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INVESTMENTS | BODIE, KANE, MARCUS
Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin
CHAPTER 15
The Term Structure of Interest
Rates
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15-2
• 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.
Overview of Term Structure
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15-3
Figure 15.1 Treasury Yield Curves
See
Treasury.gov Many other interesting links, for example:
stockcharts.com
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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.
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15-5
Table 15.1 Prices and Yields to Maturities on Zero-Coupon Bonds ($1,000 Face Value)
tt
ytm
CashFlow
1Price
These prices are in the form:
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15-6
Example 15.1 Valuing Coupon Bonds
• Value a 3 year, 10% coupon bond using
discount rates from Table 15.1:
• Price = $1082.17
• YTM = 6.88%
• 6.88% is less than the 3-year rate of 7%.
32 07.1
1100$
06.1
100$
05.1
100$Price
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15-7
Two Types of Yield Curves
Pure Yield Curve
• The pure yield curve
uses stripped or zero
coupon Treasuries.
• The pure yield curve
may differ
significantly from the
on-the-run yield
curve.
On-the-run Yield Curve
• The on-the-run yield
curve uses recently
issued coupon bonds
selling at or near par.
• The financial press
typically publishes on-
the-run yield curves.
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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+r1 1+r2
(1+y2)2
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Figure 15.2 Two 2-Year Investment Programs
1+r1 1+r2
(1+y2)2
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Yield Curve Under Certainty
• Buy and hold vs. rollover:
• 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 2-
year bond.
21
2
2 111 rry
21
212 111 rry
1+r1 1+r2
(1+y2)2
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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.
11...111
21 nnn rrry
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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.
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15-13
Figure 15.3 Short Rates versus Spot Rates
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1
1)1(
)1()1(
n
n
n
nn
y
yf
fn = one-year forward rate for period n
yn = yield for a security with a maturity of n
)1()1()1( 1
1 n
n
n
n
n fyy
Forward Rates from Observed Rates
(1+yn-1)n-1
1+fn
(1+yn)n
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15-15
Example 15.4 Forward Rates
• The forward interest rate is a forecast of a
future short rate.
• Example: compute forward rate for year 4:
– rate for 4-year maturity = 8%
– rate for 3-year maturity = 7%
1106.107.1
08.1
1
11
3
4
3
3
4
44
y
yf
%.f 06114