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Business
Review
Federal Reserve Bank of Philadelphia
January • February 1 9 9 6

ISSN 0 0 0 7 - 7 0 1 1

Business
Review
The BUSINESS REVIEW is published by the
Department of Research six times a year. It is
edited by Sarah Burke. Artwork is designed
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direction of Ronald B. Williams. The views
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JANUARY/FEBRUARY 1996
WHEN THE BUBBLE BURSTS:
PSYCHOLOGY OR
FUNDAMENTALS?
Lee E. Ohanian
The prices of stocks, bonds, and other
assets frequently fluctuate, and some
times these fluctuations are quite large.
Such price shifts have important eco
nomic implications, including the possi
bility that asset prices have predictive
power for the business cycle. In this ar
ticle, Lee Ohanian analyzes the volatility
of security prices and discusses whether
movements in asset prices reflect changes
in the fundamental value of the asset or
whether extreme price changes may be
associated with changes in market psy
chology.
THE CYCLICAL VOLATILITY
OF INTEREST RATES
Keith Sill
Interest rates change in response to a
variety of econom ic events, such as
changes in Fed policy, crises in financial
markets, and changes in prospects for
long-term economic growth and infla
tion. But such events are sporadic, and
interest rates show a more regular pat
tern of volatility that corresponds to the
business cycle. In this article, Keith Sill
examines some facts and theory about
the cyclical volatility of short-term and
long-term interest rates.

When the Bubble Bursts:
Psychology or Fundamentals?
Lee E. Ohanian*

J L rices for stocks, bonds, foreign exchange,
and other assets frequently exhibit large fluc
tuations on a daily and long-term basis. Per
haps the best known example of asset-price
volatility was the 500-point decline in the Dow
Jones Industrial Average on October 19,1987.
The 23 percent drop coincided with similar

*Lee Ohanian is an assistant professor of economics at
the University of Minnesota. He wrote this article while he
was on the faculty at the University of Pennsylvania and a
visiting scholar in the Research Department of the Phila
delphia Fed. Lee thanks Rick Lang, Steve Meyer, Dean
Croushore, Keith Sill, Len Nakamura, and Sally Burke for
helpful comments.

declines in the Tokyo, London, and Hong Kong
stock exchanges and was nearly twice the mag
nitude of the October 1929 crash that ushered
in the Great Depression.
October 19,1987, was not the only turbulent
day on the New York Stock Exchange in recent
history. Since 1987, there have been 16 trading
sessions in which the Dow moved at least 90
points. Extreme price volatility is not confined
to the stock market, nor is it strictly a short
term feature of the market. High variability
has characterized foreign exchange rates since
currencies were allowed to float in the early
1970s. The U.S. dollar, which rose 20 percent
between February 1984 and February 1985, fell
3

BUSINESS REVIEW

25 percent over the following year. Price vola
tility has also characterized the markets for
corporate and U.S. government debt in recent
years. Once the haven of conservative inves
tors, the bond market now frequently displays
fluctuations equal to those in the stock and
foreign exchange markets. For example, the
price of the 30-year U.S. Treasury bond rose
more than 40 percent between October 1985
and July 1986 and fell nearly 20 percent during
the first half of 1987.
These price fluctuations have important
economic implications. Recent empirical stud
ies suggest that asset prices have predictive
power for the business cycle. In particular, low
bond prices (high interest rates) tend to pre
cede recessions, and high bond prices (low
interest rates) tend to precede expansions.
There are also potentially important eco
nomic costs associated with asset-price vola
tility. In particular, substantial price volatility
will tend to increase the volatility of returns on
assets. Since investors typically dislike risk,
high volatility will tend to increase the average
rate of return on capital demanded by inves
tors; that may lead to lower investment, a
smaller capital stock, and a lower standard of
living.
This article presents an analysis of the vola
tility of security prices. The objective is to
discuss issues associated with whether move
ments in asset prices reflect changes in the
fundamental value of the asset or whether
these extreme price changes might be associ
ated with changes in market psychology that
may not be related to business conditions.
MARKET FUNDAMENTALS
There is an old debate associated with
whether asset prices correspond closely to
their fundamental values or whether market
psychology and extraneous factors can cause
prices to deviate substantially from an asset's
fundamental value. This debate has focused
on the interpretation of changes in security

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4
Federal Reserve Bank of St. Louis

JANUARY/FEBRUARY 1996

prices and their volatility. Many academic
economists have argued that security prices
efficiently reflect current and past information
and that market prices are a good approxima
tion of a security's fundamental value. Funda
mental values are often referred to as market
fundamentals.
The fundamental value of an asset is de
fined as the present value of the expected
payoff from that asset. For example, consider
a hypothetical asset that yields $1 per year for
five years. The fundamental value of this asset
would be the sum of the five yearly payoffs,
discounted by the relevant interest rate. (Dis
counting a future cash flow by an interest rate
is required because a $1 payoff in the future is
not equivalent to a $1 payoff today.) One can
use the same logic to determine the fundamen
tal value of a stock. Since the payoff from a
stock is the dividend, one measure of the fun
damental value of a stock is the sum of all
(expected) discounted future dividend pay
ments.
Market fundamentals, combined with the
efficient markets theory, provide a simple tool
for interpreting fluctuations in security prices.
According to the efficient markets theory, se
curity prices fluctuate only as investors re
spond to new information concerning changes
in market fundamentals (the discounted sum
of future cash flows).1 For example, suppose a
pharmaceutical manufacturer announces that
it has developed and tested a new product that
successfully combats cancer. The efficient mar
kets theory predicts that the price of the
company's stock would jump immediately as
investors re-evaluate the security in light of the
new information. The extent of the price in
crease reflects how the new information alters
market fundamentals. An increase of 15 per
cent in the stock price indicates that the dis

1For a readable discussion of security prices and the
efficient markets theory, see Burton Malkiel's book.

FEDERAL RESERVE BANK OF PHILADELPHIA

When the Bubble Bursts: Psychology or Fundamentals?

counted sum of expected future dividends is
15 percent higher, according to the theory.
A popular version of the efficient markets
theory states that security prices will follow a
"martingale."2The basic idea behind the mar
tingale model for security prices is that the
difference between a stock's price today and a
stock's discounted price tomorrow is com
pletely unpredictable.3 Thus, the main impli
cation of this model is that the best forecast for
tomorrow's stock price will simply be today's
price. Moreover, the efficient markets theory
implies that whatever change occurs in the
stock price tomorrow will be completely ac
counted for by new information on market
fundamentals.
This theory makes a number of predictions
for the behavior of asset prices. One important
implication of the martingale model is that
trading strategies designed to "beat the mar
ket" cannot be systematically successful. This
follow s from the fact th at for the m arting ale

model, the probability that the price of a stock
will rise in value tomorrow is the same as the
probability that the price will fall. Moreover,
this theory predicts that stocks cannot be iden
tified as under- or overvalued, nor are there
particularly good or bad times to purchase
stocks. Another strong implication of this
theory is that the dominant investment strat
egy is a very simple one: buy and hold a
diversified portfolio of assets.
This theory has been widely applied to un
derstanding movements in asset prices. Its
popularity likely reflects the fact that it pro
vides a simple way of using basic economic
theory to evaluate security prices. Also, an

2The martingale model of security prices, which has
also been called the random-walk model, comes from an
assumption that investors care only about the expected
rate of return on an asset, not the variability of the return.
te ch n ica lly , this implication is for the change in price
plus any dividend amount.

Lee E. Ohanian

important im plication of the theory— that
changes in asset prices are unpredictable—
seems to be fairly well supported by a large
body of data. However, some of the strong
assumptions embodied in the theory, such as
the risk neutrality of investors, and the fact
that some other features of the data are diffi
cult to reconcile with the theory have led to
criticisms of this model.
Some critics of the efficient markets theory
point out that the volatility of security prices
seems much too high to be justified by changes
in market fundamentals. Market traders and
many financial analysts claim that new infor
mation about market fundamentals provides
only a partial explanation of observed price
fluctuations. While they acknowledge that long
term movements in securities prices corre
spond to changes in fundamentals, they argue
that short-term fluctuations are caused by shifts
in market psychology or perhaps even by events
that have no direct b ea rin g on bu sin ess p ro s
pects or economic conditions.
BUBBLES
A bubble is defined as any deviation of an
asset's price from its fundamental value. We
can think of an asset's price as consisting of two
components: one associated with market fun
damentals and the other representing the
bubble. The bubble theory suggests that secu
rities may go through periods of under- and
overvaluation relative to fair-market values.
One reason for this may be investor overreac
tion. In the pharmaceutical example described
above, investors may be overly optimistic in
evaluating the increase in the firm's profits. Of
course, investors have strong incentives to
correctly evaluate how product developments
affect firm profitability. This reasoning sug
gests that it's unlikely that investors will con
sistently overreact to news about firms' profit
ability.
Bubbles may also reflect investors' reac
tions to factors unrelated to fundamental eco
5

