The Risk Structure and
Term Structure of Interest
Rates
Chapter 6
Risk Structure of Interest Rates
• Bonds with the same maturity have different
interest rates due to:
• Default risk
• Liquidity
• Tax considerations
Risk Structure of Interest Rates
• Default risk: probability that the issuer of a
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
Response to an Increase in Default Risk on
Corporate Bonds – Supply/Demand Application
Russian Default
Risk Structure of Interest Rates
• Liquidity: the relative ease with which an
asset can be converted into cash
• Cost of selling a bond
• Number of buyers/sellers in a bond market
• Income tax considerations
• Interest payments on municipal bonds are
exempt from federal income taxes.
Interest Rates on Municipal and Treasury Bonds
Taxes and Bond Prices
• Coupon payments on municipal bonds are
exempt from federal Income taxes
• For 28% tax bracket:
• After tax yield = (taxable yield) x (1 – tax rate)
3.60% = 5% x (1 – 0.28)
tax exempt yield
• Tax equivalent yield =
1 - tax rate
http://www.bloomberg.com/markets/ratesbonds/government-bonds/us/
Risk Structure of Long-Term Bonds in the
United States
Bond (credit) Ratings and Risk
Bond Ratings •
Moody’s and Standard and Poor’s
Ratings Groups
•
Investment Grade
•
Non-Investment – Speculative Grade
•
Highly Speculative
Bond (credit) ratings
S&P
Moody’s What it means
AAA
Aaa
Highest quality and creditworthiness
AA
Aa
Slightly less likely to pay principal + interest
A
A
Strong capacity to make payments, upper medium grade
BBB
Baa
Medium grade, adequate capacity to make payments
BB
Ba
Moderate ability to pay, speculative element, vulnerable
B
B
Not desirable investment, long term payment doubtful
CCC
Caa
Poor standing, known vulnerabilities, doubtful payment
CC
Ca
Highly speculative, high default likelihood, known reasons
C
C
Lowest rated class, most unlikely to reach investment grade
D
NR
Already defaulted on payments
No public rating has been requested
+ or - & 1,2,3 Within-class refinement of AA to CCC ratings
Credit rating & historic default frequencies
Moody’s
Rating
1985
1990
1995
2000
2006
2008
2009
2010
Aaa
0%
0%
0%
0%
0%
0%
0%
0%
Aa
0%
0%
0%
0%
0%
0%
0%
0%
A
0%
0%
0%
0%
0%
1.201%
0%
0.36%
Baa1
0%
0%
0%
0.29%
0%
0.271%
1.144%
0%
Baa2
0%
0%
0%
0%
0%
0.794%
0.74%
0%
Baa3
0%
0%
0%
0.98%
0%
0.321%
0.70%
0%
Ba1
0%
2.67%
0%
0.91%
0%
0%
2.27%
0%
Ba2
1.63%
2.82%
0%
0.66%
0.51%
0%
Ba3
3.77%
3.92%
1.72%
0.99%
0%
2.715%
4.01%
0%
B1
4.38%
8.59%
4.35%
3.63%
0.66%
1.783%
4.10%
0.85%
B2
7.41%
22.09%
6.36%
3.84%
0.50%
0.825%
8.68%
0%
B3
13.86%
28.93%
4.10%
11.72%
1.93%
3.198%
8.52%
0.56%
0.60%
0%
Default Risk – Price and YTM
• Suppose risk-free rate is 4%
• Suppose there is a company called FlimFlam that
issues one-year, 4% coupon bond, FV=$100.
• If risk free, the price of the FlimFlam bond is
$4  $100 $104
P

