pt t−1
s=1 bt−s−1,s represent income from maturing one-period bonds that were
purchased under forward contracts entered into in periods earlier than t − 1.
f
f
f
f
On the right-hand side, the terms ts=1 Qt−s,s Bt−s,s and Pt ts=1 qt−s,s bt−s,s
represent purchases of one-period bonds under forward contracts entered into
in periods earlier than t. For example, in period t, the consumer purchases
f
f
a quantity Bt−2,2 of one-period bonds at price Qt−2,2 in accordance with a
forward contract entered into in period t −2. Similarly, the consumer purchases
f
f
a quantity Bt−3,3 of one-period bonds at a price Qt−3,3 in accordance with a
forward contract entered into in period t − 3. Forward contracts entered into
in period t do not appear in the period t budget constraint because they do
not affect income or spending in period t; they do show up in future budget
constraints.
As mentioned earlier, by limiting our attention to zero-coupon bonds, it
is natural to focus on bond prices rather than interest rates. However, one
can easily recover interest rates from bond prices. Let Rt(n) denote the gross
nominal yield on a bond that sells in period t for price Q(n)
t and pays one dollar
in period t + n. On a standardized per-period basis, the yield satisfies
1/n
Rt(n) = 1/Q(n)
;
(4)
t
Rt(n) is the constant per-period interest rate that is implied by the price Q(n)
t .
Likewise, for an n-period real bond we have
1/n
rt(n) = 1/qt(n)
,
(5)
and for one-period nominal and real forward contracts entered into in period
t − s for execution in period t, we have
f
f
Rt−s,s = 1/Qt−s,s ,
(6)
and
f
f
rt−s,s = 1/qt−s,s .
(7)
Individual Optimality Conditions
The consumer chooses consumption and holdings of each asset to maximize
expected utility subject to the sequence of flow budget constraints. One way
to carry out this maximization is to form a Lagrangian from the utility function
and the sequence of budget constraints, and then use first-order conditions for
consumption and each asset. The Lagrangian is
L t = Et
∞
j =0
β j u ct+j +
(8)
A. L. Wolman: Bond Prices
t+j ·
∞
323
(n)
Q(n−1)
t+j Bt+j −1 +
t+1
n=1
Pt+j
∞
f
Bt+j −s−1,s +
s=1
(n−1) (n)
qt+j
bt+j −1
+ Pt+j
n=1
t+1
s=1
−t+j · Pt+j ct+j +
∞
(n)
Q(n)
t+j Bt+j
n=1
Pt+j
∞
f
bt+j −s−1,s + Wt+j
(n) (n)
qt+j
bt+j + Pt+j
n=1
+
t
f
f
Qt+j −s,s Bt+j −s,s +
s=1
t
f
f
qt+j −s,s bt+j −s,s + Zt+j
].
s=1
The first-order condition for consumption in period t is
u (ct ) = Pt t .
(9)
The multiplier t is the marginal utility of nominal income.
Nominal and Real Bonds
The first-order conditions for n-period nominal bonds are
(n−1)
t Q(n)
, n = 1, 2, ...
t = βEt t+1 Qt+1
(10)
where we have used the fact that an n-period bond in period t becomes an n−1
period bond in period t + 1. This expression implies that the price in period
t of an n-period discount bond is the ratio of the present value of expected
marginal utility in period t + n to marginal utility in period t :
n
Q(n)
(11)
t = β Et t+n /t , n = 1, 2, ...
To show this, first write (10) for period t + 1 :
(n−1)
t+1 Q(n)
, n = 1, 2, ...
t+1 = βEt+1 t+2 Qt+2
(12)
and substitute the result into (10), dividing both sides by t and using the law
of iterated expectations:
(n−1)
2
Q(n)
, n = 1, 2, ...
t = β Et (t+2 /t ) Qt+2
(13)
If n = 1 then we have (11), because Q(0)
t+2 = 1. If n > 1 then repeat the
(n−1)
process, substituting for t+2 Qt+2 using (10), etc. Intuitively, t Q(n)
t is the
utility cost of a bond in period t, and β n Et t+n is the expected utility benefit
from the payoff at maturity, discounted back to the present. When the agent
has optimized over bond holdings these two values are identical.8 Holding
8 The object on the right-hand side of (11) is referred to as the intertemporal marginal rate
of substitution.
324
Federal Reserve Bank of Richmond Economic Quarterly
constant the current marginal utility of a dollar (t ), a higher bond price Q(n)
t
will correspond to a higher payoff in utility terms, that is a higher u (ct+n ) or
a lower Pt+n .
For n-period real bonds the derivations are analogous, though it will be
useful to define the marginal utility of consumption as λt ≡ Pt t . The price
in period t of an n-period real discount bond is the ratio of the present value
of expected marginal utility in period t + n to marginal utility in period t:
qt(n) = β n Et λt+n /λt , n = 1, 2, ...
(14)
In contrast to the price of a nominal bond, which depends on the joint properties
of the marginal utility of consumption (λt = u (ct )) and the price level, the
price of a real bond depends only on the marginal utility of consumption.
Forward Contracts
As for forward contracts, the first-order condition for s-period-ahead oneperiod nominal forward contracts, committed to in period t is
f
Et t+s Qt,s = βEt t+s+1 .
However, because the forward price is known in period t, we can bring the
price outside the expectation operator:
f
Qt,s Et t+s = βEt t+s+1 .
(15)
Likewise for real forward contracts, we have
f
qt,s Et λt+s = βEt λt+s+1 .
(16)
From (11) and (15), the price of an n-period nominal bond is identical to
the product of the prices of a sequence of one-period forward contracts,
f
f
f
(1)
Q(n)
t = Qt Qt,1 Qt,2 · · · Qt,n−1 .
(17)
An n-period bond and a sequence of forward contracts can each be used to
provide a certain return n-periods ahead. To get a dollar in t + n using the nperiod bond, one needs to spend Q(n)
t today, whereas to get a dollar in t+n using
f
f
f
the sequence of forward contracts, one needs to spend Q(1)
t Qt,1 Qt,2 · · · Qt,n−1
today. This is easily illustrated in the two-period case: to get a dollar in t + 2
f
using forward contracts, one needs to have Qt,1 in period t + 1— for this is
the forward price of a bond which will deliver a dollar in period t + 2. In turn,
f
f
receiving Qt,1 in period t + 1 means spending Q(1)
t Qt,1 in period t on oneperiod bonds—the price of a bond that delivers a dollar in t + 1 is Q(1)
t , and
f
one needs to purchase Qt,1 of these bonds. True arbitrage would be possible
f
(2)
if Q(1)
t Qt,1 were not equal to Qt . The same reasoning holds for a long-term
real bond and a sequence of real forward contracts, so we have
f
f
f
qt(n) = qt(1) qt,1 qt,2 · · · qt,n−1 .
(18)
A. L. Wolman: Bond Prices
325
Note that from (17) or (18), the ratio in period t of the price of an n-period
bond to the price of an n − 1 period bond is equal to the forward price of
one-period bond in period t + n − 1, as of period t.
The optimality conditions (11) – (16) and the relationships between prices
of long bonds and forward contracts (17) and (18) serve as the basis for the
generalized Fisher relationship and generalized expectations theory.
3.
EXPECTATIONS THEORY OF THE TERM STRUCTURE
The standard version of the expectations theory of the term structure states that
long-term interest rates are equal to an average of expected future short-term
interest rates. We will derive a generalization of this theory, focusing on bond
prices instead of rates, and we will see that only under certain conditions does
the pure expectations theory hold. Our derivation exploits the fact that a long
bond is equivalent to a sequence of forward contracts.
From (17), the price of an n-period bond is the product of the prices
of n short-term forward contracts. Under the pure expectations hypothesis,
the price of an n-period bond is also equal to the product of the expected
prices of future short-term bonds, which we will denote by P EH (n) , for pure
expectations hypothesis:
(1)
(1)
(1)
P EHt(n) = Q(1)
Q
E
Q
·
·
·
E
Q
(19)
E
t
t
t
t
t+1
t+2
t+n−1 .
In a way that we will make more precise shortly, the pure expectations
hypothesis holds if covariances involving future bond prices and future marginal
utility are zero. It follows from the previous equation and (17) that the deviation of the price of an n-period nominal bond from P EH (n) is the product of
ratios of forward prices to expected future spot prices:
Q(n)
t
P EHt(n)
f
f
f
Q
Q
Q
t,1 t,2 · · ·
t,n−1 ,
=
(1)
Et Q(1)
Et Q(1)
t+1 Et Qt+2
t+n−1
(20)
or, in shorthand,
Q(n)
t
P EHt(n)
(1)
(1)
= Ft,1
· · · Ft,n−1
,
(21)
(1)
the j -period-ahead forward premium,
where we call Ft,j
f
(1)
Ft,j
Q
t,1 .
≡
Et Q(1)
t+1
(22)
In terms of marginal utilities, using (11) and (15), the forward premium is
(1)
Ft,j
=
Et t+j +1
Et t+j
Et
t+j +1
t+j
,
(23)
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Federal Reserve Bank of Richmond Economic Quarterly
and it is straightforward to show that the forward premium is pinned down by
the autocovariance properties of marginal utility, or equivalently by the covariance between the future short-term bond price and future marginal utility:
+1
Et t+j
t+j
t+j
(1)
Ft,j
=
(24)
t+j +1
Et t+j Et t+j
t+j +1 /t+j
t+j
,
= 1 + covt
Et t+j +1 /t+j Et t+j
(1)
Qt+j
t+j
,
= 1 + covt
,
(25)
Et t+j
Et Q(1)
t+j
with the last equality following from the law of iterated expectations.9
The deviation of the long bond price from the pure expectations hypothesis
(1)
is thus accounted for by the product of the individual forward premiums (Ft,j
),
(n)
(1)
(1)
Q(n)
× Ft,1
· · · Ft,n−1
.
(26)
t = P EHt
If each of the individual covariances that determine the forward premiums are
zero, then the pure expectations hypothesis holds. In turn, forward premiums
will be zero if the level of future nominal marginal utility is uncorrelated with
its subsequent growth rate. This will be the case, for example, if nominal
marginal utility is constant, or if it follows a random walk. Note that for
nominal bonds, risk neutrality is insufficient to drive all forward premiums
to zero; if the future price level is correlated with its subsequent growth rate,
there will be a forward premium, even if investors are risk-neutral.10 For
the case of risk aversion, the behavior of the marginal utility of consumption
is crucial for determining the forward premium as well as the inflation-risk
premium derived below. In standard models along the lines of Lucas (1978),
the marginal utility of consumption is a simple function of consumption itself.
Alternatively, one can consider more complicated specifications of u (c) , or be
entirely agnostic on the specification of u (c) . These approaches are discussed
in Section 6.
Why do the conditional covariances between future marginal utility and
the subsequent growth rate of marginal utility affect the price of a long-term
bond relative to the product of expected future short-term bond prices? Focus
(1)
on one term (Ft,j
), which is the price premium for a j -period-ahead forward
9 From the law of iterated expectations we know, for example, that E (
t
t+j +1 ) =
(1)
Et (t+j Et+j t+j +1 /t+j ) = Et (t+j Qt+j ).
10 Under risk-neutrality, expected real returns are equated across assets (forward and spot,
for example). Because the real return is the nominal return divided by the gross inflation rate,
expected nominal returns are not necessarily equated.
A. L. Wolman: Bond Prices
327
contract relative to the expected j -period-ahead spot price of a one-period
bond. If the growth rate of the marginal utility of a dollar (price of a oneperiod bond) is expected to covary positively with the level of the marginal
utility of a dollar in t + j, then you pay a premium at t + j to lock in at t
the contract that gives you a dollar at t + j + 1. In this situation, you tend to
value a dollar highly for consumption in t + j precisely when buying a bond
requires you to forego a lot of consumption. Thus, the expected spot market
looks expensive, which means that the forward price must be high as well.
Note that the j -period-ahead forward premium can be positive or negative,
and thus the overall term premium can be positive or negative. It is common
to think of long rates as incorporating a positive term premium (meaning that
1
Ft,j
< 0), but if future marginal utility of a dollar is positively correlated
with the expected growth rate of marginal utility, then forward premiums and
the term premium in rates will be negative.
Of course, we can also think about the forward-spot relationship from a
no-arbitrage perspective: agents must be indifferent between committing to
buy a one-period bond in period t + j (the forward contract) and expecting to
buy a one-period bond in the spot market at t + j :
f
Qt,j Et t+j = Et t+j Q(1)
t+j .
Expanding the right-hand side,
f
(1)
Qt,j Et t+j = Et t+j Et Q(1)
+
cov
,
Q
t
t+j
t+j
t+j .
(27)
1
Dividing both sides by Et t+j Et Q(1)
t+j replicates the expression for Ft,j above.
4.
FISHER RELATIONSHIP
The pure Fisher relationship states that nominal interest rates are equal to real
rates plus expected inflation. We will derive a generalization of this expression,
focusing again on bond prices instead of interest rates. The derivation follows
directly from manipulating the pricing equation for a nominal bond.
From the fact that the real and nominal multipliers are related by Pλtt = t ,
we can use (11) to write the price of a nominal bond (β n Et t+n /t ) as
λt+n Pt
(n)
n
Q t = β Et
,
(28)
λt Pt+n
or
λt+n
Pt
λt+n Pt
(n)
n
Qt = β Et
,
Et
+ covt
.
(29)
λt
Pt+n
λt Pt+n
This last expression can be used to decompose the price of a long-term nominal
bond into the price of a long-term real bond (qt(n) from [14]), the expectation
of the inverse of inflation (Et (Pt /Pt+n )), and a term we will call (n)
t , which
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Federal Reserve Bank of Richmond Economic Quarterly
can be thought of as the inflation-risk premium:
Q(n)
= qt(n) Et (Pt /Pt+n ) 1 + (n)
t
t
Pt /Pt+n
λt+n /λt
(n)
n
,
.
