1. Old Time Savings Bank pays 5% interest on its savings accounts. If
you deposit $2,000 in the bank and leave it there: (Do not round
intermediate calculations. Round your answers to 2
decimal places.)
Explanation
a.
Future value Year 1 = Present value × (1 + r)
= $2,000 × 1.05
= $2,100
Interest Year 1 = Future value Year 1 – Present value
= $2,100 – 2,000
= $100
b.
Future value Year 1 = Present value × (1 + r)2
= $2,100 × 12
= $2,205.00
Interest Year 2 = Future value Year 2 – Future value Year 1
= $2,205.00 – 2,100
= $105.00
c.
Future value Year 9 = Present value × (1 + r)9
= $2,000 × 19
= $3,102.66
Future value Year 10 = Present value × (1 + r)10
= $2,000 × 110
= $3,257.79
Interest Year 10 = Future value Year 10 – Future value Year 9
= $3,257.79 – 3,102.66
= $155.13
Calculator computations:
2. Compute the future value of a $170 cash flow for the following
combinations of rates and times. (Do not round intermediate
calculations. Round your answers to 2 decimal places.)
Explanation
Some values below may show as rounded for display purposes, though unrounded
numbers should be used for actual calculations.
FV = PV × (1 + r)t
a. FV = $170 × (1.07)10 = $334.42
b. FV = $170 × (1.07)20 = $657.85
c. FV = $170 × (1.03)10 = $228.47
d. FV = $170 × (1.03)20 = $307.04
Calculator computations:
3. If you earn 8% per year on your bank account, how long will it take an
account with $100 to double to $200? (Do not round
intermediate calculations. Round your answer to 2
decimal places.)
Explanation
Some values below may show as rounded for display purposes, though unrounded numbers
should be used for actual calculations.
FV = PV × (1 + r)t
$200 = $100 × 1.08t
1.08t = 2
t × ln1.08 = ln2
t = ln2 / ln1.08
t = 0.69315 / 0.07696
t = 9.01 years
Calculator computations:
4. In 1880 five aboriginal trackers were each promised the equivalent of
50 Australian dollars for helping to capture the notorious outlaw Ned
Kelley. In 1998 the granddaughters of two of the trackers claimed that
this reward had not been paid. The prime minister stated that if this was
true, the government would be happy to pay the $50. However, the
granddaughters also claimed that they were entitled to compound
interest.
Explanation
Some values below may show as rounded for display purposes, though unrounded numbers
should be used for actual calculations.
FV = PV × (1 + r)t
a.
FV = A$50 × (1.03)118 = A$1,635.92
b.
FV = A$50 × (1.06)118 = A$48,424.16
5. Your wealthy uncle established a bank account with $2,600 for you
when you were born. For the first 7 years of your life, the interest rate
earned on the account was 5%. Since then, rates have been only 3%.
Now you are 22 years old and ready to cash in. How much is in your
account? (Do not round intermediate calculations. Round
your answer to 2 decimal places.)
6. What is the present value of the following cash-flow stream if the
interest rate is 5%? (Do not round intermediate calculations.
Round your answer to 2 decimal places.)
Explanation
Some values below may show as rounded for display purposes, though unrounded
numbers should be used for actual calculations.
P
V=
C1 / (1 + r)1 + C2 / (1 + r)2 + C3 /
(1 + r)3
=
$250 / 1.05 + $450 / 1.052 +
$350 / 1.053
=$948.60
Calculator computations:
CF
0=0
CO
1=
1 FO1 =
1
CO
2=
250 FO2
= 1
CO
3=
450 FO3
= 1
I = 5
CPT NPV = 948.60
7. a. What is the value of the factory? (Do not round
intermediate calculations. Round your answer to 2
decimal places.)
b. Is the factory a good investment?
Explanation
Some values below may show as rounded for display purposes, though
unrounded numbers should be used for actual calculations.
a.
P
V=C1 / (1 + r)1 + C2 / (1 + r)2 + C3 / (1 + r)3
=
$140,000 / 1.11 + $200,000 / 1.112 +
$340,000 / 1.113
=$537,055.68
Value of
factory =
PV of cash inflows –
Cost
=
$537,055.68 –
440,000
=$97,055.68
b.
Since the PV of the cash inflows exceeds the cost, the factory is a good
investment.
Calculator computations:
CF
0=–440,000
CO
1=
140,000 FO1
= 1
CO
2=
200,000 FO2
= 1
CO
3=
340,000 FO3
= 1
I = 11
CPT NPV = 97,055.68
8. A famous quarterback just signed a contract for $16.5 million,
providing $3.3 million a year for 5 years. A less famous receiver signed
a contract for $15.5 million, providing $4 million now and $2.3 million a
year for 5 years. The interest rate is 10%.
Explanation
Some values below may show as rounded for display purposes, though unrounded numbers
should be used for actual calculations.
a.
P
V=C((1 / r) – {1 / [r(1 + r)t]})
=
$3,300,000 × ((1 / 0.10) – {1 /
[0.10(1.10)5]})
=$12,509,596.34, or $12.51 million
b.
