Chapter 6 - The Risk and Term Structure
of Interest Rates
Previous chapter, examined determination of
one interest rate. But there are many bonds
on which interest rates can and do differ.
This lecture examines relationship of various
interest rates to one another…
 Risk structure of interest rates… why
bonds with same term to maturity have
different interest rates
 Term structure of interest rates…
relationship among interest rates on bonds
with different terms to maturity
 Three theories that attempt to explain the
term structure of interest rates
Copyright © 2001, 2004 J. Reynolds - Addison Wesley
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Risk Structure of Interest Rates
 Figure 1 - Long-Term Bond Yields, 1919-99
 Interest rates on different categories of
bonds differ from one another in a given year
 Spread between the interest rates varies
over time
Interest rates on municipal bonds are above
those on U.S. Treasuries in the late 1930s,
lower thereafter
Spread between interest rates on Baa
corporate bonds and U.S. Treasuries large
during 1930-1933, smaller during 1940s-60s,
then again larger during 1970s-90s
 What factors explain this phenomena,
across different bond grades & through time?
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Default Risk
default risk - chance that issuer of bond will
default (be unable to make interest payments
or pay off the face value upon maturity)
 U.S. Treasury bonds are considered
default-free bonds, as Federal government
can always increase taxes or print money to
pay off obligations (no default risk)
firm with
substantial
losses
high default risk

risk on bonds
(Chrysler Corp., 1970s)
risk premium - spread between interest rates
on default-risk bonds and default-free bonds
 Indicates how much additional interest
individual must earn in order to be willing to
hold a risky bond
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Increase in Corporate Bond Default Risk
 Figure 2 - Response to an Increase in
Default Risk on Corp. Bonds
 Assume only two types of debt securities…
corporate long-term bonds and U.S. Treasury
bonds (T-bonds)
 Also assume that corporate bonds and the
U.S. T-bonds have the same default risk…
…bonds have identical risk & maturity
If so, equilibrium prices and interest rates
will be equal…
P1c = P1T
i1c = i1T
and the risk premium on corporate bonds will
equal zero…
i1c - i1T = 0
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Increase in Corporate Bond Default Risk
Spse corp. issuing
bonds begins to
suffer large losses
likelihood of default

on outstanding
bonds increases


corporate bonds
bid down
(  Pc )
 E(RET)corp.bonds
(return also more uncertain)
 demand for corporate bonds
shifts left (D1c to D2c)

demand for default-free
T-bonds shifts right
(D1T to D2T)
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Increase in Corporate Bond Default Risk
 Response to the increase in corporate bond
default risk comes about via theory of asset
demand
 E(RET)corp.bonds
(rel. to E(RET)Tbonds)
and