≡ β covt
t
Et (λt+n /λt ) Et (Pt /Pt+n )
(30)
Alternatively, converting to interest rates instead of bond prices, we have
⎤1/n
⎡
1
⎦ .
Rt(n) = rt(n) ⎣
(31)
(n)
Et (Pt /Pt+n ) 1 + t
If there is zero conditional covariance between the normalized growth rate
of marginal utility and the normalized inverse of inflation, then the inflationrisk premium is zero, and we recover the pure Fisher equation. In general,
though, the price of a long-term nominal bond exceeds the “Fisher price” when
the covariance between the growth of real marginal utility and the inverse of
inflation is positive. What is the intuition behind this covariance effect? The
bond pays off one dollar. If the covariance is positive, then the consumption
value of a dollar is high (low) in those states where the marginal utility of
consumption is high (low). In other words, the bond pays off well in terms
of consumption when you value consumption highly, so it is worth more than
the Fisher price.
Note that like the forward premium, the inflation-risk premium can be
positive or negative. We usually think of the inflation-risk premium in rates as
being positive, which would correspond to the price premium t being negative. However, this depends entirely on whether the inverse of the future price
level is negatively correlated with the future marginal utility of consumption,
conditional on time-t information.
5. YIELD DIFFERENTIAL AND HOLDING-PERIOD PREMIUM
Above we provided two decompositions of the price of an n-period nominal
bond. The first expressed the bond price in terms of the price of a real bond, the
expected inverse of the change in the price level, and an inflation-risk premium.
The second decomposition expressed the bond price in terms of the expected
product of the prices of future short-term bonds and the product of individual
forward premiums at each maturity. We now use these decompositions to study
the yield differential between bonds of any two maturities. In addition, in this
section we provide an intuitive explanation of the holding-period premium,
the expected differential between the return on a long-term bond sold before
it matures and the return on a shorter-term bond sold at maturity.
A. L. Wolman: Bond Prices
329
Yield Differential
Recall that the yield is the inverse of the price, and that we standardize yields
so that they are reported on a gross per-period basis:
−1/n
Rt(n) = (Q(n)
.
t )
(32)
We then can express the bond yield using the two decompositions as
−1/n
Rt(n) = qt(n) Et (Pt /Pt+n ) 1 + (n)
t
or
(1)
(1)
Rt(n) = P EHtn × Ft,1
· · · Ft,n−1
−1/n
.
(33)
(34)
Since these expressions hold for any n, the ratio of the yield on an nperiod bond to the yield on a one-period bond can be written using the Fisher
decomposition as
(1)
(1)
(n)
E
/P
q
1
+
(P
)
t
t
t+1
t
t
Rt
=
(35)
1/n
(1)
Rt
qt(n) Et (Pt /Pt+n ) 1 + (n)
t
=
rt(n)
Et (Pt /Pt+1 )
1+
1/n
(1)
rt (Et (Pt /Pt+n ))
1+
(1)
t
(n)
t
1/n .
(36)
The nominal yield curve slopes upward if some combination of the following
is true: (i) long-term real rates exceed short-term real rates, (ii) the value of
a dollar is expected to increase at a higher rate in the short term than over
the long term, and (iii) the short-term inflation-risk premium in bond prices
exceeds the long-term inflation-risk premium in bond prices.
Alternatively, we can use the perspective of the expectations hypothesis
to write the yield differential as
Rt(n)
Rt(1)
=
Q(1)
t
(n)
(1)
(1)
P EHt × Ft,1 · · · Ft,n−1
1/n
.
(37)
(1)
For n = 1, note that P EH = Q(1)
t , and, by convention, Ft,0 = 1. Long-term
rates exceed short-term rates if short-term bond prices are expected to fall or
if forward-price premiums are negative.
Holding-Period Premium
There are many ways that one can transport money or goods from period t to
period t + j. Until now, we have emphasized the comparison between buying
a j -period bond and buying a sequence of one-period bonds. Another option,
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Federal Reserve Bank of Richmond Economic Quarterly
however, is to purchase an i-period bond, where i > j, and sell the bond
(i−j )
in period t + j for the price Qt+j , which is uncertain as of period t. Of
course, the bond pricing relationships derived previously must imply that the
consumer is indifferent between these two strategies. That is,
(j )
Et t+j =
Qt
Q(i)
t
(i−j )
Et t+j Qt+j
.
(38)
The left-hand side is the expected payoff in period t+j in utility terms to buying
(j )
one j -period bond in period t for price Qt ; note that the only uncertainty
with respect to this strategy involves the marginal utility of a dollar in period
t + j . The right-hand side is the expected return in period t + j in utility terms
(j )
to spending the same amount, Qt , on an i-period bond in period t and selling
the bond in period t + j. With this strategy, there is uncertainty both about the
marginal utility of a dollar in period t + j and about the price at which once
can sell the bond in period t + j.
An intuitively appealing property similar to the pure expectations hypothesis is that the expected dollar return to these two strategies should be the
same. By now, it is probably clear that while the expected utility return must
be the same (as reflected in [38]), the expected dollar return will generally be
different for the two strategies. The dollar return to buying the j -period bond
(j )
and holding it to maturity is certain and given by 1/Qt . The expected dollar
return to buying the i-period bond and selling it in period t + j is given by
(i−j )
Et Qt+j
.
Q(i)
t
(39)
So the expected “premium” for holding an i-period bond for j periods is
(j )
(i,j )
Ht
=
(i−j )
Qt Et Qt+j
Q(i)
t
.
(40)
From (38) we can write this premium as
(i−j )
Et t+j Et Qt+j
(i,j )
Ht
=
(i−j )
Et t+j Qt+j
(41)
or
(i,j )
Ht
=
1 + covt
=
1 + covt
1
(i−j )
Qt+j
(i−j )
Et Qt+j
,
t+j
Et t+j
1
Et+j (t+i /t+j )
, Et t+jt+j
Et (t+i /t+j )
(42)
.
(43)
A. L. Wolman: Bond Prices
331
The holding-period premium is driven by the same uncertainty as the forward
premium, except over a possibly longer horizon. If future marginal utility is
positively conditionally correlated with the future price of an (i − j )-period
bond, then the i-period bond will tend to generate capital gains when they are
highly valued, so individuals will not require a high expected
return. That is,
(i,j )
the relative expected return Ht
will be low when covt
t+j
Et t+j
,
(i−j )
Qt+j
(i−j )
Et Qt+j
is positive.
Of course, we can also use our earlier derivations to express the holdingperiod premium in ways related to the pure expectations hypothesis and the
Fisher equation. Using the definition of the pure expectations hypothesis, we
have from (26)
(i−j )
(1)
(1)
Et P EHt+j × Ft+j,1
· · · Ft+j,i−j
−1
(i,j )
.
Ht
=
(1)
(1)
(1)
(1)
(1)
Et Qt+j Et Qt+j +1 · · · Et Qt+i−1 × Ft,j
· · · Ft,i−1
(44)
Very loosely speaking, this expression relates the holding-period premium to
the conditional covariance between expected future short prices and expected
future forward premiums. Analogously, using the Fisher equation, we have
from (30)
(j )
(i−j )
(i−j )
Et Pt /Pt+j 1 + t Et qt+j Pt+j /Pt+i
1 + t+j
(i,j )
Ht
.
=
f f
f
qt,j qt,j +1 · · · qt,i−1 Et (Pt /Pt+i ) 1 + (i)
t
(45)
Again, loosely speaking, this expression relates the holding-period premium
to the conditional covariance between the future inflation-risk premium, and
the pure Fisher component of the future (i − j )-period bond price.
6. APPLICATIONS OF THE THEORY
The derivations above provide a textbook-like guide to bond price decompositions from the perspective of consumption-based asset pricing theory.
As we stated at the outset, these decompositions can be a useful input into
the formulation of monetary policy, contributing to an understanding of the
term structure of real and nominal interest rates and expected inflation.11 But
any contribution to our understanding of these variables requires taking the
theory to the data, and this, in turn, requires making some assumptions about
11 We emphasize the usefulness for monetary policy, but there are other applications of the
theory. Many areas of economics emphasize the behavior of real interest rates, and financial market
practitioners use the kind of theories outlined here to aid in the pricing of interest rate derivatives.
332
Federal Reserve Bank of Richmond Economic Quarterly
the unobservable variables or λ, the marginal utility of nominal or real
consumption. According to the pure expectations hypothesis and the pure
Fisher equation, marginal utility is extraneous: armed with an estimate of
expected inflation, we can directly estimate the real rate from data on nominal
rates; likewise, armed with data on the term structure of nominal rates, we
can directly calculate the expected path of future short-term rates. Absent risk
neutrality, however, marginal utility takes center stage, for it is the covariance
of the marginal rate of substitution (marginal utility growth) with the evolution
of the price level that pins down the inflation-risk premium (see [30]); and it
is the autocovariance properties of marginal utility that determine the forward
premium (see [24]).
Researchers applying theory to data on bond prices have gone in two directions concerning the degree of structure they impose on the marginal utility
of consumption. The “pure” consumption-based approach takes a stand on the
form of u (c) in (2) and uses data on consumption and inflation to estimate the
parameters of u (c) and the stochastic processes for consumption and inflation.
The combination of estimated preference parameters and stochastic processes
then comprise a model of bond prices; most importantly, the specification of
u (c) together with data on c make marginal utility “observable.” Campbell
(1986) is a particularly accessible example of this strand of the literature, albeit an example that includes only real bonds; he uses a simple specification
of u (c) and the stochastic process for consumption and derives closed form
expressions for interest rates of all maturities. Campbell’s paper is primarily
pedagogical.12
Two recent papers that use more complicated forms of preferences, include nominal bonds and make a serious attempt to match data are Wachter
(2006) and Piazzesi and Schneider (2006).13 Wachter uses a habit-persistence
specification similar to the one Campbell and Cochrane (1999) apply to equity
pricing. She argues that the model “accounts for many features of the nominal term structure of interest rates.” Most importantly, from our perspective,
the model-based forward premiums that Wachter computes help to account for
the empirical disparity between long-term rates and the corresponding average
of expected future short-term rates—that is, the violation of the pure expectations hypothesis. However, there is still a noticeable divergence between
the actual time series for short-term rates and the path implied by Wachter’s
model. Piazzesi and Schneider use the recursive utility preference specification of Epstein and Zin (1989) and Weil (1989). They emphasize the inflationrisk premium in long-term bonds that arises when inflation brings bad news
12 See also Ireland (1996) and Sarte (1998).
13 Both Wachter (2006) and Piazzesi and Schneider (2006) use preference specifications that
are not encompassed by (2). However, for our purposes, we can view their approaches as involving
complicated specifications of u (c) .
A. L. Wolman: Bond Prices
333
about future consumption growth. Although their model is broadly consistent
with the behavior of the term structure, the short-term rates implied by their
model are substantially less volatile than the data. Regarding the approach
taken by Wachter (2006) and by Piazzesi and Schneider (2006), Campbell
(2006) writes, “The literature on consumption-based bond pricing is surprisingly small, given the vast literature given to consumption-based models of
equity markets.” We can thus expect much more work of this sort in the
coming years.
Ravenna and Seppälä (2006) is one recent example of studying the term
structure of interest rates in a consumption-based model that endogenizes
consumption—the papers mentioned in the previous two paragraphs treat consumption (and inflation) as exogenous. Ravenna and Seppälä embed the asset
pricing apparatus in a New Keynesian business cycle model. They argue that
their model accounts for the cyclical properties of interest rates and the rejections of the pure expectations hypothesis, but they do not provide time series
comparisons of data and model-generated interest rates. Given the high degree of structure required, matching the data with this approach is a daunting
task.
The second major empirical application of bond pricing theory is known
as the “no-arbitrage” or “arbitrage-free” approach. With this approach, one
avoids making any parametric assumptions about the form of u (c). What the
no-arbitrage approach does carry over from the theory laid out previously is
the idea that there exists a marginal rate of substitution that prices all bonds;
that is, (11) holds for some strictly positive random variable mt ≡ t /t−1 . A
good introduction to this approach is Backus, Foresi, and Telmer (1998), and
a recent example is Kim and Wright (2005). Those authors and many others
assume that the time-series behavior of the yield curve is driven by a small
number of latent factors. The arbitrage-free approach has the advantage of
being able to fit observed time series on bond prices quite well, thereby opening
the door to discussing relatively small changes in the term structure of real or
nominal rates. For example, Kim and Wright provide time series plots of the
forward rate along with the estimated expected short rate and the estimated
term premium. However, this approach has limitations from the perspective of
macroeconomics in that it does not provide a framework for studying the joint
determination of bond prices and macroeconomic outcomes. Indeed, Duffee
(2006) writes, “some readers . . . call this a nihilistic model of term premia.”
He views the approach in a positive light, though, as “an intermediate step in
the direction of a correctly specified economic model of premia, not an end in
itself.”
334
7.
Federal Reserve Bank of Richmond Economic Quarterly
CONCLUSION
Nominal and real interest rates are often viewed from the perspectives of
the intuitively appealing Fisher relationship and pure expectations hypothesis. Modern asset pricing theory implies that those relationships should not
be expected to hold exactly if investors are risk-averse. We have used that
theory to describe how the deviations from the Fisher relationship and the pure
expectations hypothesis depend on particular covariances. In the process, we
have meant to provide an introduction to the consumption-based modeling of
bond prices. From the standpoint of macroeconomics and monetary policy,
the value of this approach is that it allows researchers to interpret the behavior
of the term structure of real and nominal bond prices in ways that relate to
macroeconomic activity and monetary policy.
