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Chapter 8: Valuation of
Known Cash Flows:
Bonds
Objectives
Value contracts with a stream
of cash flows
Change of bond prices &
Yields across time
Copyright © Prentice Hall Inc. 2000. Author: Nick Bagley, bdellaSoft, Inc.
1
Contents
1.
2.
3.
4.
5.
6.
Using Present Value Formulas to Value
Known Cash Flows
The Basic Building Blocks: Pure
Discount Bonds
Coupon Bonds, Current Yield, and Yieldto-Maturity
Reading Bond Listings
Why Yields for the same Maturity may
differ
The Behavior of Bond Prices Over Time
2
Valuation of fixed income security:
single risk-free interest rate

Write the PV of the fixed income
security as the sum of terms
j


1



PV    pmt j * 


 1  i  
j 1 
n
1
2
 1 
 1 
 1 
 pmt1 * 
  pmt2 * 
  ...  pmtn 1 * 

1 i 
1 i 
1 i 
n 1
 1 
 pmtn * 

1 i 
3
n
US Treasury Yiled Curve, Jan 97
7.50
Annualized Yield (%)
7.00
6.50
6.00
5.50
5.00
4.50
0
5
10
15
20
25
30
Years to Maturity
4
Pure Discount Bonds,Zero
Coupon Bonds
Bonds that promise a single payment
of cash at some date in the future,
called the maturity date
5
Pure Discount Bonds


The pure discount bond is an
example of the present value of a
lump sum equation we analyzed in
Chapter 4
The yield-to-maturity on a pure
discount bond is given by the
relationship:
F  P1  i 
n
1
n
F
 i    1
P
6
Pure Discount Bonds
1
n
1
2
F
 10000 
i    1  
  1  5.41%
P
 9000 
N
I
PV
PMT
FV
2
?
5.41%
9,000
0
-10,000
7
Prices of Pure Discount Bonds and
Yields
Maturity
1 year
Price per $1 Yield
of Face Value (per year)
0.95
5.26%
2 year
0.88
6.60%
3 year
0.80
7.72
8
Valuation of a fixed income
security

A security with an annual payment
of $100 for 3 years.
V  $100  0.95  $100  0.88
 $100  0.80  $263
V  $100/1.0526  $100/(1.06 60)
2
 $100/(1.07 72)  $263
3
9
Coupon Bond
Obligates the issuer to make
periodic payments of interest
(coupon payments) to the
bondholder for the life of the
bond and then to pay the face
value at maturity
10
A 10 years coupon bond with an
annual coupon of $56, FV=$1000,
interest rate=5.6%
|__|__|__|__|__|__|__|__|__|__|
$56 56 ……………………………….. $56
$1000
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Valuation of the coupon bond
PV of the face value  $1000/(1.0 56)10
 $579.91
PV of the Annuity  ($56)(1 - 1/(1.056) )/.56
10
 $420.09
Total bond Value  $579.91  420.09  $1000
12
Valuation of the coupon bond
A year has gone by, the bond has 9
years to maturity, the interest rate had
risen to 7.6%
PV of the FV  $1000/1.076  $517.25
9
Annuity PV  ($56)(1 - 1/1.076 )/.076
9
 $355.71
Total bond value  $517.25  355.71
 $872.96
13
Discount Bond
Our Bond is selling for less than its
$1000 face value. Why? Compared
to the 7.6% market interest rate, it
pays only 5.6%, so investors are
only willing to buy it less than the
$1000 promised repayment.
14
Discount Bond
A bond that sells for less than face
value
15
Previous Example


The price of $873 is $127 less than
the face value, so the investor
would have a $127 additional gain
at maturity.
In fact the $56 coupon is $20 below
the coupon on a newly issued par
value bond, so the investor gives up
$20 per year for nine years. At
7.6%, this annuity is worth:
($20)(1 -1/1.076 )/.076  $127.04
9
16
What if interest rates had dropped
by 2% instead of rising by 2%?
PV of the FV  $1000/1.03 6  $727.38
9
Annuity PV  ($56)(1 - 1/1.036 )/.036
9
 $424.08
Total bond value  $727.38  424.08
 $1151.46
$151.46  PV of $20 per year
for 9 years at 3.6%
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Yield to Maturity
The Discount rate that makes the
present value of the bond’s stream
of promised cash payments equal to
its price.
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Coupon Rate, Current Yield, Yield to
Maturity
Coupon
Coupon Rate 
Face Value
Coupon
Current Yield 
Pr ice
1
1
Pr ice  Coupon(


2
1  ytm (1  ytm)
1
FV

)
n
n
(1  ytm)
(1  ytm)
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Bonds Trading at Par

Bond Pricing Principle #1: (Par
Bonds)
 If a bond’s price equals its face
value, then its yield-to-maturity =
current yield = coupon rate.
20
n
n


pmt   1    1 
P
1 
 F
& PF 


i   1  i    1  i 
  1  n  pmt   1  n 
pmt




P 1 

1 
 P
F


 1 i  
 1 i  
i
i




pmt pmt
i

P
F
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Bond Pricing Principle 2: Premium
Bonds
Yield to Maturity < Current Yield
<Coupon Rate
pmt pmt
P  F  ytm 

P
F
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Bond Pricing Principle 3: Discount
Bonds
Yield to Maturity > Current Yield >Coupon
Rate
pmt pmt
PF

 ytm
F
P
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Yield Relationships
0.2
0.18
0.16
coupon_y
current_y
y_t_m
0.14
Yield
0.12
0.1
0.08
0.06
0.04
0.02
0
600.00
800.00
1000.00
1200.00
1400.00
1600.00
1800.00
Price
24
Yield Relationships
0.13
Yield
coupon_y
current_y
y_t_m
0.11
0.09
0.07
800.00
1000.00
1200.00
Price
25
Why Yields for the Same Maturity May
Differ, The Effect of the Coupon Rate

1.
2.
Two different two-year coupon
bonds:
Coupon rate: 5%
Coupon rate: 10%
Maturity Price per $1
Yield
1 year
$0.961538
4%
2 year
$0.889996
6%
26
The Effect of the Coupon Rate

For the 5% coupon bond:
Price  0.961538  $50  0.889996  $1,050
 $982.57

For the 10% coupon bond:
Price  0.961538  $100  0.889996  $1,100
 $1,075.15
27
The Effect of the Coupon Rate

For the 5% coupon bond:
$982.57  50 /(1  ytm)  1,050 /(1  ytm)
 ytm  5.95%

For the 10% coupon bond:
$1075.15  100 /(1  ytm)  1,100 /(1  ytm) 2
 ytm  5.9064%
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2
The Effect of the Coupon Rate

When the yield curve is not flat,
bonds of the same maturity with
different coupon rates have
different yields to maturity.
29
The Effect of Default Risk and Taxes
A bond promising to pay $1,000 a
year from now.
 The one-year U.S. Treasury rate is
6% per year.
 If the bond is default free, its price
=$1,000/1.06=$943.40
 If subject to some default risk, its
price will be less than $943.40

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Other Effects on Bond Yields


Callability.Gives the issuer of the
bond the right to redeem it before
the final maturity date.
Convertibility. Gives the holder of a
bond issued by a corporation the
right to convert the bond into a
prespecified number of shares of
common stock.
31
Two Yield Curves (Pure Discount)
9.00%
8.00%
Yield to Maturiry
7.00%
6.00%
5.00%
4.00%
3.00%
2.00%
1.00%
0.00%
0
5
10
Years to Maturity
15
20
32
20-Year Bond Value Over Time
1060
1040
1000
980
Value
1020
960
940
920
20
15
10
Time to Maturity
5
0
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