P
V=C0 + C((1 / r) – {1 / [r(1 + r)t]})
=
$4,000,000 + $2,300,000 × ((1 / 0.10) – {1 /
[0.10(1.10)5]})
=$12,718,809.57, or $12.72 million
c.
Even though the receiver has the smaller contract amount, he is actually better paid because
the present value of his contract exceeds the present value of the quarterback’s contract.
b.
CF
0=4,000,000
CO
1=
2,300,000 FO1
= 5
I = 10
CPT NPV = 12,718,809.57
9. a. If you borrow $2,200 and agree to repay the loan in five equal
annual payments at an interest rate of 12%, what will your payment
be? (Do not round intermediate calculations. Round your
answer to 2 decimal places.)
a.
PVOA =C((1 / r) – {1 / [r(1 + r)t]})
$1,0
00 =
C × ((1 / .12) – {1 /
[.12(1.12)5]})
C=$610.30
9 b. What will your payment be if you make the first payment on the loan
immediately instead of at the end of the first year? (Do not round
intermediate calculations. Round your answer to 2
decimal places.)
b.
PVAD
=
[C((1 / r) – {1 / [r(1 + r)t]})] × (1
+ r)
$1,0
00 =
C × ((1 / .12) – {1 /
[.12(1.12)5]}) × 1.12
C=$544.91
You can also calculate this using the following:
PVAD
=
PVOA / (1
+ r)
=
$610.30 /
1.12
=$544.91
10. You believe you will need to have saved $590,000 by the time you
retire in 30 years in order to live comfortably. If the interest rate is 6%
per year, how much must you save each year to meet your retirement
goal? (Do not round intermediate calculations. Round
your answer to 2 decimal places.)
Explanation
Some values below may show as rounded for display purposes, though unrounded
numbers should be used for actual calculations.
Compute the annual savings needed to accumulate the desired retirement savings:
FV=
C × {[(1 + r)t –
1] / r}
$590,0
00=
C × [(1.0630 – 1) /
.06]
C=$7,462.86
11. A store will give you a 3.50% discount on the cost of your purchase
if you pay cash today. Otherwise, you will be billed the full price with
payment due in 1 month. What is the implicit borrowing rate being paid
by customers who choose to defer payment for the month? (Do not
round intermediate calculations. Enter your answer as a
percent rounded to 2 decimal places.)
Explanation
Some values below may show as rounded for display purposes, though unrounded
numbers should be used for actual calculations.
If you assume a purchase price of $1, then:
Cash price today = Purchase price – Discount
= $1 – (0.0350 × $1)
= $0.9650
You know the price in one month is $1 so you can compute the monthly interest rate
as:
FV = PV × (1 + r)t
$1 = $0.9650 × (1 + r)1
r= $1 / $0.9650 – 1
r= 0.0363, or 3.63%
EA
R= (1 + Monthly interest rate)12 – 1
= 1.036312 – 1
= 0.5335, or 53.35%
12. a. How much will $100 grow to if invested at a continuously
compounded interest rate of 8.5% for 9 years? (Do not round
intermediate calculations. Round your answer to 2
decimal places.)
12. b. How much will $100 grow to if invested at an annual interest rate
of 8.50% for 9 years? (Do not round intermediate
calculations. Round your answer to 2 decimal places.)
Explanation
Some values below may show as rounded for display purposes, though unrounded
numbers should be used for actual calculations.
a.
FV=C × ert
=
$100
× e(.0850 × 9)
=$214.90
b.
FV=C × (1 + r)t
=
$100 × (1 + .
0850)9.00
=$208.39
13. In April 2016 a pound of apples cost $1.46, while oranges cost
$1.10. Two years earlier the price of apples was only $1.25 a pound and
that of oranges was $.96 a pound.
Explanation
Some values below may show as rounded for display purposes, though unrounded numbers
should be used for actual calculations.
a.
FV = PV (1 + r)t
$1.4
6= $1.25 × (1 + r)2
r=
($1.46 / $1.25)1 /
2 – 1
r= .0807, or 8.07%
b.
FV = PV (1 + r)t
$1.1
0=$.96 × (1 + r)2
r=
($1.10 / $.96)1 /
2 – 1
r= .0704, or 7.04%
c.
FV=PV (1 + r)t
=
$1.46 × 1.0807(2030
– 2016)
=$4.33
d.
FV=PV (1 + r)t
=
$1.10 × 1.0704(2030
– 2016)
=$2.85
14. An engineer in 1950 was earning $5,200 a year. In 2015 she earned
$98,000 a year. However, on average, prices in 2015 were higher than
in 1950. What was her real income in 2015 in terms of constant 1950
dollars? Use the data in Table 5.8. (Round your answer to 2
decimal places.)
Explanation
Some values below may show as rounded for display purposes, though unrounded
numbers should be used for actual calculations.
Real
income =
Nominal income / Inflation
multiple
=$98,000 / (236.5 / 25)
=$10,359.41
Her real income increased by $5,159.41.
