Preview Extract
CHAPTER 2
TIME VALUE OF MONEY
(Difficulty Levels: Easy, Easy/Medium, Medium, Medium/Hard, and Hard)
PART I โ New and Revised Carryover Problems and Questions
Multiple Choice: Problems
FV of a lump sum
1.
Answer: a
EASY
What would the future value of $100 be after 5 years at 10% compound
interest?
a. $161.05
b. $134.54
c. $127.84
d. $151.29
e. $143.65
FV of a lump sum
2.
Answer: c
EASY
Suppose you have $2,000 and plan to purchase a 3year certificate of
deposit (CD) that pays 4% interest, compounded annually.
How much will
you have when the CD matures?
a. $2,324.89
b. $2,591.45
c. $2,249.73
d. $2,011.87
e. $2,854.13
FV of a lump sum
3.
Answer: e
EASY
A companyโs 2005 sales were $100 million. If sales grow at 8% per year,
how large will they be 10 years later, in 2015, in millions?
a. $190.49
b. $225.54
c. $188.32
d. $201.15
e. $215.89
FV of a lump sum
4.
Answer: b
EASY
How much would $1, growing at 5% per year, be worth after 100 years?
a. $141.05
b. $131.50
c. $164.52
d. $144.50
e. $155.94
Chapter 2: Time Value of Money
Page 17
PV of a lump sum
5.
Answer: d
EASY
Suppose a U.S. government bond promises to pay $2,249.73 three years from
now.
If the going interest rate on 3year government bonds is 6%, how
much is the bond worth today?
a. $2,011.87
b. $2,591.45
c. $2,324.89
d. $1,888.92
e. $2,854.13
PV of a lump sum
6.
Answer: a
EASY
How much would $10,000 due in 100 years be worth today if the discount
rate were 10%?
a. $0.73
b. $1.21
c. $2.49
d. $4.83
e. $6.30
PV of a lump sum
7.
Answer: c
EASY
Suppose a U.S. government bond will pay $1,000 three years from now. If
the going interest rate on 3year government bonds is 4%, how much is the
bond worth today?
a. $943.46
b. $991.43
c. $889.00
d. $907.91
e. $968.40
Interest rate on a simple lump sum investment
8.
Answer: e
EASY
The U.S. Treasury offers to sell you a bond for $613.81. No payments will
be made until the bond matures 10 years from now, at which time it will be
redeemed for $1,000. What interest rate would you earn if you bought this
bond at the offer price?
a. 5.91%
b. 6.71%
c. 7.10%
d. 5.59%
e. 5.00%
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Chapter 2: Time Value of Money
Simple growth rate
9.
Answer: b
EASY
Sims Inc. earned $1.00 per share in 2000. Five years later, in 2005, it
earned $2.00. What was the growth rate in Sims’ earnings per share (EPS)
over the 5year period?
a. 10.82%
b. 14.87%
c. 13.61%
d. 14.28%
e. 12.17%
Number of periods
10.
Answer: e
EASY
How long would it take $100 to double if it were invested in a bank that
pays 5% per year?
a. 15.27 years
b. 10.49 years
c. 11.34 years
d. 13.91 years
e. 14.21 years
Number of periods
11.
Answer: a
EASY
Addico Corp’s 2005 earnings per share were $2, and its growth rate during
the prior 5 years was 11.0% per year.
If that growth rate were
maintained, how long would it take for Addicoโs EPS to double?
a. 6.64 years
b. 6.81 years
c. 6.99 years
d. 7.13 years
e. 7.28 years
FV of an ordinary annuity
12.
Answer: c
EASY
You want to buy a condo 5 years from now, and you plan to save $3,000 per
year, beginning one year from today.
You will deposit the money in an
account that pays 6% interest. How much will you have just after you make
the 5th deposit, 5 years from now?
a. $14,764.40
b. $13,431.83
c. $16,911.28
d. $17,843.15
e. $15,119.76
Chapter 2: Time Value of Money
Page 19
FV of an ordinary annuity
13.
Answer: e
EASY
You want to go to grad school 3 years from now, and you can save $5,000
per year, beginning one year from today. You plan to deposit the funds in
a mutual fund which you expect to return 9% per year.