corporate bonds'
relative riskiness rises

corporate bonds
become
less desirable
(shift Dc left)
Treasury bonds
become
more desirable
(shift DT right)
 Recall assumption of only two bonds,
corporates and Treasuries
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Increase in Corporate Bond Default Risk
Result for Corporate Bonds:
 corporate bond price falls (P1c to P2c)
corporate bond interest rate rises (i1c to i2c)
Result for Default-free Bonds (U.S. Treasuries):
 default-free bond price rises (P1T to P2T)
default-free bond interest rate falls (i1T to i2T)
Interest Rate Spread:
 Spread between corporate & default-free
bond interest rates rises (from zero to i2c - i2T)
 Risk premium on
positive… i2c - i2T > 0
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corporate
bonds
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Default Risk and Positive Risk Premium
 A bond with default risk will always have
a positive risk premium
 An increase in a bond's default risk will
raise its risk premium
 Since default risk is so important to the size
of the risk premium, advisory firms grade, or
rate, the quality of bonds in terms of the
probability of default risk
 Table 1 - Bond Ratings by Moody's
and Standard and Poor's
 Investment-grade securities have rating of
Baa (or BBB) and above
 Junk bonds (high-yield bonds) have ratings
below Baa, and have higher default risk
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Liquidity
 Recall, a liquid asset is one that can be
quickly and cheaply converted into cash
 The more an asset is liquid, the more the
asset is desirable (c.p.)
 U.S. Treasury bonds are the most liquid of
all long-term bonds… widely traded so they
are easy to sell… and cost of selling is low
 Corporate bonds are less liquid, as it may
be more difficult to find buyers quickly
Does reduced liquidity of corporate bonds
(relative to default-free bonds) affect their
interest rates?
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Liquidity
 Figure 2 - Response to Increase in Default
Risk on Corporate Bonds
 Initially, assume both bonds are equally
liquid and similar in other attributes
(principal, term to maturity, etc.)
Result:
P1c = P1T and i1c = i1T
 However, suppose the corporate bond is
less widely traded… meaning it is less liquid
 liquidity on
corporate bond
corporate bonds  demand shifts left
(via asset demand theory)
 liquidity on
U.S. Treasury bond
Treasury bonds  demand shift right
(rel. to corp. bonds)
(asset demand theory)
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Liquidity
 Price of the less liquid corporate bond falls
( Pc) and its interest rate rises ( ic)
 Price of the more liquid Treasury bond rises
( PT) and its interest rate falls ( iT)
Lower liquidity of corporate bonds relative
to Treasury bonds increases the spread (and
therefore the risk premium)
 Risk premiums reflect not only corporate
bond's default risk, but its liquidity too
Risk premium also known as…
liquidity premium - additional interest an
individual must earn in order to be willing to
hold a less liquid asset
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Income Tax Considerations
 Risk premiums and "liquidity-risk"
premiums may explain some of the behavior
of long-term bond yields (Figure 1)
 But what explains the spread between
municipal bonds and U.S. Treasury bonds,
especially in the late 1930s?
 Interest payments on municipal bonds are
exempt from Federal income taxes… a factor
that has the same effect on municipal bond
demand as an increase in their E(RET)
municipal bond
given tax free status
 municipal bond

after-tax
expected return
(relative to Treasuries)
 shift municipal bond demand right
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Income Tax Considerations
 Figure 3 - Interest Rates on Municipal
and Treasury Bonds
 Since municipal bonds are now more
desirable, demand for them increases…
…and since Treasury bonds are now less
desirable, demand for U.S. Treasuries falls
Result: municipal bonds end up with
lower interest rates than those
on Treasury bonds
 Since T-bonds are exempt from State and
local income taxes, same reasoning applies as
to why interest rates on corporate bonds are
higher than those on U.S. Treasuries
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Term Structure of Interest Rates
Another factor that influences the interest
rate on a bond is its term to maturity…
 Bonds with identical risk, liquidity, and tax
characteristics may have different interest
rates, since time remaining to maturity differs
 Yield curve plots the yields on bonds with
differing terms to maturity, but with same
risk, liquidity and tax considerations
 When yield curve slopes upward…
long-term rates > short-term rates
 When yield curve slopes downward
(inverted)…
long-term rates < short-term rates
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Term Structure - Empirical Facts
Besides explaining why yield curves take on
different shapes (upward sloping, inverted,
flat) at different times, a reliable theory of the
term structure of interest rates must explain
the following…
1. Why interest rates on bonds of different
maturities move together over time (Fig. 4)
2. Why the case holds that when short-term
rates are low (high), yield curves are likely to
slope upward (downward, inverted)
3. Why yield curves typically slope upward
Three theories attempt to explain the term
structure of interest rates…
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Expectations Theory
 If people expect that short-term interest
rates will be 10% on average over the next
five years…
… the expectations theory predicts that the
interest rate on bonds with five years to
maturity will also be 10%
 Suppose short-term rates were expected to
rise after this five-year period, to 11% over
the next twenty years…
Then, the interest rate on twenty-year bonds
would equal 11%… higher than the rate on
five-year bonds. Why?
 Rates on bonds with different maturities
differ, since short-term rates are expected to
have different values at future dates
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Expectations Theory
Assumption: bonds are perfect substitutes
 If bonds with different maturities are
perfect substitutes, expected return on these
bonds must be equal
e.g. Consider two investment strategies…
1.
Purchase one-year bond at 9%… and upon
maturity, purchase another one-year bond at
11%
2. Purchase a two-year bond at 10% and hold
until maturity
 Both strategies have same E(RET)…
as a result, individual is indifferent between
implementing one strategy over the other
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Expectations Theory - General Form
it  ite1  ite 2    ite( n1)
int 
n
int = today's interest rate on n-period bond
it = today's interest rate on one period bond
iet+1 = interest rate on a one period bond
expected for the next period
iet+(n-1) = interest rate on a one period bond
expected for n - 1 period in future
e.g. (cont.):
i2t
=
it  ite1
n
strategy with one strategy with two
two-year bond = one-year bonds
(10%)
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=
 9%  11% 


