Chapter 3
Principles of
Corporate Finance
Tenth Edition
Valuing Bonds
Slides by
Matthew Will
McGraw-Hill/Irwin
Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.
Topics Covered
Using The Present Value Formula to Value
Bonds
How Bond Prices Vary With Interest Rates
The Term Structure of Interest Rates
Explaining the Term Structure
Real and Nominal Rates of Interest
Corporate Bonds and the Risk of Default
3- 2
Valuing a Bond
3- 3
1,000  C N
C1
C2
PV 

 ... 
1
2
N
(1  r )
(1  r )
(1  r )
Valuing a Bond
Example
 If today is October 1, 2010, what is the value of the
following bond? An IBM Bond pays $115 every
September 30 for 5 years. In September 2015 it pays an
additional $1000 and retires the bond. The bond is rated
AAA (WSJ AAA YTM is 7.5%)
Cash Flows
Sept 11
12 13 14 15
115
115 115 115 1115
3- 4
Valuing a Bond
Example continued
 If today is October 1, 2010, what is the value of the following bond? An
IBM Bond pays $115 every September 30 for 5 years. In September
2015 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
3- 5
Valuing a Bond
Example - France
 In December 2008 you purchase 100 Euros of bonds in France which
pay a 8.5% coupon every year. If the bond matures in 2012 and the
YTM is 3.0%, what is the value of the bond?
8.5
8.5
8.5
108.5
PV 



2
3
1.03 1.03 1.03 1.034
 120.44 Euros
3- 6
Valuing a Bond
Another Example - Japan
 In July 2010 you purchase 200 Yen of bonds in Japan which pay a 8%
coupon every year. If the bond matures in 2015 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
3- 7
Valuing a Bond
3- 8
Example - USA
 In February 2009 you purchase a 3 year US Government bond. The
bond has an annual coupon rate of 4.875%, paid semi-annually. If
investors demand a 0.6003% semiannual return, what is the price of the
bond?
PV 
24.375
24.375
24.375
24.375
24.375
1024.375





1.006003 1.0060032 1.0060033 1.0060034 1.0060035 1.0060036
 $1,107.95
Valuing a Bond
Example continued - USA
 Take the same 3 year US Government bond. If investors demand a 4.0%
semiannual return, what is the new price of the bond?
PV 
24.375 24.375 24.375 24.375 24.375 1024.375





2
3
4
5
1.04
1.04 1.04 1.04 1.04 1.046
 $918.09
3- 9
Year
2008
2002
1996
1990
1984
1978
1972
1966
1960
1954
1948
1942
1936
1930
1924
1918
1912
1906
1900
Yield , %
Interest Rate on 10yr Treasuries
16
14
12
10
8
6
4
2
0
3- 10
3- 11
Bond Prices and Yields
115.00
110.00
100.00
95.00
90.00
85.00
Interest Rates, %
10
9
8
7
6
5
4
3
2
1
80.00
0
Bond Price, %
105.00
3- 12
Maturity and Prices
3,000
Bond Price, ($)
2,500
30 yr bond
When the interest rate
equals the 5% coupon,
both bonds sell for
face value
2,000
1,500
1,000
3 yr bond
500
0
0
1
2
3
4
5
6
7
8
9
10
Interest Rates, %
11
12
13
14
Duration Formula
Duration 
1 PV (C1 ) 2  PV (C2 ) 3  PV (C3 )
T  PV (CT )


