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Chapter 4 PrinciplesPrinciples
ofof
CorporateCorporate
FinanceFinance
Ninth Edition
Valuing Bonds
Slides by
Matthew Will
Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved
McGraw Hill/Irwin
4- 2
Topics Covered
Using The Present Value Formula to Value Bonds
How Bond Prices Vary With Interest RatesThe Term Structure and YTMExplaining the Term StructureReal and Nominal Rates of Interest
4- 3
Valuing a Bond
NN
r
C
r
C
r
CPV
)1(
000,1...
)1()1( 22
11
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 0809 10 11 12
115 115 115 115 1115
4- 5
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 $1000 and retires the bond. The bond is rated AAA (WSJ AAA YTM is 7.5%)
84.161,1$
075.1
115,1
075.1
115
075.1
115
075.1
115
075.1
1155432
PV
4- 6
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?
Euros 33.106
038.1
105
038.1
5
038.1
5
038.1
5
038.1
5
038.1
565432
PV
4- 7
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?
Yen 57.243
045.1
216
045.1
16
045.1
16
045.1
16
045.1
165432
PV
4- 8
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?
54.973$
0248.1
1020
0248.1
20
0248.1
20
0248.1
20
0248.1
20
0248.1
2065432
PV
4- 9
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?
49.028,1$
015.1
1020
015.1
20
015.1
20
015.1
20
015.1
20
015.1
2065432
PV
4- 10
Bond Prices and Yields
Interest Rates, %
Bon
d P
rice
, %
80.00
85.00
90.00
95.00
100.00
105.00
110.00
115.00
4- 11
Duration Calculation
Year Ct PV(Ct) at 5.0%Proportion of Total Value
[PV(Ct)/V]Proportion of Total
Value Time
1 100 95.24 0.084 0.0842 100 90.7 0.08 0.163 1100 950.22 0.836 2.509
V = 1136.16 1 Duration= 2.753 years
4- 12
Duration
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
1085.74 1.00 Duration 4.424
Example (Bond 1)
Calculate the duration of our 6 7/8 % bond @ 4.9 % YTM
4- 13
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
Example (Bond 2)Given a 5 year, 9.0%, $1000 bond, with a 8.5% YTM, what is this bond’s
duration?
4- 14
Duration & Bond Prices
Interest rate, percent
Bon
d Pr
ice,
per
cent
4- 15
Maturity and Prices
0.00
50.00
100.00
150.00
200.00
250.00
0 1 2 3 4 5 6 7 8 9 10
3 yr 4% bond
30 yr 4% bond
Interest Rates, %
Bon
d P
rice
, %
4- 16
Term Structure
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
YTM (r)
Year
1981
1987 & Normal
1976
1 5 10 20 30
4- 17
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)
4- 18
Yield Curve
Spot rate %
3.5
3.7
3.9
4.1
4.3
4.5
4.7
4.9
5.1
5.3
5.5
10-O
ct-06
22-F
eb-0
8
6-Ju
l-09
18-N
ov-1
0
1-Apr-1
2
14-A
ug-1
3
27-D
ec-1
4
10-M
ay-1
6
22-S
ep-1
7
Maturity
U.S. Treasury Strip Spot Rates as of June 2006
4- 19
Yield to Maturity
ExampleA $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?
4- 20
Yield to Maturity
ExampleA $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 C1 C2 C3 C4 C5
-1078.80 105 105 105 105 1105
Calculate IRR = 8.5%
4- 21
Term Structure
What Determines the Shape of the TS?
1 - Unbiased Expectations Theory
2 - Liquidity Premium Theory
3 - Market Segmentation Hypothesis
Term Structure & Capital BudgetingCF should be discounted using Term Structure infoSince 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.
4- 22
example
1000 = 1000
(1+R3)3 (1+f1)(1+f2)(1+f3)
Spot/Forward rates
4- 23
Forward Rate Computations
(1+ rn)n = (1+ r1)(1+f2)(1+f3)....(1+fn)
Spot/Forward rates
4- 24
Example
What is the 3rd year forward rate?
2 year zero treasury YTM = 8.995
3 year zero treasury YTM = 9.660
Spot/Forward rates
4- 25
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%
Spot/Forward rates
4- 26
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%
Spot/Forward rates
4- 27
coupons paying bonds to derive rates
Spot/Forward rates
Bond Value = C1 + C2
(1+r) (1+r)2
Bond Value = C1 + C2
(1+R1) (1+f1)(1+f2)
d1 = C1 d2 = C2
(1+R1) (1+f1)(1+f2)
4- 28
example 8% 2 yr bond YTM = 9.43%10% 2 yr bond YTM = 9.43%What is the forward rate?
Step 1value bonds 8% = 975 10%= 1010
Step 2 975 = 80d1 + 1080 d2 -------> solve for d11010 =100d1 + 1100d2 -------> insert d1 & solve for d2
Spot/Forward rates
4- 29
example continuedStep 3 solve algebraic equationsd1 = [975-(1080)d2] / 80insert d1 & solve = d2 = .8350insert 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
Spot/Forward rates
4- 30
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
4- 31
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
Supply
Demand
$ Qty
r
Real r
4- 32
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.
4- 33
Debt & Interest Rates
Actual formula
)1()1(1 realnominal irr
4- 34
Inflation
1900
1912
1924
1936
1948
1960
1972
1984
1996
2006
Ann
ual I
nfla
tion,
%Annual U.S. Inflation Rates from 1900 - 2006
4- 35
Global Inflation Rates
0.00
2.00
4.00
6.00
8.00
10.00
12.00
Ave
rag
e In
flat
ion
, %
Switzer
land
Nether
lands
USA
Canada
Sweden
Norway
Austra
lia
Denmar
kUK
Irelan
d
South
Afri
ca
Avera
ge
Germ
any (
ex 192
2/23)
Belgium
Spain
Franc
e
Japa
nIta
ly
Averages from 1900-2006
4- 36
UK Bond Yields
0
2
4
6
8
10
12
14
16
18
20
Jan-
84
Jan-
85
Jan-
86
Jan-
87
Jan-
88
Jan-
89
Jan-
90
Jan-
91
Jan-
92
Jan-
93
Jan-
94
Jan-
95
Jan-
96
Jan-
97
Jan-
98
Jan-
99
Jan-
00
Jan-
01
Jan-
02
Jan-
03
Jan-
04
Jan-
05
Jan-
06
Inte
rest
rat
e, %
10 year nominal interest rate
10 year real interest rate
4- 37
T-Bills vs. Inflation (’53-’06)%
United States
4- 38
T-Bills vs. Inflation (’53-’06)%
Japan
4- 39
T-Bills vs. Inflation (’53-’06)%
Germany
4- 40
Web Resources
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
Click to access web sitesClick to access web sites
Internet connection requiredInternet connection required
