chapter 5
The Risk and Term Structure of
Interest Rates
Instructor: Xiajing Dai
preview
In our supply and demand analysis of
interest-rate behavior in Chapter 5, we
examined the determination of just one
interest rate. Yet we saw earlier that there
are enormous numbers of bonds on which the
interest rates can and do differ.
In this chapter, we complete the interestrate picture by examining the relationship of
the various interest rates to one another
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 Figure 1 shows the yields to maturity for several
categories of long-term bonds from 1919 to 2002.
 It shows us two important features of interest-rate
behavior for bonds of the same maturity:
 (1)Interest rates on different categories of bonds
differ from one another in any given year
 (2)the spread (or difference) between the interest
rates varies over time.
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Default risk
One attribute of a bond that influences its
interest rate is its risk of default, which
occurs when the issuer of the bond is unable
or unwilling to make interest payments when
promised or pay off the face value when the
bond matures.
By contrast, U.S. Treasury bonds have
usually been considered to have no default
risk because the federal government can
always increase taxes to pay off its
obligations. Bonds like these with no default
risk are called default-free bonds.
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The spread between the interest rates on
bonds with default risk and default-free
bonds, called the risk premium, indicates how
much additional interest people must earn in
order to be willing to hold that risky bond
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Conclusion
A bond with default risk will always have a
positive risk premium, and an increase in
its default risk will raise the risk premium.
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 Because default risk is so important to the size of
the risk premium, purchasers of bonds need to know
whether a corporation is likely to default on its
bonds.
 Two major investment advisory firms, Moody’s
Investors Service and Standard and Poor’s
Corporation, provide default risk information by
rating the quality of corporate and municipal bonds
in terms of the probability of default
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Liquidity
The differences between interest rates on
corporate bonds and Treasury bonds (that is,
the risk premiums) reflect not only the
corporate bond’s default risk but its liquidity,
too.
This is why a risk premium is more accurately
a “risk and liquidity premium,” but convention
dictates that it is called a risk premium.
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Income Tax Considerations
Why is it, then, that these bonds have had lower
interest rates than U.S. Treasury bonds for at least
40 years, as indicated in Figure 1? The explanation
lies in the fact that interest payments on municipal
bonds are exempt from federal income taxes, a
factor that has the same effect on the demand for
municipal bonds as an increase in their expected
return.
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Summary
 The risk structure of interest rates is explained by
three factors: default risk, liquidity, and the income
tax treatment of the bond’s interest payments.
 As a bond’s default risk increases, the risk premium
on that bond (the spread between its interest rate
and the interest rate on a default-free Treasury
bond) rises.
 The greater liquidity of Treasury bonds also explains
why their interest rates are lower than interest
rates on less liquid bonds.
 If a bond has a favorable tax treatment, its
interest rate will be lower.
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Term Structure of Interest Rates
 A plot of the yields on bonds with differing
terms to maturity but the same risk, liquidity,
and tax considerations is called a yield curve, and
it describes the term structure of interest rates
for particular types of bonds.
 Yield curves can be classified as upward-sloping,
flat, and downward-sloping (the last sort is often
referred to as an inverted yield curve).
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A good theory of the term structure of interest
rates must explain the following three important
empirical facts:
 1. As we see in Figure 4, interest rates on bonds of
different maturities move together over time.
 2. When short-term interest rates are low, yield
curves are more likely to have an upward slope; when
short-term interest rates are high, yield curves are
more likely to slope downward and be inverted.
 3. Yield curves almost always slope upward, as in the
“Following the Financial News” box.
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Three theories have been put forward to explain the
term structure of interest rates; that is, the
relationship among interest rates on bonds of
different maturities reflected in yield curve
patterns:
 (1) the expectations theory
 (2) the segmented markets theory
 (3) the liquidity premium theory
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Expectation Theory
 The expectations theory of the term structure
states the following commonsense proposition: The
interest rate on a long-term bond will equal an
average of short-term.
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 The key assumption behind this theory is that
buyers of bonds do not prefer bonds of one
maturity over another, so they will not hold any
quantity of a bond if its expected return is less than
that of another bond with a different maturity.
Bonds that have this characteristic are said to be
perfect substitutes
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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
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Expected return from strategy 2
(1 + i2t)(1 + i2t) – 1
1 + 2(i2t) + (i2t)2 – 1
=
1
1
Since (i2t)2 is extremely small, expected return is
approximately 2(i2t)
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(1 + it)(1 + iet+1) – 1 1 + it + iet+1 + it(iet+1) – 1
=
1
1
Since it(iet+1) is also extremely small, expected return is
approximately
it + iet+1
From implication above expected returns of two
strategies are equal: Therefore
2(i2t) = it + iet+1
Solving for i2t
it + iet+1
i2t =
2
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More generally for n-period bond:
int =
it + iet+1 + iet+2 + ... + iet+(n–1)
n
In words: Interest rate on long bond = average short rates
expected to occur over life of long bond
Numerical example:
One-year interest rate over the next five years 5%, 6%, 7%, 8%
and 9%:
Interest rate on two-year bond:
(5% + 6%)/2 = 5.5%
Interest rate for five-year bond:
(5% + 6% + 7% + 8% + 9%)/5 = 7%
Interest rate for one to five year bonds:
5%, 5.5%, 6%, 6.5% and 7%.
