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ECON7200
Economics of Financial Markets
Lecture 4
The Risk and Term Structure of Interest Rates
Learning Objectives
• Identify and explain three factors explaining the risk structure of
interest rates.
• List and explain the three theories of why interest rates vary across
maturities.
• Bonds with the same maturity have different interest rates due to:
• Default risk
• Liquidity
• Tax considerations
4
LearnX – Risk Structure of Interest Rates
• Default risk: probability that the issuer of the bond is unable or
unwilling to make interest payments or pay off the face value
• U.S. Treasury bonds are considered default free (government can
raise taxes).
• Risk premium: the spread between the interest rates on bonds
with default risk and the interest rates on (same maturity)
Treasury bonds
5
Figure: Response to an Increase in Default Risk on Corporate Bonds
Step 1. An increase in default risk shifts the demand curve for corporate
bonds left . . .
Step 2. and shifts the demand curve for Treasury bonds to the right . . .
Step 3. which raises the price of Treasury bonds and lowers the price of
corporate bonds, and therefore lowers the interest rate on Treasury
bonds and raises the rate on corporate bonds, thereby increasing the
spread between the interest rates on corporate versus Treasury bonds.
(a) Corporate bond market (b) Default-free (U.S. Treasury) bond market
Sc
Quantity of Corporate Bonds Quantity of Treasury Bonds
Price of Bonds, P Price of Bonds, P
P 1c
P2
c
D2
c D1
c
P2
T
1
TP
D1T
ST
D2T
2
Ti
i 2
c
Risk
Premium
6
Table: Bond Ratings by Moody’s, Standard
and Poor’s, and Fitch
Moody’s Rating Agency S&P Fitch Definitions
Aaa AAA AAA Prime Maximum Safety
Aa1 AA+ AA+ High Grade High Quality
Aa2 AA AA Blank
Aa3 AA– AA– Blank
A1 A+ A+ Upper Medium Grade
A2 A A Blank
A3 A– A– Blank
Baa1 BBB+ BBB+ Lower Medium Grade
Baa2 BBB BBB Blank
Baa3 BBB– BBB– Blank
Ba1 BB+ BB+ Noninvestment Grade
7
Table: Bond Ratings by Moody’s, Standard
and Poor’s, and Fitch
Moody’s Rating Agency S&P Fitch Definitions
Ba2 BB BB Speculative
Ba3 BB– BB– Blank
B1 B+ B+ Highly Speculative
B2 B B Blank
B3 B– B– Blank
Caa1 CCC+ CCC Substantial Risk
Caa2 CCC — In Poor Standing
Caa3 CCC– — Blank
Ca — — Extremely Speculative
C — — May Be in Default
— — D Default
8
Game: Fit the Category
9
Rating Agency S&P Definitions
AAA Prime Maximum Safety
AA+ High Grade High Quality
AA Blank
AA– Blank
A+ Upper Medium Grade
A Blank
A– Blank
BBB+ Lower Medium Grade
BBB Blank
BBB– Blank
BB+ Noninvestment Grade
• Apple
• Microsoft
• Tesla
• Australia Treasury Bond
Game: Fit
the Category
10
Rating Agency
S&P
Definitions
AAA Prime Maximum
Safety
AU Treasury Bond
Microsoft
AA+ High Grade High
Quality
Apple
BBB Tesla
Crises and the Baa–Treasury Spread
• Starting in August 2007, the collapse of the subprime mortgage
market led to large losses among financial institutions.
• Many investors began to doubt the financial health of corporations
with low credit ratings such as Baa.
• Even the reliability of the ratings themselves!
• The perceived increase in default risk for Baa bonds made them less
desirable at any given price.
• The same thing happened during the pandemic starting in March
2020.
11
12
Baa bonds
U.S.
