Chapter 6
The Risk and Term Structure of
Interest Rates
This chapter examines the relationship
among different interest rates and why
they differ.
We examine the sources and causes of
fluctuations interest rates relative to
one another and look at a number of
theories that explain these fluctuations.
Risk structure of interest rates
Why do bonds with same term to
maturity have different interest
rates? The relationship among
these interest rates is called the risk
structure of interest rates
Term structure of interest rates
A bond’s term to maturity also
affects its interest rate.
The relationship among interest
rates on bonds with different terms
to maturity is called the term
structure of interest rates.
Risk Structure of Interest
Rates
The relationship among interest
rates with same term to maturity.
Default risk
one attribute of a bond that influences
its interest rate.
The chance that the issuer of the bond
will default, that is, unable to make
interest payments or pay off the face
value when the bond matures.
Default-free bonds
. Treasury bonds have usually been
considered to have no default risk.
Bonds like these with no default risk are
called default-free bonds.
Risk premium
The spread between the interest rates
on bonds with default risk and default-
free bonds. It indicates how much
additional interest people must earn to
be willing to hold a risky bond.
A bond with default risk will always
have a positive risk premium, and an
increase in its default risk will raise the
risk premium.
Liquidity
Another attribute of a bond that
influences its interest rate. The more
liquid an asset is, the more desirable it is.
The spread between the interest rates on
corporate bonds and Treasury bonds
reflect not only the corporate bond’s
default risk but its liquidity too.
This is why a risk premium is sometimes
called a liquidity premium, or a “risk
and liquidity premium’’.
Income tax consideration
The interest rate of a bond that is not
subject to taxation will be lower than
that subject to taxation, given other
things being the same.
Term Structure of interest
Rates
The relationship among interest rates
on bonds with different terms to
maturity, but same risk, liquidity, and
tax consideration.
Another factor that influences
interest rate on a bond is its
term to maturity.
Yield curve
A plot of the yields on bonds with
differing terms to maturity but the
same risk, liquidity, and tax
consideration. It describes the term
structure of interest rates for particular
types of bonds, such as government
bonds.
Yield curves can be classified as
upward-sloping, flat, and downward
-sloping (inverted yield curve).
Upward-sloping yield curves
It means that long-term interest rates
are above the short-term interest
rates.
Flat yield curves
It means that long-term interest rates
are the same as the short-term interest
rates.
Downward-sloping yield curves
It means that long-term interest rates
are below the short-term interest
rates.
A good theory of the term structure of
interest rates must explain the following
three important empirical facts:
1. 1. Interest rates on bonds of different maturities Interest rates on bonds of different maturities
move together over time. move together over time.
2. When short-term interest rates are low, yield 2. When short-term interest rates are low, yield
curves are more likely to have an upward- curves are more likely to have an upward-
slope; when short-term interest rates are high, slope; when short-term interest rates are high,
yield curves are more likely to have an yield curves are more likely to have an
downward-slope. downward-slope.
3. Yield curves almost always slop upward. 3. Yield curves almost always slop upward.
The theories that explain the term
structure of interest rates are
1. The expectations theory: It can explain
the first two facts, but not the third fact.
2. The segmented markets theory: It can
explain the third fact, but not the first
two facts.
3. The liquidity premium theory: It can
explain all three facts.
The expectations theory
The proposition: The interest rate on a
long-term bond will equal an average of
short-term interest rates that people
expect to occur over the life of the long-
term bond.
Why interest rates on bonds with
different maturities differ is that short
term interest rates are expected to have
different values at future dates.
Assumptions behind this theory
1. Buyers of bonds are to maximize their
rates of return on investment.
2. Capital will flow freely between markets
of long-term and short-term bonds. So
these two markets are interdependent.
3. Buyers of bonds do not prefer bonds of
one maturity over another.
4. Bonds are perfect substitutes.
5. The expected return on these bonds must
be equal.
Example
Let us consider the following
two investment strategies:
1. Purchase a one-year bond, and when it
matures in one year, purchase another
one-year bond.
2. Purchase a two-year bond and hold it
until maturity.
= today’s(time )interest rate on a
one-period bond
= interest rate on a one-period
bond expected for next period(time
)
= today’s(time )interest rate on
the two-period bond
The rate of return on the first
strategy
The rate of return on the second
strategy
These two rates of return should be
equal, so we have
The interest rate of on an n-
period bond must equal
Numerical Example
If the one-year interest rate over the next
five years is expected to be 5, 6, 7, 8, and 9
percent, then the interest rate on the two-
year bond would be
The interest rate on the five-year
bond would be
The rising trend in expected short-
term interest rates produces an
upward-sloping yield curve along
which interest rates rise as maturity
lengthens.
When the yield curve is upward-sloping, the
expectations theory suggests that short-
term interest rates are expected to rise in
the future.
When the yield curve is downward-sloping,
the expectations theory suggests that short-
term interest rates are expected to fall in
the future.
When the yield curve is flat, the
expectations theory suggests that short-
term interest rates will remain the same in
the future.
Explanation of the first fact
Because the long-term interest rate is
the average of expected short-term
interest rates, so when short-term
interest rates rise, the long-term interest
rate will also rise. They move together.
Explanation of the second fact
When short-term interest rates are low,
people generally expect them to rise to
some normal level in the future, and the
average of future expected short-term
rates is high relative to the current short-
term rates. Therefore the long-term
interest rates will be above current short-
term rates, then the yield curve would be
upward sloping.