Under these
conditions, how much will you have just after you make the 3rd deposit, 3
years from now?
a. $18,349.15
b. $16,110.34
c. $17,513.68
d. $17,976.84
e. $16,390.50
FV of an annuity due
14.
Answer: b
EASY
You want to buy a condo 5 years from now, and you plan to save $3,000 per
year, beginning immediately. You will make 5 deposits in an account that
pays 6% interest. Under these assumptions, how much will you have 5 years
from today?
a. $16,110.34
b. $17,925.96
c. $17,513.68
d. $15,976.84
e. $18,349.15
FV of an annuity due
15.
Answer: d
EASY
You want to go to grad school 3 years from now, and you can save $5,000
per year, beginning immediately.
You plan to deposit the funds in a
mutual fund which you expect to return 9% per year.
Under these
conditions, how much will you have just after you make the 3rd deposit, 3
years from now?
a. $14,976.84
b. $16,110.34
c. $17,513.68
d. $17,865.65
e. $18,349.15
PV of an ordinary annuity
16.
Answer: a
EASY
What is the PV of an ordinary annuity with 5 payments of $3,000 if the
appropriate interest rate is 5%?
a. $12,988.43
b. $13,431.83
c. $14,764.40
d. $17,843.15
e. $15,119.76
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Chapter 2: Time Value of Money
PV of an ordinary annuity
17.
Answer: c
EASY
You have a chance to buy an annuity that pays $1,000 at the end of each
year for 5 years. You could earn 6% on your money in other investments
with equal risk. What is the most you should pay for the annuity?
a. $3,324.89
b. $2,591.45
c. $4,212.36
d. $2,011.87
e. $3,854.13
PV of an ordinary annuity
18.
Answer: e
EASY
Your father is about to retire, and he wants to buy an annuity that will
provide him with $50,000 of income per year for 20 years, beginning a year
from today. The going rate on such annuities is 6%. How much would it
cost him to buy such an annuity today?
a. $488,349.15
b. $416,110.34
c. $517,513.68
d. $615,976.84
e. $573,496.06
PV of an annuity due
19.
Answer: b
EASY
What is the PV of an annuity due with 5 payments of $3,000 at an interest
rate of 5%?
a. $11,110.34
b. $13,637.85
c. $12,513.68
d. $14,976.84
e. $15,349.15
PV of an annuity due
20.
Answer: d
EASY
You have a chance to buy an annuity that pays $1,000 at the beginning of
each year for 5 years.
You could earn 6% on your money in other
investments with equal risk.
What is the most you should pay for the
annuity?
a. $2,854.13
b. $2,591.45
c. $3,324.89
d. $4,465.11
e. $5,011.87
Chapter 2: Time Value of Money
Page 21
PV of an annuity due
21.
Answer: a
EASY
Your father is about to retire, and he wants to buy an annuity that will
provide him with $50,000 of income a year for 20 years, with the first
payment coming immediately. The going rate on such annuities is 6%. How
much would it cost him to buy the annuity today?
a. $607,905.82
b. $416,110.34
c. $517,513.68
d. $615,976.84
e. $488,349.15
Payments on an ordinary annuity
22.
Answer: c
Suppose you inherited $200,000 and invested it at 6% per year.
could you withdraw at the end of each of the next 15 years?
EASY
How much
a. $24,764.40
b. $23,431.83
c. $20,592.55
d. $17,843.15
e. $15,119.76
Payments on an ordinary annuity
23.
Answer: e
EASY
Your father has $500,000 and wants to retire.
He expects to live for
another 20 years, and to be able to earn 8% on his invested funds. How
much could he withdraw at the end of each of the next 20 years and end up
with zero in the account?
a. $55,119.76
b. $53,431.83
c. $54,764.40
d. $47,843.15
e. $50,926.10
Payments on an annuity due
24.
Answer: b
EASY
Your father has $500,000 and wants to retire.
He expects to live for
another 20 years, and he also expects to earn 8% on his invested funds.
