CHAPTER 15
The Term Structure of Interest
Rates
INVESTMENTS | BODIE, KANE, MARCUS
McGraw-Hill/Irwin
Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.
15-2
Overview of Term Structure
• The yield curve is a graph that
displays the relationship between
yield and maturity.
• Information on expected future
short term rates can be implied
from the yield curve.
INVESTMENTS | BODIE, KANE, MARCUS
15-3
Figure 15.1 Treasury Yield Curves
See
Many other interesting links, for example:
Treasury.gov
stockcharts.com
INVESTMENTS | BODIE, KANE, MARCUS
15-4
Bond Pricing
• Yields on different maturity bonds are not all
equal – there is a term structure.
• We need to consider each bond cash
flow as a stand-alone zero-coupon bond.
• The value of the bond should be the sum
of the values of its parts.
• Bond stripping and bond reconstitution
offer opportunities for arbitrage.
INVESTMENTS | BODIE, KANE, MARCUS
15-5
Table 15.1 Prices and Yields to Maturities
on Zero-Coupon Bonds ($1,000 Face Value)
These prices are in the form:
CashFlowt
Price 
t
1  ytm 
INVESTMENTS | BODIE, KANE, MARCUS
15-6
Example 15.1 Valuing Coupon Bonds
• Value a 3 year, 10% coupon bond using
discount rates from Table 15.1:
$100 $100 $1100
Price 


2
3
1.05 1.06
1.07
• Price = $1082.17
• YTM = 6.88%
• 6.88% is less than the 3-year rate of 7%.
INVESTMENTS | BODIE, KANE, MARCUS
15-7
Two Types of Yield Curves
Pure Yield Curve
On-the-run Yield Curve
• The pure yield curve • The on-the-run yield
uses stripped or zero
curve uses recently
coupon Treasuries.
issued coupon bonds
selling at or near par.
• The pure yield curve
may differ
• The financial press
significantly from the
typically publishes onon-the-run yield
the-run yield curves.
curve.
INVESTMENTS | BODIE, KANE, MARCUS
15-8
Yield Curve Under Certainty
• Suppose you want to invest for 2 years:
– Buy and hold a 2-year zero
-or– Rollover a series of 1-year bonds
• Equilibrium (or no arbitrage) requires
that both strategies provide the same
return.
1+r
1+r
1
2
(1+y2)2
INVESTMENTS | BODIE, KANE, MARCUS
15-9
Figure 15.2 Two 2-Year Investment
Programs
(1+y2)2
1+r1
1+r2
INVESTMENTS | BODIE, KANE, MARCUS
15-10
Yield Curve Under Certainty
• Buy and hold vs. rollover:
1  y2 
2
 1  r1  1  r2 
1  y2  1  r1  1  r2 
1
2
1+r1
1+r2
(1+y2)2
• Next year’s 1-year rate (r2) is just
enough to make rolling over a series of
1-year bonds equal to investing in the 2year bond.
INVESTMENTS | BODIE, KANE, MARCUS
15-11
Spot Rates vs. Short Rates
• Spot rate – the rate that prevails today for
a given maturity
• Short rate – the rate for a given maturity
(e.g. one year) at different points in time.
• A spot rate is the geometric average of its
component short rates.
yn  1  r1  1  r2  ...  1  rn  n  1
1
INVESTMENTS | BODIE, KANE, MARCUS
15-12
Short Rates and
Yield Curve Slope
• When next year’s
short rate, r2 , is
greater than this
year’s short rate,
r1, the yield curve
slopes up.
– May indicate
market expects
rates to rise.
• When next year’s
short rate, r2 , is
less than this
year’s short rate,
r1, the yield curve
slopes down.
– May indicate
market expects
rates to fall.
INVESTMENTS | BODIE, KANE, MARCUS
15-13
Figure 15.3 Short Rates versus Spot Rates
INVESTMENTS | BODIE, KANE, MARCUS
15-14
Forward Rates from Observed Rates
(1  yn )
(1  f n ) 
n 1
(1  yn 1 )
n
fn = one-year forward rate for period n
yn = yield for a security with a maturity of n
(1  yn 1 )
n 1
(1  f n )  (1  yn )
n
1+fn
(1+yn-1)n-1
(1+yn)n
INVESTMENTS | BODIE, KANE, MARCUS
15-15
Example 15.4 Forward Rates
• The forward interest rate is a forecast of a
future short rate implied by the market.
• Example: compute forward rate for year 4:
– rate for 4-year maturity = 8%
– rate for 3-year maturity = 7%

