Chapter 4:
Net Present Value
4.1
a.
Future Value
= C
0
(1+r)
T
= $1,000 (1.05)
10
=
$1,628.89
b.
Future Value
= $1,000 (1.07)
10
=
$1,967.15
c.
Future Value
= $1,000 (1.05)
20
=
$2,653.30
d.
Because interest compounds on interest already earned, the interest earned in part (
c
),
$1,653.30 (=$2,653.30 - $1,000) is more than double the amount earned in part (
a
),
$628.89 (=$1,628.89).
4.2
The present value, PV, of each cash flow is simply the amount of that cash flow discounted back
from the date of payment to the present.
For example in part (
a
), discount the cash flow in year 7
by seven periods, (1.10)
7
.
a.
PV(C
7
)
= C
7
/ (1+r)
7
= $1,000 / (1.10)
7
=
$513.16
b.
PV(C
1
) = $2,000 / 1.10
=
$1,818.18
c.
PV(C
8
) = $500 / (1.10)
8
=
$233.25
4.3
The decision involves comparing the present value, PV, of each option.
Choose the option with
the highest PV.
Since the first cash flow occurs 0 years in the future, or today, it does not need to
be adjusted.
PV(C
0
)
=
$1,000
Since the second cash flow occurs 10 years in the future, it must be discounted back 10 years at
eight percent.
PV(C
10
)
= C
10
/ (1+r)
10
= $2,000 / (1.08)
10
=
$926.39
Since the present value of the cash flow occurring today is higher than the present value of
the cash flow occurring in year 10, you should take the $1,000 now.
4.4
Since the bond has no interim coupon payments, its present value is simply the present value of
the $1,000 that will be received in 25 years. Note that the price of a bond is the present value of
its cash flows.
P
0
= PV(C
25
)
= C
25
/ (1+r)
25
= $1,000 / (1.10)
25
=
$92.30
The price of the bond is $92.30.
B-49

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*Sign up*4.5
The future value, FV, of the firm’s investment must equal the $1.5 million pension liability.
FV
= C
0
(1+r)
27
To solve for the initial investment, C
0
, discount the future pension liability ($1,500,000) back 27
years at eight percent, (1.08)
27
.
$1,500,000 / (1.08)
27
= C
0
=
$187,780.23
The firm must invest $187,708.23 today to be able to make the $1.5 million payment.
4.6
The decision involves comparing the present value, PV, of each option.
Choose the option with
the highest PV.
a.
At a discount rate of zero, the future value and present value of a cash flow are always
the same.
There is no need to discount the two choices to calculate the PV.
PV(Alternative 1)
=
$10,000,000
PV(Alternative 2)
=
$20,000,000
Choose Alternative 2 since its PV, $20,000,000, is greater than that of Alternative 1,
$10,000,000.
b.
Discount the cash flows at 10 percent.
Discount Alternative 1 back one year and
Alternative 2, five years.
PV(Alternative 1)
= C / (1+r)
= $10,000,000 / (1.10)
1
=
$9,090,909.10
PV(Alternative 2)
= $20,000,000 / (1.10)
5
=
$12,418,426.46
Choose Alternative 2 since its PV, $12,418,426.46, is greater than that of Alternative
1, $9,090,909.10.
c.

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