Treasury
bonds
Spread
July 2007 6.63% 4.78% 1.85%
October
2008 9.43% 3.98% 5.45%
February
2020 3.6% 1.5% 2.1%
March
2020 4.3% 0.9% 3.4%
Figure: Interest Rates on Municipal and Treasury
Bonds
Step 1. Tax-free status shifts the demand for municipal bonds to the right . . .
Step 2. and shifts the demand for Treasury bonds to the left . . .
Step 3. with the result that municipal bonds end up with a higher price and a
lower interest rate than on Treasury bonds.
(a) Market for municipal bonds (b) Market for Treasury bonds
2
mP
1
mD
2
TP
1
TD
Price of Bonds, P Price of Bonds, P
Quantity of Municipal Bonds Quantity of Treasury Bonds
1
mP 1TP
Sm
2
mD
ST
2
TD
13
Effects of the Tax Changes on Bond Interest Rates
• In 2013, Congress approved legislation favored by the Obama
administration to increase the income tax rate on high-income
taxpayers from 35% to 39%.
• Consistent with supply and demand analysis, the increase in income
tax rates for wealthy people helped to lower the interest rates on
municipal bonds relative to the interest rate on Treasury bonds.
• In 2017, Congress approved a decrease the income tax rate on high-
income taxpayers from 39.6% to 37%.
• Increase the interest rates on municipal bonds relative to the interest rate on
Treasury bonds.
14
Term Structure of
Interest Rates
• Bonds with identical risk, liquidity, and tax
characteristics may have different interest
rates because the time remaining to
maturity is different.
15
Figure: Movements over Time of Interest Rates on U.S. Government
Bonds with Different Maturities
Sources: Federal Reserve Bank of St. Louis FRED database: http://research.stlouisfed.org/fred2/ 16
LearnX – Yield Curve
• Yield curve: a plot of the yield on bonds
with differing terms to maturity but the
same risk, liquidity and tax considerations.
• A yield curve shows the relationship
between interest rate and years to maturity.
• Upward-sloping: long-term rates are above
short-term rates
• Flat: short- and long-term rates are the same
• Inverted: long-term rates are below short-term
rates
17
Interest Rates
Years to Maturity
LearnX –
Term
Structure of
Interest
Rates
• The term structure of interest rates is the relationship
between interest rates and different maturities.
• 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 and
be inverted.
3. Yield curves almost always slope upward.
18
LearnX – Term Structure of Interest Rates
• Three theories to explain the three 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.
• Preferred habitat theory supports liquidity premium
theory.
19
Expectations
Theory: An
Example
20
We
consider
the
following
two
investment
strategies
Purchase a one-year bond, and
when it matures in one year,
purchase another one-year bond;
Purchase a two-year bond and
hold it until maturity.
Because both strategies must have the same
expected return, the interest rate on the two-
year bond must equal the average of the two
one-year interest rates.
Expectations Theory:
An Example
• Let the current rate on one-year bond be
6%.
• You expect the interest rate on a one-year
bond to be 8% next year.
• Then the expected return for buying two
one-year bonds averages (6% + 8%)/2 =
7%.
• The interest rate on a two-year bond must
be 7% for you to be willing to purchase it.
21
Expectations Theory: More General Case
22
• We consider an investment of
$1.
• Let denote today’s interest
rate on a one-period bond.
• Let denote interest rate
on a one-period bond
expected for next period.
• Let denote today’s interest
rate on a two-period bond.
• Expected return in the two-
period bond and holding it for
the 2 periods
• Since is very small,
Expectations Theory
23
• It two one-period bonds are
bought with the $1
investment,
• Since is very small,
• Both bonds will be held only if the
expected returns are equal
• So, శభ
• More generally,
Expectations Theory: Numerical Example
Year 1 Year 2 Year 3 Year 4 Year 5
5% 6% 7% 8% 9%
5% 5% + 6%
2
= 5.5%
5%+ 6%+ 7%
3
= 6%
5% + 6% + 7%+ 8%
4
= 6.5% 7% (see below)
On the five-year bond, it will be
7% ହ%ା%ା%ା଼%ାଽ%
ହ
The relationship between the one-year interest rates and two, three, four, and
five-year bond’s interest rate is
24
Recall: Three Facts
25
Interest rates on bonds
with different maturities
move together over time.
