CHAPTER 6
The Risk and Term
Structure of
Interest Rates
LEARNING OBJECTIVES
1. Identify and explain the three factors affecting the risk
structure of interest rates
2. List and explain the three theories of why interest rates vary
across different maturities
RISK STRUCTURE OF INTEREST RATES
▪Risk structure of interest rates: the relationship among
interest rates on bonds with the same term to maturity
LONG-TERM BOND YIELDS, 1919-2020
RISK STRUCTURE OF INTEREST RATES
▪Risk structure of interest rates: the relationship among
interest rates on bonds with the same term to maturity
▪Bonds with the same maturity have different interest rates
due to:
▪Default risk
▪Liquidity
▪Income tax considerations
RISK STRUCTURE OF INTEREST RATES
▪Default risk: probability that the issuer of the bonds is unable or
unwilling to make interest payments or pay off the face value
▪U.S. Treasury bonds considered default free (government can
always pay off its obligations)
▪Risk premium: the spread between interest rates on bonds with
default risk and interest rates on default-free bonds (of the same
maturity)
▪Indicates how much additional interest people must earn to be
willing to hold the risky bond
RESPONSE TO AN INCREASE IN DEFAULT RISK ON
CORPORATE BONDS
Treasury bonds become
more attractive which
increases the demand for
Treasury bonds
𝑆𝑇
Price of Bonds
Price of Bonds
Step 1. An increase in
default risk decreases the
demand for corporate
bonds
𝑆𝐶
𝑖2𝑇
Risk
Premium
𝑃1𝐶
…raising the price
of Treasury bonds
& decreasing the
interest rate
𝑃2𝑇
𝑃1𝑇
𝑖2𝐶
𝑃2𝐶
…lowering the price
of corporate bonds
& increasing the
interest rate
→ increasing the spread between the two
𝐷2𝐶
𝐷1𝑐
Quantity of Corporate Bond
Corporate Bond Market
𝐷1𝑇
𝐷2𝑇
Quantity of Treasury Bond
Default-free Bond Market
RESPONSE TO AN INCREASE IN DEFAULT RISK ON
CORPORATE BONDS
Treasury bonds become
more attractive which
increases the demand for
Treasury bonds
𝑆𝑇
Price of Bonds
Price of Bonds
Step 1. An increase in
default risk decreases the
demand for corporate
bonds
𝑆𝐶
𝑐
𝑖2
𝑖2𝑇
Risk
Premium
𝑃1𝐶
𝑖2𝐶
𝑃2𝐶
…lowering the price
of corporate bonds
& increasing the
interest rate
𝑃2𝑇
𝑐𝑃1𝑇
𝑇
𝑖1 = 𝑖1
→ increasing the spread between the two
𝐷2𝐶
𝐷1𝑐
Quantity of Corporate Bond
Corporate Bond Market
𝑇
𝑖2
Risk
Premium
…raising the price
of Treasury bonds
& decreasing the
interest rate
𝐷1𝑇
𝐷2𝑇
Quantity of Treasury Bond
Default-free Bond Market
BOND RATINGS
▪Purchasers of bonds need to know whether a corporation is
likely to default on its bonds
▪Default risk is important for the risk premium
▪Credit-rating agencies: investment advisory firms that rate the
quality of corporate & municipal bonds in terms of their
probability of default
▪3 largest agencies – Moddy’s Investor Service, Standard and
Poor’s Corporation, and Fitch Ratings
BOND RATINGS BY MOODY’S, STANDARD & POOR’S,
AND FITCH
Moody’s
S&P
Fitch
Definitions
Moody’s
S&P
Fitch
Definitions
Aaa
AAA
AAA
Prime Maximum Safety
Ba2
BB
BB
Speculative
Aa1
AA+
AA+
High Grade High Quality
Ba3
BB-
BB-
Aa2
AA
AA
B1
B+
B+
Aa3
AA-
AA-
B2
B
B
A1
A+
A+
B3
B-
B-
A2
A
A
Caa1
CCC+
CCC
Substantial Risk
A3
A-
A-
Caa2
CCC
-
In Poor Standing
Baa1
BBB+
BBB+
Caa3
CCC-
-
Baa2
BBB
BBB
Ca
-
-
Extremely Speculative
Baa3
BBB-
BBB-
C
-
-
May Be in Default
Ba1
BB+
BB+
-
-
D
Default
Upper Medium Grade
Lower Medium Grade
Noninvestment Grade
Highly Speculative
RISK STRUCTURE OF INTEREST RATES
▪Liquidity: the relative ease with which an asset can be converted to cash
▪More liquid = _____ desirable
▪U.S. Treasury bonds are the most liquid – ________ traded
