Chapter 5: The Time Value of Money
5.1
a.
$1,000
×
1.05
10
= $1,628.89
b.
$1,000
×
1.07
10
= $1,967.15
c.
$1,000
×
1.05
20
= $2,653.30
d.
Interest compounds on the interest already earned.
Therefore, the interest earned
in part c, $2653.30, is more than double the amount earned in part a, $628.89.
5.2
a.
$1,000 / 1.1
7
= $513.16
b.
$2,000 / 1.1 = $1,818.18
c.
$500 / 1.1
8
= $233.25
5.3
You can make your decision by computing either the present value of the $2,000 that you
can receive in ten years, or the future value of the $1,000 that you can receive now.
Present value:
$2,000 / 1.08
10
= $926.39 is less than $1000 today;
Future value:
$1,000
×
1.08
10
= $2,158.92 is greater than $2000 in 10 years.
Either calculation indicates you should take the $1,000 now.
5.4
Since this 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: As will be discussed in the next
chapter, the present value of the payments associated with a bond is the price of that bond.
PV = $1,000 /1.1
25
= $ 92.30
5.5
PV = $1,500,000 / 1.04
27
= $520,224.86
5.6
a.
At a discount rate of zero, the future value and present value are always the same.
Remember, FV = PV (1 + r)
t
.
If r = 0, then the formula reduces to FV = PV.
Therefore, the values of the alternatives are $10,000,000 and $20,000, 000,
respectively.
You should choose the second option.
b.
Option one:
$10,000,000 / 1.1 = $9,090,909
Option two:
$20,000,000 / 1.1
5
= $12,418,426
Choose the second option.
c.
Option one:
$10,000,000 / 1.2 = $8,333,333
Option two:
$20,000,000 / 1.2
5
= $ 8,037,551
Choose the first option.
d.
You are indifferent at the rate that equates the PVs of the two alternatives.
You
know that rate must fall between 10% and 20% because the option you would
choose differs at these rates.
Let r be the discount rate that makes you indifferent
between the options.
$10,000,000 / (1 + r) = $20,000,000 / (1 + r)
5
(1 + r)
4
= $20,000,000 / $10,000,000 = 2
1 + r = 1.189207115
r = 0.189207115 = 18.92%
5.7
PV of Joneses’ offer = $150,000 / (1.1)
3
= $ 112,697.22
Answers to EndofChapter Problems
B21
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Since the PV of Joneses’ offer is less than Smiths’ offer of $115,000, you should choose
Smiths’ offer.
5.8
a.
P
0
= $1,000 / 1.08
20
= $214.55
b.
P
10
= 1000/(1.08)
10
= $463.19
c.
P
15
= 1000/(1.08)
5
= $680.58
5.9
PV = $5,000,000 / 1.12
10
= $1,609,866.18
5.10
a.
The cost of investment is $900,000.
PV of cash inflows = $120,000 / 1.12 + $250,000 / 1.12
2
+ $800,000 / 1.12
3
=
$875,865.53.
Since the PV of cash inflows is less than the cost of investment,
you should not make the investment.
b.
NPV
= $900,000 + $875,865.53
= $24,134.47
c.
NPV
= $900,000 + $120,000 / 1.11 + $250,000 / 1.11
2
+ $800,000 / 1.11
3
=
$4,033.18
Since the NPV is negative, so you should not make the investment.
5.11
NPV = ($340,000 + $10,000) + ($100,000  $10,000) / 1.1 + $90,000 /1.1
2
+ $90,000 / 1.1
3
+ $90,000 / 1.1
4
+ $100,000 / 1.1
5
=
$2,619.98
Since the NPV is negative, you should not buy it.
If the relevant discount rate is 9 percent,
NPV
= $350,000 + $90,000 / 1.09 + $90,000 / 1.09
2
+ $90,000 / 1.09
3
+ $90,000 /
1.09
4
+ $100,000 / 1.09
5
= $ 6,567.93
Since the NPV is positive, you should buy it.
5.12
a.
NPV = $90,000 / 1.1
5
 $60,000 = $4,117
b.
Find r that makes zero NPV.
$90,000 / (1+r)
5
 $60,000 = $0
(1+r)
5
= 1.5
r = 0.08447 = 8.447%
5.14
80,000 / (1+r)
12
= 10,000;
(1 + r)
12
= 8;
(1 + r) = 1.18919; r = 18.92%
5.15
The $1,000 you invest at the end of year 1 will earn 12% for 6 years.
The $1,000 that
you invest at the end of the second year, will earn interest for 5years, and so on.
This,
the account will have a balance of
$1,000 (1.12)
6
+ 1,000 (1.12)
5
+ 1,000 (1.12)
4
+ 1,000 (1.12)
3
=
6,714.61
5.16
a.
$1,000 (1.08)
3
= $1,259.71
b.
$1,000 [1 + (0.08 / 2)]
2
×
3
= $1,000 (1.04)
6
= $1,265.32
c.
$1,000 [1 + (0.08 / 12)]
12
×
3
= $1,000 (1.00667)
36
= $1,270.39
d.
$1,000e
0.08 x 3
= $1,271.25
e.
The future value increases because of the compounding.
The account is earning
interest on interest.
Essentially, the interest is added to the account balance at
the end of every compounding period.
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 Fall '08
 Wood
 Time Value Of Money, EAY

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