REFERENCES
Backus, David, Silverio Foresi, and Christopher Telmer. 1998.
“Discrete-Time Models of Bond Pricing.” Available at:
http://pages.stern.nyu.edu/˜dbackus/tuck.ps (accessed September 13,
2006).
Bernanke, Ben S., and Michael Woodford. 1997. “Inflation Forecasts and
Monetary Policy.” Journal of Money, Credit and Banking 29 (4–2):
653–84.
Campbell, John Y. 1986. “Bond and Stock Returns in a Simple Exchange
Model.”Quarterly Journal of Economics 101 (4): 785–804.
Campbell, John Y. 2006. Discussion of Monika Piazzesi and Martin
Schneider, “Equilibrium Yield Curves.” Available at:
www.nber.org/books/macro21/campbell7-24-06comment.pdf. (accessed
September 13, 2006).
Campbell, John Y., and John Cochrane. 1999. “By Force of Habit: A
Consumption-Based Explanation of Aggregate Stock Market Behavior.”
Journal of Political Economy 107 (2): 205–51.
Campbell, John Y., and Robert Shiller. 1991. “Yield Spreads and Interest
Rate Movements: A Bird’s Eye View.” Review of Economic Studies 58
(3): 495–514.
Cochrane, John. 2001. Asset Pricing. Princeton, NJ: Princeton University
Press.
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Duffee, Gregory R. 2006. “Are Variations in Term Premia Related to the
Macroeconomy?” Available at:
http://faculty.haas.berkeley.edu/duffee/duffee premia macro.pdf
(accessed September 13, 2006).
Epstein, Larry G., and Stanley E. Zin. 1989. “Substitution, Risk Aversion,
and the Temporal Behavior of Consumption and Asset Returns: A
Theoretical Framework.” Econometrica 57 (4): 937–69.
Fisher, Irving. [1896] 1997. Appreciation and Interest. Reprinted in vol. 1 of
The Works of Irving Fisher, ed. William J. Barber. London: Pickering
and Chatto.
Hicks, John R. 1939. Value and Capital. Oxford, UK: Oxford University
Press.
Humphrey, Thomas M. 1983. “The Early History of the Real/Nominal
Interest Rate Relationship.” Federal Reserve Bank of Richmond
Economic Review 69 (3): 2–10.
Ireland, Peter. 1996. “Long-Term Interest Rates and Inflation: A Fisherian
Approach.” Federal Reserve Bank of Richmond Economic Quarterly 82
(1): 21–35.
Kim, Don H., and Jonathan H. Wright. 2005. “An Arbitrage-Free
Three-Factor Term Structure Model and the Recent Behavior of
Long-Term Yields and Distant-Horizon Forward Rates.” Federal Reserve
Board Finance and Economics Discussion Series 2005-33 (August).
Lucas, Robert E., Jr. 1978. “Asset Prices in an Exchange Economy.”
Econometrica 46 (6): 1429–45.
Lutz, Friedrich A. 1940. “The Strucure of Interest Rates.” Quarterly Journal
of Economics 55 (1): 36–63.
Mundell, Robert. 1963. “Inflation and Real Interest.” Journal of Political
Economy 71 (3): 280–3.
Piazzesi, Monika, and Martin Schneider. 2006. “Equilibrium Yield Curves.”
NBER Macroeconomics Annual Vol. 21. Available at:
http://www.nber.org/books/macro21/piazzesi-schneider7-22-06.pdf.
(accessed September 13, 2006).
Ravenna, Federico, and Juha Seppälä. “Policy and Rejections of the
Expectations Hypothesis.” Available at:
http://ic.ucsc.edu/%7Efravenna/home/mopoeh4.pdf (accessed
September 13, 2006).
Sargent, Thomas J. 1987. Dynamic Macroeconomic Theory. Cambridge,
MA: Harvard University Press.
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Sarte, Pierre-Daniel G. 1998. “Fisher’s Equation and the Inflation Risk
Premium in a Simple Endowment Economy.” Federal Reserve Bank of
Richmond Economic Quarterly 84 (4): 53–72.
Wachter, Jessica A. 2006. “A Consumption-Based Model of the Term
Structure of Interest Rates.” Journal of Financial Economics 79 (2):
365–99.
Watson, Mark W. 1999. “Explaining the Increased Variability in Long-Term
Interest Rates.” Federal Reserve Bank of Richmond Economic Quarterly
85 (4): 71–96.
Weil, Philippe. 1990. “Nonexpected Utility in Macroeconomics.” Quarterly
Journal of Economics 105 (1): 29–42.
Wood, John H. 1964. “The Expectations Hypothesis, The Yield Curve and
Monetary Policy.” Quarterly Journal of Economics 78 (3): 457–70.
Stark Optimal Fiscal
Policies and Sovereign
Lending
Pierre-Daniel G. Sarte
C
apital income taxes are a salient feature in the taxation schemes of
many modern countries, though countries make distinctions between
capital gains and income earned by capital. Most countries include
rents received on capital in the income calculation for each taxpayer. When
the rates are averaged over the 1965–1996 period, average capital income tax
rates can be moderately high in industrially advanced nations, ranging from
24.1 percent in France to 54.1 percent in the United Kingdom. When averaged
across the first six years of the 1990s, average capital income tax rates for the
same countries are 25 percent and 47.7 percent, respectively. Over the same
periods, the United States had average capital income taxes of 40.1 percent
(1965–1996) and 39.7 percent (1990–1996) (see Domeij and Heathcote 2004).
Following the work of Judd (1985), and Chamley (1986), much of the
literature on optimal taxation has argued that it is not efficient to tax capital
in the long run. As shown in Atkeson, Chari, and Kehoe (1999), this policy
prescription is relatively robust in the sense that it holds whether agents are
heterogenous or identical, the economy’s growth rate is endogenous or exogenous, and the economy is open or closed.1 At the same time, however,
the notion that long-run capital taxation is inefficient arises in settings where
We wish to thank Andreas Hornstein, Juan Carlos Hatchondo, Alex Wolman, and Nashat Moin
for their comments. The views expressed in this article do not necessarily represent those
of the Federal Reserve Bank of Richmond, the Board of Governors of the Federal Reserve
System, or the Federal Reserve System. Any errors are my own.
1 Exceptions to this recommendation include Correia (1996), and Jones, Manuelli, and Rossi
(1997), who show that the optimal long-run tax on capital differs from zero when other factors of
production are either untaxed or not taxed optimally. As pointed out in Erosa and Gervais (2001),
capital income taxes in an overlapping generations environment are not just distortionary, they
involve some redistribution among agents. Hence, optimal steady state capital income taxes need
not be zero in such a framework, as shown in the early work of Atkinson and Sandmo (1980), and
later Garriga (1999), as well as in Erosa and Gervais (2002) with age-dependent taxes. Finally,
Federal Reserve Bank of Richmond Economic Quarterly Volume 92/4 Fall 2006
337
338
Federal Reserve Bank of Richmond Economic Quarterly
optimal policies are often extreme at shorter horizons. For instance, when the
capital stock is sufficiently large and no restrictions are placed on the capital
income tax, it is optimal for the government to raise all revenues through a
single capital levy at date 0 and never again tax either capital or labor. For that
reason, Chamley (1986) imposes a 100 percent exogenous upper bound on the
capital income tax, which Chari, Christiano, and Kehoe (1994) show can be
motivated by assuming that households have the option of holding onto their
capital, subject to depreciation, rather than renting it to firms. With an upper
bound imposed on capital income tax rates, optimal fiscal policy continues to
be surprisingly stark, with the optimal capital tax rate set at confiscatory levels
for a finite number of periods, after which the tax takes on an intermediate
value between 0 and 100 percent for one period, and is zero, thereafter. This
article points out that government lending to households, which is seldom
observed in practice, plays a crucial role in generating extreme optimal fiscal
policies. Absent a domestic debt instrument, more moderate capital and labor
tax rates emerge as optimal, although the capital tax rate does converge to
zero, asymptotically.
Without an upper bound imposed on capital income taxes, it is easy to see
why a single capital levy at date 0 is optimal in the environment studied by
Chamley (1986). Since the capital stock is fixed at date 0, the initial capital
levy amounts to a single lump sum tax and no distortions are ever imposed
on resource allocations over time. The economy, therefore, achieves a firstbest optimum. That said, the presence of a debt instrument plays a key role
in the implementation of a single initial capital levy. In particular, such an
instrument allows the government to front-load all taxes in the initial period
(equal to the net present discounted value of future government expenses),
lend the proceeds to the private sector, and finance government expenditures
from interest revenue on the loans. Thus, Chamley’s model never generates
government debt but generates government surpluses which are lent back to
households.
Although Chamley’s single initial capital levy allows for first-best allocations to be achieved, such a taxation scheme has evidently little to do with the
kinds of policies one observes in practice. In particular, almost all countries
continue to rely on distortional tax systems to finance their public expenditures. This article highlights the fact that environments that limit ex ante government lending are more apt to generate nonzero optimal distortional taxes at
all dates. In particular, we provide an analysis of Chamley’s taxation problem
without a policy instrument that allows for sovereign lending. In that case, the
government cannot carry the proceeds from a large initial tax to future
see Aiyagari (1995) for an environment with idiosyncratic uninsurable shocks where optimal capital
income taxes are not zero in the long run.
P. D. Sarte: Fiscal Policies and Sovereign Lending
339
periods, and the single capital levy prescribed in Chamley (1986) cannot be
implemented.2
For the purposes of this article, we think of restrictions on sovereign lending as a (rudimentary or ad hoc) way of getting rid of extreme taxation policies
in the short run.3 More generally, gaining a better understanding of institutional or agency constraints that endogenously limit the kinds of contracts the
government can write with households seems central in generating optimal
fiscal policies that more closely resemble those observed in practice. It may
be helpful, for instance, to consider in greater depth the kinds of frictions that
may limit or impede government lending. At first glance, such frictions are
not necessarily obvious. In particular, any potential commitment problems
on the part of households (to repay their loans) should easily be overcome by
suitable punishment such as the garnishment of wages. There have also been
times in U.S. history, (such as during World War II), where the government
directly owned privately operated capital.4
Formally, the taxation problem we study, introduced by Kydland and
Prescott (1980), discusses the time inconsistency of optimal policy. While
matters related to time inconsistency lie beyond the scope of this article, we
use the insights of Kydland and Prescott (1980), as well as more recent work by
Marcet and Marimon (1999), to present the solution to this problem in terms
of stationary linear difference equations that can be solved using standard
numerical methods. While Kydland and Prescott (1980) show how the taxation
problem they consider can be written (and solved) as a recursive dynamic
program, the article does not ultimately present properties of the solution
either in the short run or long run.
What do optimal tax rates look like when one exogenously prohibits
sovereign lending? Numerical simulations carried out in this article indicate
that capital income tax rates are never set either at 100 percent or zero at any
point during the transition to the long-run equilibrium. Furthermore, we find
that labor income is subsidized in the first few periods. This feature of optimal
fiscal policy drives up labor supply and allows household consumption to be
everywhere above its long-run equilibrium along its transition path. Finally,
we provide straightforward proof of the optimality of zero long-run capital
2 In this case, what matters is a government’s ability to bequeath revenue-generating assets to
its successor that, potentially, render the use of future taxes unnecessary. Thus, optimal confiscatory
short-run capital taxes would behave as stated in Chamley (1986) in environments in which the
government can lend directly to firms. Alternatively, one can also imagine optimal allocations
emerging in an environment in which the government directly owned the capital stock and had no
disadvantage in operating production directly.
3 With this restriction in place, an exogenously imposed upper bound on capital income tax
rates is no longer necessary. Moreover, the upper limit of 100 percent imposed by Chamley (1986)
is not helpful in creating moderate optimal tax rates since this limit turns out to be binding in
the short run.
4 See McGrattan and Ohanian (1999).
340
Federal Reserve Bank of Richmond Economic Quarterly
taxes that does not rely on the primal approach used in Chamley (1986) and
summarized in Ljungqvist and Sargent (2000, chapter 12).
This article is organized as follows. Section 1 provides a brief summary
of findings in the literature on optimal fiscal policy in the presence of domestic
lending and borrowing. In Section 2, we describe the economic environment
under consideration. Section 3 presents the Ramsey problem associated with
the analysis of optimal fiscal policy. Salient properties of the long-run Ramsey
equilibrium are discussed in Section 4. Section 5 gives a numerical characterization of the transitional dynamics of optimal tax rates. Section 6 offers
concluding remarks.
1. A BRIEF DESCRIPTION OF STARK FISCAL POLICIES
WITH SOVEREIGN LENDING
This section describes the kinds of extreme optimal fiscal policies that have
been described in the optimal taxation literature. Beginning with Chamley
(1986), we have already seen in the introduction that a single capital levy at
date 0, if feasible, allows the economy to achieve first-best allocations. The
problem, of course, is that the associated capital income tax rate might well
exceed 100 percent, in which case one might interpret the tax as not only
applying to capital income but more directly to the capital stock. Whether
this policy is feasible ultimately depends on if the initial capital stock is large
enough to finance the net present discounted value of future government expenditures. If so, the necessary revenue is raised entirely through the levy
at date 0, and lent back to households with the proceeds from the loans used
to finance the stream of government expenditures over time. In that sense, a
need for strictly positive distortional taxes never arises.
When capital income tax rates are restricted to be at most 100 percent,
Chari, Christiano, and Kehoe (1994) show that it is optimal to set tax rates at
their upper limit for a finite number of periods, after which the capital tax rate
takes on an intermediate value and is zero, thereafter. The intuition underlying
this result relates directly to the distortional nature of capital income tax rates.
In particular, having the capital tax rate positive in some period t > 0 distorts
savings decisions, and thus, private capital allocations, in all prior periods.