BUSINESS REVIEW

nomic and business conditions. Hypotheti
cally, individual investors may rush into the
stock market because they believe everyone
else is making money in the market. In this
case, they prefer to buy stocks immediately
rather than miss an excellent buying opportu
nity. As a result, the anticipation of rising
prices becomes a self-fulfilling prophecy, and
market participants enjoy profits that may not
necessarily reflect favorable business prospects.
For example, investors know that the out
come of the Super Bowl played each January
has had a good track record in predicting the
course of that year's stock-market performance.
When a National Football Conference team
has won, the stock market has frequently in
creased considerably over the year, while a
win for an American Football Conference team
often presages a lower stock market. Even
though the outcome of a football game has
little, if any, effect on overall business condi
tions, the business press and investor publica
tions often cite this correspondence. As long as
some investors are perceived to act on this
statistic, others also may buy in anticipation of
this higher demand and rising prices. If enough
investors behave this way, prices rise and ex
pectations become self-fulfilling.
Certain types of bubbles can be difficult to
explain in a sensible way. They are similar to
Ponzi schemes and chain letters in that partici
pants will benefit from the game as long as
others can be found who are eager to play the
game. Of course, Ponzi schemes crash as soon
as individuals believe it will be difficult to find
others willing to participate. Similarly, some
types of bubbles imply that dramatic declines
in security prices are the result of investors
finally realizing that rising prices may never be
justified on economic grounds. At that point,
investors try to sell their assets and prices
drop: the bubble bursts.
While certain types of bubbles seem to be
inconsistent with rational behavior, there is a
class of bubbles called rational bubbles.4 A ra

JANUARY/FEBRUARY 1996

tional bubble reflects a self-fulfilling belief
among rational investors that an asset's price
depends on variables unrelated to market fun
damentals. In this context, a rational investor
is an individual who efficiently uses relevant
information for assessing the value of a secu
rity. Within the bubbles framework, the fact
that investors are rational means that while
bubbles can exist, obvious profit opportunities
cannot. This simply means that if an easy profit
opportunity were available, a rational investor
would exploit it and quickly eliminate the
opportunity. In other words, for simple types
of bubbles, the expected rate of return on a
security must be the same whether or not the
price includes a bubble.
This means that one key feature of a rational
bubble is that the evolution of the bubble over
time is restricted to rule out easy profit oppor
tunities. For example, a situation in which all
investors expect a security to double in price
between today and tomorrow, but fall back to
its original value the following day would not
constitute a rational bubble. In this case, every
one would rationally want to sell the security
tomorrow, so that the price would fall before
the following day. Alternatively, an asset could
be overpriced 20 percent relative to its funda
mental value and, thus, could exhibit a rational
bubble, as long as both the fundamental value
and the bubble component are expected to
grow at the same rate. For example, suppose
that market fundamentals for a security were
expected to grow at 5 percent per year forever.
The price of this security would have a rational
bubble if the bubble component also grew at 5
percent per year. In this case, the rate of return
on the security with the bubble component

4A large literature has analyzed rational bubbles. This
review provides an analysis of some very simple examples.
For an extensive review of this literature, see the Journal o f
Economic Perspectives, Spring 1990, Symposium on Bubbles,
pp. 13-102.

FEDERAL RESERVE BANK OF PHILADELPHIA

When the Bubble Bursts: Psychology or Fundamentals?

Lee E. Ohanian

erns, where trades were made without margin
limits and, presumably, the flow of spirits
facilitated transactions. Interestingly, specula
tion apparently spread to common bulbs unaf
fected by the mosaic virus. In the first week of
February 1637, prices peaked, and common
bulb prices rose 20-fold in one month. Then,
prices fell dramatically. While historical data
from this period are sketchy at best, Peter
Garber of Brown University has estimated that
common bulb prices lost about 95 percent of
their peak values just three months after the
crash. A century later, the bulbs were virtually
worthless. The strikingly colored Semper
HISTORICAL EPISODES
Augustus bulb, which traded for about $60,000
OF DRAMATIC PRICE MOVEMENTS
(in current dollars) in February 1637, com
A number of historical episodes of extreme manded just 50 cents in 1739.
Tulipmania was a costly lesson for the Dutch.
price movements have been interpreted as
bubbles. While these episodes and the circum Unfortunately, the British did not learn from
stances surrounding them bear little resem this episode. In 1711, some holders of short
blance to modern financial markets, they are term British government war debt agreed to
interesting to analyze, since they may be help exchange that debt for equity shares in a new
government-chartered, joint-stock company
ful in understanding current experience.
Perhaps the most famous episode occurred called the South Sea Company. In return, the
in 17th century Holland with an unlikely asset: company received a perpetual annuity paying
diseased tulip bulbs. Tulipmania, as it is often 6 percent annually on the same face value of
called, began quietly when a nonfatal virus, debt that had been exchanged. The South Sea
known as a mosaic, attacked tulip bulbs. The Company was also given a monopoly on all
effect of the virus was to produce a variegated trade to the South Seas. Although initial trad
flower of brilliant stripes and colors. The virus ing was fraught with mistakes and a war with
affected only a relatively small number of Spain shut off most trading opportunities, the
bulbs, and these bulbs became highly prized price of the stock rose modestly. By 1719, it
appeared that peace with Spain was at hand,
by collectors.
As the prices of the mosaic bulbs began to and as a result, prospects for the South Sea
rise rapidly, investors as well as horticulturists Company looked better than ever.
In 1720, many additional holders of govern
began acquiring them. The increased demand
for the bulbs resulted in even higher bulb ment debt traded the debt to the South Sea
prices and large profits for existing owners. Company in exchange for new stock. The com
Charles Mackay, who described this episode pany was expected to consolidate the debt and
in his book, noted that "nobles, citizens, farm receive a steady stream of interest payments
ers, mechanics, seamen, footmen, maid-ser on the government obligation. At this point,
v an ts, even chim ney sw eeps and old the stock's price rose from 130 pounds to 300
pounds per share. After Parliament approved
clotheswomen dabbled in bulbs."
By 1635, tulipmania had engulfed the coun this plan, a new stock offering at 300 pounds
try. Futures markets sprang up in local tav quickly shot up to 340. Fights among investors

would be identical to the rate of return on the
security without a bubble.
Bubble interpretations have been popular
with professional investors and the financial
press for many years. In his introduction to
Charles Mackay's Memoirs o f Extraordinary
Popular Delusions and the Madness o f Crowds, the
noted investor Bernard Baruch wrote, "All
economic movements, by their very nature,
are motivated by crowd psychology...Men
think in herds; it will be seen that they go mad
in herds, while they only recover their senses
slowly, and one by one."

BUSINESS REVIEW

JANUARY/FEBRUARY 1996

eager to buy the offering were common. The
Critics of the efficient markets theory point
next offering came out at 400, and the next at out that the theory cannot account for ob
500, with an option to buy at just 10 percent served volatility in security prices. Neverthe
margin. When the stock hit 800, half of the less, the implication of the theory that changes
members of the House of Lords and the House in asset prices are unpredictable has received
a fair amount of
of
Com m ons
em pirical sup
plunged in. Soon
PLEASE DON'T EAT THE TULIPS!
port. For ex
the price hit 1000
ample, a num
pounds
per
ber of experi
share. At this
In his book, Charles Mackay relates an anecdote that
m ents
have
point, the direc
shows just how seriously the Dutch took their tulips.
been
conducted
tors of the com
Mackay describes an incident in which a young sailor
in which stock
pany began sell
notified a merchant of the arrival of a shipment of new
ing, w hich re
p
ortfolios
goods. For bringing the news, the sailor was summarily
picked by Wall
sulted in rapid
rewarded with a breakfast of herring. It so happened that
the sailor noticed the ideal condiment for his herring, an
liq u id atio n of
Street's leading
onion, perched on the merchant's counter and helped
South Sea shares.
money manag
himself to it. To the merchant's— and ultimately the
ers were com
Parliament ul
sailor's— distress, the "onion" was actually a prized
pared over time
timately passed
Semper Augustus bulb. The merchant pressed charges,
against a port
the Bubble Act,
and the unwitting felon spent several months in prison.
folio chosen by
w hich p ro h ib
Of course, who knows how much— if any— of this
throwing darts
ited the issuing
story is true. Peter Garber, for one, points out that an
at
a stock page
of stock certifi
astute merchant would hardly leave such a valuable
from
the Wall
cates by compa
object lying around, especially within easy reach of a
Street
Journ al.
nies. So strong
random guest. Nonetheless, it underlines the frenzy cre
The martingale
was the British
ated by the speculation in tulip bulbs in 17th century
Holland.
model predicts
aversion to a re
that portfolios
peat bubble that
chosen at ran
this law was in
force for the next century: British companies dom should perform, on average, about the
were not allowed to issue stock until 1825.5*
same as those chosen by portfolio managers. In
many of these experiments, random picks do
BUBBLES VS. MARKET FUNDAMENTALS: just as well as many of Wall Street's leading
EVIDENCE FROM MODERN TIMES
traders.
Moreover, critics of the bubble theory point
Although not accepted universally, many
economists agree that prices during these his out that technical analysis, which is the prac
torical periods reflect some bubble compo tice of trying to identify systematic patterns in
nent. Are bubble explanations of extreme price security price movements, should be useful in
movements confined to just a few historical choosing securities if bubbles are present. The
episodes, or might bubbles be relevant for basic idea is to plot security prices over time
and use past price behavior to predict future
today's financial markets?
prices. Patterns often considered important
for predicting future price movements include
the "inverted head and shoulders," "triple top
5See Charles Kindleberger's book for a more in-depth
treatment of the South Sea bubble.
double bottoms," and "piercing necklines." In