2


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Expectations Theory and the Yield Curve
e.g. Spse the one-year interest rate over next
five years is expected to be 5, 6, 7, 8, and 9%.
Expectations theory says that the interest rate
on the two-year bond would be…
5%  6%
 5.5%
2
while the interest rate for the five-year bond
would be…
5%  6%  7%  8%  9%
 7%
5
Similar calculations for all other years
indicate that the one- to five-year interest
rates are 5.0, 5.5, 6.0, 6.5, and 7.0%,
respectively.
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Expectations Theory and the Yield Curve
Rising trend in expected short-term interest
rates produces an upward sloping yield
curve… along which interest rates rise as
maturity lengthens
 Expectations theory explains why the term
structure of interest rates changes at different
times
 Upward sloping yield curve indicates that
short-term interest rates are expected to rise
(on average) in future
 Downward sloping (inverted) yield curve
indicates that short-term rates are expected to
fall (on average) in the future
 Flat yield curve indicates short-term rates
not expected to change (on average) in future
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Expectations Theory and the Yield Curve
 Expectations theory also explains why
interest rates on bonds of different maturities
move together over time…
If short-term rates rise, the expectations on
future rates will also rise…
…and since long-term rates are just an
"average" of expected future short-term rates,
a rise in short-term rates will also raise longterm rates (they move together over time)
 One drawback to the expectations theory is
that it doesn't explain why yield curves
usually slope upward…
…the theory gives equal weight to why the
yield curve would slope upward, slope
downward (inverted) or even be flat
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Segmented Markets Theory
 The segmented markets theory of the term
structure sees markets for different maturity
bonds as completely separate and segmented
Assumption: bonds of different maturities
are not substitutes
 Interest rate for each bond (with different
maturity) determined by the supply and
demand for that bond… and only that bond
…with no effect from expected returns on
other bonds (with different maturities)
 Investors have strong preferences for bonds
of one maturity but not for another… they are
only interested in E(RET)'s for assets in
which they have preferences for
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Segmented Markets Theory & Yield Curve
e.g. Investors who have a short holding
period prefer to hold short-term bonds.
Conversely, if investor was putting funds
away for young child to go to college,
preference is to hold longer-term bonds
 If most investors have short desired holding
periods (reasonable assumption), then they
will prefer bonds with shorter maturities that
have less interest-rate risk
 Segmented markets theory explains why
yield curves typically slope upwards…
demand for short-term bonds are higher than
that for long-term bonds, so…
PST bonds > PLT bonds  iLT bonds > iST bonds
()
()
()
()
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Segmented Markets Theory - Drawbacks
 While the segmented markets theory can
explain why yield curves tend to slope up, it
cannot explain the other empirical facts…
 Since the market for bonds (of different
maturities) are segmented, there is no reason
for interest rates on bonds of different
maturities to affect one another (but
empirically they do move together over time)
 Segmented Markets Theory also does not
explain why yield curves tend to slope
upward (downward, inverted) when shortterm rates are low (high)
One theory explains all aspects of the term
structure of interest rates…
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Liquidity Premium Theory
 Liquidity premium theory states that the
interest rate on a long-term bond will equal…
…average of short-term interest rates
expected to occur over the life of the bond
(expectations theory)
…plus a liquidity (term) premium that
responds to the supply/demand conditions for
that bond (segmented markets theory)
Assumption: bonds of different maturities