Hence, front-loading capital income taxes, by having the associated tax rates
set at their upper limit from date 0 to some finite date t > 0, distorts the
least number of investment periods. This intuition is only partially complete
in that household preferences also play an important role in determining the
horizon over which capital income is initially taxed. When preferences are
separable in consumption and leisure, it is not optimal to tax capital after the
initial period, although labor taxes may be positive at all dates (see Chari,
Christiano, and Kehoe 1994). Xie (1997) shows that when preferences are
logarithmic in consumption less leisure, it is optimal never to tax labor while
P. D. Sarte: Fiscal Policies and Sovereign Lending
341
capital income tax rates (Chari, Christiano, and Kehoe 1994) hit their upper
bound for a finite number of periods and are zero, thereafter.
All of the above policies have in common a radical character and a lack
of resemblance to more moderate capital tax rates in practice (i.e., capital
tax rates that are neither set at confiscatory rates nor zero). However, a key
part underlying the mechanics of these policies relates to the fact that the
government is able to build large negative debt holdings by having the capital
income tax rate hit its upper limit over some initial period of time, date 0 to
date t > 0. In Xie (1997), it is apparent that once these negative debt holdings
are large enough to finance the remaining net present discounted value of
government expenditures, then no distortional taxes need ever be set again. In
essence, date t is then analogous to date 0 in Chamley (1986).
The following sections examine the problem of optimal taxation initially
posed by Chamley (1986), but without the policy instrument that allows the
government to accumulate large negative debt holdings. Absent this instrument, numerical simulations suggest that it is possible to have more moderate
taxes on capital and labor emerge as optimal at every date, without any bounds
necessarily imposed on either capital or labor income tax rates. Since the restriction on sovereign lending takes away the usefulness of building a large
negative debt position on the part of the government, setting capital tax rates at
confiscatory levels is no longer warranted. More importantly, this observation
suggests further consideration of the role of institutional or agency constraints
that prevent the government from confiscating capital income for an extended
period of time, frictions that limit sovereign lending in practice, and how these
constraints help shape optimal fiscal policy more generally.
2.
ECONOMIC ENVIRONMENT
Consider an economy populated by infinitely many households whose preferences are given by
⎡
⎤
1+ γ1
∞
1−σ
n
−
1
c
U=
− ν t 1 ⎦ , σ > 0, γ > 0,
βt ⎣ t
(1)
1
−
σ
1+ γ
t=0
where ct and nt denote household consumption and labor effort at date t,
respectively, and β ∈ (0, 1) is a subjective discount rate.
A single consumption good, yt , is produced using the technology
, 0 < α < 1,
yt = ktα n1−a
t
(2)
where kt denotes the date t stock of private capital. Capital can be accumulated
over time and evolves according to
kt+1 = it + (1 − δ)kt ,
(3)
342
Federal Reserve Bank of Richmond Economic Quarterly
where δ ∈ (0, 1) denotes the depreciation rate and it represents household
investment. Production can be used for either private or government consumption, or to increase the capital stock,
ct + it + gt = ktα n1−α
t,
(4)
where {gt }∞
t=0 is an exogenously given sequence of public expenditures.
As in Chamley (1986), the government finances its purchases using timevarying linear taxes on labor income and capital income. We denote these
tax rates by τ nt and τ kt , respectively. At each date, the government’s budget
constraint is given by
τ kt rt kt + τ nt wt nt = gt ,
(5)
where rt and wt are the market rates of return to capital and labor. The leftand right-hand sides of (5) represent sources and uses of government revenue,
respectively.
There exists a large number of homogenous small size firms that act com∞
petitively. Taking the sequences of prices {rt }∞
t=0 and {wt }t=0 as given, each
firm maximizes profits and solves
− r t kt − w t n t .
max ktα n1−α
t
kt , nt
(6)
The implied first-order conditions equate prices to their corresponding marginal
products, rt = αktα−1 n1−α
= α yktt and wt = (1 − α)ktα n−α
= (1 − α) nytt .
t
t
At each date, households decide how much to consume and save in the
form of private capital investment, as well as how much labor effort to provide. Taking the sequences of government expenditures, {gt }∞
t=0 , and tax
rates, {τ nt , τ kt }∞
,
as
given,
these
households
maximize
lifetime
utility
subject
t=0
to their budget constraint,
⎡
⎤
1+ γ1
∞
1−σ
n
−1
c
max
−ν t 1⎦
βt ⎣ t
(PH )
ct , nt , kt+1
1
−
σ
1+ γ
t=0
subject to
ct + kt+1 = (1 − τ kt )rt kt + (1 − τ nt )wt nt + (1 − δ)kt ,
k0 > 0 given.
(7)
The first-order necessary conditions implied by problem (7) yield a static
equation describing households’ optimal labor-leisure choice,
1
νntγ = ct−σ (1 − τ nt )wt ,
(8)
as well as a standard Euler equation describing optimal consumption allocations over time,
−σ
(9)
(1 − τ kt+1 )rt+1 + 1 − δ .
ct−σ = βct+1
P. D. Sarte: Fiscal Policies and Sovereign Lending
343
The constraints (5) and (7), together with the optimality conditions (8) and (9)
and the expression for prices given above, describe our economy’s decentralized allocations over time.
3. THE RAMSEY PROBLEM
Having described the decentralized behavior of households and firms, we now
tackle the problem of choosing policy optimally. Thus, consider a benevolent government that, at date 0, is concerned with choosing a sequence of
tax rates that maximize household welfare given the exogenous sequence of
government spending. In choosing policy, this government takes as given the
behavior of households and firms. We further assume that, at date 0, the government can credibly commit to any sequence of policy actions. The problem
faced by this benevolent planner is to maximize (1) subject to the constraints
(5) and (7), and households’ optimality conditions (8) and (9), where prices
are given by marginal products.5
We can address the policy problem described at the start of this section by
solving the following Lagrangian,
⎡
⎤
1+ γ1
∞
1−σ
n
−
1
c
max
L=
−ν t 1⎦
βt ⎣ t
(PR )
1
−
σ
1
+
ct , nt , τ kt , τ nt , kt+1
γ
t=0
+
+
∞
t=0
∞
−σ
β t μ1t βct+1
(1 − τ kt+1 )rt+1 + 1 − δ − ct−σ
β t μ2t τ kt rt kt + τ nt wt nt − gt
t=0
+
∞
β t μ3t (1 − τ kt )rt kt + (1 − τ nt )wt nt + (1 − δ)kt − ct − kt+1
t=0
+
∞
t
β μ4t
ct−σ (1
−
τ nt )wt
1
γ
− νnt
,
t=0
where the Lagrange multipliers μj t , j = 1, ..., 4, are all nonnegative at the
optimum.
The first-order necessary conditions associated with problem (10) that are
related to the optimal choices of ct , nt , and τ kt are as follows:
5 It is tempting at this point to simply solve a Lagrangian corresponding to the policy problem we have just described. The exact way in which to write this Lagrangian, however, is not
immediately clear. To apply Lagrangian methods to this constrained maximization problem, and in
particular, to interpret the Lagrange multipliers associated with constraints (5), (7), (8), and (9) as
nonnegative, one must first write these constraints as inequalities that define convex sets. See the
Appendix for details.
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Federal Reserve Bank of Richmond Economic Quarterly
ct : ct−σ − σ μ1t−1 ct−σ −1 [(1 − τ kt )rt + 1 − δ] + σ μ1t ct−σ −1
(10)
−μ3t − σ μ4t ct−σ −1 (1 − τ nt )wt = 0, t > 0,
with
(11)
c0−σ − σ μ10 c0−σ −1 − μ30 − σ μ40 c0−σ −1 (1 − τ n0 )w0 = 0 at t = 0,
1
∂wt
γ
k ∂rt
n
+ τ t (wt + nt
)
nt : −νnt + μ2t τ t kt
∂nt
∂nt
∂rt
∂wt
+ (1 − τ nt )(wt + nt
)
+μ3t (1 − τ kt )kt
∂nt
∂nt
1−γ
∂wt
ν
∂rt
− nt γ + μ1t−1 ct−σ (1 − τ kt )
+μ4t ct−σ (1 − τ nt )
∂nt
γ
∂nt
= 0, t > 0,
(12)
with
1
∂r0
∂w0
+ τ n0 (w0 + n0
)
−νn0γ + μ20 τ k0 k0
∂n0
∂n0
∂r0
∂w0
k
n
+ (1 − τ 0 )(w0 + n0
)
+μ30 (1 − τ 0 )k0
∂n0
∂n0
∂w0
ν 1−γ
+μ40 c0−σ (1 − τ n0 )
− n0 γ
∂n0
γ
= 0, at t = 0,
(13)
and
τ kt : −μ1t−1 ct−σ + (μ2t − μ3t )kt = 0, t > 0,
(14)
μ20 − μ30 = 0, at t = 0.
(15)
with
The fact that the above first-order conditions differ at t = 0 and t > 0 suggests
an incentive to take advantage of initial conditions in the first period only, with
the promise never to do so in the future. It is exactly in this sense that the
optimal policy is not time consistent. Once date 0 has passed, a planner at date
t > 0 who re-optimizes would want to start with choices for consumption,
labor effort, and capital taxes that differ from what was chosen for that date at
time 0.
It should be clear that the incentives identified by Chamley (1986) continue
to be present in our model economy. Consider that the difference between
equations (14) and (15), which governs the optimal choice of τ t at dates t = 0
and t > 0, and involves an additional term in (14),
−μ1t−1 uc (ct ) < 0.
(16)
P. D. Sarte: Fiscal Policies and Sovereign Lending
345
This term originates from the Euler constraint in problem (10),
βuc (ct ) [(1 − τ t )rt + 1 − δ] = uc (ct−1 ), and corresponds to the reduction
in the after-tax real return to investment made at date t − 1 which is created by
an increase in the tax rate at time t. Consequently, in committing to a tax rate
in a given period t > 0, the government takes into account the implied substitution effect on investment decisions undertaken in the preceding period. Of
course, at date t = 0, no such distortion exists since history commences on
that date with a predetermined capital stock, k0 . In choosing τ 0 , therefore, the
government is free to ignore its effects on previous investment decisions that
can be thought of as “sunk”; and there exists some incentive for the optimal
sequence of tax rates to begin with a high tax in period 0 relative to all other
dates.
A central insight in Kydland and Prescott (1980) is that despite the time
inconsistency problem we have just mentioned, it is actually possible to collapse equations (10) through (15) into a set of stationary difference equations
∀t ≥ 0. This requires interpreting the lagged Lagrange multiplier μ1t−1 as a
predetermined variable with initial condition μ1t−1 = 0 at t = 0.
The remaining first-order conditions associated with problem (10) determining the optimal choice of labor income taxes and private capital are,
respectively,
τ nt : (μ2t − μ3t )nt − μ4t ct−σ = 0, t ≥ 0,
(17)
and
−σ
kt+1 : μ1t βct+1
(1 − τ kt+1 )
τ kt+1 (rt+1
∂rt+1
+ βμ2t+1
∂kt+1
∂rt+1
∂wt+1
n
+ kt+1
) + τ t+1 nt+1
− μ3t + βμ3t+1
∂kt+1
∂kt+1
(1 − τ kt+1 )[rt+1 + kt+1
∂rt+1
∂wt+1
] + (1 − τ nt+1 )nt+1
+1−δ
∂kt+1
∂kt+1
−σ
+βμ4t+1 ct+1
(1 − τ nt+1 )
(18)
∂wt+1
= 0, t ≥ 0.
∂kt+1
4. THE STATIONARY RAMSEY EQUILIBRIUM
With the optimality conditions (10) through (18) in hand, we first turn to longrun properties of optimal taxes and revisit the notion that it is not efficient
to tax capital in the long run. We do so, however, without any reference to
the primal approach that is standard in the literature, but rely instead on the
simple first-order conditions we have just derived. To this end, we define the
long-run equilibrium of the Ramsey problem as follows:
346
Federal Reserve Bank of Richmond Economic Quarterly
Definition: A stationary Ramsey equilibrium is a ninetuple (c, n, k, τ n ,
τ k , μ1 , μ2 , μ3 , μ4 ) that solves the government budget constraint (5), households’ budget constraint (7), the optimality condition for labor effort (8), and
the Euler equation (9), as well as the first-order conditions associated with
problem (10), equations (10), (12), (14), (17), and (18), all without time subscripts.
It is straightforward to show that in a stationary Ramsey equilibrium,
equations (14), (17), and (18) imply that
τ k μ2 βr − μ3 1 − β (1 − τ k )r + 1 − δ = 0.
(18)
From the Euler equation in the stationary equilibrium, it follows that 1 −
β[α(1 − τ k ) yk + 1 − δ] = 0. Hence equation (18) above reduces to
τ k μ2 βr = 0.
(19)
Now, we have that either τ k > 0 or τ k = 0. Suppose first that τ k > 0. Then,
it must be the case that μ2 = 0. From equation (14), this would mean that
μ1 = −μ3 kcσ ,
which implies that μ1 = μ3 = 0 since μ1 and μ3 are both nonnegative. However, in that case, all Lagrange multipliers are zero in the steady state and
c−σ = 0 from equation (10), which cannot be a solution because household
utility would be unbounded. Hence, τ k > 0 cannot be a solution, and therefore, τ k = 0. As in Chamley (1986), it is optimal not to tax capital in the
long run. From the budget constraint, this implies that the steady state tax on
labor is essentially determined by the extent of government expenditures. For
instance, if government spending was a constant fraction, φ of output in the
long run, we would have the optimal tax on labor income in the long run to
φ
be simply τ n = 1−α
.
