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Chapter 6: How to Value Bonds and Stocks
6.1
The price of a pure discount (zero coupon) bond is the present value of the par. Even
though the bond makes no coupon payments, the present value is found using
semiannual compounding periods, consistent with coupon bonds. This is a bond pricing
convention. So, the price of the bond for each YTM is:
a.
Price = $1,000/1 + 0.025
20
= $610.27
b.
Price
= $1,000/1 + 0.05
20
= $376.89
c.
Price = $1,000/1 + 0.075
20
= $235.41
6.2.
The price of any bond is the PV of the interest payment, plus the PV of the par value. Notice this
problem assumes semiannual coupon. The price of the bond at each YTM will be:
a.
P = $40({1 – [1/(1 + 0.04)]
40
} / 0.04) + $1,000[1 / (1 + 0.04)
40
]
P = $1,000.00
When the YTM and the coupon rate are equal, the bond will sell at par.
b.
P = $40({1 – [1/(1 + 0.05)]
40
} / 0.05) + $1,000[1 /1 + 0.05
40
]
P = $828.41
When the YTM is greater than the coupon rate, the bond will sell at a
discount.
c.
P = $40({1 – [1/(1 + 0.03)]
40
} / 0.03) + $1,000[1 /1 + 0.03
40
]
P = $1,231.15
When the YTM is less than the coupon rate, the bond will sell at a
premium.
We would like to introduce shorthand notation here. Rather than write (or
type, as the case may be) the entire equation for the PV of an annuity , it is
common to abbreviate the equations as:
t
r
Α
= ({1 – [1/(1 + r)]
t
} / r )
which stands for Present Value Interest Factor of an Annuity
This abbreviation is short hand notation for the equations in which the
interest rate and the number of periods are substituted into the equation
and solved. We will use this shorthand notation in the remainder of the
solutions key.
6.3.
Here we are finding the YTM of a semiannual coupon bond. The bond price
equation is:
Price = $970 = $43
20
r
Α
+ $1,000 /1+r
20
Since we cannot solve the equation directly for r, using a spreadsheet, a financial calculator, or
trial and error, we find:
r = 0.04531 or 4.531%
Answers to End–of–Chapter Problems
B–41
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View Full Document Since the coupon payments are semiannual, this is the semiannual interest rate.
The
YTM is
the APR of the bond, so:
YTM = 2 x 4.531% = 9.06%
6.4.
Here we are finding the YTM of semiannual coupon bonds for various maturity lengths. The
bond price equation is:
P = C
t
r
Α
+ $1,000/1+r
t
Miller Corporation bond:
P
0
= $40
26
03
.
0
Α
+ $1,000/1+0.03
26
= $1,178.77
P
1
= $40
24
03
.
0
Α
+ $1,000/1+0.03
24
= $1,169.36
P
3
= $40
20
03
.
0
Α
+ $1,000/1+0.03
20
= $1,148.77
P
8
= $40
10
03
.
0
Α
+ $1,000/1+0.03
10
= $1,085.30
P
12
= $40
2
03
.
0
Α
+ $1,000/1+0.03
2
= $1,019.13
P
13
= $1,000
Modigliani Company bond:
P
0
= $30
26
04
.
0
Α
+ $1,000/1+0.04
26
= $840.17
P
1
= $30
24
04
.
0
Α
+ $1,000/1+0.04
24
= $847.53
P
3
= $30
20
04
.
0
Α
+ $1,000/1+0.04
20
= $864.10
P
8
= $30
10
04
.
0
Α
+ $1,000/1+0.04
10
= $918.89
P
12
= $30
2
04
.
0
Α
+ $1,000/1+0.04
24
= $981.14
P
13
= $1,000
All else held equal, the premium over par value for a premium bond declines as maturity
approaches, and the discount from par value for a discount bond declines as maturity approaches.
This is called “pull to par.” In both cases, the largest percentage price changes occur at the
shortest maturity lengths.
Also, notice that the price of each bond when no time is left to maturity is the par value, even
though the purchaser would receive the par value plus the coupon payment immediately. This is
because we calculate the clean price of the bond.
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This note was uploaded on 07/18/2010 for the course ECONMICS ECM359 taught by Professor Matazi during the Summer '10 term at University of Toronto Toronto.
 Summer '10
 matazi

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