are substitutes,… but they
are not perfect substitutes
 E(RET) on a bond influences the E(RET)
on another bond of a different maturity…
…but the investor now can prefer one
bond over another
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Liquidity Premium Theory
 Shorter-term bonds preferred as they bear
less interest-rate risk
Result: offer investors a positive premium to
induce them to hold longer-term bonds
it  ite1  ite 2    ite( n1)
int 
 lnt
n
where lnt = liquidity premium for n-period
bond at time t
liquidity premium - extra return required to
induce lenders to lend long term rather than
short term
 Fig. 5 - Relationship Between the Liquidity
Premium and Expectations Theory
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Liquidity Premium Theory
 Liquidity premium is always positive…
and grows as the term to maturity increases
Result: Yield curve implied by the LPT is
always above the yield curve that is
implied by the ET(and, it has a steeper slope)
e.g. (cont.) From before, suppose one-year
interest rate over next five years is expected
to be 5, 6, 7, 8, and 9%.
Now, let investor's preferences for holding
short-term bonds indicate liquidity premiums
for one- to five-year bonds as 0, 0.25, 0.5,
0.75, and 1.0%, respectively.
Recall, that's the extra return needed to
compensate investors for holding longer-term
debt securities
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Liquidity Premium Theory
Previous equation states that the interest rate
on the two-year bond would be…
5%  6%
 0.25%  5.75%
2
while interest rate for 5 year bond would be…
5%  6%  7%  8%  9%
 1%  8%
5
Similar calculations for other years indicate
that the one- to five-year interest rates are
5.0, 5.75, 6.5, 7.25, and 8.0%, respectively.
Comparing results from previous example,
the LPT produces yield curves that slope
more steeply upward… due to investors'
preferences for short-term bonds
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LPT and Validation of Empirical Facts
Liquidity premium theory is consistent with
all three empirical facts…
1. LPT explains why interest rates on
different maturity bonds move together
over time
 short-term rates
 short-term rates 
in the future
(on average)
  long-term rates
 first term in 
 the LPT equation 
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LPT and Validation of Empirical Facts
2. LPT explains why the case holds that
when short-term rates are low (high),
yield curves are likely to slope upward
(downward, inverted)
 Investors generally expect short-term rates
to rise when they are low…
…so the average of future expected shortterm rates will be high relative to the current
short-term rate
 With the additional boost of a positive
liquidity premium, long-term rates will be
substantially above current short-term rates
(LPT equation)
Result: the yield curve will exhibit
a steep, upward slope
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LPT and Validation of Empirical Facts
3. LPT explains why yield curves typically
slope upward
 Liquidity premium rises with a bond's
maturity due to investor's preferences for
shorter-term bonds
LPT and Shapes of the Yield Curve
 Finally, the LPT allows us to predict future
short-term interest rates just by observation
of the yield curve (Fig. 6)
yield curve shape
steep, rising
moderately steep
flat
inverted
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direction of future
short-term rates
rise
small rise or fall
moderate fall
sharp fall
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LPT Summary
 LPT combines features of the expectations
theory and the segmented market theory…
…by asserting that a long-term interest rate
will equal the sum of a liquidity premium and
the average of short-term interest rates
expected to occur of the life of the bond
 LPT explains the three empirical facts:
1. interest rates on different maturities
tend to move together over time
2. when short-term rates are low (high),
yield curves are likely to slope
upward (downward, inverted)
3. yield curves usually slope upward
 LPT also allows prediction of the
movements of short-term rates in the future
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