How much could he withdraw at the beginning of each of the next 20 years
and end up with zero in the account?
a. $53,431.83
b. $47,153.80
c. $54,764.40
d. $47,843.15
e. $45,119.76
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Chapter 2: Time Value of Money
Payments on an annuity due
25.
Answer: d
EASY
Suppose you inherited $200,000 and invested it at 6% per year. How much
could you withdraw at the beginning of each of the next 15 years?
a. $17,843.15
b. $13,431.83
c. $14,764.40
d. $19,426.94
e. $15,119.76
Years to deplete an ordinary annuity
26.
Answer: a
EASY
Your father has $500,000 invested at 8%, and he now wants to retire. He
wants to withdraw $50,000 at the end of each year, beginning at the end of
this year. How many years will it take to exhaust his funds, i.e., run
the account down to zero?
a. 20.91 years
b. 18.49 years
c. 11.34 years
d. 13.91 years
e. 15.27 years
Years to deplete an annuity due
27.
Answer: c
EASY
Your father has $500,000 invested at 8%, and he now wants to retire. He
wants to withdraw $50,000 at the beginning of each year, beginning
immediately. How many years will it take to exhaust his funds, i.e., run
the account down to zero?
a. 11.34 years
b. 18.49 years
c. 17.54 years
d. 13.91 years
e. 15.27 years
Interest rate implicit in an annuity
28.
Answer: e
EASY
You just won the state lottery.
The state gives you the choice of
$1,000,000 today or a 20year annuity of $75,000, with the first payment
coming one year from today.
What rate of return is built into the
annuity?
a. 5.91%
b. 6.71%
c. 7.10%
d. 5.59%
e. 4.22%
Chapter 2: Time Value of Money
Page 23
Interest rate implicit in an annuity
29.
Answer: b
EASY
Your girlfriend just won the Power Ball lottery. She has the choice of
$10,000,000 today or a 30year annuity of $500,000, with the first payment
coming today. What rate of return is built into the annuity?
a. 2.71%
b. 3.08%
c. 4.10%
d. 3.59%
e. 3.91%
Interest rate implicit in an annuity
30.
Answer: d
EASY
Assume that you own an annuity that will pay you $10,000 per year for 10
years, with the first payment being made today. Your girlfriend’s father
offers to give you $45,000 for the annuity. If you sell it, what rate of
return would your girlfriendโs father earn on his investment?
a. 25.59%
b. 26.71%
c. 17.10%
d. 24.63%
e. 22.91%
PV of an annuity due
31.
Answer: a
EASY
You own an oil well that will pay you $25,000 per year for 8 years, with
the first payment being made today.
If you think a fair return on the
well is 7%, how much should you ask if you decide to sell it?
a. $159,732
b. $116,110
c. $217,513
d. $315,976
e. $288,349
PV of an ordinary annuity plus an ending payment
32.
Answer: c
EASY
Whatโs the present value of a 6year ordinary annuity of $1,000 per year
plus an additional $1,500 at the end of Year 6 if the interest rate is 6%?
a. $5,324.89
b. $5,591.45
c. $5,974.77
d. $6,011.87
e. $4,854.13
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Chapter 2: Time Value of Money
PV of a perpetuity
33.
Answer: e
EASY
Whatโs the present value of a perpetuity that pays $100 per year if the
appropriate interest rate is 6%?
a. $2,854.13
b. $2,591.45
c. $1,324.89
d. $1,011.87
e. $1,666.67
Rate of return on a perpetuity
34.
Answer: b
EASY
Whatโs the rate of return you would earn if you paid $1,500 for a
perpetuity that pays $105 per year?
a. 6.71%
b. 7.00%
c. 7.30%
d. 5.59%
e. 5.91%
Dollar payments on a perpetuity
35.
Answer: d
EASY
What annual payment would you have to receive in order to earn an 8% rate
of return on a perpetuity that cost $1,500?
a. $127.84
b. $134.54
c. $151.29
d. $120.00
e. $143.65
PV of an uneven cash flow stream
36.