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When the Bubble Bursts: Psychology or Fundamentals?

general, these approaches have not signifi
cantly outperformed randomly chosen strate
gies or buy-and-hold strategies.
Nevertheless, several observations from the
stock market do challenge efficient markets
explanations. One of the best known patterns
is the January effect, which refers to the first
two weeks of January when stock returns tend
to be unusually high. This is also a period when
stocks of smaller companies, such as those that
tend to trade on the over-the-counter market,
outperform larger, well-known issues. While
selling stocks because of end-of-the-year tax
considerations may play a role in explaining
the January effect, it cannot completely ac
count for the anomaly. The January effect was
present in the United States even before in
come taxes.
Some economists have made another obser
vation that challenges the market fundamen
tals theory: the underpricing of initial public
offerings (IPOs). An IPO is the initial sale of
equity shares in a company that was privately
held. Brokers allocate the initial offerings of
shares to customers, and after the initial offer
ing, these shares are traded on public ex
changes. For many IPOs, the initial rate of
return is enormous.6 In a 1988 paper, Roger
Ibbotson, Jody Sindelar, and Jay Ritter re
ported that between 1977 and 1987, the aver
age initial return, which is defined as the per
centage increase from the offering price to the
end-of-first-day bid price, is over 20 percent.
On an annualized basis, this rate of return
would be in the neighborhood of over 1000
percent.
These enormous returns suggest to some
observ ers th at the shares are in itially
underpriced. There does not appear to be a

6For example, in August 1995, Netscape, a company
that produces software for the Internet, had an IPO with an
offering price of $28 on Tuesday and closed at $58.25 on
Wednesday.

Lee E. Ohanian

generally accepted theory of this observation,
and it is somewhat puzzling as to why issuing
firms would agree to deal with underwriters
who underprice the security.7
TESTING FOR BUBBLES AND EXCESS
VOLATILITY IN ASSET MARKETS
The tulipmania and the South Sea bubbles
are striking examples of how prices may di
verge from fundamental values. Many econo
mists think it unlikely that similar episodes
could occur today. If there are bubble or
nonfundamental components in asset prices,
chances are they will be much less dramatic
and harder to distinguish from market funda
mentals.
Until recently, claims that prices were out of
line with market fundamentals were conjec
tures, substantiated by little more than anec
dotal evidence. However, recently developed
statistical tests may help shed some light on the
debate. A number of tests have been devel
oped, and two widely used tests will be dis
cussed here.
Robert Shiller of Yale University developed
and implemented one popular test that has
been used to evaluate whether prices are con
sistent with market fundamentals. Shiller con
structed an economic model of the fundamen
tal price of an asset. The test compares the
volatility of the observed security price with
the volatility of the fundamental price. These
tests are typically called variance bounds tests,
since the basic idea is to determine whether the
observed variability of market price is consis
tent with the observed variability of market
fundamentals.
For stocks, the model assumes that the price
an investor would be willing to pay today
depends on the total return (the dividend and
price appreciation) he expects to receive from

7For additional discussion of asset market anomalies,
see Richard Thaler's 1992 book.

BUSINESS REVIEW

the stock tomorrow. In turn, the price in the
following period depends on the dividend and
price appreciation he expects to receive two
periods from now, and so forth. This logic
implies that the fundamental price of a stock
today will depend on all expected future divi
dends adjusted by an appropriate discount
rate (interest rate). This analysis suggests that
today's share price is a predictor of future
returns. If the market price is consistent with
market fundamentals, the share price should
equal market fundamentals. In this case, the
volatility of predicted cash flows (the market
fundamentals price) cannot exceed the volatil
ity of actual cash flows (the returns). Using
data on dividends and prices, we can compare
the historical volatility of the predicted cash
flows to the actual cash flows.
A constructed series represents the sum of
discounted dividends from stocks listed in the
Standard & Poor's 500 graphed against the
price of the S&P 500 since 1871 (Figure 1).
Clearly, stock prices are many times more
volatile than the present value of discounted
dividends. Given the relatively stable history
of dividends over the last century, market
fundamentals, constructed this way, clearly
cannot account for the extreme volatility of
asset prices. One interpretation is that stock
prices are too volatile relative to observed
changes in cash flows and that some factor
unrelated to business conditions is responsible
for the bulk of asset price fluctuations.
However, there are some important caveats
associated with interpreting these tests. First,
there is no unique way to determine how
investors discount future cash flows. The typi
cal procedure carried out in these tests (and in
Figure 1) is to assume that the discount factor
(interest rate) is constant, which may not be
true. Second, we cannot observe people's ex
pectations of future dividends directly, so we
must infer them. It is common to simply as
sume that today's stock price is exactly equal
to the future discounted sum of dividends. But

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JANUARY/FEBRUARY 1996

this practice leads to difficulties in evaluating
whether market fundamentals are consistent
with price data. Instead, Robert Flood, Robert
Hodrick, and Paul Kaplan, in a 1986 paper,
suggested that apparent violations of variance
bounds tests reflect errors in the model. That
is, the test depends on the underlying eco
nomic model being correct. Of course, this is a
very strong assumption, and test results may
simply reflect misspecification of the economic
model. While there may be bubble compo
nents to asset prices, this type of test will not
likely resolve the debate.
An alternative approach for testing whether
variations in security prices are consistent with
variations in market fundamentals is to deter
mine whether the trend rate of growth in the
asset price is similar to that in market funda
mentals. Specifically, if market fundamentals
are growing at a slower rate than the price of
the corresponding asset, we may reasonably
conclude that prices include a particular type
of bubble component. This procedure can be
used to detect the presence of bubbles that
grow continuously over time.
In 1985, Jam es H am ilton and Charles
Whiteman, and in 1988, Behzad Diba and
Herschel Grossman conducted tests along these
lines. To determine whether market prices
grow at a faster rate than market fundamen
tals, we must evaluate the trends in the data.
First, we test the data on annual stock prices
and annual dividends to see if there are trends.
If both series have trends, the series are
"differenced." For example, to calculate the
differenced data for market prices, subtract
the price of the asset last year from its price this
year.
The differenced data for market prices and
dividends are then tested for trends. If both of
these differenced series have trends, the series
are differenced again, and the trend tests are
repeated . This process of su ccessiv ely
differencing the data continues until the trans
formed data do not have trends. If market
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Lee E. Ohanian

When the Bubble Bursts: Psychology or Fundamentals?

FIGURE 1

Detrended Stock Prices and the Present Value
of Detrended Dividends

1871 -1994
Index

P is the real Standard & Poor's Composite Stock Price Index, detrended by a long-run exponential growth factor. P* is
the discounted present value of real dividends, detrended by the long-run exponential growth factor. Real values are
calculated by dividing nominal values by the wholesale price index.
Source: Shiller, Market Volatility, Figure 5.1, updated by author.

prices must be differenced more times than
market fundamentals, we may reasonably con
clude that a bubble is present in market prices.
This analysis for dividends and stock price
data, which appears in Figure 2, offers evi
dence that both prices and dividends have
trends, but when differenced once, both do
not. This implies that prices over this period
have not grown consistently faster than divi
dends and provides evidence against the no

tion that stock prices have included a growing
bubble component.
Although the analysis presented here was
conducted with data only from the stock mar
ket, these same tests can be used to evaluate
data from the bond and foreign exchange mar
kets. Briefly, the nature of these data are quite
similar to data from the stock market. Like
stocks, the variability of bond prices and ex
change rates seems to be high relative to marll

JANUARY/FEBRUARY 1996

BUSINESS REVIEW

FIGURE 2

Stock Prices and Dividends

1871 -1994

1871

1881

1891

1901

1911

1921

1931

1941

1951

1961

1971

1981

1991

"Stock prices" is a logarithmic index of the real Standard & Poor's Composite Stock Price Index. "Dividends" is a
logarithmic index of the real dividends on the real Standard & Poor's Composite Stock Price Index. Real values are
calculated by dividing nominal values by the wholesale price index.
Source: Author's calculations from data in Standard & Poor's Security Price Index Record.

ket fundamentals. Moreover, there don't ap
pear to be any differences in the trend behavior
of market fundamentals and prices for either
bonds or foreign exchange.
CONCLUSION
The extreme volatility of security prices has
been a source of considerable interest since
financial assets have traded in organized mar
kets. It is important to distinguish between

http://fraser.stlouisfed.org/
12
Federal Reserve Bank of St. Louis

market fundamentals and bubbles when ana
lyzing the volatility of any security. If there are
dramatic changes in fundamental economic
factors, we would expect to see highly volatile
security prices. If the volatility of security prices
is considerably greater than the volatility of
underlying business conditions, or if asset
prices tend to grow much faster than the asset's
associated cash flows, price movements may
reflect a bubble component.
FEDERAL RESERVE BANK OF PHILADELPHIA

When the Bubble Bursts: Psychology or Fundamentals?