When short-term interest rates are high,
people generally expect them to fall to
some normal level in the future, and the
average of future expected short-term
rates is low relative to the current short-
term rates. Therefore the long-term
interest rates will be below current short
-term rates, then the yield curve would
be downward sloping.
Failure to explain the third fact
Short-term interest rates are as likely to
fall as they are to rise, so the yield curve
should be flat rather than upward
sloping most of the time.
Segmented Markets Theory
Proposition
Markets for different-maturity bonds are
completely separate and segmented.
The interest rate is determined by the
supply of and demand for that bond with
no effects from expected returns on other
bonds with other maturities.
Assumptions behind this theory
1. Buyers of bonds have strong
preferences for bonds of one maturity but
not for others, and are concerned only
with the expected returns on the bonds
they prefer.
2. Bonds of different maturities are not
substitutes at all.
3. Capital will not flow between markets
of long-term and short-term bonds. So
these two markets are separate and
segmented.
4. Buyers of bonds will prefer short-term
bonds over long-term bonds due to risk
and liquidity consideration.
Explanation of the third fact
Because buyers of bonds prefer to hold
short-term bonds over long-term bonds,
so the demand for short-term bonds is
higher than that for long-term bonds.
The short-term interest rates are lower
than long-term interest rates, and the
yield curve is upward sloping.
Failure to explain the first and
second facts
Because sort-term bonds market and long-term Because sort-term bonds market and long-term
bonds market are independent, so are their bonds market are independent, so are their
interest rates. Therefore short-term interest interest rates. Therefore short-term interest
rates and long-term interest rates will not move rates and long-term interest rates will not move
together. There is no correlation between short-together. There is no correlation between short-
term and long-term interest rates. So when shortterm and long-term interest rates. So when short
-term interest rates are low or high, the demand -term interest rates are low or high, the demand
for and supply of long-term bonds can not be for and supply of long-term bonds can not be
expected to change in certain direction, therefore expected to change in certain direction, therefore
the yield curves’ slope can not be predicted the yield curves’ slope can not be predicted
.
The Liquidity Premium Theory
Proposition
The interest rate on a long-term bond
will equal an average of short-term
interest rates expected to occur over
the life of the long-term bond plus a
liquidity premium ( term premium)
that responds to supply and demand
conditions for that bond.
Assumptions behind this theory
1. Buyers of bonds prefer bonds of one
maturity over bonds of others.
2. Bonds are imperfect substitutes.
3. The expected return on one bond does
influence the expected return on a bond
of a different maturity.
4. Investors will be willing to buy bonds
that do not have the preferred maturity
only if they earn a somewhat higher
expected return.
5. If investors prefer short-term bonds
over long-term bonds, they might be
willing to hold short-term bonds even
though they have a lower expected
return.
6. Investors would have to be paid a
positive liquidity premium to be willing
to hold a long-term bond.
The liquidity premium theory
modifies the expectations theory by
adding a positive liquidity premium
to the equation that describes the
relationship between long- and short-
tem interest rates
Liquidity premium theory states that
a positive liquidity ( term) premium
must be offered to buyers of longer-
term bonds to compensate them for
their increased interest rate risk.
Because the liquidity premium is
always positive and grows as the
term to maturity increases, the
yield curve implied by the liquidity
premium theory is always above
the yield curve implied by the
expectations theory and has a
steeper slope.
Example
One year interest rate over the next five
year is expected to be 5, 6, 7, 8, and 9
percent. Investors prefer to hold short-
term bonds so that the term premiums
for one- to five-year bonds are 0, ,
, , and 1 percent, respectively.
The interest rate for the five-year bond
would be
Explanation of the first fact
Because the long-term interest rate is the
average of short-term interest rates plus a
term premium, so when short-term
interest rates rise, their average will rise
too. Therefore long-term interest rate will
also rise. They move together.
Explanation of the second fact
When short-term interest rates are low,
people generally expect them to rise to
some normal level in the future, and the
average of future expected short-term
rates is high relative to the current short-
term rates. Therefore the long-term
interest rates will be above current short-
term rates, then the yield curve would be
upward sloping.
When short-term interest rates are high,
people generally expect them to fall to
some normal level in the future, and the
average of future expected short-term
rates is low relative to the current short-
term rates. Therefore the long-term
interest rates will be below current short
-term rates, then the yield curve would
be downward sloping.
Explanation of the third fact
Because buyers of bonds prefer to hold
short-term bonds over long-term bonds,
so the liquidity premium rises with a
bond’s maturity. Even if the short-term
interest rates are expected to stay the
same on average in the future, long-term
interest rates will be above short-term
interest rates, and the yield curves will
typically slope upward.
The inverted yield curves appear if the
expected short-term rates are expected to
fall so much in the future that the average
of the expected short-term rates is well
below the current short-term rate even
with a positive liquidity premium is added
to this average. The resulting long-term
rate will still be below the current short-
term interest rate.
The liquidity premium theory allows
us to predict future short-term
interest rates by looking at the slope
of the yield curve.
1. A steeply rising yield curve indicates
that short term interest rates are
expected to rise in the future.
2. A moderately steep yield curve
indicates that short-term interest rates
are not expected to rise or fall in the
future.
3. A flat yield curve indicates that short-
term interest rates are expected to fall
moderately in the future.
4. An inverted yield curve indicates that
short-term interest rates are expected to
all sharply in the future.
Recent Evidence on the Term
Structure
Recent research shows that the term
structure contains quite a bit of
information for the very short-term and
the long-term rates, but not for the
intermediate term rates.