Answer: a
EASY
At a rate of 8%, what is the present value of the following cash flow
stream? $0 at Time 0; $100 at the end of Year 1; $300 at the end of Year
2; $0 at the end of Year 3; and $500 at the end of Year 4?
a. $717.31
b. $625.54
c. $788.32
d. $701.15
e. $690.49
PV of an uneven cash flow stream
37.
Answer: c
EASY/MEDIUM
An investment promises the following cash flow stream: $1,000 at Time 0;
$2,000 at the end of Year 1 (or at T=1); $3,000 at the end of Year 2; and
$5,000 at the end of Year 3.
At a discount rate of 5%, what is the
present value of the cash flow stream?
a. $9,324.89
b. $9,591.45
c. $9,945.04
d. $9,011.87
Chapter 2: Time Value of Money
Page 25
e. $9,854.13
FV of a lump sum, semiannually
38.
Whatโs the future value of $2,000 after 3
interest rate is 8%, compounded semiannually?
Answer: e
years
if
EASY/MEDIUM
the
appropriate
Answer: b
EASY/MEDIUM
a. $2,854.13
b. $2,781.45
c. $2,324.89
d. $2,011.87
e. $2,530.64
PV of a lump sum, semiannually
39.
Whatโs the present value of $2,000 discounted back 3
appropriate interest rate is 8%, compounded semiannually?
years
if
the
a. $1,110.34
b. $1,580.63
c. $1,413.68
d. $1,976.84
e. $1,349.15
FV of an uneven cash flow stream
40.
Answer: d
MEDIUM
At a rate of 8%, what is the future value of the following cash flow
stream? $0 at Time 0; $100 at the end of Year 1; $300 at the end of Year
2; $0 at the end of Year 3; and $500 at the end of Year 4?
a. $907.91
b. $991.43
c. $943.46
d. $975.89
e. $968.40
Interest rate built into an uneven cash flow stream
41.
Answer: a
MEDIUM
An investment costs $1,000 (CF at T = 0) and is expected to produce cash
flows of $50 at the end of each of the next 5 years, then an additional
lump sum payment of $1,000 at the end of the 5th year.
What is the
expected rate of return on this investment?
a. 5.0%
b. 6.7%
c. 7.1%
d. 5.5%
e. 5.9%
Page 26
Chapter 2: Time Value of Money
Interest rate built into an uneven cash flow stream
42.
Answer: c
MEDIUM
An investment costs $500 and is expected to produce cash flows of $50 at
the end of Year 1, $60 at the end of Year 2, $70 at the end of Year 3, and
$516 at the end of Year 4.
What rate of return would you earn if you
bought this investment?
a. 11.1%
b. 12.7%
c. 10.0%
d. 9.5%
e. 10.9%
FV of a lump sum, monthly
43.
Whatโs the future value of $2,000 after
interest rate is 8%, compounded monthly?
Answer: e
3
years
if
the
MEDIUM
appropriate
a. $2,854.13
b. $2,491.45
c. $2,324.89
d. $2,011.87
e. $2,540.47
PV of a lump sum, monthly
Answer: b
MEDIUM
44.
3
if
Whatโs the present value of $2,000 discounted back
appropriate interest rate is 8%, compounded monthly?
years
the
a. $1,491.45
b. $1,574.51
c. $1,324.89
d. $1,011.87
e. $1,854.13
APR vs. effective annual rate
45.
Answer: d
MEDIUM
Credit card issuers must by law print their Annual Percentage Rate (APR)
on their monthly statements.
If the APR is stated to be 15%, with
interest paid monthly, what is the EFF% on the card?
a. 15.59%
b. 16.71%
c. 17.10%
d. 16.08%
e. 12.91%
Nominal vs. effective annual rate
46.
Answer: a
MEDIUM
If a bank pays a 6% nominal rate, with monthly compounding, on deposits,
what effective annual rate does the bank pay?
a. 6.17%
b. 6.71%
c. 5.10%
d. 6.59%
Chapter 2: Time Value of Money
Page 27
e. 5.91%
Interest charges, simple interest
47.
Answer: c
MEDIUM
Columbus Corp. borrowed $10,000 at a rate of 8%, simple interest, with
interest paid at the end of each month.