The notion that it is optimal to set capital income tax rates to zero in the
long run is independent of whether government lending takes place. This is
relatively easy to see within our framework when no upper bound is imposed
on the capital income tax rate. In that case, the government budget constraint
(5) reads as
τ kt rt kt + τ nt wt nt + bt+1 = gt + (1 + rtb )bt ,
(20)
where bt denotes one-period government bonds that are perfectly substitutable
with capital, and rtb is the return on bonds from period t − 1 to t. Government
lending takes place when bt < 0. Moreover, the household budget constraint
becomes
ct + kt+1 + bt+1 = (1 − τ kt )rt kt + (1 − τ nt )wt nt + (1 − δ)kt + (1 + rtb )bt . (21)
Substituting these modified constraints in problem (10), the planner now also
has to decide how much sovereign lending will take place. A simple arbitrage
P. D. Sarte: Fiscal Policies and Sovereign Lending
347
equation (obtained from the modified household problem) dictates that in the
decentralized equilibrium, 1 + rtb = (1 − τ kt+1 )rt+1 + 1 − δ. Hence, the
first-order condition associated with the optimal choice of bt+1 in the Ramsey
problem is
(μ2t − μ3t ) − β[(1 − τ kt+1 )rt+1 + 1 − δ](μ2t+1 − μ3t+1 ) = 0 ∀t ≥ 0. (22)
It is now easy for us to show that capital income taxes are zero in the long run. In
fact, with no upper bound imposed on the capital income rate, τ kt = 0 ∀t > 0.
To see this, observe that equations (15) and (22) imply that μ2t − μ3t = 0
∀t ≥ 0. It follows from (14) that μ1t−1 = 0 ∀t ≥ 0 and from (17) that μ4t = 0
∀t ≥ 0. Substituting these results into equation (18) gives
−σ
rt+1 + 1 − δ ∀t ≥ 0,
ct−σ = βct+1
which is simply the household’s Euler equation (9) when τ kt = 0 ∀t > 0.
When an upper bound is imposed on the capital income tax rate, τ kt ≤ 1
∀t ≥ 0, it is still the case that τ kt = 0 ∀t > 0 when preferences are separable
in consumption and leisure, and that limt→∞ τ kt = 0, otherwise. Proof of the
latter results is more difficult to see using our Lagrangian formulation, but is
nicely presented in Erosa and Gervais (2001).
5. TRANSITIONS TO THE STEADY STATE
Even in the absence of an instrument that allows government lending to households, we saw in the previous section that the optimal fiscal policy with commitment prescribes zero capital taxes in the long run. In the short and medium
run, however, capital income tax rates are not as extreme as predicted in
a model with a debt instrument. Compared to the environment studied by
Chari, Christiano, and Kehoe (1994) for instance, where capital income tax
rates are set at their upper bound up to some date t and are zero thereafter,
capital income tax rates in our framework approach confiscatory rates only in
the initial period and then decline monotonically over time. Labor income tax
rates are also moderate at every point along the transition.
To illustrate these points, we carry out a numerical simulation of our economy when fiscal policy is determined optimally. The parameters we use are
standard and selected along the lines of other studies in quantitative general
equilibrium theory. A time period represents a quarter and we assume a 6.5
percent annual real interest rate, β = 0.984, and a 10 percent capital depreciation rate, δ = 0.025. We set the intertemporal elasticity of substitution, 1/σ ,
to 1/2 and the Frisch elasticity of labor supply, γ , to 1.25. The share of private
capital in output in the United States is approximately 33 percent so we assign
a value of α = 1/3. Finally, we fix the share of government expenditures in
output at 0.20.
To compute the transitional dynamics associated with optimal capital and
labor income tax rates, we replace the optimality conditions in Section 3 with
348
Federal Reserve Bank of Richmond Economic Quarterly
log-linear approximations around the stationary Ramsey equilibrium. The
solution paths for the state and co-state variables are then computed using
techniques described in Blanchard and Kahn (1980) or in King, Plosser, Rebelo
(1988). The resulting system of linearized equations possesses a continuum of
solutions, but only one of these is consistent with the transversality condition
associated with the household problem.
Figure 1 depicts transitions to the stationary Ramsey equilibrium when the
initial capital stock is set at its long-run level. In other words, Figure 1 shows
transitions to the steady state when restarting the problem. In Panel A, we
can see that capital income tax rates start near confiscatory rates in the initial
period but quickly fall within 10 quarters to a more moderate range, at less
than 35 percent.6 Thus, the notion that capital income tax rates are optimally
higher in the initial periods remain, but these rates are within a moderate
range for the greater part of the transition. More specifically, in contrast to
Chari, Christiano, and Kehoe (1994), capital income tax rates are never set at
either 100 percent or zero at any point during the transition.7 Interestingly,
the optimal fiscal policy suggests subsidizing labor income in the first few
periods, after which labor income taxes monotonically rise to their steady
state. Because labor income represents 2/3 of total output in our calibrated
economy, and because government expenditures account for 20 percent of
output, the labor income tax rate approaches 30 percent asymptotically as
capital income tax rates approach zero.
The initial subsidization of labor income generates an increase in labor
input, shown in Figure 1, Panel D, along the transition to the steady state.
As a result, household consumption is everywhere above its long-run level on
its way to the steady state. Moreover, the optimal fiscal policy is such that
households are able to front-load consumption.
6.
CONCLUSION
In this article, we highlighted that environments in which ex ante government
lending is limited are more apt to generate optimal distortional taxes at all
dates that do not share the stark character of those typically presented in the
literature. Absent an instrument that allows households to borrow from the
government, the government cannot carry the proceeds from a large initial tax
to future periods, and the single capital levy prescribed in Chamley (1986)
cannot be implemented.
6 The captial stock is fixed in period 0. It then decreases slowly while capital income tax
rates are relatively high and converges back to its long-run level.
7 Chari, Christiano, and Kehoe (1994) present an actual numerical solution to the problem
without linearizing. The linearization in our case involves an approximation, but the fact that optimal taxes do not involve corner solutions in our framework does not depend on the approximation.
P. D. Sarte: Fiscal Policies and Sovereign Lending
349
Figure 1 Transitions to the Steady State, k0 kss
Panel B
Labor Income Tax Rates
30
70
20
Percent
Percent
Panel A
Capital Income Tax Rates
90
50
10
5
30
0
-5
10
0
-15
0
20
40
60
80
100 120 140 160 180
0
20
40
60
Time
80
100 120 140 160 180
Time
Panel D
Labor Input
Panel C
Consumption
22
16
14
18
Percent Dev. from SS
Percent Dev. from SS
12
14
10
6
2
10
8
6
4
2
0
-2
0
20
40
60
80
100 120 140 160 180
Time
0
20
40
60
80
100 120 140 160 180
Time
For an economy whose capital stock is initially below its long-run level,
we have shown that capital income tax rates are never set at either 100 percent or zero at any point during the transition to the long-run equilibrium.
Furthermore, our analysis has highlighted that labor income is subsidized in
the first few periods. This feature of optimal fiscal policy gave rise to increased labor supply and allowed household consumption to be everywhere
above its long-run equilibrium along its transition path. As in Chamley (1986),
however, even without a debt instrument, our analysis continued to prescribe
zero-capital income tax rates in the long run.
We interpret our findings to suggest that a better understanding of institutional or agency constraints that prevent the government from confiscating
capital income for an extended period of time, as well as commitment problems associated with household borrowing, may be central in explaining the
character of observed fiscal policies.
350
Federal Reserve Bank of Richmond Economic Quarterly
APPENDIX
Marcet and Marimon (1999) point out that equalities associated with feasibility constraints can generally (and in this case) be replaced with weak inequalities, with an appeal to nonsatiated preferences thereby guaranteeing that
the feasibility constraints rewritten as such will also be satisfied with equality.
However, an analogous argument for the equality constraints (8) and (9) is less
obvious. Because of the equality signs in (8) and (9), the set of allocations
satisfying these equations is not convex.
Consider the Euler equation (9). Marcet and Marimon (1999) show that
it is possible to rewrite this constraint as a weak inequality in such a way that,
in the optimum of the new problem, this weak inequality is satisfied as a strict
equality. One can be sure, therefore, that the optimum subject to the weak
inequality constraint is the same as that subject to the strict equality (9), and
that one is actually solving the problem of interest.
We now provide a brief description of the arguments presented in Marcet
and Marimon (1999) but refer the reader to the paper for the formal proofs.
The question is whether to write the inequality associated
with (9) as ≤ or
−σ
≥. Consider the case ≥ first, in which ct−σ ≥ βct+1
(1 − τ kt+1 )rt+1 + 1 − δ .
The authors show that writing the inequality constraint in this way actually
makes the first-best allocation feasible, so that the solution would be the unconstrained optimum, which is not the same as the Ramsey equilibrium. Hence,
this option does not yield a solution equivalent to the solution under equation
(9).
Next, consider the
case kwhere the inequality constraint is written as ≤
−σ
−σ
so that ct ≤ βct+1 (1 − τ t+1 )rt+1 + 1 − δ . Writing the inequality in this
way reproduces the household’s first-order condition if the household faced
the constraint kt ≤ k t , where k t is an upper bound imposed on households’
capital position. In other words, the modified Euler equation corresponds
to a setting where the policy instruments available to the planner now include the ability to set an upper limit on capital accumulation, k t . In that
case, Marcet and Marimon (1999) then show that the planner will actually
choose allocations where the constraint kt ≤ k t does not bind, since the
equilibrium with distortional taxes is associated with too little capital relative to the full
This implies that the government will act so that
optimum.
−σ
ct−σ ≤ βct+1
(1 − τ kt+1 )rt+1 + 1 − δ is satisfied with equality, and the optimum is then the same as the Ramsey equilibrium. Similar arguments can be
made regarding constraint (8).
P. D. Sarte: Fiscal Policies and Sovereign Lending
351
REFERENCES
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Markets, Borrowing Constraints, and Constant Discounting.” Journal of
Political Economy 103 (6): 1158–75.
Atkeson, A., V. V. Chari, and P. J. Kehoe. 1999. “Taxing Capital Income: A
Bad Idea.” Federal Reserve Bank of Minneapolis Quarterly Review 23
(3): 3–17.
Atkinson, A., and A. Sandmo. 1980. “Welfare Implications of the Taxation of
Savings.” The Economic Journal. 90 (359): 529–49.
Blanchard, O., and C. M. Kahn. 1980. “The Solution of Linear Difference
Models under Rational Expectations.” Econometrica 48 (5): 1305–12.
Chamley, C. 1986. “Optimal Taxation of Capital Income in General
Equilibrium with Infinite Lives.” Econometrica 54 (3): 607–22.
Chari, V. V., L. Christiano, and P. J. Kehoe. 1994. “Optimal Fiscal Policy in a
Business Cycle Model.” Journal of Political Economy 102 (4): 617–52.
Correia, I. H. 1996. “Should Capital Income be Taxed in the Steady State?”
Journal of Public Economics 60 (1): 147–51.
Domeij, D., and J. Heathcote. 2004. “Factor Taxation with Heterogenous
Agents.” International Economic Review 45 (2): 523–54.
Erosa, A., and M. Gervais. 2001. “Optimal Taxation in Infinitely-Lived
Agent and Overlapping Generations Models: A Review.” Federal
Reserve Bank of Richmond Economic Quarterly (87) 2: 23–44.
Erosa, A., and M. Gervais. 2002. “Optimal Taxation in Life-Cycle
Economies.” Journal of Economic Theory 105 (2): 338–69.
Garriga, C. 1999. “Optimal Fiscal Policy in Overlapping Generations
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Kydland, F. E., and E. C. Prescott. 1980. “Dynamic Optimal Taxation,
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Ljungqvist, L., and T. J. Sargent. 2000. Recursive Macroeconomic Theory.
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Not Your Father’s Credit
Union
John R. Walter
P
roponents of the credit union industry have viewed credit unions as especially well-suited to making small-value, uncollateralized consumer
loans since the time the first United States credit union was formed
in 1909. At that time, other financial institutions such as banks and savings
and loans were focused on alternative types of lending, specifically lending
to businesses as well as to consumers with collateral. For consumers who
lacked real estate or other simple-to-value collateral—in other words for unsecured borrowers—borrowing options were limited, at least for loans from
depositories.
The high fixed costs of extending a small-value loan, and the significant
risk of making an unsecured consumer loan to a borrower about whom the
lender had little information on creditworthiness, meant that a lofty interest
rate was necessary in order for the lender to cover costs. Such rates often
exceeded usury ceilings, maximum interest rates set by state laws. Still, many
borrowers were willing to pay high rates, even if it meant illegal borrowing.
A 1911 estimate indicated that in larger towns and cities, one in five workers
borrowed from illegal lenders (Calder 1999, 118).
Credit unions developed in this environment, allowing individuals to borrow at fairly low interest rates (Whitney 1922, 40–54). When consumers could
band together into groups based upon a common bond—i.e., a shared characteristic such as working for the same company—they could substitute their
knowledge of one another’s creditworthiness for collateral. Credit unions
offered creditworthy borrowers the opportunity to differentiate themselves
from less creditworthy individuals in an era before the advent of nationwide
credit bureaus. Credit union members were differentiated from unknown
The author benefited greatly from comments from Kevin Bryan, Huberto Ennis, Robert Hetzel,
and Ned Prescott. Able research assistance was provided by Kevin Bryan and Nashat Moin.
The views expressed herein are not necessarily those of the Federal Reserve Bank of Richmond
or the Federal Reserve System.
Federal Reserve Bank of Richmond Economic Quarterly Volume 92/4 Fall 2006
353
354
Federal Reserve Bank of Richmond Economic Quarterly
borrowers because of in-depth information among the group of members about
the character and economic prospects of one another. In 1932, the average
size of a U.S. credit union was only 187 members, so intimate knowledge
among the membership was feasible in the early decades of the industry (U.S.
