Old Time Savings Bank pays 3% interest on its savings accounts. If you
deposit $2,100 in the bank and leave it there: (Do not round
intermediate calculations. Round your answers to 2 decimal
places.)
a. How much interest will you earn in the first year?
b. How much interest will you earn in the second year?
c. How much interest will you earn in the 10th year?
Explanation
a.
Future value Year 1 = Present value × (1 + r)
= $2,100 × 1.03
= $2,163
Interest Year 1 = Future value Year 1 – Present value
= $2,163 – 2,100
= $63
b.
Future value Year 2 = Present value × (1 + r)2
= $2,100 × 1.032
= $2,227.89
Interest Year 2 = Future value Year 2 – Future value Year 1
= $2,227.89 – 2,163
= $64.89
c.
Future value Year 9 = Present value × (1 + r)9
= $2,100 × 1.039
= $2,740.02
Future value Year 10 = Present value × (1 + r)10
= $2,100 × 1.0310
= $2,822.22
Interest Year 10 = Future value Year 10 – Future value Year 9
= $2,822.22 – 2,740.02
= $82.20
Calculator computations:
a.
Enter 1 3 –2,100
N I/Y PV PM
T F
V
Solve for 2,163
b.
Enter 2 3 –2,100
N I/Y PV PM
T FV
Solve for 2,227.89
c.
Enter 9 3 –2,100
N I/Y PV PM
T FV
Solve for 2,740.02
Enter 10 3 –2,100
N I/Y PV PM
T FV
Solve for 2,822.22
Compute the future value of a $200 cash flow for the following combinations
of rates and times. (Do not round intermediate calculations.
Round your answers to 2 decimal places.)
a. r = 8%; t = 10 years
b. r = 8%; t = 20 years
c. r = 4%; t = 10 years
d. r = 4%; t = 20 years
: 03_04_2021_QC_CS-256151
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 = $200 × (1.08)10 = $431.78
b. FV = $200 × (1.08)20 = $932.19
c. FV = $200 × (1.04)10 = $296.05
d. FV = $200 × (1.04)20 = $438.22
Calculator computations:
a.
Enter 10 8 –200
N I/Y P
V PM
T FV
Solve for 431.78
b.
Enter 20 8 –200
N I/Y P
V PM
T FV
Solve for 932.19
c.
Enter 10 4 –200
N I/Y P
V PM
T FV
Solve for 296.05
d.
Enter 20 4 –200
N I/Y P
V PM
T FV
Solve for 438.22
If you earn 8% per year on your bank account, how long will it take an account
with $105 to double to $210? (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
$210 = $105 × 1.08t
1.08t = 2
t × ln1.08 = ln2
t = ln2 / ln1.08
t = 0.69315 / 0.07696
t = 9.01 years
Calculator computations:
Enter 8 –105 210
N I/Y P
V PM
T F
V
Solve for 9.01
In 1880 five aboriginal trackers were each promised the equivalent of 100
Australian dollars for helping to capture the notorious outlaw Ned Kelley. In
2002 the granddaughters of two of the trackers claimed that this reward had
not been paid. The Victorian prime minister stated that if this was true, the
government would be happy to pay the $100. However, the granddaughters
also claimed that they were entitled to compound interest.
a. How much was each granddaughter entitled to if the interest rate was
3%? (Do not round intermediate calculations. Round your
answer to 2 decimal places.)
b. How much was each entitled to if the interest rate was 6%? (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
a.
FV = A$100 × (1.03)122 = A$3,682.49
b.
FV = A$100 × (1.06)122 = A$122,268.78
a.
Enter 122 3 –100
N I/Y P
V PM
T FV
Solve for 3,682.49
b.
Enter 122 6 –100
N I/Y P
V PM
T FV
Solve for 122,268.78
Your wealthy uncle established a $1,800 bank account for you when you were
born. For the first 9 years of your life, the interest rate earned on the account
was 4%. Since then, rates have been only 2%. Now you are 23 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.)
Explanation
You earned compound interest of 4% for 9 years and 2% for 14 years. Your $1,800 has grown to:
$1,800 × (1.04)9 × (1.02)14 = $3,380.45
What is the present value of the following cash-flow stream if the interest rate
is 4%? (Do not round intermediate calculations. Round your
answer to 2 decimal places.)
YearCash Flow
1 $ 110
2 310
3 210
Explanation
Some values below may show as rounded for display purposes, though unrounded numbers should
be used for actual calculations.
PV=C1 / (1 + r)1 + C2 / (1 + r)2 + C3 / (1 + r)3
=($110 / 1.04) + ($310 / 1.042) + ($210 / 1.043)
$105.77 + $286.61 + $186.69
=$579.07
Calculator computations:
CF0=0
CO1=110 FO1 = 1
CO2=310 FO2 = 1
CO3=210 FO3 = 1
I = 4
CPT NPV = 579.07
A factory costs $430,000. You forecast that it will produce cash inflows of
$135,000 in year 1, $195,000 in year 2, and $330,000 in year 3. The discount
rate is 12%.