The bank uses a 360day year.
How much interest would the firm have to pay in a 30day month?
a. $27.84
b. $34.54
c. $66.67
d. $51.29
e. $43.65
Fractional time periods
48.
Answer: e
MEDIUM
Suppose you deposited $5,000 in a bank account that pays 6% with daily
compounding and a 360day year.
How much could you withdraw after 7
months, assuming each month has 30 days?
a. $5,854.13
b. $5,591.45
c. $5,324.89
d. $5,011.87
e. $5,178.08
Loan amortization: payment
49.
Answer: b
MEDIUM
Suppose you borrowed $25,000 at a rate of 8% and must repay it in 4 equal
installments at the end of each of the next 4 years. How large would your
payments be?
a. $7,691.45
b. $7,548.02
c. $7,324.89
d. $7,011.87
e. $7,854.13
Loan amortization: interest
50.
Answer: d
MEDIUM
Suppose you borrowed $25,000 at a rate of 8% and must repay it in 4 equal
installments at the end of each of the next 4 years. How much interest
would you have to pay in the first year?
a. $2,081.87
b. $2,591.45
c. $2,324.89
d. $2,000.00
e. $2,854.13
Page 28
Chapter 2: Time Value of Money
Comparing the effective cost of two bank loans
51.
Answer: a
MEDIUM
Bank A offers to lend you $10,000 at a nominal rate of 7%, compounded
monthly. The loan (principal plus interest) must be repaid at the end of
the year. Bank B also offers to lend you the $10,000, but it will charge
8%, with interest due at the end of the year. What is the difference in
the effective annual rates charged by the two banks?
a. 0.77%
b. 1.71%
c. 1.10%
d. 1.59%
e. 0.91%
Mortgage payments
52.
Answer: c
MEDIUM
You are buying your first house for $220,000, and are paying $30,000 as a
down payment. You have arranged to finance the remaining $190,000 30year
mortgage with a 7% nominal interest rate and monthly payments. What are
the equal monthly payments you must make?
a. $1,513
b. $1,110
c. $1,264
d. $1,976
e. $1,349
Loan amortization: principal repayment
53.
Answer: e
MEDIUM/HARD
Suppose you borrowed $25,000 at a rate of 8% and must repay it in 4 equal
installments at the end of each of the next 4 years. By how much would
you reduce the amount you owe in the first year?
a. $5,349
b. $6,110
c. $6,513
d. $4,976
e. $5,548
Loan amortization: ending balance
54.
Answer: b
MEDIUM/HARD
Suppose you borrowed $25,000 at a rate of 8% and must repay it in 4 equal
installments at the end of each of the next 4 years. How much would you
still owe at the end of the first year, after you have made the first
payment?
a. $21,110
b. $19,452
c. $18,513
d. $18,976
e. $19,049
Chapter 2: Time Value of Money
Page 29
Retirement planning
55.
Answer: d
MEDIUM/HARD
Your sister turned 30 today, and she is planning to save $3,000 per year
for retirement, with the first deposit to be made one year from today.
She will invest in a mutual fund, which she expects to provide a return of
10% per year. She plans to retire 35 years from today, when she turns 65,
and she expects to live for 30 years after retirement, to age 95. Under
these assumptions, how much can she spend in each year after she retires?
Her first withdrawal will be made at the end of her first retirement year.
a. $78,976
b. $91,110
c. $88,513
d. $86,250
e. $83,049
Nonannual compounding
56.
Answer: a
MEDIUM/HARD
You just deposited $5,000 in a bank account that pays a 12% nominal
interest rate, compounded monthly. If you also add another $10,000 to the
account one year (12 months) from now and another $15,000 to the account
two years from now, how much will be in the account three years (36
months) from now?
a. $36,753.57
b. $33,431.83
c. $34,764.40
d. $37,843.15
e. $35,119.76
Lifetime subscription vs. annual payments
57.
Answer: c
MEDIUM/HARD
Your subscription to Making Money Monthly is about to run out. You plan
to take the magazine for the rest of your life, and you can renew it by
paying $100 per year, beginning immediately, or you can get a lifetime
subscription for $1,000, payable immediately. Assuming you can earn 7% on
your capital and the annual renewal rate will remain constant, how many
years must you live to make the lifetime subscription the better buy?
