The Term Structure
of Interest Rates
CHAPTER 15
Overview of Term Structure
• Information on expected future short term rates
can be implied from the 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
Figure 15.1 Treasury Yield Curves
Bond Pricing
• Yields on different maturity bonds are not all
equal
– Need to consider each bond cash flow as a
stand-alone zero-coupon bond when valuing
coupon bonds
Table 15.1 Yields and Prices to Maturities on ZeroCoupon Bonds ($1,000 Face Value)
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.
Sample Bonds
A
Maturity
4 years
Coupon Rate
6%
Par Value
1,000
Cash Flow in 1-3
60
Cash Flow in 4
1,060
Assuming Annual compounding
B
4 years
8%
1,000
80
1,080
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
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
Solving For Yield to Maturity
Bond A
Bond Price
YTM
Bond B
Price
YTM
978.54
6.63%
1,047.56
6.61%
Yield Curve Under Certainty
• An upward sloping yield curve is evidence that
short-term rates are going to be higher next year
(1  y2 ) 2  (1  r1 ) x(1  r2 )
1  y2   (1  r1 ) x(1  r2 )
1
2
• When next year’s short rate is greater than this
year’s short rate, the average of the two rates is
higher than today’s rate
Figure 15.2 Two 2-Year Investment Programs
Figure 15.3 Short Rates versus Spot Rates
Forward Rates from Observed Rates
(1  yn ) n
(1  f n ) 
n 1
(1  yn 1 )
fn = one-year forward rate for period n
yn = yield for a security with a maturity of n
(1  yn ) n  (1  yn1 ) n1 (1  f n )
Example 15.4 Forward Rates
4 yr = 8.00%
3yr = 7.00%
(1.08)4 = (1.07)3 (1+fn)
(1.3605) / (1.2250) = (1+fn)
fn = .1106 or 11.06%
fn = ?
Downward Sloping Spot Yield Curve
Example
Zero-Coupon Rates Bond Maturity
12%
1
11.75%
2
11.25%
3
10.00%
4
9.25%
5
Forward Rates for Downward Sloping
Y C Example
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
Interest Rate Uncertainty
• What can we say when future interest rates are
not known today
• Suppose that today’s rate is 5% and the expected
short rate for the following year is E(r2) = 6%
then:
(1  y2 )2  (1  r1 ) x[1  E (r2 )]  1.05 x1.06
• The rate of return on the 2-year bond is risky for
if next year’s interest rate turns out to be above
expectations, the price will lower and vice versa
Interest Rate Uncertainty Continued
• Investors require a risk premium to hold a longerterm bond
• This liquidity premium compensates short-term
investors for the uncertainty about future prices
Theories of Term Structure
• Expectations
• Liquidity Preference
– Upward bias over expectations
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
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
Figure 15.4 Yield Curves
Interpreting the Term Structure
• If the yield curve is to rise as one moves to longer
maturities
– A longer maturity results in the inclusion of a
new forward rate that is higher than the
average of the previously observed rates
– Reason:
• Higher expectations for forward rates or
• Liquidity premium
Figure 15.5 Price Volatility of Long-Term
Treasury Bonds
Figure 15.6 Term Spread: Yields on 10-Year
Versus 90-Day Treasury Securities