01
Yield curves tend to have
an upward slope when
short-term interest rates
are low and be inverted
when short-term rates
are high.
02
Yield curves usually slope
upward.
03
Expectations Theory
26
• Remember ଶ௧
ାశభ
ଶ
• Why interest rates on bonds with different maturities move together
over time (fact 1). Historically if short-term interest rates increase today,
they will tend to be higher in the future.
• Why yield curves tend to slope up when short-term rates are low, and
slope down when short-term rates are high (fact 2). If short-term
interest rates start at a low level, people expect that it will rise in a near
future.
Expectations theory explains:
Cannot explain why yield curves usually slope upward (fact 3)
Segmented
Markets
Theory
• Bonds of different maturities are not
substitutes at all.
• The interest rate for each bond with a
different maturity is determined by the
demand for and supply of that bond.
• Investors have preferences for bonds of
one maturity over another.
• The expected return from holding a bond
of one maturity has no effect on the
demand for a bond of another maturity.
27
LearnX – Segmented Markets Theory and Fact 3
• 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).
• Remember that a price of bond and the interest rate have
negative relationship.
• The demand for the long-term bond is typically lower than the
demand for the short-term bond.
• Long-term bond has a lower price.
• Long-term bond has a higher interest rate.
28
LearnX – Segmented Markets Theory and
Facts 1 & 2
• Fact 1
• Because in views the market for bonds of different maturities as
completely segmented, there is no reason that a rise in the interest
rate on a bond of one maturity would affect the interest rate on a
bond of another maturity.
• Fact 2
• Because it is not clear how the demand for and supply of short-
versus long-term bonds change with the level of short-term interest
rates, the theory does not explain why yield curves tend to slope
upward when short-term interest rates are low.
29
LearnX – Liquidity Premium Theory
• 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 and demand
conditions for that bond.
• 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.
• Short-term securities are less risky compared to long-term rates due
to the difference in maturity dates. Therefore investors expect a
premium, or risk premium for investing in the risky security.
30
LearnX – Preferred Habitat Theory
• Investors have a preference for bonds of one maturity over another.
• They will be willing to buy bonds of different maturities only if they
earn a somewhat higher expected return.
• Risk-averse investors are likely to prefer short-term bonds over
longer-term bonds.
31
LearnX – Liquidity Premium Theory
32
• Let denote the liquidity premium for the
-period bond at time .
• Then
• Note that is always positive and rises
with the term to maturity.
Term to maturity
Liquidity premium ௧
Figure 5 The Relationship Between the Liquidity Premium (Preferred Habitat)
and Expectations Theory
Liquidity Premium (Preferred Habitat) Theory
Yield Curve
0 5 10 15 20 25 30
Years to Maturity, n
Interest
Rate, int
Expectations Theory
Yield Curve
Liquidity
Premium, lnt
33
LearnX – under Liquidity Premium & Preferred Habitat Theories
Pattern (a) Short-term interest rates are expected
to rise.
LearnX – under Liquidity Premium & Preferred Habitat Theories
Pattern (b) Short-term interest rates are not expected to
rise or fall much in the future.
When future short term interest rates are
expected to stay constant
• Compare a 4-year bond and a 1-year bond using liquidity premium
theory:
• If future short term interest rates are expected to neither rise nor fall,
. As a result
• However, since , , and the difference is the liquidity
premium .
• Using the same analogy, it follows that .
• If we plot, we will see that the yield curve is slightly upward when
short term interest rates are expected to neither rise nor fall.
36
LearnX – Liquidity Premium Theory
Pattern (c) Short-term interest rates are expected to fall
moderately.