The episodes of Dutch tulipmania and the
British South Sea bubble provide dramatic
examples of how prices may have deviated
from fundamental values. Anecdotal evidence
from recent periods provides no clear answer
to the question of whether price movements
may be due to bubbles. A number of statistical
procedures have been developed to investi
gate these questions directly, and these tests
have been applied to stock market data. Unfor
tunately, these tests often rely on assumptions
that make interpretation of results very diffi
cult. Test results that show differences be
tween security prices and market fundamen
tals may be due to bubble components, but

Lee E. Ohanian

they may also reflect errors in the model for
market fundamentals. That is, a researcher
may find evidence in favor of bubbles, but this
may simply be due to the fact that the model
for market fundamentals is wrong.
Since market fundamentals are generically
unobservable, it will always be difficult, if not
impossible, to analyze data on asset prices and
determine whether price movements can be
entirely reconciled with movements in market
fundamentals. We are left with the interesting
observation that there are historical variations
in asset prices that, at least, do not appear to be
consistent with variations in underlying busi
ness conditions.

References
Diba, Behzad T., and Herschel I. Grossman. "Explosive Rational Bubbles in Stock Prices?" American
Economic Review, June 1988, pp. 520-30.
Flood, Robert P., Robert J. Hodrick, and Paul Kaplan. "An Evaluation of Recent Evidence on Stock Market
Bubbles," National Bureau of Economic Research Working Paper 1971, Cambridge, MA, 1986.
Garber, Peter M. "Tulipmania," Journal o f Political Economy, 97, June 1989, pp. 535-60.
Hamilton, James D., and Charles H. Whiteman. "The Observable Implications of Self-Fulfilling Expec
tations," Journal o f Monetary Economics, 16, November 1985, pp. 353-73.
Ibbotson, Roger G., Jody L. Sindelar, and Jay R. Ritter.
Corporate Finance, Summer 1988, pp. 37-45.

"Initial Public Offerings," Journal o f Applied

Kindleberger, Charles P. Manias, Panics, and Crashes: A History o f Financial Crises. New York: Basic Books,
Inc., 1978.
Mackay, Charles. Memoirs o f Extraordinary Popular Delusions and the Madness o f Crowds. London: Bentley,
1841.
Malkiel, Burton G. A Random Walk Down Wall Street: Updated fo r the 1990s Investor, 5th ed. New York:
Norton, 1991.
Shiller, Robert J. Market Volatility. Cambridge, MA: MIT Press, 1989.
Thaler, Richard H. The Winner's Curse: Paradoxes and Anomalies o f Economic Life. New York: The Free Press,
1992.

The Federal Reserve Bank of Philadelphia
Conference on

Expectations in Economics:
In Honor of the 50th Anniversary of the Livingston Survey

October 3 & 4 ,1 9 9 6
Philadelphia, Pennsylvania

The Research Division of the Federal Reserve Bank of Philadelphia is sponsoring a conference on
Expectations in Economics: In Honor of the 50th Anniversary of the Livingston Survey to be held at
the Philadelphia Fed on October 3 and 4,1996. The purpose of the conference is to bring together
practitioners and researchers who are engaged in both theoretical and empirical work on expecta
tions, including research using data from surveys of expectations.

CALL FOR PAPERS
If you are interested in presenting your research at the conference, please send a completed
paper or detailed abstract by April 1 to the address below. Please note that we will not publish the
papers presented at the conference; we will, however, publish a summary of the conference in our
Business Review. We will pay modest honoraria and provide travel expenses for paper presenters.

Send papers or abstracts by April 1 to:
Dean Croushore
Research Department
Federal Reserve Bank of Philadelphia
Ten Independence Mali
Philadelphia, PA 19106-1574
e-mail address: croushor@frbphil.org

FEDERAL RESERVE BANK OF PHILADELPHIA

The Cyclical Volatility of Interest Rates
Keith Sill*

JL he variability of short-term and long
term interest rates is a prominent feature of the
economy. Interest rates change in response to
a variety of economic events, such as changes
in Fed policy, crises in domestic and interna
tional financial markets, and changes in the
prospects for long-term economic growth and
inflation. However, economic events such as
these tend to be irregular. There is a more
regular variability of interest rates associated

*Keith Sill is an economist in the Research Department
of the Philadelphia Fed.

with the business cycle, the expansions and
contractions that the economy experiences over
time. For example, short-term interest rates
rise in expansions and fall in recessions. Long
term interest rates do not appear to co-vary
much with the level of economic output.
The term cyclical volatility o f interest rates
refers to the variability of interest rates over
periods that correspond to the length of the
typical business cycle. In this article, we will
examine some facts and theory about the cycli
cal volatility of short-term and long-term in
terest rates. Why should we care about interest
rate volatility? How do short-term and long
15

BUSINESS REVIEW

term interest rates behave over the business
cycle? What determines the cyclical volatility
of interest rates associated with different
maturities of government bonds? These ques
tions are important to ask and answer as we
seek a fuller understanding of the dynamics of
the business cycle in market economies.
WHY DOES INTEREST RATE
VOLATILITY MATTER?
The variability of interest rates affects deci
sions about how to save and invest. Investors
differ in their willingness to hold risky assets
such as stocks and bonds. When the returns to
holding stocks and bonds are highly volatile,
investors who rely on these assets to provide
for their consumption face a relatively large
chance of having low consumption at any given
time. For example, before retirement, people
receive a steady stream of income that helps to
buffer the changes in wealth associated with
changes in the returns on their investment
portfolios. This steady return from working
helps them maintain a relatively steady level of
consumption. After retirement, people no
longer have the steady stream of income from
working (though it will, in part, be replaced by
pension income and Social Security), so a less
volatile investment portfolio is called for. The
lower volatility of investment returns allows
retirees to maintain a relatively even level of
consumption over time. Young investors, who
are saving for retirement, are better able to
absorb the risks of holding assets with highly
volatile prices and returns. They can weight
their portfolio more heavily toward risky stocks
and bonds because they are receiving a steady
return from working. For holding these riskier
assets, the young investor will be rewarded
with a higher average return on the invest
ment.
Just as individuals care about managing
risk in their investment portfolios, so do firms.
To manage risk, firms must pay attention to
interest rate volatility and the composition of

JANUARY/FEBRUARY 1996

their portfolios. Many business firms hold
portfolios containing large numbers of assets
and, thus, are interested in quantifying the risk
of losing large sums of money. As risks in the
economy change, the expected gains and losses
from the investment portfolio change. Mea
suring this risk involves knowing how volatile
prices of and returns on assets are, as well as
how the returns on different assets change
together over time. The volatility of interest
rates is likely to be an important component in
quantifying risk and guiding the investment
decisions of these institutions.
Interest rate volatility also has implications
for how the prices of certain types of assets are
determined. Options are assets that give inves
tors the right, but not the obligation, to buy
(call options) or sell (put options) other assets
(such as stocks or bonds) at a prespecified
price at or before some prespecified time in the
future. For options purchased on interest-bear
ing securities, modern finance theory demon
strates that the option price depends on the
volatility of returns on the underlying asset.
The volatility of interest rates is related to the
volatility of returns on these assets.
Thus, interest rates and their volatility have
important implications for how both individu
als and firms make investment decisions. These
investment decisions are part of the process
w hereby resou rces are allocated in the
economy. To begin, we'll briefly discuss how
bond prices, interest rates, and maturities of
bonds are related and how interest rates can be
determined from bond prices.
INTEREST RATES, BOND PRICES,
AND THE TERM STRUCTURE
There is a very close connection between
bond prices and interest rates. We will focus on
interest rates calculated from prices of traded
U.S. government securities and show how the
interest rate on a particularly simple type of
security can be derived solely from its price.
We focus on yields derived from U.S. governFEDERAL RESERVE BANK OF PHILADELPHIA