▪Corporate bonds don’t have as many bonds being traded for any one
corporation
▪Can illustrate effect of liquidity just like in slide #7
▪Lower liquidity of corporate bonds relative to Treasury bonds
▪…increases the risk (& liquidity) premium
RISK STRUCTURE OF INTEREST RATES
▪Income tax considerations
▪Interest payments on municipal bonds are ________ from
federal income taxes
▪Increases the after-tax expected return & makes municipal
bonds ______ desirable
INTEREST RATES ON MUNICIPAL AND TREASURY BONDS
Price of Bonds
Price of Bonds
𝑆𝑚
𝑃2𝑚
𝑃1𝑚
Result:
municipal
bonds have a
higher price &
a lower interest
rate than
Treasury bonds 𝑃𝑇
1
𝑆𝑇
…and shifts the
demand for Treasury
bonds to the left…
𝑃2𝑇
Tax-free status shifts the demand for
municipal bonds to the right…
𝐷1𝑚
𝐷2𝑚
Quantity of Municipal Bonds
Market for Municipal Bonds
𝐷2𝑇
𝐷1𝑇
Quantity of Treasury Bonds
Market for Treasury Bonds
TERM STRUCTURE OF INTEREST RATES
▪Term structure of interest rates: the relationship among
interest rates on bonds with different terms to maturity
▪Bonds with identical risk, liquidity, & tax characteristics may
have different interest rates because the time remaining to
maturity is different
TERM STRUCTURE OF INTEREST RATES
▪Yield curve: a plot of the yields on bonds with differing terms to
maturity but the same risk, liquidity, & tax considerations
▪Describes the term structure of interest rates for particular types of
bonds
▪Upward-sloping: long-term interest rates are _______ short-term rates
▪Most common
▪Flat: short- and long-term rates are the ________
▪Inverted (downward-sloping): long-term rates are _______ short-term
rates
TREASURY YIELD CURVE AS OF AUGUST 2021
TERM STRUCTURE OF INTEREST RATES
▪The theory of the term structure of interest rates must explain the
following facts:
1. Interest rates on bonds of different maturities move together
over time
MOVEMENT OVER TIME OF INTEREST RATES ON US
GOVERNMENT BONDS WITH DIFFERENT MATURITIES
TERM STRUCTURE OF INTEREST RATES
▪The theory of the term structure of interest rates must explain the
following facts:
1. 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 rates are high, yield curves are more likely to slope
downward & be inverted
3. Yield curves almost always slope upward
TERM STRUCTURE OF INTEREST RATES
▪Three theories to explain these facts:
1. Expectations theory explains the first two facts but not the third
2. Segmented markets theory explains the third fact but not the
first two
3. Liquidity premium theory combines the two theories to explain
all three facts
EXPECTATIONS THEORY
▪Theory states: the interest rate on a long-term bond will equal the
average of the short-term interest rates that people expect to occur
over the life of the long-term bond
▪Key assumption: Buyers of bonds do not prefer bonds of one
maturity over another; they will not hold any quantity of a bond if
its expected return is less than that of another bond with a different
maturity
▪Bond holders consider bonds with different maturities with equal
expected returns to be perfect substitutes
EXPECTATIONS THEORY
▪Example:
▪Let the current rate on a one-year bond be 6%
▪You expect the interest rate on a one-year bond to be 8% next year
▪You buy the 2 one-year bonds
▪The expected return over the two years will average:
▪The interest rate on a two-year bond must be ____ for you to be
willing to purchase it
EXPECTATIONS THEORY
▪For an investment of $1
▪𝑖𝑡 = today’s interest rate on a one-period bond
𝑒
▪𝑖𝑡+1
= interest rate on a one-period bond expected for next period
▪𝑖2𝑡 = today’s interest rate on the two-period bond
