CHAPTER 15
The Term
Structure of
Interest Rates
Investments, 8th edition
Bodie, Kane and Marcus
Slides by Susan Hine
McGraw-Hill/Irwin
Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.
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
15-2
Figure 15.1 Treasury Yield Curves
15-3
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
15-4
Table 15.1 Yields and Prices to
Maturities on Zero-Coupon Bonds
($1,000 Face Value)
15-5
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
15-6
Figure 15.2 Two 2-Year Investment
Programs
15-7
Figure 15.3 Short Rates versus Spot
Rates
15-8
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 )
15-9
Example 15.4 Forward Rates
4 yr = 8.00%
3yr = 7.00%
fn = ?
(1.08)4 = (1.07)3 (1+fn)
(1.3605) / (1.2250) = (1+fn)
fn = .1106 or 11.06%
15-10
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
15-11
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
15-12
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
15-13
Interest Rate Uncertainty Continued
• Investors require a risk premium to hold a
longer-term bond
• This liquidity premium compensates shortterm investors for the uncertainty about future
prices
15-14
Theories of Term Structure
• Expectations
• Liquidity Preference
– Upward bias over expectations
15-15
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
15-16
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 shortterm rates
15-17
Figure 15.4 Yield Curves
15-18
Figure 15.4 Yield Curves (Concluded)
15-19
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
15-20
Figure 15.5 Price Volatility of Long-Term
Treasury Bonds
15-21
Figure 15.6 Term Spread: Yields on 10Year Versus 90-Day Treasury Securities
15-22
Forward Rates as Forward Contracts
• In general, forward rates will not equal the
eventually realized short rate
– Still an important consideration when trying to
make decisions :
• Locking in loan rates
15-23
Figure 15.7 Engineering a Synthetic
Forward Loan
15-24