The Term Structure
of Interest Rates
Chapter 15
Overview of Term Structure
The relationship between yield to maturity
and maturity.
Information on expected future short term
rates can be implied from yield curve.
The yield curve is a graph that displays the
relationship between yield and maturity.
Three major theories are proposed to explain
the observed yield curve.
15-2
1-2
Yield Curves
Yields
Upward
Sloping
Flat
Downward
Sloping
Maturity
15-3
1-3
Expected Interest Rates (Table 15.1)
Expected One-Year Rates in Coming Years
Year
Interest Rate
0 (today)
8%
1
10%
2
11%
3
11%
15-4
1-4
Pricing of Bonds using Expected Rates
1
PVn 
(1  r1 )(1  r2 )...(1  rn )
PVn = Present Value of $1 in n periods
r1 = One-year rate for period 1
r2 = One-year rate for period 2
rn = One-year rate for period n
15-5
1-5
Bond Prices using Expected Rates
Time to Maturity Price of Zero* Yield to Maturity
1
$925.93
8.00%
2
841.75
8.995
3
758.33
9.660
4
683.18
9.993
* $1,000 Par value zero
15-6
1-6
Forward Rates from Observed Rates
(1  yn )
(1  f n ) 
(1  yn 1 ) n 1
n
fn = one-year forward rate for period n
yn = yield for a security with a maturity of n
n 1
(1  yn )  (1  yn1 ) (1  f n )
n
15-7
1-7
Example of Forward Rates using Table 15.2
4 yr = 9.993
3yr = 9.660
fn = ?
(1.0993)4 = (1.0966)3 (1+fn)
(1.46373) / (1.31870) = (1+fn)
fn = .10998 or 11%
Note: this is expected rate that was used
in the prior example.
15-8
1-8
Downward Sloping Spot Yield Curve
Zero-Coupon Rates Bond Maturity
12%
1
11.75%
2
11.25%
3
10.00%
4
9.25%
5
15-9
1-9
Forward Rates Downward Sloping Y C
1yr Forward Rates
1yr
[(1.1175)2 / 1.12] - 1
= 0.115006
2yrs [(1.1125)3 / (1.1175)2] - 1 = 0.102567
3yrs [(1.1)4 / (1.1125)3] - 1
=
0.063336
4yrs [(1.0925)5 / (1.1)4] - 1
=
0.063008
15-10
1-10
Theories of Term Structure
Expectations
Liquidity Preference
Upward bias over expectations
Market Segmentation
Preferred Habitat
15-11
1-11
Expectations Theory
Observed long-term rate is a function of
today’s short-term rate and expected
future short-term rates.
Long-term and short-term securities are
perfect substitutes.
Forward rates that are calculated from
the yield on long-term securities are
market consensus expected future
short-term rates.
15-12
1-12
Liquidity Premium Theory
Long-term bonds are more risky.
Investors will demand a premium for the risk
associated with long-term bonds.
The yield curve has an upward bias built into
the long-term rates because of the risk
premium.
Forward rates contain a liquidity premium and
are not equal to expected future short-term
rates.
15-13
1-13
Liquidity Premiums and Yield Curves
Yields
Observed Yield
Curve
Forward Rates
Liquidity
Premium
Maturity
15-14
1-14
Liquidity Premiums and Yield Curves
Yields
Observed Yield
Curve
Forward Rates
Liquidity
Premium
Maturity
15-15
1-15
Market Segmentation and Preferred Habitat
Short- and long-term bonds are traded in distinct
markets.
Trading in the distinct segments determines the
various rates.
Observed rates are not directly influenced by
expectations.
Preferred Habitat:
Modification of market segmentation
Investors will switch out of preferred maturity
segments if premiums are adequate.
15-16
1-16
Using Spot Rates to Price Coupon Bonds
A coupon bond can be viewed as a series of
zero coupon bonds.
To find the value each payment is discount at
the zero coupon rate.
Once the bond value is found, one can solve
for the yield.
It’s the reason that similar maturity and
default risk bonds sell at different yields to
maturity.
15-17
1-17
Sample Bonds
A
B
Maturity
4 years 4 years
Coupon Rate
6%
8%
Par Value
1,000
1,000
Cash Flow in 1-3
60
80
Cash Flow in 4
1,060
1,080
Assuming Annual compounding
15-18
1-18
Price Using Spot Rates Bond A
Period
Spot
Rate
Cash
Flow
PV of
Flow
1
.05
60
57.14
2
.0575
60
53.65
3
.063
60
49.95
4
.067
1,060
817.80
Total
978.54
15-19
1-19
Price Using Spot Rates Bond B
Period
Spot
Rate
Cash
Flow
PV of
Flow
1
.05
80
76.19
2
.0575
80
71.54
3
.063
80
66.60
4
.067
1,080
833.23
Total
1,047.56
15-20
1-20
Solving For Yield to Maturity
Bond A
Bond Price
YTM
Bond B
Price
YTM
978.54
6.63%
1,047.56
6.61%
15-21
1-21