 $100
1.04
1.04
Default Risk
Suppose 5% probability FlimFlam goes bankrupt – you get nothing
Expected Value of FlimFlam bond payment
Possibilities
Payoff
Probability
Payoff x
Probability
Full Payment
$104
$0
.95
.05
$98.80
$0
Default
•Expect to receive $98.80 one-year from now.
•Discount at risk-free rate = $98 .80
1.04
•P = $95
 $95
Default Risk Premium
• We can calculate the probability of
repayment from the interest rates.
• Let 1+k be the return on a one-year
corporate debt and 1+ i be the return on a
one-year default risk-free treasury.
1 i
p
• The probability of repayment is
1 k
• the probability of default is 1 – p
• The probability of repayment:
1.04
 0.95
1.0947
Default Risk
Suppose 10% probability FlimFlam goes bankrupt – you get nothing
Expected Value of FlimFlam bond payment
Possibilities
Payoff
Probability
Payoff x
Probabilities
Full Payment
$104
$0
.90
.10
$93.60
$0
Default
•Expect to receive $93.60 one-year from now.
•Discount at risk-free rate =
$93.60
 $90
1.04
•Yield = ($104 / $90) -1 = .1555 or 15.55%
•Default risk premium = 15.55% - 4% = 11.55%.
Bond Ratings and Risk
• Increased risk reduces bond demand.
• The resulting shift to the left causes a decline in
equilibrium price and an increase in the bond
yield.
• Bond Yield = U.S. Treasury Yield
+ Default Risk Premium
• Risk spread or default risk premium =
Bond Yield - U.S. Treasury Yield
Information Content of Interest Rates:
Risk Structure
• When the economy starts to slow, this puts a
strain on private firms.
• A slower economy means a higher default
probability
• Risk Spreads increase.
Information Content of Interest Rates: Risk Structure
Risk spread = Baa Corporate minus 10-year Treasury
Term Structure of Interest Rates
Definition of the Term Structure:
The relationship among bonds with the same
risk, liquidity and tax characteristics but different
maturities is called the term structure of interest
rates.
Yield Curve:
A plot of the term structure, with the yield to
maturity on the vertical axis and the time to
maturity on the horizontal axis.
http://finance.yahoo.com/bonds/composite
_bond_rates?desktop_view_default=true
Term Structure of Interest Rates
Term Structure of Interest Rates
http://stockcharts.com/index.html
Term Structure of Interest Rates:
Facts to Explain
1. Interest rates (Yields) on different
maturities tend to move together
2. Yields on short-term bond are more
volatile than yields on long-term bonds
3. Long-term yields tend to be higher than
short-term yields.
•
Also want to explain the fact that yield
curves can be inverted.
Movements over Time of Interest Rates on U.S.
Government Bonds with Different Maturities
Sources: Federal Reserve: www.federalreserve.gov/releases/h15/data.htm.
Three Theories to Explain the Three Facts
1. Pure Expectations Theory explains the
first two facts but not the third
2. Segmented Markets Theory explains
fact three but not the first two
3. Liquidity Premium Theory combines
the two theories to explain all three
facts
Pure Expectations Theory
• The interest rate on a long-term bond will
equal an 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.
• Bonds of different maturities are considered to be
perfect substitutes
Expectations Theory Notation
i1t 
interest rate on 1-year bond today (t).
i2t 
interest rate on 2-year bond today (t).
int 
interest rate on n-year bond today (t).
i1t 1 
interest rate on 1-year bond, 1-year from
today (t+1).
i1et 1 
Expected interest rate on 1-year bond, 1-year
from today (t+1).
i1et n 
Expected interest rate on 1-year bond, n-years
from today (t+n).
A Note on Averages
• Geometric average of
i1t and
((1  i1t )(1  i1t 1 ))
1/ 2
• Arithmetic average =
i1t 1 =
1
i1t  i1t 1
2
Expectations Theory:
• Let the current interest rate on one-year bond
(i1t) be 6%.
• You expect the interest
rate
on
a
one-year
bond next year ( i1e t 1 ) to be 9%.
• Then the expected return from buying 2 oneyear bonds averages (6% + 9%)/2 = 7.5%.
• Under the Expectations Theory the current
