Principles
of
Corporate
Finance
Chapter 4
Valuing Bonds
Ninth Edition
Slides by
Matthew Will
McGraw Hill/Irwin
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
4- 2
Topics Covered
Using The Present Value Formula to Value
Bonds
How Bond Prices Vary With Interest Rates
The Term Structure and YTM
Explaining the Term Structure
Real and Nominal Rates of Interest
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4- 3
Valuing a Bond
1,000  C N
C1
C2
PV 


...

1
2
N
(1  r )
(1  r )
(1  r )
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4- 4
Valuing a Bond
Example
 If today is October 1, 2007, what is the value of the
following bond? An IBM Bond pays $115 every
September 30 for 5 years. In September 2012 it pays an
additional $1000 and retires the bond. The bond is rated
AAA (WSJ AAA YTM is 7.5%)
Cash Flows
Sept 08 09 10 11 12
115
115 115 115 1115
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Valuing a Bond
Example continued
 If today is October 1, 2007, what is the value of the following bond? An
IBM Bond pays $115 every September 30 for 5 years. In September
2012 it pays an additional $1000 and retires the bond. The bond is rated
AAA (WSJ AAA YTM is 7.5%)
115
115
115
115
1,115
PV 




2
3
4
1.075 1.075 1.075 1.075 1.0755
 $1,161.84
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Valuing a Bond
Example - Germany
 In July 2006 you purchase 100 Euros of bonds in Germany which pay a
5% coupon every year. If the bond matures in 2012 and the YTM is
3.8%, what is the value of the bond?
5
5
5
5
5
105
PV 





2
3
4
5
1.038 1.038 1.038 1.038 1.038 1.0386
 106.33 Euros
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Valuing a Bond
Another Example - Japan
 In July 2006 you purchase 200 Yen of bonds in Japan which pay a 8%
coupon every year. If the bond matures in 2011 and the YTM is 4.5%,
what is the value of the bond?
16
16
16
16
216
PV 




2
3
4
1.045 1.045 1.045 1.045 1.0455
 243.57 Yen
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Valuing a Bond
Example - USA
 In July 2006 you purchase a 3 year US Government bond. The bond has
an annual coupon rate of 4%, paid semi-annually. If investors demand a
2.48% return on 6 month investments, what is the price of the bond?
20
20
20
20
20
1020
PV 





2
3
4
5
1.0248 1.0248 1.0248 1.0248 1.0248 1.02486
 $973.54
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Valuing a Bond
Example continued - USA
 Take the same 3 year US Government bond. The bond has an annual
coupon rate of 4%, paid semi-annually. If investors demand a 1.50%
return on 6 month investments, what is the new price of the bond?
PV 
20
20
20
20
20
1020