1  y4 
1 f4 
3
1  y3 
4
4
1.08

 1.1106
3
1.07
f 4  11.06%
INVESTMENTS | BODIE, KANE, MARCUS
15-16
Interest Rate Uncertainty
• Suppose that today’s rate is 5% and the
expected short rate for the following
year is E(r2) = 6%. The value of a 2-year
zero is:
$1000
1.051.06
 $898.47
• The value of a 1-year zero is:
$1000
 $952.38
1.05
INVESTMENTS | BODIE, KANE, MARCUS
15-17
Interest Rate Uncertainty
• The investor wants to invest for 1 year.
– Buy the 2-year bond today and plan to
sell it at the end of the first year for
$1000/1.06 =$943.40.
or:
– Buy the 1-year bond today and hold to
maturity.
INVESTMENTS | BODIE, KANE, MARCUS
15-18
Interest Rate Uncertainty
• What if next year’s interest rate is
more (or less) than 6%?
–The actual return on the 2-year
bond is uncertain!
INVESTMENTS | BODIE, KANE, MARCUS
15-19
Interest Rate Uncertainty
• Investors require a risk premium to
hold a longer-term bond.
• This liquidity premium
compensates short-term investors
for the uncertainty about future
prices.
INVESTMENTS | BODIE, KANE, MARCUS
15-20
Theories of Term Structure
• Expectations
–Forward rates come from market
consensus
• Liquidity Preference
–Upward bias over expectations due
to premium the market requires
INVESTMENTS | BODIE, KANE, MARCUS
15-21
Expectations Theory
• Observed long-term rate is a function
of today’s short-term rate and
expected future short-term rates.
(1  y2 )  (1  y1 )(1  f 2 )
2
(1  y2 )  (1  y1 )(1  Er2 )
2
• fn = E(rn) and liquidity premiums are
zero.
INVESTMENTS | BODIE, KANE, MARCUS
15-22
Liquidity Premium Theory
• Long-term bonds carry more risk;
therefore, fn generally exceeds E(rn)
• The excess of fn over E(rn) is the
liquidity premium
• The yield curve has an upward bias
built into the long-term rates because
of the liquidity premium
INVESTMENTS | BODIE, KANE, MARCUS
15-23
Figure 15.4 Yield Curves - A
INVESTMENTS | BODIE, KANE, MARCUS
15-24
Figure 15.4 Yield Curves - B
INVESTMENTS | BODIE, KANE, MARCUS
15-25
Figure 15.4 Yield Curves - C
INVESTMENTS | BODIE, KANE, MARCUS
15-26
Figure 15.4 Yield Curves - D
INVESTMENTS | BODIE, KANE, MARCUS
15-27
Interpreting the Term Structure
• The yield curve reflects expectations of
future interest rates.
• The forecasts of future rates are clouded by
other factors, such as liquidity premiums.
• An upward sloping curve could indicate:
– Rates are expected to rise
– And/or
– Investors require large liquidity premiums
to hold long term bonds.
INVESTMENTS | BODIE, KANE, MARCUS
15-28
Interpreting the Term Structure
• The yield curve is a good predictor of the
business cycle.
– Long term rates tend to rise in anticipation
of economic expansion.
– Inverted yield curve may indicate that
interest rates are expected to fall and
signal a recession.
INVESTMENTS | BODIE, KANE, MARCUS
15-29
Figure 15.6 Term Spread: Yields on 10-year
vs. 90-day Treasury Securities
INVESTMENTS | BODIE, KANE, MARCUS
15-30
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
INVESTMENTS | BODIE, KANE, MARCUS
15-31
Figure 15.7 Engineering a Synthetic
Forward Loan
INVESTMENTS | BODIE, KANE, MARCUS