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?
multiple choice
Yes Correct
No
: 02_07_2020_QC_CS-199502, 09_26_2020_QC_CS-230950
Explanation
Some values below may show as rounded for display purposes, though unrounded numbers should
be used for actual calculations.
a.
PV =C1 / (1 + r)1 + C2 / (1 + r)2 + C3 / (1 + r)3
=$135,000 / 1.12 + $195,000 / 1.122 + $330,000 / 1.123
=$510,876.00
NPV=-$430,000 + $510,876.00 = 80,876.00
b.
The PV exceeds the cost of the factory, so the investment is attractive.
Calculator computations:
CF0=–430,000
CO1=135,000 FO1 = 1
CO2=195,000 FO2 = 1
CO3=330,000 FO3 = 1
I = 12
CPT NPV = 80,876.00
A famous quarterback just signed a $14 million contract providing $3.5 million
a year for 4 years. A less famous receiver signed a $10.6 million 4-year
contract providing $3 million now and $3.2 million a year for 4 years. The
interest rate is 8%.
a. What is the PV of the quarterback's contract? (Do not round
intermediate calculations. Enter your answer in millions
rounded to 2 decimal places.)
b. What is the PV of the receiver's contract? (Do not round
intermediate calculations. Enter your answer in millions
rounded to 2 decimal places.)
c. Who is better paid?
multiple choice
Quarterback
Receiver Correct
Explanation
Some values below may show as rounded for display purposes, though unrounded numbers should
be used for actual calculations.
a.
PV=C((1 / r) – {1 / [r(1 + r)t]})
=$3,500,000 × ((1 / 0.08) – {1 / [0.08(1.08)4]})
=$11,592,443.94, or $11.59 million
b.
PV=C0 + C((1 / r) – {1 / [r(1 + r)t]})
=$3,000,000 + $3,200,000 × ((1 / 0.08) – {1 / [0.08(1.08)4]})
=$13,598,805.89, or $13.60 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.
Calculator computations:
Enter 4 8 -3,500,000
N I/Y PV PMT F
V
Solve for 11,592,443.94
b.
CF0=3,000,000
CO1=3,200,000 FO1 = 4
I = 8
CPT NPV = 13,598,805.89
a. If you borrow $1,300 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.)
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.)
Explanation
Some values below may show as rounded for display purposes, though unrounded numbers should
be used for actual calculations.
a.
PVOA =C((1 / r) − {1 / [r(1 + r)t]})
$1,300=C × ((1 / 0.12) − {1 /
[0.12(1.12)5]})
C=$360.63
b.
PVAD =[C((1 / r) − {1 / [r(1 + r)t]})] × (1 + r)
$1,300=C × ((1 / 0.12) − {1 / [0.12(1.12)5]}) ×
1.12
C=$321.99
You can also calculate this using the following:
If the first payment is made immediately instead of in a year, the annuity factor will be greater by a
factor of 1.12. Therefore:
C × (3.6048 × 1.12) = $1,300 ⇒⇒ C = PMT = $321.99
With the annuity due, you are paying each payment one year sooner which means that you will owe
less interest on the loan. Thus, the annual loan payment with the annuity due will be less than the
annual payment with the ordinary annuity.
Calculator computations:
Calculator payments set to "END".
Enter 5 12 −1,300
N I/Y PV PM
T F
V
Solve for 360.63
Calculator payments set to "BGN".
Enter 5 12 −1,300
N I/Y PV PM
T F
V
Solve for 321.99
Explanation
Some values below may show as rounded for display purposes, though unrounded numbers
should be used for actual calculations.
a.
First, determine the amount of savings required on the date of retirement:
PV = C((1 / r) − {1 / [r(1 + r)t]})
= $40,000 × ((1 / 0.07) − {1 / [0.07(1.07)25]})
= $466,143.33
Now compute the annual savings needed to accumulate the needed retirement savings:
FV = C × {[(1 + r)t − 1] / r}
$466,143.33 = C × [(1.0750 − 1) / 0.07]
C= $466,143.33 / [(1.0750 − 1) / 0.07]
C= $1,146.64
b.
The first step is to determine the present value of the cash needed for both college and retirement
at Time 0:
PV = $70,000 / 1.0720 + $466,143.33 / 1.0750
= $33,913.85
Now determine the annual savings for 50 years that equates to that present value:
PV = C((1 / r) − {1 / [r(1 + r)t]})
$33,913.85 = C × ((1 / 0.07) − {1 / [0.07(1.07)50]})
C=$33,913.85 / ((1 / 0.07) − {1 /
[0.07(1.07)50]})
C = $2,457.39
Calculator computations:
a.
Savings needed on retirement date:
Enter 25 7 −40,000
N I/Y PV PMT F
V
Solve for 466,143.33
Annual savings needed to fund retirement:
Enter 50 7 46,6143.33
N I/Y P
V PMT FV
Solve for 1,146.64
b.