1.015 1.0152 1.0153 1.0154 1.0155 1.0156
 $1,028.49
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Bond Prices and Yields
115.00
110.00
Bond Price, %
105.00
100.00
95.00
90.00
85.00
10
9
8
7
6
5
4
3
2
1
0
80.00
Interest Rates, %
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Duration Calculation
Year
Ct
PV(Ct) at 5.0%
Proportion of Total Value
[PV(Ct)/V]
1
2
3
100
100
1100
95.24
90.7
950.22
V = 1136.16
0.084
0.08
0.836
1
McGraw Hill/Irwin
Proportion of Total
Value Time
0.084
0.16
2.509
Duration= 2.753 years
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Duration
Example (Bond 1)
Calculate the duration of our 6 7/8 % bond @ 4.9 % YTM
Year CF
PV@YTM
% of Total PV% x Year
1
68.75 65.54
.060
0.060
2
68.75 62.48
.058
0.115
3
68.75 59.56
.055
0.165
4
68.75 56.78
.052
0.209
5
68.75 841.39
.775
3.875
1.00
Duration 4.424
1085.74
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Duration
Example (Bond 2)
Given a 5 year, 9.0%, $1000 bond, with a 8.5% YTM, what is
this bond’s duration?
Year CF
PV@YTM
1
90
82.95
.081
0.081
2
90
76.45
.075
0.150
3
90
70.46
.069
0.207
4
90
64.94
.064
0.256
5
1090
724.90
.711
3.555
1019.70
1.00 Duration= 4.249
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% of Total PV% x Year
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Bond Price, percent
Duration & Bond Prices
Interest rate, percent
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Maturity and Prices
250.00
Bond Price, %
200.00
3 yr 4% bond
30 yr 4% bond
150.00
100.00
50.00
0.00
0
1
2
3
4
5
6
7
8
9
10
Interest Rates, %
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Term Structure
YTM (r)
1981
1987 & Normal
1976
1
5
10
20
30
Year
Spot Rate - The actual interest rate today (t=0)
Forward Rate - The interest rate, fixed today, on a loan made
in the future at a fixed time.
Future Rate - The spot rate that is expected in the future
Yield To Maturity (YTM) - The IRR on an interest bearing
instrument
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Yield To Maturity
All interest bearing instruments are priced
to fit the term structure
This is accomplished by modifying the asset
price
The modified price creates a New Yield,
which fits the Term Structure
The new yield is called the Yield To
Maturity (YTM)
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Yield Curve
U.S. Treasury Strip Spot Rates as of
June 2006
Spot rate %
5.5
5.3
5.1
4.9
4.7
4.5
4.3
4.1
3.9
3.7
22
-S
ep
-1
7
ay
-1
6
10
-M
27
-D
ec
-1
4
14
-A
ug
-1
3
1Ap
r-1
2
18
-N
ov
-1
0
6Ju
l- 0
9
22
-F
eb
-0
8
10
-O
ct
-0
6
3.5
Maturity
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Yield to Maturity
Example
A $1000 treasury bond expires in 5 years.
It pays a coupon rate of 10.5%. If the
market price of this bond is 107.88, what is
the YTM?
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Yield to Maturity
Example
A $1000 treasury bond expires in 5 years. It pays
a coupon rate of 10.5%. If the market price of this
bond is 107.88, what is the YTM?
C0
-1078.80
C1
C2
C3
C4
C5
105
105
105
105
1105
Calculate IRR = 8.5%
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Term Structure
What Determines the Shape of the TS?
1 - Unbiased Expectations Theory
2 - Liquidity Premium Theory
3 - Market Segmentation Hypothesis
Term Structure & Capital Budgeting
 CF should be discounted using Term Structure info
 Since the spot rate incorporates all forward rates, then you
should use the spot rate that equals the term of your project.
 If you believe in other theories take advantage of the
arbitrage.
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Spot/Forward rates
example
1000
(1+R3)3
McGraw Hill/Irwin
=
1000
(1+f1)(1+f2)(1+f3)
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Spot/Forward rates
Forward Rate Computations
(1+ rn)n = (1+ r1)(1+f2)(1+f3)....(1+fn)
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Spot/Forward rates
Example
What is the 3rd year forward rate?
2 year zero treasury YTM = 8.995
3 year zero treasury YTM = 9.660
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Spot/Forward rates
Example
What is the 3rd year forward rate?
2 year zero treasury YTM = 8.995
3 year zero treasury YTM = 9.660
Answer
FV of principal @ YTM
2 yr 1000 x (1.08995)2 = 1187.99
3 yr 1000 x (1.09660)3 = 1318.70
IRR of (FV1318.70 & PV=1187.99) = 11%
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Spot/Forward rates