interest rate on a two-year (i2t) bond must be
7.5% for you to be willing to purchase that
bond.
• Why?
Example: 2 year investment horizon
•
•
•
•
•
Strategy 1:
Invest $1,000 for 2-years at 8%:
Ending Balance = (1+0.08)2($1,000) = $1,166.40
Strategy 2:
Invest $1,000 1-year at 6% and expect 9% one
year later:
• Ending Balance = (1 +0.06)(1+0.09)($1,000) =
$1,155.40
• Come out $11 ahead with Strategy 1.
• What happens to S and D?
Expectations Theory ( Math)
1. Return from a 2-year bond over 2 years
(1  i2t )(1  i2t ) 1
2. Return from a 1-yr bond and then another 1-yr bond
(1 i1t )(1 i1te 1 ) - 1
3. If one and two year bonds are perfect substitutes, then:
(1 i2t )(1 i2t )  (1 i1t )(1 i )
e
1t 1
Term Structure of Interest Rates:
Expectations Theory
From:
(1 i 2t )(1 i 2t )  (1 i1t )(1 i
e
1t 1
)
We can derive the following arithmetic approximation:
i1t  i
i 2t 
2
e
1t 1
Which says the long-term interest rate = average of
current and expected future short-term interest rates.
Here is how we get the approximation:
Expected return over the two periods from investing $1 in the
two-period bond and holding it for the two periods
(1 + i2t )(1 + i2t )  1
 1  2i2t  (i2t ) 2  1
 2i2t  (i2t ) 2
Since (i2t ) 2 is very small
the expected return for holding the two-period bond for two periods is
2i2t
Here is how we get the approximation:
If two one-period bonds are bought with the $1 investment
(1  it )(1  ite1 )  1
1  it  ite1  it (ite1 )  1
it  ite1  it (ite1 )
it (ite1 ) is extremely small
Simplifying we get
it  i
e
t 1
Expectations Theory
Both bonds will be held only if the expected returns are equal
2i2t  it  ite1
it  ite1
i2t 
2
The two-period rate must equal the average of the two one-period rates
For bonds with longer maturities
int 
it  ite1  ite 2  ...  ite( 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
Actual math: No Approximation
(1 i 2t )(1 i 2t )  (1 i1t )(1 i
e
1t 1
(1 i 2t )  [(1 i1t )(1 i
2
e
1t 1
(1 i 2t )  [(1 i1t )(1 i
e
1t 1
i2t  [(1  i1t )(1 i
e
1t 1
)]
1/2
)]
)] 1
1/2
This is a geometric average
)
Expectations Hypothesis - Arithmetic Average
int 
i1t  i
e
1t 1
i
e
1t  2
 ....  i
e
1t  n 1
n
In words: The interest rate on a bond with n years to maturity at time t is the
average of the n expected future one-year rates.
Numerical example:
One-year interest rate over the next five years 5%, 6%, 7%, 8% and 9%:
Interest rate on a two-year bond:
(5% + 6%)/2 = 5.5%
This is the only interest rate that is
known at time t
Interest rate for a five-year bond:
(5% + 6% + 7% + 8% + 9%)/5 = 7%
Interest rate for one, two, three, four and five-year bonds are:
5%, 5.5%, 6%, 6.5% and 7%.
Expectations Hypothesis
int 
i1t  i
e
1t 1
i
e
1t  2
 ....  i
e
1t  n 1
n
Another example:
One-year interest rate over the next five years 7%, 6%, 5%, 4% and 3%:
Interest rate on a two-year bond:
(7% + 6%)/2 = 6.5%
Interest rate for a five-year bond:
(7% + 6% + 5% + 4% + 3%)/5 = 5%
Interest rate for one, two, three, four and five-year bonds:
7%, 6.5%, 6%, 5.5% and 5%.
Recall the Fisher Equation: i = r + πe
• Holding r constant:
• If inflation is expected to rise in the future, expected
one-year interest rates will rise and the yield curve
will slope upward.
• If inflation is expected to fall in the future, expected
one-year interest rates will fall and the yield curve
will slope downward.
• If inflation is expected to remain the same in the
future, expected one-year interest rates will remain
the same and the yield curve will be flat.
Term Structure of Interest Rates:
Expectations Theory
Using the Pure Expectations Theory to Solve for
Expected 1-year (forward) Interest rates
From the formula for the yield on a 2-year bond:
e
it  ite1
i
t 1  2i2t  it
i2t 
2
From the formula for the yield on a 3-year bond:
i
it  ite1  ite 2
i3t 
3
e
t 2
i
 3i3t  (it  i )
e
t 1
In general:
e
t ( n1)
i
 nint  (n  1)i(n1)t
e
t 2
i
 3i3t  2i2t
Actual math: No Approximation
(1  i2t )(1  i2t )  (1  i1t )(1  i
e
1t 1
(1 i
e
1t 1
(1  i2t )
)
1  i1t
(1  i2t )

