Solutions to Chapter 4
The Time Value of Money
1.
a.
$100/(1.08)
10
= $46.32
b.
$100/(1.08)
20
= $21.45
c.
$100/(1.04)
10
= $67.56
d.
$100/(1.04)
20
= $45.64
2.
a.
$100
×
(1.08)
10
= $215.89
b.
$100
×
(1.08)
20
= $466.10
c.
$100
×
(1.04)
10
= $148.02
d.
$100
×
(1.04)
20
= $219.11
3.
$100
×
(1.04)
113
= $8,409.45
$100
×
(1.08)
113
= $598,252.29
4.
With simple interest, you earn 4% of $1,000 or $40 each year.
There is no interest on
interest.
After 10 years, you earn total interest of $400, and your account accumulates to
$1,400.
With compound interest, your account grows to: $1,000
×
(1.04)
10
= $1480.24
Therefore $80.24 is interest on interest.
5.
PV = $700/(1.05)
5
= $548.47
6.
Present Value
Years
Future Value
Interest Rate*
a.
$400
11
$684
%
0
.
5
1
400
684
)
11
/
1
(
=
−
⎥
⎦
⎤
⎢
⎣
⎡
b.
$183
4
$249
%
0
.
8
1
183
249
)
4
/
1
(
=
−
⎥
⎦
⎤
⎢
⎣
⎡
c.
$300
7
$300
%
0
1
300
300
)
7
/
1
(
=
−
⎥
⎦
⎤
⎢
⎣
⎡
To find the interest rate, we rearrange the basic future value equation as follows:
4-1

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FV = PV
×
(1 + r)
t
⇒
r =
1
PV
FV
)
t
/
1
(
−
⎥
⎦
⎤
⎢
⎣
⎡
7.
You should compare the present values of the two annuities.
Discount Rate
10-year,
$1,000 annuity
15-year,
$800 annuity
a.
5%
$7,721.73
$8,303.73
b.
20%
$4,192.47
$3,740.38
c.
When the interest rate is low, as in part (a), the longer (i.e., 15-year) but smaller
annuity is more valuable because the impact of discounting on the present value of
future payments is less significant.
8.
$100
×
(1 + r)
3
= $115.76
⇒
r = 5.0%
$200
×
(1 + r)
4
= $262.16
⇒
r = 7.0%
$100
×
(1 + r)
5
= $110.41
⇒
r = 2.0%
9.
PV = ($200/1.06) + ($400/1.06
2
) + ($300/1.06
3
) = $188.68 + $356.00 + $251.89 = $796.57
10.
In these problems, you can either solve the equation provided directly, or you can use your
financial calculator, setting: PV = (
−
)400, FV = 1000, PMT = 0, i as specified by the
problem. Then compute n on the calculator.
a.
$400
×
(1.04)
t
= $1,000
⇒
t = 23.36 periods
b.
$400
×
(1.08)
t
= $1,000
⇒
t = 11.91 periods
c.
$400
×
(1.16)
t
= $1,000
⇒
t = 6.17 periods
11.
APR
Compounding
period
Effective
annual rate
a.
12%
1 month
(m = 12/yr)
1.01
12
−
1 = 0.1268 = 12.68%
b.
8%
3 months
(m = 4/yr)
1.02
4
−
1 = 0.0824 = 8.24%
c.
10%
6 months
(m = 2/yr)
1.05
2
−
1 = 0.1025 = 10.25%
4-2