Department of Health, Education, and Welfare 1965, 1).
With this knowledge in hand, the credit union loan committee could make a
low-risk and, therefore, low-interest loan to a credit union member. Borrowing
members received low interest rate loans, and the owners of the credit union—
also members—were repaid with interest, which then was returned to members
in the form of interest (also called dividends) on deposits (also called shares) in
the credit union. Since members knew one another, and failure to repay a loan
harmed fellow members and damaged the defaulting member’s reputation in
the group, there was social pressure to repay, thus reducing the likelihood of
default, and consequently providing another explanation for low interest rates
on loans from credit unions.
The advent of nationwide credit bureaus, which provided in-depth information on the creditworthiness of most consumers, and the growth of credit
card lending as well as loan products that allowed borrowers to inexpensively employ real estate collateral for small loans—for example, home equity
lines—the necessity of credit unions as producers of common bond-driven information on creditworthiness declined. In order to continue to prosper and
grow, which many did successfully, credit unions were required to evolve from
their original structure. To do so, the industry shifted toward mortgage and
credit card lending and, to a smaller degree, toward business lending. The
industry was also granted liberalized membership rules by its regulator and
later by statute.
As the credit union industry evolved and many credit unions became more
bank-like in their product offerings, the banking industry expressed growing
concern over a perceived credit union advantage. Specifically, credit unions
are exempt from federal income taxes, while banks pay such taxes. Credit
union proponents have responded that the tax exemption remains appropriate because credit unions continue to be nonprofit, member-owned financial
institutions, and because they face restrictions not placed on banks. The
significance of the controversy was illustrated in November 2005 when congressional hearings were held to examine the credit union tax-exemption and
its justifications in the face of changes to the credit union industry (U.S. House
of Representatives 2005).
How do banks and credit unions differ, and what is the history behind
these differences? How have these differences changed over time and why?
Observers of the credit union tax exemption controversy might ask such questions. This article provides a history of the evolution of credit unions and the
market forces behind the evolution, without taking a stand on the merits of
either side of the tax debate.
J. R. Walter: Credit Unions
1.
355
EARLY HISTORY OF THE CREDIT UNION INDUSTRY
The first credit union in the United States, St. Mary’s Cooperative Credit Association, was chartered in Manchester, New Hampshire, in 1909 (National
Credit Union Administration 2006a). This and other similar credit unions
developed to fill a niche in the loan market that was largely unfilled by existing banks and savings institutions. The niche that they filled was that of
unsecured, small dollar, consumer loans—for example, for the payment of a
medical bill or to purchase a home appliance.
Lending institutions along the lines of United States credit unions first
developed in Europe. Cooperative banks or people’s banks appeared in Germany in the 1850s (Moody and Fite 1984, 1–10). These spread to a number
of other countries, and then to Canada, at the turn of the 20th century, before
coming to the United States.
Initial Leaders of the Credit Union Industry
A number of individuals are credited with the introduction and spread of
credit unions in the United States. Three of the most important are Alphonse
Desjardins, Pierre Jay, and Edward A. Filene.
Alphonse Desjardins, a Canadian journalist and politician, organized one
of the first cooperative banks in Canada, La Caisse Populaire de Levis. Caisse
Populaire loosely translates to people’s credit society or people’s bank (Moody
and Fite 1984, 12–15). Caisse, located in the city of Levis, Quebec, opened
for business in January 1901, and was modeled after European cooperative
banks which Desjardins had been studying for years. Caisses grew rapidly in
Canada, thanks to Desjardin’s advocacy, numbering 150 by 1914 (Moody and
Fite 1984, 17).
In late 1908, Desjardins helped organize the first credit union in the United
States, at the request of the priest of St. Mary’s Parish in Manchester. He also
consulted with Pierre Jay and Edward Filene as they pushed forward the spread
of credit unions in the United States following the formation of St. Mary’s.1
Pierre Jay, Banking Commissioner of Massachusetts, became familiar
with the cooperative banking concept soon after 1900 (Moody and Fite 1984,
22–23). As banking commissioner, he recognized a lack of lenders willing to
make small-value loans to consumers. His interest led him to contact Alphonse
Desjardins. In July 1908, Jay traveled to Ottawa to meet with Desjardins and
learn first-hand about les caisses populaires in Canada (Flannery 1974, 14).
In April 1909, Jay persuaded the Massachusetts legislature to pass legislation
that allowed cooperative banks, named credit unions, to be chartered in that
1 While St. Mary’s was created in late 1908, it did not receive a charter to operate from
the legislature of New Hampshire until early 1909 (Whitney 1922, 16).
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state. Then, as now, most states prohibited the gathering of deposits by firms
lacking a charter that specifically grants it this power.
Edward Filene was a Boston merchant (of Filene’s Basement fame) who
traveled widely and witnessed successful agricultural cooperative lenders in
India. These lenders gathered small deposits from Indian farmers, members of
the lending cooperative, and made small-value loans to their members (Moody
and Fite 1984, 21). Because of his business experience in the United States,
Edward Filene, like Jay, perceived a dearth of small-value lenders and wished
to bring the cooperative bank to the United States.
While Desjardins and Jay were responsible for the first steps in establishing
credit unions, Filene created momentum to produce growth in the industry over
the coming several decades. His efforts included lobbying legislators to pass
credit union legislation, speaking frequently on the benefits of cooperative
lending, and funding organizations to promote the expansion of credit unions.
Credit union growth was fairly slow between 1909, when the first U.S.
credit union opened, and 1920 (Moody and Fite 1984, 32–72). Growth of
credit unions began to pick up in the mid-1920s, as an increasing number of
states passed credit union laws (Moody and Fite 1984, 73). By 1925, credit
union laws had passed in 26 states. In 1925, there were 176 credit unions; this
count had grown to 868 in 1929 (U.S. Bureau of the Census 1975, 1,049) and
2,500 by 1934, when the Federal Credit Union Act created federal credit union
charters (U.S. Treasury Department 1997, 16). According to Moody and Fite
(1984, 55–108), much of the legislation that made this growth possible was
the result of Edward Filene’s lobbying efforts or the result of efforts financed
by him.
Beyond Credit Unions
Credit unions were not the only response to limited borrowing opportunities
for consumers. The Russell Sage Foundation was formed, based on funding
provided by Russell Sage’s widow (Sage was a railroad baron who died in
1906). The foundation soon began advocating for the liberalization of usury
laws for small-value loans. Low usury ceilings in a number of states meant
that small-value consumer loans could not be made profitably. Out of this
advocacy grew a movement to encourage all states to adopt uniform smallvalue loan regulations that would provide for fairly high usury ceilings on small
consumer loans, while at the same time implementing consumer protection
rules. Approximately two-thirds of the states passed the uniform laws by the
time the Foundation ended its push for such laws in the 1940s (Carruthers,
Guinnane, and Lee 2005).
In addition to the effort of the Russell Sage Foundation, Morris Plan
banks were formed around the country based on a bank founded in Norfolk,
Virginia. The original Morris Plan bank was chartered by Virginia in 1910 at
J. R. Walter: Credit Unions
357
the request of Arthur J. Morris. It gathered deposits from and made smallvalue unsecured installment loans to consumers. Morris lowered losses and,
therefore, was able to charge fairly low interest rates, by requiring several
cosigners for all consumer loans. Further, he sidestepped usury ceilings to
some degree by charging loan fees and deducting interest payments from the
loan when it was initially made (Giles 1951, 81–86). Ultimately, Morris
Plan banks were called industrial banks, which remain today in the form of
industrial loan corporations, one of the few bank types that can be owned by
commercial firms (Walter 2006).2
2.
COMMON BOND LENDING
Given existing lenders’ focus on other types of lending, the early 20th century
consumer lending market was ripe for the development of financial institutions
that could act as low-cost providers of small-value, unsecured consumer loans.
Commercial banks, mutual savings banks, and savings and loans focused their
lending efforts in other directions.
Pierre Jay noted the absence of providers of small-value, unsecured loans
in Massachusetts, and foresaw the operating characteristics of credit unions:
What are known in Massachusetts as “cooperative banks,” are excellent
examples of the success of cooperation between savers and borrowers in
an important but limited field. Their members are almost entirely salary
and wage earners, who have joined together for systematic saving and
for the owning of their homes. But inasmuch as all of their loans are
secured, either on real estate or on the cash value of their shares, there is
no need for selecting the members, no scrutiny is made of the object of
loans made on the cash value of shares, and but little interest is taken by
the members in the management of the institution. Furthermore they are
of no service to those who have neither real estate nor shares as security,
many of whom undoubtedly have legitimate need for loans. (quoted in
Moody and Fite 1984, 24–25)
2 The early 20th century saw not only the development of the credit union industry, but also
the growth of other cooperative business organizations, i.e., organizations in which customers are
also owners or members. For example, agricultural cooperatives, which began to spread in the
United States in the 1870s, grew rapidly between 1890 and 1920. The number of agricultural
cooperatives peaked, at 14,000 in the 1920s (Frederick 1997). Consumer cooperatives gained popularity in the 1920s, and cooperatives also formed to provide rural electric and telephone service,
encouraged by the Rural Electrification Administration, which passed in 1935 (Frederick 1997).
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Relationships As a Substitute for Collateral and
Credit Ratings
Credit unions were small, with, as noted earlier, on average only 187 members
in 1932. Further, they were formed based on a common bond, where members
knew one another. In other words, members of a common bond likely knew,
at least better than other potential lenders, the economic prospects of the
borrowers who were fellow members.
Alphonse Desjardins noted the benefit of common bond lending when
discussing exceedingly low loan losses at La Caisse Populaire. He claimed
that losses were low because, “The main security is the fact that the association
is working within a small area and that everybody knows each other” (Moody
and Fite 1984, 16). Pierre Jay had an eye toward the members of a common
bond, monitoring one another while discussing the lack of such monitoring
when loans are secured by real estate or deposits (shares), when he said, “there
is no need for selecting members, no scrutiny is made of the object of loans.”
At the time, nationwide consumer credit rating agencies did not exist,
though there were credit rating agencies in some of the larger cities. Because
of the lack of these agencies, it was difficult for lenders to differentiate between
creditworthy and non-creditworthy consumers. The common bond acted as a
substitute, allowing its members to differentiate good credit risks from poor
credit risks.
Monitoring Within the Common Bond
Not only were members of small credit unions, or the credit committee, likely
to know something of the financial circumstances of their fellow members,
they had incentive to deny loans to poor credit risks and to monitor borrowers.
Federal deposit insurance coverage was not granted to credit unions until 1970.
A few states had state insurance schemes for credit unions, but in general,
depositors were subject to losses when borrowers failed to repay (Flannery
1974, 26). Depositors faced the danger that if losses were large enough they
might lose principal on their deposits.
The view that common bond lending contained this monitoring advantage
was expressed by Alphonse Desjardins when he noted that “a second security
[in addition to members’ knowledge of one another’s creditworthiness] is that
everybody is interested by being a shareholder” (Moody and Fite 1984, 16).
In effect, common bond lending used social pressure from associates instead of the threat of confiscation of collateral, to ensure loan repayment. Since
these associates suffered loss due to a member’s default, the social pressure
to repay was likely significant. The borrower knows that if he does not repay,
he is costing his associates lost interest earnings and perhaps lost principal.
A 1922 Department of Labor study of early credit unions reported that
loans were typically made without collateral, but with careful scrutiny of
J. R. Walter: Credit Unions
359
the character of the borrower by the credit committee—typically the loan
decision-making body in credit unions (Whitney 1922, 33). Small loans were
made on the signature of the borrower only, while loans over $50 required the
endorsement of two other credit union members. According to this 1922 study,
typical interest rates charged by credit unions ranged from 6 to 12 percent per
annum (Whitney 1922, 40–54).3
Still, the incentive of any individual member to keep an eye on another
borrower is somewhat muted. The cost of one member’s efforts to monitor
another member’s financial condition is borne only by that member, but the
benefit if loans are properly denied, or remedial action is taken, is shared by
all members in the form of higher returns on their credit union deposits.4
Reputational Incentive to Repay
While members’ incentive to monitor one another’s loans in order to protect
credit union earnings may be muted, there is another reason to expect borrowers within a common bond group to repay. Members of a credit union are
likely to have many valuable interactions with one another, outside of interactions through the credit union. Failure to repay a loan is likely to damage
the reputation of the defaulting borrower and undercut these interactions.5
For instance, members of a credit union who share the common bond of
working for the same company may worry that defaulting on a credit union
loan will decrease their opportunities for advancement. Their co-workers,
including supervisors, would be likely to learn of the default, undercutting the
defaulting worker’s reputation.
Further, a credit union member who shares a close common bond with
fellow members might be concerned that his default could damage his reputation and limit social interactions with members. For example, the credit
union borrower might believe that members will view a default as a sign of
dishonesty and, therefore, be less willing to include him in confidences.
3 The author of the 1922 study notes that some credit union loans were made on collateral
such as deposits in the credit union, stocks, or real estate collateral (Whitney 1922, 40).
4 Group lending, whereby loans are made to members of a group and all members of the
group are jointly liable for each other’s debts, can provide strong incentive to members to monitor and discipline one another to ensure repayment. Group lending, which, like common bond
lending, uses social connections to encourage repayment, has received wide attention as a means
of inexpensively lending to individuals for whom little formal information on creditworthiness is
available. Group lending and a number of interesting examples as practiced in developing countries
are discussed and explored in Prescott (1997).
5 Besley and Coate (1995) develop a model which is used to explore the ways “social connectedness” can encourage debt repayment.
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Members of a common bond will be aware of the potential damage to
one’s reputation. As a result, they will be less likely to default because of
the potential damage to other interactions with fellow members of a common
bond.