Present value of college education:
Enter 20 7 70,000
N I/Y PV PM
T FV
Solve for 18,089.33
Present value of required retirement savings:
Enter 50 7 466,143.33
N I/Y PV PM
T FV
Solve for 15,824.52
Total savings needed at Time 0 = $18,089.33 + 15,824.52 = $33,913.85
Annual savings needed to fund both college and retirement:
Enter 50 7 33,913.85
N I/Y PV PMT F
V
Solve for 2,457.39
A store will give you a 4.50% 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 – (0.0450 × $1)
= $0.9550
Suppose the purchase price is $1. If you pay today, you get the discount and pay only $0.9550. If
you wait a month, you pay $1. Thus, you can view the deferred payment as saving a cash flow of
$0.9550 today but paying $1 in a month. That makes the $0.9550 the principal in this situation, and
the $0.0450 the interest payment, for which we can calculate an interest rate against the principal.
Therefore, the monthly rate is:
FV = PV × (1 + r)t
$1 = $0.9550 × (1 + r)1
r= $1 / $0.9550 – 1
r= 0.0471, or 4.71%
EAR = (1 + Monthly interest rate)12 – 1
= 1.047112 – 1
= 0.7376, or 73.76%
Calculator computations:
Enter 1 –0.9550 1
N I/Y PV PM
T F
V
Solve for 4.71
A store will give you a 4.50% 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 – (0.0450 × $1)
= $0.9550
Suppose the purchase price is $1. If you pay today, you get the discount and pay only $0.9550. If
you wait a month, you pay $1. Thus, you can view the deferred payment as saving a cash flow of
$0.9550 today but paying $1 in a month. That makes the $0.9550 the principal in this situation, and
the $0.0450 the interest payment, for which we can calculate an interest rate against the principal.
Therefore, the monthly rate is:
FV = PV × (1 + r)t
$1 = $0.9550 × (1 + r)1
r= $1 / $0.9550 – 1
r= 0.0471, or 4.71%
EAR = (1 + Monthly interest rate)12 – 1
= 1.047112 – 1
= 0.7376, or 73.76%
Calculator computations:
Enter 1 –0.9550 1
N I/Y PV PM
T F
V
Solve for 4.71
In October 2017 a pound of apples cost $1.61, while oranges cost $1.25. Two
years earlier the price of apples was only $1.40 a pound and that of oranges
was $1.11 a pound.
a. What was the annual compound rate of growth in the price of apples? (Do
not round intermediate calculations. Enter your answer as a
percent rounded to 2 decimal places.)
b. What was the annual compound rate of growth in the price of
oranges? (Do not round intermediate calculations. Enter your
answer as a percent rounded to 2 decimal places.)
c. If the same rates of growth persist in the future, what will be the price of
apples in 2030? (Do not round intermediate calculations. Round
your answer to 2 decimal places.)
d. If the same rates of growth persist in the future, what will be the price of
oranges in 2030? (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.
Inflation is captured in a single number, the inflation rate per year, so that we can compare across
different products and different time frames. Here we can call the annual inflation rate “i” and
calculate the average for each separate product based on the change in price over many years. To
get the answer, we treat the growth in the price of the product like the growth in any asset, and
isolate the rate at which it has grown:
a.
FV = PV (1 + r)t
$1.61= $1.40 × (1 + r)2
r= ($1.61 / $1.40)1 / 2 – 1
r= 0.0724, or 7.24%
b.
FV = PV (1 + r)t
$1.25=$1.11 × (1 + r)2
r=($1.25 / $1.11)1 / 2 – 1
r= 0.0612, or 6.12%
c.
FV=PV (1 + r)t
=$1.61 × 1.072413
=$3.99
d.
FV=PV (1 + r)t
=$1.25 × 1.061213
=$2.71
Calculator computations:
a.
Enter 2 –1.40 1.61
N I/Y P PM F
V T V
Solve for 7.24
b.
Enter 2 –1.11 1.25
N I/Y P
V PM
T F
V
Solve for 6.12
c.
Enter 13 7.24 –1.61
N I/Y P
V PM
T F
V
Solve for 3.99
d.
Enter 13 6.12 –1.25
N I/Y P
V PM
T F
V
Solve for 2.71
An engineer in 1950 was earning $5,600 a year. In 2017 she earned $94,000
a year. However, on average, prices in 2017 were higher than in 1950. What
was her real income in 2017 in terms of constant 1950 dollars? Use the data
in Table 5.8. (Round your answer to 2 decimal places.)
: 06_07_2021_QC_CS-266572
Explanation
Some values below may show as rounded for display purposes, though unrounded numbers should
be used for actual calculations.
Here our goal is to compare the actual 1950 salary to the current salary, to learn if the current salary
grew at the real rate of CPI, or a rate higher or lower. To do that we discount today’s salary to 1950
using the change in the CPI since 1950; we do this by dividing the ending CPI by the beginning CPI;
the resulting answer, is divided into the ending salary. Hence, we have:
Real income=Nominal income / Inflation multiple
=$94,000 / (248.0 / 25)
=$9,475.81
Her real income increased by $3,875.81.