6-24
Expectations Hypothesis explains Fact 1
that short and long rates move together
1. Short rate rises are persistent
2. If it  today, iet+1, iet+2 etc.   average of
future rates   int 
3. Therefore: it   int , i.e., short and long
rates move together
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Explains Fact 2
1. When short rates are low, they are expected to rise to normal
level, and long rate = average of future short rates will be
well above today’s short rate: yield curve will have steep
upward slope
2. When short rates are high, they will be expected to fall in
future, and long rate will be below current short rate: yield
curve will have downward slope
Doesn’t explain Fact 3 that yield curve usually has upward
slope
Short rates as likely to fall in future as rise, so average of
future short rates will not usually be higher than current short
rate: therefore, yield curve will not usually slope upward
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Segmented Markets Theory
Key Assumption: Bonds of different maturities are not
substitutes at all
Implication: Markets are completely segmented:
interest rate at each maturity determined separately
Explains Fact 3 that yield curve is usually upward
sloping
People typically prefer short holding periods and thus
have higher demand for short-term bonds, which have
lower interest rates than long bonds
Does not explain Fact 1 or Fact 2 because assumes
long and short rates determined independently
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Liquidity Premium (Preferred Habitat)
Theories
Key Assumption: Bonds of different maturities are
substitutes, but are not perfect substitutes
Implication: Modifies Expectations Theory with
features of Segmented Markets Theory
Investors prefer short rather than long bonds  must
be paid positive liquidity (term) premium, lnt, to hold
long-term bonds
Results in following modification of Expectations
Theory
int =
it + iet+1 + iet+2 + ... + iet+(n–1)
n
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+ lnt
Relationship Between the Liquidity Premium
(Preferred Habitat) and Expectations
Theories
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• The implication of figure 5
The liquidity premium is always positive and
typically grows as the term to maturity
increases
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Numerical Example
1. One-year interest rate over the next five years:
5%, 6%, 7%, 8% and 9%
2. Investors’ preferences for holding short-term bonds,
liquidity
premiums for one to five-year
bonds:
0%, 0.25%, 0.5%, 0.75% and 1.0%.
Interest rate on the two-year bond:
(5% + 6%)/2 + 0.25% = 5.75%
Interest rate on the five-year bond:
(5% + 6% + 7% + 8% + 9%)/5 + 1.0% = 8%
Interest rates on one to five-year bonds:
5%, 5.75%, 6.5%, 7.25% and 8%.
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Liquidity Premium (Preferred Habitat)
Theories: Term Structure Facts
Explains all 3 Facts
Explains Fact 3 of usual upward sloped yield
curve by investors’ preferences for short-term
bonds
Explains Fact 1 and Fact 2 using same
explanations as expectations hypothesis
because it has average of future short rates as
determinant of long rate
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Market
Predictions
of Future
Short Rates
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Answers in brief
• 1. bond with a C rating should have a higher
interest rate because it has a higher default
risk, which reduces its demand and raises its
interest rate relative to that on the Baa bond.
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• 3. During business cycle booms, fewer corporations
go bankrupt and there is less default risk on
corporate bonds, which lowers their risk premium.
Similarly, during recessions, default risk on
corporate bonds increases and their risk premium
increases.
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• 5. If yield curves on average were flat, this
would suggest that the risk premium on
long-term relative to short-term bonds would
equal zero and we would be more willing to
accept the expectations hypothesis.
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7.
• (a) yield curve would be upward- and then
downward-sloping
• (b) yield curve would be downward- and then up
ward-sloping
• If people prefer shorter-term bonds over longer-term
bonds, the yield curve tend to be even more upward
sloping because long-term bonds would then have a
positive risk premium.
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• 9. The steep upward-sloping yield curve at shorter
maturities suggests that short-term interest rates are
expected to rise in the near future .
• The downward slope for longer maturities indicates
that short-term interest rates are eventually expected
to fall sharply.
• Since interest rates and expected inflation move
together, the yield curve suggests that the market
expects inflation to rise moderately in the near future
but fall later on.
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• 11. The government guarantee will reduce the
default risk on corporate bonds, making them
more desirable relative to Treasury securities.
The increased demand for corporate bonds
and decreased demand for Treasury
securities will lower interest rates on
corporate bonds and raise them on Treasury
bonds.
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• 13. Abolishing the tax-exempt feature of
municipal bonds would make them less
desirable relative to Treasury bonds. The
resulting decline in the demand for municipal
bonds and increase in demand for Treasury
bonds would raise the interest rates on
municipal bonds, while the interest rates on
Treasury bonds would fall.
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• 15. The slope of the yield curve would fall
because the drop in expected future short
rates means that the average of expected
future short rates falls so that the long rate
falls.
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