3.
FACTORS DIFFERENTIATING CREDIT UNIONS FROM
OTHER DEPOSITORIES
Credit unions share many features common to all depository institutions: they
gather savings and transaction deposits, make various types of loans, and
employ branch networks, ATMs, telephone service departments, and Web
sites to provide services to their customers. They are insured by an agency of
the federal government, receive a charter to operate, and are closely regulated.
Beyond these similarities, credit unions are distinct. They differ from most
depositories in that they are mutually owned, rather than stockholder-owned.
Another distinction is their greater concentration on unsecured lending, though
the growth of credit card lenders has meant that credit unions are less distinct
here. While other depositories take deposits from and make loans to anyone
meeting creditworthiness requirements, credit unions face restrictions limiting
whom they may accept as customers. Last, credit unions have the unique
advantage of being exempt from federal taxes.
Mutual Ownership
State credit union laws and the Federal Credit Union Act require that credit
unions be cooperative or mutually owned organizations. This implies that
a credit union’s members, meaning its depositors, are also its owners. As
a result, depositors have the right to vote for the members of the board of
directors of the credit union, who, in turn, hire managers who run the day-today operations of the credit union.
Credit unions do not raise equity by selling shares in the equity markets,
as do nonmutual corporations. Instead, credit unions raise equity by retaining
earnings. Credit union depositors, as owners, are due not only the repayment of
their deposits and any interest owed to them, but also the credit union’s equity,
meaning funds that might remain after all liabilities are repaid if the credit
union is dissolved. Consequently, credit union depositors are the residual
claimants to the credit union’s assets, like corporate shareholders, explaining
J. R. Walter: Credit Unions
361
why credit union deposits are often called “shares.” Depositors in most other
depository institutions are due only the repayment of deposits plus interest.6
Uncollateralized Consumer Lending
Beyond their ownership, credit unions were originally set apart from other
depositories by their emphasis on small-value, nonmortgage loans to individuals and households, meaning uncollateralized loans for household expenses
and the purchase of consumer durables. For example, in the early 1920s, the
average size loan made by credit unions in Massachusetts and NewYork—two
states in which numerous credit unions operated—was $272 ($3,092 in current dollar terms) in Massachusetts and $169 ($1,720) in New York (Whitney
1922, 55).
With the exception of some securities investments, credit union assets
were devoted almost completely to loans to individuals. The earliest data on
the types of loans credit unions made comes from a 1948 survey of federally
chartered credit unions conducted by the Bureau of Federal Credit Unions.
The survey indicated that the largest categories were loans for the purchase
of automobiles to consolidate loans for current living expenses and medical
expenses (U.S. Bureau of Federal Credit Unions 1948, 4).
In contrast to credit unions, savings and loans and savings banks focused
on mortgage lending and commercial banks on business lending. In the case
of mutual savings banks, as of 1910, mortgages and investments in securities
accounted for 89 percent of assets (U.S. Bureau of the Census 1975, 1,046).
Figures on types of lending by savings and loans is reported starting with
1929, when mortgages accounted for 75 percent of all assets (U.S. Bureau of
the Census 1975, 1,047). Data on types of loans and assets for commercial
banks are reported beginning with 1939. During that year, 91 percent of bank
assets were accounted for by loans to businesses, securities investments, cash
holdings, and mortgage lending (Board of Governors 1959, 34–35). In each
of these cases, uncollateralized loans to individuals were no more than 25
percent of assets.
Until 1977, federal credit unions were largely prohibited from making real
estate loans, though some states allowed such lending by state-chartered credit
6 Another group of depository institutions, mutual savings banks, which traditionally focused
on home mortgage lending, are like credit unions, also owned by depositors.
The advantages and disadvantages of mutual ownership relative to stockholder ownership have
been studied widely, especially in the insurance industry, where both types of ownership are prevalent. Generally these studies find that the choice is driven by conflicts of interest between owners
and managers on one hand, and owners and customers on the other. Examples of such studies
are Mayers and Smith (2002), Mayers and Smith (1986), and Fama and Jensen (1983).
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unions. The shift into mortgage lending after 1977 contributed to growth in
loan size at credit unions, such that in March 2006 the average credit union
loan was $10,903 (National Credit Union Administration 2006b).
During the 1940s, credit unions had little opportunity to compete for
mortgage loans (as discussed later), but by 2006, such loans accounted for
32 percent of all assets. Nevertheless, credit unions continue to exhibit considerably greater emphasis on nonmortgage loans to individuals, which as of
March 2006, accounted for 31 percent of all of their assets, compared to 8
percent for commercial banks, savings banks, and savings and loans (Federal
Deposit Insurance Corporation 2006, 5).
Restricted Customer Base
An additional important factor differentiating credit unions from other depositories is that they have traditionally been required by state and federal law
to choose their members from one group. Specifically, a credit union must
focus on a group sharing one of three forms of affinity: (1) a common bond of
association such as a fraternal or religious organization, (2) an occupation, or
(3) residence in a single community, neighborhood, or rural district. In other
words, unlike banks, savings and loans, and savings banks, credit unions face
restrictions that limit with whom they may do business.
The apparent goal of this limitation is to protect the health of credit unions,
as members are more likely to repay loans given that failure to repay harms
fellow members of their own group (Robbins 2005, 3). While this repayment
benefit may flow from the common bond requirement, there is also a serious
disadvantage. Credit unions bound to accept members only from one employer
or community are likely to suffer from a severe lack of diversification. Limited
diversification may have contributed to high failure rates among small credit
unions in the late 1970s and 1980s (Wilcox 2005, 2–3).
Tax Exemption
Tax exemption is the final, and most controversial, difference separating credit
unions from other depositories. Credit unions are free from federal income
taxes. All of a credit union’s net earnings, meaning earnings after paying
expenses, will be paid out to its owners (depositors) as interest and dividends or
retained by the credit union as capital; none goes to federal taxes. In contrast,
before a bank can make dividend payments to its owners (shareholders) or
retain capital, it must pay a portion of its net earnings as federal income
taxes.7
7 Since credit unions do not raise capital by issuing equity to investors, they are at a disadvantage compared to banks, which can quickly raise additional capital in the equity markets.
J. R. Walter: Credit Unions
363
While commercial banks have never been tax exempt and mutual savings
and loans and mutual savings banks lost their tax-exempt status in the Revenue
Act of 1951, credit unions remain tax exempt even though this status has
been questioned by representatives of other portions of the financial services
industry.
Section 501(c)(1) of the Internal Revenue Code grants tax-exempt status
to federal credit unions. Section 501(c)(14) of the Code does the same for
state-chartered credit unions as long as they do not have capital stock, are
mutuals, and are nonprofit. In 1998, legislators explained and affirmed credit
unions’ tax-exempt status in the Credit Union Membership Access Act. The
act stated that credit unions are tax exempt because they “are member-owned,
democratically operated, not-for-profit organizations generally managed by
volunteer boards of directors and because they have the specified mission
of meeting the credit and savings needs of consumers, especially persons of
modest means.” (U.S. House of Representatives, Committee on Ways and
Means 2005).
Credit unions are granted an advantage relative to their competitors because they are tax exempt. One measure of the size of this advantage is the
total amount of taxes avoided. In a 2001 study, the Department of Treasury
estimated that federal tax revenues of between $1.3 billion and $1.6 billion
per year were lost due to credit unions’ tax-exempt status (U.S. Department
of the Treasury 2001a).
If credit unions paid this $1.6 billion to their depositors in the form of
greater interest on deposits, the advantage would amount to about 30 basis
points ($1.6 billion divided by $550 billion in total credit union deposits equals
.29 percent). Alternatively, if they passed it along in lower loan interest rates,
the basis point advantage would be about the same (as the dollar amount of
loans and deposits are about the same).
Still, it seems likely that at least part of the tax advantage move in other
directions. Some of the advantage is likely to be shifted toward capital holdings
at credit unions and higher salaries and expenses.
While tax exemption may provide as much as a 30 basis point advantage
that can be used to attract customers to a credit union, the advantage is undercut by two other considerations. The first consideration is that many small
banks are organized as Subchapter S corporations. Within certain limits, S
corporations pay no federal income taxes and can, like credit unions, devote all
earnings either to dividends and interest, retained capital, or some of all three.
As of March 2006, about 2,400 FDIC-insured institutions (commercial banks,
Instead, credit unions increase equity only by retaining earnings. The inability to quickly add to
their equity can be a disadvantage if they wish to grow rapidly. This disadvantage, compared to
banks, is offset somewhat because credit union earnings are not diminished by taxes, leaving more
earnings that might be retained.
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savings and loans, and savings banks), or 27 percent of all FDIC-insured institutions, were organized as S corporations (Federal Deposit Insurance Corporation 2006, 4–5).
The second consideration that limits, somewhat, the advantage of credit
unions over non-tax-exempt banks is the current fairly low dividend tax rate
of 15 percent. As an example, assume a taxed bank earns $100 million dollars, pays federal taxes (of $35 million at the top federal corporate tax rate
of 35 percent), and pays out as dividends all of its remaining earnings. Its
shareholders then pay taxes on the dividends at the current dividend tax rate
of 15 percent. Ultimately, bank shareholders receive $55 million in earnings.
In contrast, shareholders (customers) in credit unions receive $65 million
after taxes. While they do not face double taxation as corporate shareholders do, since the credit union pays no taxes before paying them interest or
dividends, the tax rate on payments to credit union shareholders is the individual tax rate, which can be as high as 35 percent. The bottom line is that
the tax advantage may be smaller than one might imagine.8
4.
SIZE RELATIVE TO BANKS
As of December 2005, there were about 9,000 credit unions, 3,600 of which
were state-chartered and 5,400 of which were federal credit unions (Credit
Union NationalAssociation 2006). In sum, these 9,000 credit unions held $700
billion in assets. In comparison, there were 8,800 banks, savings and loans,
and savings banks, together holding $10.9 trillion in assets. Several banks,
individually, have assets that exceed the total assets of the sum of all credit
unions. Nevertheless, with 87 million members as of 2005, credit unions are
important competitors in the market for consumer loans and deposits (Credit
Union National Association 2006).
Like banks, credit unions come in a wide range of sizes. Still, compared
to banks, a much higher proportion of credit unions are very small. Approximately 46 percent of all credit unions are smaller than $10 million in assets
(Credit Union National Association 2006). Only 79 banks, or 1 percent of all
banks, have assets below this threshold. In part, this size difference reflects the
long-standing focus of credit unions on retail deposits and loans compared to
banks’ focus on lending to firms and holding their deposits. Commercial loans
account for only 2.5 percent of all credit union assets, though this amount
has grown fairly rapidly in recent years. This contrasts with banks where
8 Because these calculations of amounts received by bank and credit union shareholders are
made assuming the highest tax rates (corporate and individual rates), they cannot be considered
accurate for all shareholders. For example, credit union shareholders in lower tax brackets (presumably a majority of shareholders) will receive higher after-tax earnings.
J. R. Walter: Credit Unions
365
Figure 1 Average Credit Union Size and Percentage Growth in Size
80
40
Average Size
Growth Rate
35
60
30
50
25
40
20
30
15
20
10
10
5
0
0
Growth Rate (%)
Real Assets (Millions $)
70
-5
-10
1945
1955
1965
1975
1985
1995
2005
Year
Notes: Commercial bank data are from FDIC. Credit union data are from Credit Union
National Association.
commercial loans amount to about 22 percent of total assets (Credit Union
National Association 2006).
Nevertheless, there are some large credit unions, even by bank standards.
For example, 104 credit unions had assets greater than $1 billion as of June
2005 (Callahan and Associates 2006, 44–46).
While compared to banks, most credit unions are small, as Figure 1 shows.
Yet, the size of the average credit union has grown significantly over the years.
Figure 2 demonstrates that, relative to banks, credit union assets have grown
fairly consistently. They increased from less than 1 percent of bank assets in
1944 to about 3.5 percent in 1981. Growth was especially rapid relative to
banks during the 1980s and early 1990s, as membership restrictions were first
being eased and banks were experiencing a widespread decrease in profits.
More recently, growth has been less robust in relation to bank assets (see
Figure 2).
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Federal Reserve Bank of Richmond Economic Quarterly
Figure 2 Credit Union Assets as a Percent of Bank Assets
9
8
7
Percent
6
5
4
3
2
1
0
1945
1955
1965
1975
1985
1995
2005
Year
Notes: Commerical bank data are from FDIC. Credit union data are from Credit Union
National Association.
5.
REGULATORY FRAMEWORK
Historical Development
The first credit union in the United States operated under a special charter
granted specifically to that credit union by the New Hampshire legislature in
1909.9 As noted earlier, over a period of years following the 1909 chartering
of the first credit union, other states passed laws giving credit-union-chartering
authority to state banking agencies. Credit unions were required to seek state
charters, otherwise they would have run afoul of state laws stipulating that deposits may be accepted only by entities holding government-issued charters.
9 A 1922 study of cooperative lending associations notes that there were several credit unionlike financial institutions operating even before 1909. For example, a cooperative association was
created for employees of the Boston Globe in 1892. It gathered deposits from employees and
made small-value loans to them. The Boston Globe Cooperative Association, and several other
similar associations, operated without legal authority, leading the Massachusetts banking commissioner, Pierre Jay, to call for a law authorizing these activities (Whitney 1922, 17).
J. R. Walter: Credit Unions
367
These early charter laws typically specified rules such as how credit unions
might be founded, from whom they might gather deposits (credit union members), to whom they might make loans (typically only to members), who might
become members (those sharing a community or occupational bond), and how
an individual might become a member (by purchasing at least one share in
the credit union) (Whitney 1922, 21–29). State banking agencies not only
chartered credit unions, but also oversaw their operations, including sending
examiners to review their activities (Whitney 1922, 48).