1
1  i1t
2
e
1t 1
i
2
)
Term Structure Facts and the Expectations
Theory
Expectations Theory Explains:
1. Interest Rates of different maturities tend to move
together
- long term interest rates are averages of expected future
short-term interest rates.
2. Yields on short-term bond are more volatile than
yields on long-term bonds –
- long term interest rates are averages of expected future
short-term interest rates.
But Expectations Theory does not explain:
3. Long-term yields tend to be higher than short-term
yields.
Segmented Market Theory
• Bonds of different maturities are not
perfect substitutes for each other.
Segmented Markets Hypothesis
• Assumptions:
• Investors have specific preferences about
the maturity or term of a security.
• Investors do not stray from their preferred
maturity.
Segmented Markets Hypothesis
• The slope of the yield curve is explained by
different demand and supply conditions for
bonds of different maturities.
• If the yield curve slopes up, it does so
because the demand for short term bonds is
relatively greater than the demand for long
term bonds.
• Short term bonds have a higher price and a
lower yield as a result of the relatively
greater demand. So the yield curve slopes
upward.
Segmented Markets Hypothesis
Price
Price
S
S
P2s
P1l
P1s
D2s
P2l
D1l
D1s
0
Quantity of Short-term Bonds
D2l
0
Quantity of Long-term Bonds
Upward Sloping Yield Curve
Segmented Markets Hypothesis
• The segmented markets hypothesis
explains why….
• Yield curves typically slope upward.
• On average, investors prefer bonds with
shorter maturities that have less interest
rate risk.
• Therefore, the demand for short term
bonds is relatively greater than the demand
for long-term bonds
Segmented Markets Hypothesis
• But, the segmented markets hypothesis does
not explain why…
• Interest rates on different maturities move
together.
• The segmented markets hypothesis assumes that
short and long markets are completely segmented.
Liquidity Premium Theory of the Term
Structure of Interest Rates
• Yield curve upward slope is explained by the fact
that long-term bonds are riskier than short-term
bonds.
• Bondholders face both inflation risk and interest
rate risk.
• The longer the term of the bond, the greater
both types of risk.
• Investors need to be compensated for the
greater risk.
Term Structure of Interest Rates
Liquidity Premium Theory
int 
i1t  i
e
1t 1
i
e
1t  2
 .... i
n
Pure Expectations Theory:
average of expected future short-term rates
(explains facts 1&2)
e
1t  n 1
 RPn
Liquidity or Risk
Premium
(explains fact 3)
Numerical Example
Term in years (n)
1
2
3
4
5
One year interest rate
expectations
5%
6%
7%
8%
9%
Liquidity premium
0%
0.25%
0.5%
0.75%
1.0%
Pure expectations
predicted n-year bond
interest rates
5%
5.5%
6%
6.5%
7%
Actual n-year bond
interest rates,
accounting for liquidity
preference
5%
 5%  6% 


2


5.75%
 5%  6%  7%   5  6  7  8 % 


 
4

3


6.5%
7.25%
 56789 
%

5


8%
Relationship Between the Liquidity Premium and
Expectations Theories
(if short term interest rates are
expected to remain constant)
Information Content of Interest Rates:
Term Structure
• When the yield curve slopes down,
it is called inverted
• An inverted yield curve
is a very valuable forecasting tool
• It signals an economic downturn
Information Content of Interest Rates:10-year T-bond
compared to 3-month T- bill
Market
Predictions
of Future
Short
Rates
The actual math is a lot more
interesting. Refer to the note on
“Term Structure and Forward
Interest Rates.”