Keith Sill

The Cyclical Volatility of Interest Rates

merit securities because these assets are backed
by the full faith and credit of the government
and, therefore, have virtually no default risk.
The U.S. government issues securities of
many different maturities: the maturity is the
length of time until the final payment on the
security is made by the issuer. Treasury bonds
are fixed -cou p on secu rities w ith in itial
maturities of more than 10 years. Treasury
notes are fixed-coupon securities with initial
maturities of from two to 10 years. Treasury
bills are securities that are sold at a discount
from face value and have initial maturities of a
year or less.
If we know a bond's current price and the
payments that the bondholder will receive
over the course of the bond's life, we can
calculate the implied interest rate on the bond.
This interest rate, called yield-to-maturity,

equates the current price of the bond to the
present value of the bond's payment stream.1*
The relationship between the maturity of bonds
and the interest rates implied by bond prices is
called the term structure of interest rates. A
plot of the relationship between interest rates
and bond maturity, called the yield curve, can
take a variety of shapes (Figure 1). Typically,
interest rates on short-term bonds are lower
than interest rates on long-term bonds, in which
case the yield curve is upward sloping, as
shown in the figure for the fourth quarter 1987.
But sometimes the yield curve inverts, in which
case interest rates on short-term bonds are

1For more detail on how yield-to-maturity is calcu
lated, see my article in the July/A ugust 1994 Business
Review.

FIGURE 1

Term Structure of Interest Rates
Interest Rate

BUSINESS REVIEW

higher than interest rates on long-term bonds,
as shown in the figure for the third quarter
1989.
The shape of the typical yield curve shows
that interest rates often vary with maturity.
We might also suspect that the volatility of
interest rates varies with maturity. But before
we turn to how volatility is measured and how
v olatility is related to m atu rity , le t's clarify the

relationship between interest rates and the
price of a particularly simple type of bond.
Interest Rates and Bond Prices. Interest
rates on certain types of bonds can be derived
solely from the bonds' price and maturity.
Let's look at a particular type of bond called a
discount, or zero-coupon, bond. A discount
bond sells at a discount from its face value and
makes no interest payments over its lifetime.
When the bond matures, the bondholder re
ceives the bond's face value. For example, a
one-year Treasury bill with a face value of
$10,000 is a discount bond that promises to pay
the holder $10,000 in one year's time. Such a
bond may sell for a current price of $9434, in
which case the implied interest rate on the
bond is 6 percent (($10,000-$9434)/$9434 =
.06). Clearly, as the current price of the bond
changes, the implied interest rate will change.
For example, suppose the current price of the
bond falls to $9009. Then the implied interest
rate on the bond is 11 percent (($10,000-$9009) /
$9009 = .11). So, as the price of the bond falls,
the interest rate rises; as the price rises, the
interest rate falls.2
The U.S. Treasury does not issue discount
bonds with maturities greater than one year.

2There is a simple relationship between the interest
rate on a discount bond and the price of the bond. Suppose
the price today of a bond that pays off $100 in five years is
$75. The five-year interest rate on the bond is 33.3 percent
[(100-75)/75 = .333]. The average annual interest rate on
the bond is 5.9 percent since [$75 x (1.059)5 = $100], Gen
eralizing this idea, if the interest rate is r on a bond paying
$1 in j years, the current price of the bond is $1 / (l+r)>.

JANUARY/FEBRUARY 1996

However, financial market participants create
pure discount bonds from long-term, coupon
paying Treasury bonds by "stripping" the cou
pon (semiannual interest) payments from the
principal payment and selling the components
as separate discount securities. In February
1985, the Treasury announced the STRIPS (sepa
rate trading of registered interest and princi
pal o f securities) p ro g ram , which facilitated
the "stripping" of long-term Treasury bonds.
Under the STRIPS program, all newly issued
Treasury bonds and notes with maturities of
10 years or longer are eligible for stripping.
The prices of these pure discount bonds can be
found in publications such as the Wall Street
Journal.
Since there is a clearly defined relationship
between interest rates and prices for discount
bonds, we need to refer to only one of these
elements, not both. When we consider dis
count bond prices, we can easily derive the
implied interest rates. Similarly, when we talk
about the volatility of discount bond prices, we
will easily be able to make inferences about the
volatility of interest rates.
Trends and Cycles in Interest Rates. We
can plot the interest rate on discount bonds
with a 10-year maturity from 1959 to 1990
(Figure 2).3 Notice that, overall, the interest
rate tended to rise from 1959 to the early 1980s,
after which it generally declined.
From Figure 2 we can discern two types of
variability in interest rates and hence in dis
count bond prices: long term and short term.
Long-term variability refers to broad trends in
interest rates, such as the upward trend until
the early 1980s and the downward trend since
then. Short-term variability refers to how in
terest rates vary around these long-term swings.
Since our focus is on the business-cycle volatil
ity of interest rates, we would like to remove

3The data plotted in Figure 2 are yields on discount
bonds from the dataset compiled by McCulloch and Kwon.

FEDERAL RESERVE BANK OF PHILADELPHIA

Keith Sill

The Cyclical Volatility of Interest Rates

FIGURE 2

Ten-Year Interest Rate and Long-Term Trend

that part of interest rate volatility associated
with swings of longer duration than the typical
business cycle.
The National Bureau of Economic Research
defines minor cycles as recurrent fluctuations
lasting from two to four years and major cycles
as recurrent fluctuations lasting about eight
years. Figure 2 clearly shows long-term trends
in interest rates that are of greater duration
than typical business-cycle lengths. In Figure 2
we've plotted a long-run trend that is fitted to
the interest rate data. This long-term trend is
chosen in such a way that it removes the
swings in interest rates associated with peri
ods longer than about eight years.4* The re
maining short-run variability then corresponds
more closely to variability that is part of the
business-cycle movement in interest rates. We
will define the difference between the actual
interest rate and the long-run trend as the

cyclical component of the interest rate (Figure
3). The long swings in interest rates have been
taken out, and all the variability in interest
rates is around zero because this figure plots
deviations from the long-term trend. When the
interest rate is zero in Figure 3, we are on the
long-term trend line in Figure 2. Henceforth,
when we refer to interest rate volatility, we
will be referring to the variability of this shortrun component.
MEASURING THE VOLATILITY
OF INTEREST RATES
We will measure interest rate volatility us
ing a statistic called the standard deviation.

4The fitted trend is calculated using the HodrickPrescott filter. More details on how this filter works are
provided in the paper by Robert King and Sergio Rebelo.

JANUARY/FEBRUARY 1996

BUSINESS REVIEW

FIGURE 3

Cyclical Component of Ten-Year Interest Rate
Percent
3.0

2.0

1.0

0.0

1.0

2.0

The standard deviation measures how dis
persed a variable is around its average value.
If the standard deviation is high, observations
on a variable tend to be far away from the
variable's average value. If the standard de
viation is low, observations on the variable
tend to be clustered around the average value.
Therefore, as the standard deviation increases,
there is a greater chance that we will see large
changes in the value of the variable.5
The volatility of interest rates can be calcu5The standard deviation is calculated as the square root
of the variance of a variable. Suppose we have n observa
tions on a variable {X1,X2,...Xn}. Denote the average value
of X by X. The variance of X is then the average of the
squared deviations of X from its mean: V(x) = [ (X,-X)2 +
(X2-X)2 + ... (Xn-X)2]/n . The standard deviation of X is the
positive square root of V(x).

lated over the entire term structure of interest
rates: we simply use historical data to calculate
the standard deviation of interest rates for
each maturity. Table 1 presents the relation
ship between interest rate volatility and the
maturity of bonds as well as the standard
deviation of the associated prices for discount
bonds.6 The table shows that short-term inter
est rates are more volatile than long-term in
terest rates and that long-term discount bond

6More specifically, we calculate the standard deviation
of detrended yields and logarithms of bond prices using
quarterly data over the period 1959:Q1 to 1990:Q1. We use
the logarithm of the discount bond price because it is
proportional to the yield-to-maturity of the bond, with the
factor of proportionality equal to the maturity of the bond.
The interest rates and bond prices are detrended using the
Hodrick-Prescott filter.