Example
Two years from now, you intend to begin a
project that will last for 5 years. What
discount rate should be used when
evaluating the project?
2 year spot rate = 5%
7 year spot rate = 7.05%
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Spot/Forward rates
coupons paying bonds to derive rates
Bond Value =
C1 +
(1+r)
C2
(1+r)2
Bond Value =
C1 +
(1+R1)
C2
(1+f1)(1+f2)
d1 =
C1
(1+R1)
McGraw Hill/Irwin
d2 =
C2
(1+f1)(1+f2)
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Spot/Forward rates
example
8% 2 yr bond YTM = 9.43%
10% 2 yr bond YTM = 9.43%
What is the forward rate?
Step 1
value bonds
8% = 975
10%= 1010
Step 2
975 = 80d1 + 1080 d2 -------> solve for d1
1010 =100d1 + 1100d2 -------> insert d1 & solve for d2
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Spot/Forward rates
example continued
Step 3 solve algebraic equations
d1 = [975-(1080)d2] / 80
insert d1 & solve = d2 = .8350
insert d2 and solve for d1 = d1 = .9150
Step 4
Insert d1 & d2 and Solve for f1 & f2.
.9150 = 1/(1+f1)
.8350 = 1 / (1.0929)(1+f2)
f1 = 9.29%
f2 = 9.58%
PROOF
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Debt & Interest Rates
Classical Theory of Interest Rates
(Economics)
developed by Irving Fisher
Nominal Interest Rate = The rate you actually
pay when you borrow money
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Debt & Interest Rates
Classical Theory of Interest Rates (Economics)
 developed by Irving Fisher
Nominal Interest Rate = The rate you actually pay when you
borrow money
Real Interest Rate = The theoretical rate you pay when you
borrow money, as determined by supply and demand
r
Supply
Real r
Demand
$ Qty
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Debt & Interest Rates
Nominal r = Real r + expected inflation (approximation)
Real r is theoretically somewhat stable
Inflation is a large variable
Q: Why do we care?
A: This theory allows us to understand the Term Structure of
Interest Rates.
Q: So What?
A: The Term Structure tells us the cost of debt.
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Debt & Interest Rates
Actual formula
1  rnominal  (1  rreal )  (1  i)
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Inflation
Annual Inflation, %
Annual U.S. Inflation Rates from 1900 - 2006
00
19
McGraw Hill/Irwin
12
19
24
19
36
19
48
19
60
19
72
19
84
19
96
19
06
20
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Sw
it
Ne zer
th lan
er d
la
nd
s
US
Ca A
na
Sw da
ed
No en
r
Au wa
st y
De ra lia
nm
ar
k
UK
So Ire
G
ut lan
er
h
d
m
Af
an
ric
A
y
(e ve a
r
x
19 age
22
/
Be 2 3)
lg
iu
m
Sp
a
Fr in
an
ce
Ja
pa
n
Ita
ly
Average Inflation, %
4- 35
Global Inflation Rates
Averages from 1900-2006
12.00
10.00
McGraw Hill/Irwin
8.00
6.00
4.00
2.00
0.00
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UK Bond Yields
20
18
16
10 year nominal interest rate
Interest rate, %
14
12
10
8
10 year real interest rate
6
4
2
McGraw Hill/Irwin
Jan-06
Jan-05
Jan-04
Jan-03
Jan-02
Jan-01
Jan-00
Jan-99
Jan-98
Jan-97
Jan-96
Jan-95
Jan-94
Jan-93
Jan-92
Jan-91
Jan-90
Jan-89
Jan-88
Jan-87
Jan-86
Jan-85
Jan-84
0
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T-Bills vs. Inflation (’53-’06)
United States
16.00
14.00
T Bill Return
12.00
Inflation
10.00
%
8.00
6.00
4.00
2.00
2005
2003
2001
1999
1997
1995
1993
1991
1989
1987
1985
1983
1981
1979
1977
1975
1973
1971
1969
1967
1965
1963
1961
1959
1957
1955
1953
0.00
-2.00
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T-Bills vs. Inflation (’53-’06)
Japan
25.00
20.00
T Bill Return
Inflation
%
15.00
10.00
5.00
2005
2003
2001
1999
1997
1995
1993
1991
1989
1987
1985
1983
1981
1979
1977
1975
1973
1971
1969
1967
1965
1963
1961
1959
1957
1955
1953
0.00
-5.00
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4- 39
T-Bills vs. Inflation (’53-’06)
Germany
12.00
10.00
T Bill Return
8.00
Inflation
%
6.00
4.00
2.00
2005
2003
2001
1999
1997
1995
1993
1991
1989
1987
1985
1983
1981
1979
1977
1975
1973
1971
1969
1967
1965
1963
1961
1959
1957
1955
1953
0.00
-2.00
-4.00
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Web Resources
Click to access web sites
Internet connection required
www.finpipe.com
www.investinginbonds.com
www.investorguide.com
http://finance.yahoo.com
http://money.cnn.com/markets/bondcenter
www.federalreserve.gov
www.stls.frb.org
www.ustreas.gov
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