Chapter 6
The Risk Structure and Term Structure of Interest
Rates
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 equivalent yield = rate tax - 1
yieldexempt tax
http://www.bloomberg.com/markets/rates-bonds/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) ratingsS&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 Already defaulted on payments
NR No public rating has been requested
+ or - & 1,2,3 Within-class refinement of AA to CCC ratings
Credit rating & historic default frequenciesMoody’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% 0.60% 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%
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
100$04.1
104$
04.1
100$4$
P
Default Risk
Expected Value of FlimFlam bond payment
Possibilities Payoff Probability Payoff x Probability
Full Payment $104 .95 $98.80Default $0 .05 $0
95$04.1
80.98$
Suppose 5% probability FlimFlam goes bankrupt – you get nothing
•Expect to receive $98.80 one-year from now.
•Discount at risk-free rate =
•P = $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.
• The probability of repayment is
• the probability of default is 1 – p
• The probability of repayment:
1
1
ip
k
1.040.95
1.0947
Default Risk
Expected Value of FlimFlam bond payment
Possibilities Payoff Probability Payoff x Probabilities
Full Payment $104 .90 $93.60Default $0 .10 $0
Suppose 10% probability FlimFlam goes bankrupt – you get nothing
•Expect to receive $93.60 one-year from now.
•Discount at risk-free rate =
•Yield = ($104 / $90) -1 = .1555 or 15.55%
•Default risk premium = 15.55% - 4% = 11.55%.
90$04.1
60.93$
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 RatesDefinition 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
1ti interest rate on 1-year bond today (t).
2ti interest rate on 2-year bond today (t).
nti interest rate on n-year bond today (t).
1 1ti
1 1eti
interest rate on 1-year bond, 1-year from today (t+1).
Expected interest rate on 1-year bond, 1-year from today (t+1).
1et ni Expected interest rate on 1-year bond, n-years
from today (t+n).
A Note on Averages
• Geometric average of and =
• Arithmetic average =
1ti 1 1ti
1/ 21 1 1((1 )(1 )) 1t ti i
1 1 1
2t ti i
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 ( ) to be 9%.
• Then the expected return from buying 2 one-year 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?
1 1eti
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)1)(1( 2t2t ii
2. Return from a 1-yr bond and then another 1-yr bond
1-))(1(1 e11t1t ii
3. If one and two year bonds are perfect substitutes, then:
))(1(1))(1(1 e11t1t2t2t iiii
Term Structure of Interest Rates:Expectations Theory
From: ))(1i(1)i)(1i(1 e11t1t2t2t i
We can derive the following arithmetic approximation:
Which says the long-term interest rate = average of current and expected future short-term interest rates.
2
iii
e11t1t
2t
Here is how we get the approximation:
2 2
22 2
22 2
22
Expected return over the two periods from investing $1 in the
two-period bond and holding it for the two periods
(1 + )(1 + ) 1
1 2 ( ) 1
2 ( )
Since ( ) is very small
the expected re
t t
t t
t t
t
i i
i i
i i
i
2
turn for holding the two-period bond for two periods is
2 ti
Here is how we get the approximation:
1
1 1
1 1
1
1
If two one-period bonds are bought with the $1 investment
(1 )(1 ) 1
1 ( ) 1
( )
( ) is extremely small
Simplifying we get
et t
e et t t t
e et t t t
et t
et t
i i
i i i i
i i i i
i i
i i
Expectations Theory
2 1
12
Both bonds will be held only if the expected returns are equal
2
2The two-period rate must equal the average of the two one-period rates
For bonds with longer maturities
et t t
et t
t
t tnt
i i i
i ii
i ii
1 2 ( 1)...
The -period interest rate equals the average of the one-period
interest rates expected to occur over the -period life of the bond
e e et t ni i
nn
n
Actual math: No Approximation
)])(1i(1[)i(1 e11t1t
22t i
))(1i(1)i)(1i(1 e11t1t2t2t i
1)])(11([ 1/2e11t1t2 iii t
1/2e11t1t2t )])(1i(1[)i(1 i
This is a geometric average
Expectations Hypothesis - Arithmetic Average
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%
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%.
n
iiiii
ent
et
ett
nt1121111 ....
This is the only interest rate that is known at time t
Expectations Hypothesis
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%.
n
iiiii
ent
et
ett
nt1121111 ....
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
i
21
2
ett
t
iii
ttet iii 21 2
321
3
et
ett
t
iiii
ttet iii 232 23
tnnte
nt innii )1()1( )1(
In general:
)(3 132ettt
et iiii
From the formula for the yield on a 2-year bond:
From the formula for the yield on a 3-year bond:
Using the Pure Expectations Theory to Solve for Expected 1-year (forward) Interest rates
Actual math: No Approximation
)1)(1()1)(1( 11122etttt iiii
t
t
i
ii
1
22e
11t 1
)1()(1
11
)1(
1
22e
11t
t
t
i
ii
Term Structure Facts and the Expectations TheoryExpectations 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
0 0
S S
P2s
P1s P1
l
P2l
D1s
D2s
D1l
D2l
Quantity of Short-term Bonds 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
n
ent
et
ett
nt RPn
iiiii
1121111 ....
Liquidity or Risk Premium
(explains fact 3)
Pure Expectations Theory: average of expected future short-term rates
(explains facts 1&2)
Numerical Example
Term in years (n)
1 2 3 4 5One year interest rate
expectations5% 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% 5.75% 6.5% 7.25% 8%
5% 6%
2
5% 6% 7%
3
5 6 7 8%
4
5 6 7 8 9%
5
Relationship Between the Liquidity Premium and Expectations Theories
(if short term interest rates areexpected 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.”
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