FEDERAL RESERVE BANK OF PHILADELPHIA

Keith Sill

The Cyclical Volatility of Interest Rates

prices are more volatile than short-term dis
count bond prices.7
In describing the cyclical volatility of inter
est rates we would like to know not just how
much interest rates vary but also how they
vary with the state of the economy. During
recessions, real output is declining; during
expansions, it's rising. We can get an idea of
the behavior of interest rates over the business
cycle by evaluating how interest rates and the
level of real output co-vary over the business
cycle. The correlation coefficient is a measure
of the strength of the co-variation between two
variables, and it can take on values between

7There is a direct relationship between bond-price vola
tility and the volatility of the interest rate on the bond.
Using the relationship between bond prices and interest
rates in footnote 2, it can be shown that the standard
deviation of the interest rate on a j-period bond is approxi
mately equal to the standard deviation of the logarithm of
the j-period bond price divided by j.

minus one and one. When the correlation coef
ficient between two variables is positive and
close to one, the two variables track each other
closely and move in the same direction: when
one variable is high, the other variable is very
likely to be high. If the correlation coefficient is
negative and close to one, the two variables
track each other closely but move in opposite
directions: when one variable is high, the other
is likely to be low. When the correlation coef
ficient is zero, the two variables do not track
each other closely in either direction.
The cyclical component of short-term inter
est rates has a positive contemporaneous cor
relation with the cyclical component of real
output. So when current output falls, short
term interest rates tend to fall, and when cur
rent real output rises, short-term interest rates
tend to rise (Table 2). The strength of this
correlation between output and interest rates
tends to decline as the maturity of the bonds
increases. By the time we get to bonds with 10-

TABLE 1

Interest Rate and Bond-Price Volatility
1959:Q1 - 1990:Q1
Time to Maturity

Standard Deviation of Detrended
Interest Rate

Standard Deviation of Detrended
Discount Bond Price

quarter

.0032

quarters

.0031

quarters

.0030

.0063
.0092

year

.0029

.0117

years

.0025

.0205

years

.0021

.0426

10 years

.0018

.0735

Standard deviations are calculated from the term structure data in McCulloch and Kwon (1993). Standard deviation is
of the logarithm of discount bond prices (see footnote 5).

BUSINESS REVIEW

JANUARY/FEBRUARY 1996

TABLE 2

Correlations of Interest Rates and Real GDP
1959:Q1 - 1990:Q1
Correlation of Detrended Interest Rate in
Period t with Detrended Output in Period:
Time to Maturity

t-1

t+1

1 quarter

.42

.35

.11

2 quarters

.41

.34

.09

3 quarters

.40

.32

.08

1 year

.37

.29

.05

2 years

.27

.20

-.02

5 years

.11

.04

-.15

10 years

.02

-.05

-.21

Output is measured as the logarithm of real GDP. Interest rate data are from the term structure data in McCulloch and
Kwon (1993).

year maturity, the contemporaneous correla
tion is negative, though quite small. This im
plies that there is little co-variation between
the cyclical movements in current real output
and the cyclical movements in long-term inter
est rates. These facts can be expressed by
saying that short-term interest rates are
procyclical and long-term interest rates are
acyclical. The results in Table 2 suggest signifi
cant business-cycle variability in short-term
interest rates but relatively little business-cycle
variability in long-term interest rates.
The last column of Table 2 shows that the
correlation between current interest rates and
real output one quarter into the future is posi
tive for short-term and negative for long-term
interest rates. This fact suggests that upward
movements in short-term interest rates are
associated with upward movements in nearterm output, but that higher long-term interest
rates forecast lower near-term output. The

first column of Table 2 shows the correlation
between current interest rates and the level of
real output one quarter in the past. These
correlations suggest that increases in current
output are associated with increases in future
interest rates.
We can also make some deductions about
the shape of the yield curve over the business
cycle using the data in Table 2. We have seen
that short-term interest rates tend to move up
when output moves up but that the correlation
tends to decline as the maturity of the bond
increases. Thus, when current output rises, the
yield curve tends to flatten, since short-term
interest rates tend to rise and long-term inter
est rates move relatively little. Similarly, when
current output declines, the yield curve tends
to steepen, since short-term interest rates tend
to fall with output and long-term interest rates
tend to remain about the same.
We have seen how the volatility of interest
FEDERAL RESERVE BANK OF PHILADELPHIA

Keith Sill

The Cyclical Volatility of Interest Rates

ity, and the deviations of the interest rate on
10-year bonds from the trend line have been
large and persistent.
In fact, the results in Table 4 show that
interest rates at all maturities may have been
more variable since that time. The table shows
the standard deviation of interest rates using
the same data, but the sample is divided into
two subsamples: from first quarter 1959 to first
quarter 1979 and from second quarter 1979 to
first quarter 1990. We see that interest rates at
all maturities have been more volatile since
1979. This result suggests the possibility that
some structural change in the economy has
affected the variability of interest rates and
bond prices.8

rates changes with maturity and how interest
rates move in relation to real output over the
business cycle. But how are interest rates on
bonds of different maturities related to each
other? In general, interest rates on bonds of
different maturities are highly correlated with
each other, with the highest correlations occur
ring between bonds of similar maturities (Table
3). Let's take the case of the interest rate on a
security with one-quarter maturity. We see
that the one-quarter interest rate is most highly
correlated with the interest rate on a bond with
two-quarter maturity, and that the correlation
declines, though remains strong, as we com
pare bonds w ith in creasin g ly d ifferen t
maturities. These correlations suggest a ten
dency for the entire yield curve to shift up and
down, while allowing for the possibility that
the shape of the yield curve can change.
Finally, if we re-examine Figure 3, we might
suspect that the measured volatility of interest
rates depends on the period we're looking at.
Since the late 1970s, long-term interest rates
appear to have shown more short-run variabil

8For short-term interest rates in particular, higher vola
tility after 1979 may reflect a change in the way that the
Federal Reserve implements monetary policy. After late
1979, and especially between late 1979 and late 1982,
monetary policy placed less emphasis on smoothing short
term interest rates. Thus, after 1979, short-term interest
rates were more likely to reflect changes in the state of the
economy.

TABLE 3

Cross Correlations of Detrended Interest Rates
1959:Q1 - 1990:Q1
Maturity
1 quarter

1 qtr

2 qtr

3 qtr

1 year

2 years

5 years

10 years

1.0

2 quarters

.99

3 quarters

.97

.99

1 year

.96

.99

2 years

.90

.94

.96

.98

5 years

.79

.84

.87

.90

.96

10 years

.69

.75

.78

.81

.90

1.0
1.0
1.0
1.0
1.0
.98

1.0

Interest rate data are from McCulloch and Kwon (1993).

JANUARY/FEBRUARY 1996

BUSINESS REVIEW

TABLE 4

Interest Rate and Bond-Price Volatility
Time to Maturity

Standard Deviation of Detrended
Interest Rate

Standard Deviation of Detrended
Discount Bond Price

59:1-79:1

79:2-90:1

59:1-79:1

79:2-90:1

1 quarter

.0026

.0040

.0026

.0040

2 quarters

.0026

.0040

.0051

.0079

3 quarters

.0025

.0039

.0075

.0116

1 year

.0024

.0037

.0094

.0148

2 years

.0019

.0034

.0151

.0275

5 years

.0014

.0030

.0278

.0608

10 years

.0011

.0027

.0442

.1080

Standard deviations are calculated from the term structure data in McCulloch and Kwon (1993). Logarithms of discount
bond prices are taken before the standard deviation is calculated (see footnote 5).

WHAT DETERMINES
INTEREST RATE VOLATILITY?
The postwar data imply that prices of long
term discount bonds are more variable than
those of short-term discount bonds and that
long-term interest rates, measured by yield-tomaturity, are less volatile than short-term rates.
In addition, we find that short-term interest
rates are procyclical, while long-term interest
rates vary little with current output. What
economic factors influence interest rate vari
ability? If we can isolate some economic deter
minants of the levels of interest rates and bond
prices, we will be well on our way to finding
determinants of this variability.