A federal charter for credit unions was created in 1934. At that time,
39 states had laws authorizing credit unions and there were 2,028 state credit
unions (Giles 1951, 106; U.S. Bureau of the Census 1975, 1,049). The Federal
Credit Union Act, enacted on June 26, 1934, created a federal agency to grant
charters so that credit unions could be formed in any state regardless of whether
that state offered charters to credit unions. The federal agency was placed in
the Farm Credit Administration (Moody and Fite 1984, 120).
Over the years the agency responsible for federal credit unions was housed
in government entities. It was moved to the FDIC in April 1942 and then in
July 1948 to the Federal Security Agency—later renamed the Department of
Health, Education, and Welfare (Croteau 1956, 121).
Current Regulatory Environment
In March 1970, legislation was enacted to create an independent agency, the
National Credit Union Administration (NCUA) (Moody and Fite 1984, 304).
The NCUA charters and supervises all federal credit unions. State credit
unions continue to be chartered and examined by state agencies.
Also in 1970, credit unions gained federal deposit insurance when the
Federal Credit Union Act was amended in October of that year to create the
National Credit Union Share Insurance Fund (NCUSIF). The Fund is overseen
by the NCUA. All federal credit unions must join the fund and state-chartered
credit unions may choose to join.
The amendment that created the NCUSIF passed in spite of the objections
of the credit union industry, which argued that insurance premiums would
be high and losses from credit union failures had been quite low historically
(Moody and Fite 1984, 304). While opposed in the beginning, insurance is
now almost universal; only 191 state-chartered credit unions, out of the total
of 3,600 such credit unions, operate without NCUSIF coverage (Callahan and
Associates 2006, 19).
6. ADVANTAGES OF COMMON BOND LENDING DIMINISH
In the second half of the 20th century, a number of factors combined to slowly
undercut the advantages of common bond lending relative to other forms of
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lending. First, nationwide credit-reporting agencies emerged and expanded
in the 1970s, lowering the cost of lending to consumers with whom the lender
had no direct contact. Credit-reporting agencies are companies that gather and
sell information about individual consumers, information that can be used to
predict the likelihood that a consumer will repay a debt. Second, in the 1980s
and 1990s the home equity line of credit was developed and became widely
available, allowing consumers to easily tap their home equity for small purchases. Third, credit card lending developed as a convenient, and reasonably
low-cost means of making small, unsecured loans. And fourth, the advent of
deposit insurance eliminated some of the social pressure that might be brought
to bear on members who failed to repay loans.
Growing Availability of Creditworthiness Information
Until the 1970s, credit information was gathered largely only on a local basis, and on just a limited number of consumer variables. For example, Trans
Union, one of several nationwide credit-reporting agencies, entered the creditreporting business in 1969 by purchasing a local credit-reporting agency, and
then expanding nationwide, and developing the computer technology to store
and retrieve broad, detailed records on millions of consumers over the next
several years (Trans Union 2006b). Before computers were employed, information was necessarily limited to a few items on each consumer, and was
difficult to retrieve quickly.
Today, three credit-reporting agencies—Equifax Credit Information Services, Trans Union, and Experian Information Solutions—gather and maintain
information that lenders consider predictive of debt repayment on the majority of United States consumers. Such information can include the consumer’s
debt and bill repayment history, bankruptcies, liens, number of loans and lines
of credit, amount owed on each, home address, and employer name (Trans
Union 2006a). For a fee, a reporting agency will supply this information about
a consumer to a lender considering making a loan to that consumer, as well as a
score that grades the consumer’s creditworthiness based on the information. A
lender might gather scores and the underlying information from two or three of
the agencies before deciding to make a loan. Because many borrowers know
that a default will mean a lowered credit score and higher interest rates on future loans, the presence of these credit agencies provides an incentive to repay.
Because of the low cost and ease with which lenders can gather consumer creditworthiness information, and the repayment incentive their presence provides,
the relative advantages of creditworthiness knowledge gained by maintaining
direct contact with the borrower through a common bond and the motivation
to repay produced by common bond relationships are reduced.
J. R. Walter: Credit Unions
369
Home Equity Lines of Credit
The spread of home equity lines of credit (HELOCs) also tended to undercut
the advantage of common bond lending. Prior to the mid-1980s spread of
home equity loans, it was difficult for a consumer to make use of the equity
built up in his or her house as collateral for a loan (Canner, Durkin, and Luckett
1998, 242). Second mortgages were often taken on homes to make a large,
one-time purchase. But closing and other transaction costs made such loans
expensive to use for making smaller and more frequent consumer purchases.
The development of HELOCs greatly expanded the ability of consumer
borrowers to tap the equity in their homes and, thereby, collateralize
borrowings for occasional consumer expenditures.10 With a HELOC, the
consumer bears the closing costs only once, but then can borrow as frequently
as desired by simply writing a check against the HELOC. Further, such loans
are far less risky than uncollateralized loans, and, therefore, allow lenders
to offer consumers a lower rate of interest. The growing availability of HELOCs greatly diminished the need to employ the collateral substitution ability
provided by common bond lending, at least for consumers with home equity.
Credit Card Lending Use Expands
Credit card lending makes use of a cost savings similar to that provided by
HELOC lending. Once the credit card lending arrangement is established,
costs of drawing down the credit card line are minimal for the lender and for
the consumer. The credit card company makes an initial credit decision, based
largely upon information from credit-reporting agencies, on whether to grant
a loan to a consumer, at what rate, and how large a line. The consumer is then
free to draw on the line in whatever increments he or she chooses, and to pay
it down over a period largely determined by the consumer.
Credit card availability grew rapidly following a 1978 Supreme Court decision, which allowed credit card lenders to avoid state usury ceilings (Athreya
2001, 11–15). Because credit cards offer low-transaction-cost lending, based
on detailed creditworthiness information from the files of credit reporting
agencies, credit card lending offered an additional strong alternative to common bond lending.
Credit Unions Receive Federal Deposit Insurance
The 1970 application of federal deposit insurance to credit unions also tended
to undercut an advantage of common bond lending. Before federal deposit
10 Weinberg (2005) provides a thorough analysis of the growth in recent decades of consumer
debt, the causes of this growth, and its consequences.
370
Federal Reserve Bank of Richmond Economic Quarterly
insurance was extended to credit unions, credit union members had reason to
monitor one another’s loan repayment. Any defaults increased the probability
of the failure of the credit union and losses to its depositors. Sharing a common
bond also meant that borrowers were likely to have frequent contact with
one another and, therefore, the opportunity to bring pressure on delinquent
borrowers. Additionally, frequent contact implies that a borrower might face
significant embarrassment if he or she defaults.11
But once credit unions gained federal deposit insurance, the incentive
to monitor repayment was reduced. With insurance in place, members are
protected against loss of principal and interest if a default or several defaults
cause the failure of the credit union. If the default is too small to cause the
failure of the credit union but leads to a reduction in interest payments, a
competitive market for deposits means that members can simply shift their
deposits to another depository, paying an unreduced interest rate.
7.
CREDIT UNIONS EVOLVE
As the benefits of common bond lending declined for reasons discussed previously, the credit union industry moved in directions that tended to deemphasize this form of lending. As long as common bonds were important
to consumer lending, one would expect credit unions to operate with fairly
small memberships in order to take advantage of the tighter bonds present in
smaller groups. Once the advantage of the common bond is undercut, credit
unions should tend to move toward the size of other depositories that are not
focused on common bond lending. Further, one would expect credit unions
to begin to place greater emphasis on other forms of lending. This movement
is evident (1) in the growing average size of credit unions, encouraged to a
degree by liberalizations that broadened the potential membership base of any
given credit union, and (2) the move of credit unions into real estate, credit
card, and business lending.
Common Bond Requirement Loosened
The strict common bond requirement for credit unions ended in 1982. In that
year, the NCUA began to allow single credit unions to draw members from
multiple groups. At the time, the failure rate of credit unions was at a historically high level. Failures were likely driven, in part, by the inability of credit
unions to achieve diversification due to tight common bond membership limitations (Wilcox 2005, 1, 3). For example, a credit union with a membership
11 Athreya (2004) finds evidence that fear of embarrassment or stigma remains today as an
important incentive to repay. His finding contrasts with an often-cited view that the stigma has
diminished.
J. R. Walter: Credit Unions
371
comprised completely of employees of one firm is likely to face heavy losses
if the employer goes out of business.
While in 1998 the Supreme Court determined that the NCUA had overstepped its statutory limits when it approved multiple groups for one credit
union, this decision was nullified almost immediately by legislation. In 1998,
shortly after the Supreme Court decision, Congress passed, and President
Clinton signed, the Credit Union Membership Access Act, which authorized
individual credit unions to serve multiple groups, within some restrictions.
The credit union industry had lobbied aggressively for passage of this liberalization (Gill 1998).
The move to allow multiple groups along with mergers of credit unions,
which were frequent in the 1980s, offered credit unions the opportunity to
enlarge their membership base and size. As seen in the growth rates plotted
in Figure 1, year-after-year growth in the average size of credit unions increased, in constant dollars, at historically high rates between 1982 and 1987,
following the liberalization of membership requirements. Overall, average
size increased from $7.5 million in 1981 to $77.7 million in 2005; though the
annual growth rate after 1987 is about the same as it was before 1982.
In 1920, the average number of members per credit union was 200. By this
measure, credit unions remained fairly small, even in 1960, with an average of
598 members per credit union. By 2005, the average credit union membership
had risen to 10,000. Regarding lending decisions, there is probably little
opportunity to use special knowledge gained from direct contact between
members.12 Further, it seems likely that embarrassment and social pressure
will play a less important role in institutions with thousands of members than
in small institutions in which members are known to one another.
Credit Unions Gain Mortgage Lending Powers
Federal credit unions gained the authority to make long-term (up to 30 years)
mortgage real estate loans in 1977 with amendments made to the Federal
Credit Union Act by Public Law 95-22. Before these amendments passed,
federal credit unions were limited to providing short-term mortgage loans
such as second mortgages and mobile home loans (Credit Union National
Association 1978, 12).
State-chartered institutions in a number of states had this authority prior
to 1977. Consequently, at the beginning of 1978, credit unions held $2.55
billion in first mortgages, which accounted for 6.2 percent of all credit union
12 Karlan (2005) finds that for members of a group-lending organization in Peru, loans are
more frequently repaid by members of groups who live close to one another and who are more
culturally similar. This result occurs because such individuals are better able to monitor one another.
372
Federal Reserve Bank of Richmond Economic Quarterly
loans. Since 1977, first mortgage real estate lending has grown in importance
as a percentage of credit union lending. As of 2005, mortgages amounted to
32 percent of all loans (Credit Union National Association 2006). As common
bond lending became less necessary, credit unions moved more extensively
into real estate lending.
Growth of Credit Card Lending
Similarly, credit card lending has grown in importance for credit unions, so that
today such lending accounts for a large percentage of all unsecured lending by
credit unions. Approximately 54 percent of credit unions offer credit cards,
and the percentage rises from very low at the smallest credit unions to 98
percent at the largest (Credit Union National Association 2005). Since this
type of lending is most important for the largest credit unions, (those with the
largest membership) it seems unlikely that common bond lending can play
much of a role in this new category of credit union assets.
Expanded Business Lending
During the last several years, credit unions expanded their business lending
significantly, doubling the amount of such loans over the three years before
2005 (Credit Union National Association 2006). Specifically, business loans
increased from $8.3 billion, in 2002, to $16.9 billion, or 2.5 percent of credit
union assets, in 2005. In comparison, in 2005, business loans accounted for
22 percent of total assets at commercial banks (FDIC 2005). Since these loans
are typically collateralized and made for business purposes, they are beyond
the historical credit union emphasis on uncollateralized consumer lending.
Still, the extent of business lending by credit unions is limited. The 1998
Credit Union Membership Access Act set limits on credit union lending to
businesses. Specifically, the sum of all business loans of a federal credit union
may not exceed 1.75 percent of its net worth or 12.25 percent of its total assets,
whichever is the lesser (National Credit Union Administration 2004, Section
723.16).
8.
CONCLUSION
Credit unions once focused almost entirely on small-value, unsecured consumer lending. Other depositories, specifically commercial banks, savings
banks, and savings and loans, emphasized lending to businesses or making
large-dollar-value consumer loans based on real estate collateral. For unsecured consumer lending, the credit union model of taking deposits from and
lending to a tightly knit group of members of a community or organization
J. R. Walter: Credit Unions
373
was an ideal business model in the early part of the 20th century, filling a niche
in the marketplace at that time.
But the financial marketplace changed greatly in the second half of the 20th
century. Computer databases and networks allowed the inexpensive collection
and dissemination of consumer creditworthiness information. Home ownership spread and products developed that allowed consumers to borrow against
their home equity for repeated small-value loans. Further, the availability and
use of credit cards as a vehicle for unsecured consumer loans expanded immensely. The combination of these factors required credit unions to change
as well.
Credit unions shifted away from their heavy emphasis on unsecured consumer loans into mortgage lending, other forms of collateralized consumer
lending (such as auto loans), credit card lending, and business lending. Today,
unsecured consumer loans account for only about 10 percent of all credit union
loans. Auto loans are responsible for 37 percent, and mortgages comprise most
of the remainder (Credit Union National Association 2005).
Additionally, while the very small credit union was well-suited to common
bond lending, small size became less advantageous as common bond lending
lost its advantage over the last several decades. Concurrently, the average
number of members and size of credit unions increased significantly. Nevertheless, a large portion of credit unions remain quite small in comparison to
commercial banks. By evolving with changes in the marketplace, the credit
union industry remained healthy and grew, both in terms of dollar amounts
and relative to other depositories.
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@FedFRASER