▪𝑖𝑡 = interest rate on a one-period bond
𝑒
▪𝑖𝑡+1
= interest rate on a one-period bond
expected for next period
EXPECTATIONS THEORY
▪𝑖2𝑡 = interest rate on the two-period bond
▪Expected return from investing $1 in the two-period bond & holding it
for the two periods:
1 + 𝑖2𝑡 1 + 𝑖2𝑡 − 1
= 1 + 2𝑖2𝑡 + 𝑖2𝑡
2
= 2𝑖2𝑡 + 𝑖2𝑡
2
−1
▪Since 𝑖2𝑡 2 is very small, the expected return for holding the two-period
bond for two periods is 2𝑖2𝑡
▪𝑖𝑡 = interest rate on a one-period bond
EXPECTATIONS THEORY
𝑒
▪𝑖𝑡+1
= interest rate on a one-period bond
expected for next period
▪𝑖2𝑡 = interest rate on the two-period bond
▪If two one-period bonds are bought with the $1 investment:
𝑒
1 + 𝑖𝑡 1 + 𝑖𝑡+1
−1
𝑒
𝑒
1 + 𝑖𝑡+1
+ 𝑖𝑡 + 𝑖𝑡 𝑖𝑡+1
−1
𝑒
𝑒
𝑖𝑡 + 𝑖𝑡+1
+ 𝑖𝑡 𝑖𝑡+1
𝑒
▪𝑖𝑡 𝑖𝑡+1
is extremely small
𝑒
▪Simplifying we end up with: 𝑖𝑡 + 𝑖𝑡+1
EXPECTATIONS THEORY
▪Both bonds will be held only if the expected returns are equal
𝑒
2𝑖2𝑡 = 𝑖𝑡 + 𝑖𝑡+1
𝑖2𝑡
𝑒
𝑖𝑡 + 𝑖𝑡+1
=
2
▪The two-period rate must equal the average of the two one-period rates
▪For bonds with longer maturities:
𝑖𝑛𝑡 =
𝑒
𝑒
𝑒
𝑖𝑡 + 𝑖𝑡+1
+ 𝑖𝑡+2
+ ⋯ + 𝑖𝑡+(𝑛−1)
𝑛
▪The n-period interest rate equals the average of the one-period interest
rates expected to occur over the n-period life of the bond
𝑖𝑛𝑡 =
𝑒
𝑒
𝑒
𝑖𝑡 + 𝑖𝑡+1
+ 𝑖𝑡+2
+ ⋯ + 𝑖𝑡+(𝑛−1)
𝑛
EXPECTATIONS THEORY
▪Example: Suppose the one-year interest rates over the next five
years are expected to be 5%, 6%, 7%, 8%, and 9%.
1. What would the interest rate be on the three-year bond?
2. What would the interest rate be on the five-year bond?
EXPECTATIONS THEORY
▪Expectations theory explains:
▪Why the term structure of interest rates changes at different
times
▪Why interest rates on bonds with different maturities move
together over time (Fact 1)
▪Why yield curves tend to slope up when short-term rates are low
and slope down when short-term rates are high (Fact 2)
▪Can’t explain why yield curves usually slope upward (Fact 3)
SEGMENTED MARKETS THEORY
▪Theory sees markets for different-maturity bonds as separate &
segmented
▪Key Assumption: bonds of different maturities are not substitutes
▪The interest rate for each bond with a different maturity is determined
by the demand for & supply of that bond
▪Investors have preferences for bonds of one maturity over another
▪If investors generally prefer bonds with shorter maturities that have
less interest-rate risk, then this explains why yield curves usually slope
upward (Fact 3)
LIQUIDITY PREMIUM THEORY
▪Theory states: 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 that responds to supply &
demand conditions for that bond
▪Key Assumption: bonds of different maturities are partial (not perfect)
substitutes
▪The expected return on one bond DOES influence the expected return
on a bond of a different maturity
▪Investors prefer one bond maturity over another → usually shorterterm as these bear less interest-rate risk
LIQUIDITY PREMIUM THEORY
▪Liquidity premium theory is written as:
𝑖𝑛𝑡 =
𝑒
𝑒
𝑒
𝑖𝑡 + 𝑖𝑡+1
+ 𝑖𝑡+2
+ ⋯ + 𝑖𝑡+(𝑛−1)
+ 𝑙𝑛𝑡
𝑛
▪𝑙𝑛𝑡 = liquidity (term) premium for the n-period bond at time t
▪Always positive & rises with the term to maturity of the bond, n
PREFERRED HABITAT THEORY
▪Assumption: investors prefer bonds of one maturity over bonds of
another – a particular bond maturity (“preferred habitat”) in which
they prefer to invest
▪Investors are willing to buy bonds of different maturities only if
they earn a somewhat higher expected return
▪Investors are likely to prefer short-term bonds over longer-term
bonds
▪Only willing to hold long-term bonds if they have higher
expected returns than short-term bonds