Determination of Short-Term Interest
Rates. A standard economic model will help
us think about how the interest rate on short
term discount bonds is determined. Let's con
sider the case of a discount bond that will pay
off $100 with certainty in one year. Suppose a
prospective bond buyer expects her real in

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24
Federal Reserve Bank of St. Louis

come over the coming year to be higher than
usual (real income refers to income adjusted
for any change in the general level of prices
over time). In that case, she has less of an
incentive to increase her savings by purchas
ing a bond today. In fact, she may well decide
to borrow against some of her expected in
crease in income. If all prospective bond pur
chasers expect higher real income over the
coming year, demand for current one-year
bonds will fall, and their prices will fall as well,
which means that the one-year interest rate
will rise. On the other hand, investors may
decide to hedge against the risk of lower future
income by purchasing bonds today that pro
vide a guaranteed future payoff.
If current real output (and thus aggregate
real income) is low, investors may expect fu
ture output to be low, because there is some
persistence to output movements. Hence, a
downward movement in current output is
consistent with a downward movement in
FEDERAL RESERVE BANK OF PHILADELPHIA

The Cyclical Volatility of Interest Rates

current short-term interest rates if people ex
pect output and income in the near future to be
low as well. This theory is consistent with
procyclical movement in short-term interest
rates.
The yield curve tends to flatten when out
put is high and tends to steepen when output
is low. Suppose we are currently in a boom, but
people expect a recession in one year. Inves
tors may buy one-year bonds to hedge the risk
of low future income, and they may pay for
these bonds, in part, by cashing in their shorterterm assets. This portfolio reallocation tends
to lower one-year interest rates and raise
shorter-term interest rates, thus leading to a
flatter yield curve. Empirical studies have
found that the shape of the yield curve does
help predict recessions and expansions.9
Expected inflation is also a determinant of
interest rates. Consider again the case of a
discount bond that pays $100 with certainty in
one year. Suppose now that prospective bond
purchasers expect inflation to rise over the
coming year. When inflation rises, the current
price of one-year bonds will fall because inves
tors realize that their dollars buy less when
prices rise. For example, if the price of a cup of
coffee one year from now is $1, bondholders
can buy 100 cups of coffee with the $100 that
the bond pays off. But if the price of a cup of
coffee is expected to rise to $1.05, bondholders
will be able to buy only 95 cups of coffee. To be
compensated for the loss in purchasing power,
investors must get a higher dollar return on
their investments. Thus, bond prices will fall
and interest rates will rise when expected in
flation rises.
This model suggests that when expected
income or expected inflation rises, bond prices
will fall. This fall in bond prices translates into
higher interest rates. So, when we think about
how short-term interest rates are determined,

9See the article by Campbell R. Harvey.

Keith Sill

we want to think about people's forecasts for
real income growth and inflation. Any current
economic variables that help to predict real
income growth and inflation will help to deter
mine current short-term bond prices and inter
est rates.
D eterm ination of Long-Term Interest
Rates. Long-term interest rates can be linked
to short-term interest rates by the expectations
theory of the term structure. This theory says
that long-term interest rates are equal to an
average of expected short-term interest rates
plus a risk premium.10 The risk premium ac
counts for the co-variation over time of vari
ables like income growth and inflation that
could influence the level of interest rates.
The logic of the expectations theory of bond
prices is most clearly seen in an example in
which we ignore the risk premium. Take the
case of an investor who has a two-year invest
ment horizon. The investor can purchase a
two-year bond, or he can purchase a one-year
bond today and, when that bond matures,
purchase another one-year bond. The expected
return on these alternative investment strate
gies should be equal. Since there is a direct
relationship between interest rates on bonds
and bond prices, the expectations theory also
links long-term discount bond prices to ex
pected short-term discount bond prices over
the life of the long-term bond.
In terms of expected future short-term bond
prices, the same variables that affect short
term bond prices basically determine long
term bond prices and interest rates. Thus,
expected future income growth and expected
inflation are also determinants of long-term
bond prices, but now the forecasts of income
growth and inflation are for further in the
future. It is still the case that if, over the life of

10For more detail on the expectations theory and risk
premiums, see my article in the Ju ly / August 1994 Business
Review.

BUSINESS REVIEW

the bond, expected future income growth or
expected future inflation rises, long-term in
terest rates will rise. Including a risk premium
does not alter these basic conclusions about the
determinants of interest rates. However, the
risk premium can be an additional source of
variability for interest rates because it picks up
some indirect effects of income growth and
inflation on interest rates, as well as other risk
factors.
This model helps us think about why long
term interest rates co-vary less with current
output than do short-term interest rates. Cur
rent movements in real output are much more
closely correlated with output movements in
the near future than they are with output
movements in the far future. Since the pay
ment stream on a long-term bond extends
further out into the future than that on a short
term bond, long-term interest rates are less
likely to have a strong co-variation with cur
rent output movements.
Determinants of Interest Rate Volatility.
The same basic economic factors that deter
mine interest rates and the prices of bonds also
determine the volatilities of interest rates and
bond prices. This economic model suggests
that expected real income growth and expected
inflation determine bond prices and interest
rates. It follows then that the volatility of ex
pected real income growth and the volatility of
expected inflation, as well as the correlation
between the two, determine the volatility of
interest rates and bond prices.
The reasoning behind this conclusion is
straightforward. Take the case of real income
growth. We saw above that if real income
growth is expected to be high, current bond
prices will fall and interest rates will rise. The
higher real income growth is expected to be,
the higher interest rates will be. Thus, large
changes in expected real income growth are
associated with large changes in interest rates.
When real income growth has high volatility,
large changes in real income growth occur

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26
Federal Reserve Bank of St. Louis

JANUARY/FEBRUARY 1996

more frequently, and hence large changes in
current bond prices and interest rates occur
more frequently. When large changes in inter
est rates occur more often, interest rates are
more volatile. Similar reasoning holds for the
case of inflation. When large changes in ex
pected inflation occur, large changes in cur
rent bond prices and interest rates occur also.
So, more volatile inflation translates into more
volatile bond prices and interest rates.11
W hat determ ines how volatile income
growth and inflation will be? One factor is
monetary policy. Take the case of monetary
policy and inflation. Economists generally be
lieve that a persistent inflation has its root
causes in monetary policy, in particular, how
fast the money supply grows relative to real
income growth. If growth of the money supply
is excessive, inflation is likely to be high. If we
take growth of the money supply as the pri
mary determinant of inflation, highly volatile
growth in the money supply can lead to vola
tile inflation. This does not mean that every
change in the money supply necessarily leads
to a change in inflation. Rather, if, on average,
money supply growth becomes more volatile,
inflation can become more volatile as well. As
we have seen, the model then suggests that
bond prices and interest rates will also be more
volatile.
Monetary policy could also have an effect
on real income, although economists disagree
on the mechanism by which this occurs. One
theory is that workers write contracts with
their employers that fix a nominal wage rate
over the contract period. Workers and firms

11Higher volatility of income growth and inflation sug
gests that price volatilities for both short-term and long
term bonds will increase. Long-term volatility remains
higher than short-term volatility because investors who
buy long-term bonds have to make forecasts about future
variables that are not relevant for determining the prices
of short-term bonds.

FEDERAL RESERVE BANK OF PHILADELPHIA

The Cyclical Volatility of Interest Rates

negotiate the contracted wage based, in part,
on their expectations of what inflation will be
over the contract period. Since monetary policy
affects inflation, this requires workers and
firms to forecast what monetary policy will be
over this same period. If monetary policy and
the price level turn out to be different from
what workers and firms expected when they
wrote the contract, employment and output
could be affected because firms' demand for
workers depends on the real wage rate that
must be paid. If nominal wages are fixed by a
contract and prices rise unexpectedly, real
wages fall, and firms demand more workers
and produce more output. If prices fall unex
pectedly, real wages rise, firms lay off work
ers, and output falls. Thus, variability of the
money supply, through its impact on prices,
could have an impact on the variability of real
income.
We can point to many other factors, besides
monetary policy, as potentially influencing the
volatility of output and inflation. For example,
variability in weather can affect agricultural
output as well as production in the economy.
Changes in productivity due to the introduc
tion of new technologies can influence the
variability of output and inflation as well. A
whole class of economic models, called realbusiness-cycle models, attempts to account for
output volatility over the business cycle. These
models assume that shocks to productivity are
the main cause of business cycles.12 Shocks to
current productivity affect peoples' forecasts
of the future course of the economy and thereby
affect their expectations about economic vari
ables like real income growth and inflation.
The more persistent productivity shocks are,
the greater their effect on long-term interest
rates will be, since output and inflation far into
the future will be affected.

12See the article by Satyajit Chatterjee in the September/O ctober 1995 Business Review.