 ... 
PV
PV
PV
PV
duration
Modified Duration  volatility (%) 
1  yield
3- 13
3- 14
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
Proportion of Total
Value Time
0.084
0.16
2.509
Duration= 2.753 years
3- 15
Duration
Example (Bond 1)
Calculate the duration of our 6 7/8 % bond @ 4.9 % YTM
Year
CF
1
PV@YTM
% of Total PV
% x Year
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
1085.74
1.00
Duration 4.424
3- 16
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
% of Total PV
% x Year
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
Bond Price, percent
Duration & Bond Prices
Interest rate, percent
3- 17
Interest Rates
 Short- and long-term interest rates do not always move in parallel.
Between September 1992 and April 2000 U.S. short-term rates rose
sharply while long term rates declined.
3- 18
Term Structure of Interest Rates
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
3- 19
Maturity
2037 Aug 15
2035 Aug 15
2033 Aug 15
2031 Aug 15
2029 Aug 15
2027 Aug 15
2025 Aug 15
2023 Aug 15
2021 Aug 15
2019 Aug 15
2017 Aug 15
2015 Aug 15
2013 Aug 15
2011 Aug 15
2009 Aug 15
Spot rates (%)
Yield Curve
3- 20
U.S. Treasury Strip Spot Rates as of February 2009
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
Law of One Price
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)
3- 21
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?
3- 22
3- 23
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
-1078.80 105
C1
C2
C3
C4
105
105
105
1105
Calculate IRR = 8.5%
C5
Term Structure
What Determines the Shape of the Term Structure?
Expectations Theory
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.
3- 24
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
3- 25
3- 26
Inflation Rates
Annual rates of inflation in the United States from 1900–2008.
25
15
10
5
-5
-10
-15
2008
2002
1996
1990
1984
1978
1972
1966
1960
1954
1948
1942
1936
1930
1924
1918
1912
1906
0
1900
Annual Inflation (%)
20
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, %
Global Inflation Rates
Averages from 1900-2006
12.00
10.00
8.00
6.00
4.00
2.00
0.00
3- 27
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.
3- 28
Debt & Interest Rates
Actual formula
1  rnominal  (1  rreal )  (1  i)
3- 29
1-Jan-09
1-Oct-07
1-Jul-06
1-Apr-05
1-Jan-04
1-Oct-02
1-Jul-01
1-Apr-00
1-Jan-99
1-Oct-97
1-Jul-96
1-Apr-95
16
1-Jan-94
1-Oct-92
1-Jul-91
1-Apr-90
6
1-Jan-89
1-Oct-87
1-Jul-86
1-Apr-85
1-Jan-84
Interest rate (%)
UK Bond Yields
3- 30
20
18
10 year nominal interest rate
14
12
10
8
10 year real interest rate
4
2
0
Govt. Bills vs. Inflation (’53-’08)
United Kingdom
30.00
25.00
Inflation
20.00
%
15.00
T-Bill Returns
10.00
5.00
0.00
3- 31
Govt. Bills vs. Inflation (’53-’08)
United States
16.00
14.00
12.00
Inflation
10.00
%
8.00
6.00
4.00
2.00
0.00
-2.00
T-Bill Returns
3- 32
Govt. Bills vs. Inflation (’53-’08)
Germany
12.00
10.00
T-Bill Returns
8.00
%
6.00
4.00
2.00
0.00
-2.00
-4.00
Inflation
3- 33
Bond Ratings
 Key to bond ratings. The highest-quality bonds are rated triple A.
Bonds rated triple B or above are investment grade. Lower-rated bonds
are called high-yield, or junk, bonds.
3- 34
-1
Years
0.67
1.38
1.9
1.18
1.08
1.21
0.75
0.83
-0.01
-0.05
0.25
0.79
4
0.58
0.44
0.42
0.22
0.5
0.22
0.24
0.4
Yield spread between corporate
and government bonds, %
Yield Spread
3- 35
Yield spreads between corporate and 10-year
Treasury bonds.
7
6
5
Spread on Baa bonds
3
2
1
Spread on Aaa bonds
0
Prices and Yields
 Prices and yields of a sample of corporate bonds,
December 2008.
Source: Bond transactions reported on FINRA’s TRACE service:
http://cxa.marketwatch.com/finra/BondCenter
3- 36
Web Resources
Click to access web sites
Internet connection required
http://cxa.marketwatch.com/finra/BondCenter
www.ft.com
www.smartmoney.com
www.wsj.com
www.finpipe.com
www.investinginbonds.com
www.investorguide.com
http://money.cnn.com/markets/bondcenter
www.federalreserve.gov
www.stls.frb.org
www.ustreas.gov
3- 37