THE RELATIONSHIP BETWEEN THE LIQUIDITY
PREMIUM & EXPECTATIONS THEORY
𝑖𝑛𝑡 =
𝑒
𝑒
𝑒
𝑖𝑡 + 𝑖𝑡+1
+ 𝑖𝑡+2
+ ⋯ + 𝑖𝑡+(𝑛−1)
𝑛
LIQUIDITY PREMIUM THEORY
▪Example: Suppose the one-year interest rates over the next five years are
expected to be 5%, 6%, 7%, 8%, and 9%. Investors prefer short-term
bonds, so liquidity premiums for one- to five- year bonds are 0%, 0.25%,
0.5%, 0.75%, and 1%.
1. What would the interest rate be on the three-year bond?
2. What would the interest rate be on the five-year bond?
+ 𝑙𝑛𝑡
LIQUIDITY PREMIUM THEORY & PREFERRED
HABITAT THEORY – 3 FACTS
1. Interest rates on different maturity bonds move together over time; explained by
the first term in the equation
2. Yield curves tend to slope upward when short-term rates are low…
a. Explained by the liquidity premium added to the higher expected short-term
average
2. & to be inverted when short-term rates are high
a. Explained by the liquidity premium added to the lower expected short-term
average
3. Yield curves typically slope upward; explained by a larger liquidity premium as
the term to maturity lengthens due to investor preferences of short-term bonds
YIELD CURVES & THE MARKET’S EXPECTATIONS OF FUTURE
SHORT-TERM INTEREST RATES ACCORDING TO THE LIQUIDITY
PREMIUM THEORY
Short-term rates
are expected to
rise in the future
Short-term rates
aren’t expected
to change much
in the future
Short-term rates
are expected to
fall moderately
in the future
Short-term rates
are expected to
fall sharply in
the future
SUMMARY
▪ Bonds with the same maturity will have different interest rates because of three factors: default
risk, liquidity, and tax considerations. The greater a bond’s default risk, the higher its interest rate
relative to the interest rates of other bonds; the greater a bond’s liquidity, the lower its interest
rate; and bonds with tax-exempt status will have lower interest rates than they otherwise would.
The relationship among interest rates on bonds with the same maturity that arises because of
these three factors is known as the risk structure of interest rates
▪ Three theories of the term structure provide explanations of how interest rates on bonds with
different terms to maturity are related. The expectations theory views long-term interest rates as
equaling the average of future short-term interest rates expected to occur over the life of the
bond. By contrast, the segmented markets theory treats the determination of interest rates for
each bond’s maturity as the outcome of supply and demand in that market only. Neither of these
theories by itself can explain the fact that interest rates on bonds of different maturities move
together over time and that yield curves usually slope upward
SUMMARY
▪ The liquidity premium (preferred habitat) theory combines the features of the other
two theories, and by so doing is able to explain the facts just mentioned. The liquidity
premium (preferred habitat) theory views long-term interest rates as equaling the
average of future short-term interest rates expected to occur over the life of the
bond plus a liquidity premium. The liquidity premium (preferred habitat) theory
allows us to infer the market’s expectations about the movement of future short-term
interest rates from the yield curve. A steeply upward-sloping curve indicates that
future short-term rates are expected to rise; a mildly upward-sloping curve, that
short-term rates are expected to stay the same; a flat curve, that short-term rates
are expected to decline slightly; and an inverted yield curve, that a substantial
decline in short-term rates is expected in the future.