Keith Sill

Economic Models and Interest Rate Vola
tility. This economic model for determining
bond prices and interest rates suggests that
investors' expectations of future real income
growth and inflation are the primary determi
nants of current bond prices and interest rates.
There are, of course, other determinants of
interest rates and interest rate volatility in the
economy. But we can try to assess how well
this view of interest rate determination ex
plains the interest rate volatility that we ob
serve in the actual economy.
One approach to assessing how well a model
performs is to use the model to simulate inter
est rates and then compare the properties of
the simulated interest rates to the properties of
actual interest rates. For example, we can set
up models and use them to simulate price data
on discount bonds of various maturities. We
can then calculate the standard deviation of
these simulated data and compare it to the
standard deviation of discount bond prices
implied from the interest rates we observe in
the economy. We can also examine how the
simulated bond prices and interest rates co
vary with simulated output and compare the
correlations to the correlations we find in the
actual data. In this way, we can assess the
ability of the model to account for the cyclical
volatility of interest rates.
SIMULATION RESULTS
In my 1994 working paper, I present an
exercise similar to the one following. Briefly, in
the model, expected real income growth and
expected money growth determine current
discount bond prices and yields. Expected
money growth is assumed to be the primary
determinant of inflation. The model also re
quires some input on investor characteristics,
such as how willing investors are to undertake
risky investments. Table 5 shows the variabil
ity of the bond prices and yields simulated by
one particular version of the model and repro
duces the variability of bond prices and yields
27

JANUARY/FEBRUARY 1996

BUSINESS REVIEW

TABLE 5

Yield and Bond-Price Volatility From Model Simulations
Time to Maturity

Standard Deviation of Detrended
Interest Rate

Standard Deviation of Detrended
Discount Bond Price

Simulated

Actual

Simulated

Actual

1 quarter

.00362

.00322

.00362

.00322

2 quarters

.00332

.00315

.00663

.00629

3 quarters

.00303

.00308

.00909

.00923

1 year

.00278

.00293

.01112

.01172

2 years

.00202

.00256

.01619

.02051

5 years

.00100

.00213

.02005

.04265

10 years

.00051

.00184

.02045

.07348

Standard deviations are of the yields and logarithms of discount bond prices (see footnote 5). Actual discount bond prices
are calculated from the term structure data in McCulloch and Kwon (1993).

derived from actual interest rate data for com
parison.13
The model generates data in which volatil
ity of interest rates falls but bond price volatil
ity rises with the maturity of the bond. Out to
a maturity of about one year, the variability of
the simulated bond yields and prices matches
the variability of the data fairly closely. At a
m aturity of three m onths, the m odel
overpredicts the volatility of bond prices and
yields about 12 percent. At a maturity of one
year, the model underpredicts the volatility of
bond prices and yields about 5 percent. These
results suggest that the variability of income

13The model also replicates some of the correlations
between discount bond prices and output as well as some
features of the correlation patterns of bond prices.

growth and money growth account for a sub
stantial portion of the variability of short-term
discount bond prices and hence of short-term
interest rates.
For longer maturities, the variability of simu
lated bond prices and yields underpredicts the
volatility of actual yields and implied prices of
discount bonds by a progressively larger
amount. When we look at the historical data,
the variability of implied prices for a discount
bond with 10-year maturity is about 23 times
larger than the variability of short-term dis
count bond prices. But in the simulated data,
the variability of 10-year discount bond prices
is only about five times greater than the vari
ability of short-term bond prices.
Many reasons might explain why the growth
of the money supply and the growth of real
income do not account for much of the vari
FEDERAL RESERVE BANK OF PHILADELPHIA

Keith Sill

The Cyclical Volatility of Interest Rates

ability of long-term bond prices and yields.
The basic model is designed to highlight the
business-cycle variability of interest rates, and
as we have seen, long-term interest rates do
not appear to have a large business-cycle com
ponent. In addition, the model is very simple,
and so it is missing some important elements
found in actual economies. For example, the
model does not account for the fact that differ
ent people have different beliefs about the
future course of the economy or that people are
continually learning about the economic envi
ronment. Changes in fiscal and monetary poli
cies may induce greater volatility in interest
rates than the simple economic model accounts
for. The expectations theory may be an inad
equate model of the term structure of interest
rates. Despite difficulties such as this, the
model's implication that real income growth
and money supply growth are factors that help
to determine the volatilities of interest rates
and discount bond prices does find some sup

port, especially for shorter maturities, when
we compare the model with actual data.
CONCLUSION
We have seen that the volatility of interest
rates depends on the maturity of the underly
ing bond: long-term interest rates are less vari
able than short-term interest rates. Short-term
interest rates are procyclical while long-term
interest rates co-vary little with movements in
output over the business cycle. Economic
theory suggests that both the level and volatil
ity of interest rates should be tied to economic
variables such as income growth and inflation.
Simulation results suggest that the volatility of
both income growth and money growth ac
counts for a large portion of the volatility of
short-term discount bond prices. However,
these same economic variables by themselves
are able to account for only a small fraction of
the volatility of long-term discount bond prices.

References
Chatterjee, Satyajit. "Productivity and the American Business Cycle," Federal Reserve Bank of
Philadelphia Business Review (September/October 1995).
Harvey, Campbell R. "Term Structure Forecasts Economic Growth," Financial Analysts Journal
(M ay/June 1993).
King, Robert G., and Sergio T. Rebelo. "Low Frequency Filtering and Real Business Cycles,"
Journal o f Economic Dynamics and Control, 17 (1993), pp. 207-31.
McCulloch, J. Huston, and H.C. Kwon. "U.S. Term Structure Data, 1957-1991," Working Paper
93- 6, Ohio State University (1993).
Sill, Keith. "Money, Output, and the Cyclical Volatility of the Term Structure," Working Paper
94- 14, Federal Reserve Bank of Philadelphia (July 1994).
Sill, Keith. "Managing the Public Debt," Federal Reserve Bank of Philadelphia Business Review
(July/August 1994).

Philadelphia / RESEARCH
Working Papers
The Philadelphia Fed's Research Department occasionally publishes working papers based on
the current research of staff economists. These papers, dealing with virtually all areas within
economics and finance, are intended for the professional researcher. The papers added to the
Working Papers series thus far this year are listed below. To order copies, please send the
number of the item desired, along with your address, to WORKING PAPERS, Department of
Research, Federal Reserve Bank of Philadelphia, 10 Independence Mall, Philadelphia, PA 19106.
For overseas airmail requests only, a $3.00 per copy prepayment is required; please make checks
or money orders payable (in U.S. funds) to the Federal Reserve Bank of Philadelphia. A list of
all available papers may be ordered from the same address.

95-1

Satyajit Chatterjee and Dean Corbae, "Valuation Equilibria with Transactions Costs"

95-2/R

Sherrill Shaffer, "Structural Screens in Stochastic Markets" (supersedes Working
Paper No.
92-23)

95-3

Franklin Allen and Douglas Gale, "A Welfare Comparison of Intermediaries and
Financial Markets in Germany and the U.S."

95-4

Franklin Allen and Douglas Gale, "Financial M arkets, Interm ediaries, and
Intertemporal Smoothing"

95-5

Gregory P. Plopper, "The Dynamics of the Exchange Rate Under a Crawling Peg
Regime: A Game Theory Approach"

95-6

Franklin Allen and Douglas Gale, "Universal Banking, Intertemporal Risk Smooth
ing, and European Financial Integration"

95-7

Paul Calem and Michael Stutzer, "The Simple Analytics of Observed Discrimination
in Credit Markets"

95-8

Joseph Hughes, William Lang, Loretta Mester, and Choon-Geol Moon, "Recovering
Technologies That Account for Generalized Managerial Preferences: An Application
to Non-Risk-Neutral Banks"

95-9

Ana Castaeda, Javier Diaz-Gimenez, and Jose-Victor Rios-Rull, "Unemployment
Spells and Income Distribution Dynamics"

95-10

Paul Calem and Loretta J. Mester, "Consumer Behavior and the Stickiness of Credit
Card Interest Rates" (Supersedes No. 92-24/R)

95-11

Richard Voith, "Parking, Transit, and Employment in a CBD"

FEDERAL RESERVE BANK OF PHILADELPHIA

95-12

Gary Gorton and Richard Rosen, "Banks and Derivatives"

95-13

Sherrill Shaffer, "Translog Bias Under Declining Average Costs"

95-14

Alberto Trejos and Randall Wright, "Toward a Theory of International Currency: A
Step Further"

95-15

Gerald Carlino and Robert DeFina, "The Differential Effects of Monetary Policy
Shocks on Regional Economic Activity"

95-16

Paul Calem, "Mortgage Credit Availability in Low- and Moderate-Income Minority
Neighborhoods: Are Information Externalities Critical?"

95-17

Leonard I. Nakamura, "New Directions in Information and Screening in Real Estate
Finance"

95-18

William J. Stull, "Is High School Economically Relevant for Noncollege Youth?"

95-19

James McAndrews and George Wasilyew, "Simulations of Failure in a Payment
System"

95-20

Keith Sill, "An Empirical Investigation of Money Demand: Evidence from a Cash-InAdvance Model"

95-21

Leonard Nakamura, "Is U.S. Economic Performance Really That Bad?"

95-22

Laurence Ball and Dean Croushore, "Expectations and the Effects of Monetary
Policy"

95-23

Sherril Shaffer, "The Discount Window

95-24

Stephen Morris and Hyun Song Shin, "Informational Events That Trigger Currency
Attacks"

95-25

Robert H. DeFina, Thomas C. Stark, and Herbert E. Taylor, "The Long-Run Variance
of Output and Inflation Under Alternative Monetary Policy Rules"

95-26

Bernardino Adao and Theodosios Temzelides, "Beliefs, Competition, and Bank
Runs"

95-27

Theodosios Temzelides, "Evolution, Coordination, and Banking Panics"

and Credit Availability"

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