Round up if necessary to obtain a whole number of years. (Hint: Be sure
to remember that you are solving for how many years you must live, not for
how many payments must be made.)
a. 16 years
b. 14 years
c. 15 years
d. 17 years
e. 13 years
Page 30
Chapter 2: Time Value of Money
Comparing the effective cost of two bank loans
58.
Answer: e
MEDIUM/HARD
Bank A offers to lend you $10,000 at a nominal rate of 6%, simple
interest, with interest paid monthly.
Bank B offers to lend you the
$10,000, but it will charge 7%, simple interest, with interest paid at the
end of the year. What is the difference in the effective annual rates
charged by the two banks?
a. 1.17%
b. 1.12%
c. 0.91%
d. 1.28%
e. 0.83%
Retirement planning
59.
Answer: b
MEDIUM/HARD
It is now January 1, 2005. Tom and Jerry are cousins who were both born
on January 1, 1975. Both turned 30 today.
Their grandfather gave Tom
$4,000 on his 25th birthday, January 1, 2000, putting the funds into a
trust that will be paid to Tom on his 70th birthday, January 1, 2045.
Each year since 2000, the grandfather put an additional $4,000 in the
account on Tom’s birthday, and the grandfather’s own trustee will continue
making the $4,000 payments until January 1, 2045, when a 46th and final
$4,000 contribution will be made on Tom’s 70th birthday. The grandfather
wants Tom to work, not to be a “trust fund baby,” but he also wants to
insure that Tom is well provided for in his old age.
The grandfather has until now has been disappointed with Jerry, hence has
not given him anything, but they recently reconciled, and the grandfather
has decided to make an equivalent provision for Jerry. He will make the
first payment to a trust for Jerry today, and he has instructed his
trustee to make additional annual payments each year until January 1,
2045, when the 41st and final payment will be made. If both trusts earn
an annual return of 10%, how much must the grandfather put into Jerry’s
trust annually to enable him to receive the same amount as Tom on January
1, 2045, when they reach age 70?
a. $6,110
b. $6,492
c. $7,513
d. $5,976
e. $8,349
Chapter 2: Time Value of Money
Page 31
Saving to start a business
60.
Answer: d
HARD
After graduation, you plan to work for Mega Corporation for 10 years and
then start your own business. You expect to save $5,000 a year for the
first 5 years and $10,000 annually for the following 5 years, with the
first deposit being made a year from today.
The first $10,000 will be
deposited at the end of Year 5. In addition, your grandfather just gave
you a $20,000 graduation gift which you will deposit immediately. If the
account earns 8% compounded annually, how much will you have when you
start your business 10 years from now?
a. $185,976
b. $116,110
c. $217,513
d. $144,944
e. $128,349
Cash flow required to provide a given rate of return
61.
Answer: a
HARD
You have been offered a 7year investment at a price of $50,000. It will
pay $5,000 at the end of Year 1, $10,000 at the end of Year 2, and $15,000
at the end of Year 3, plus a fixed but currently unspecified cash flow, X,
at the end of Years 4 through 7. The payer is essentially riskless, so
you are sure the payments will be made, and you regard 9% as an
appropriate rate of return on riskless 7year investments. What cash flow
must the investment provide at the end of each of the final 4 years, that
is, what is X?
a. $10,158.58
b. $13,431.83
c. $14,764.40
d. $17,843.15
e. $15,119.76
Page 32
Chapter 2: Time Value of Money
Saving for college
62.
Answer: c
HARD
Nathan and Stephanie are saving for their daughter’s college education.
Their daughter, Paige, is now 8 years old and will be entering college 10
years from now (t = 10).
College tuition and expenses at State U. are
currently $16,000 a year and are expected to increase at a rate of 4% a
year. They expect Paige to graduate in 4 years (if Paige wants to go to
graduate school, she’s on her own). Tuition and other costs will be due
at the beginning of each school year (at t = 10, 11, 12, and 13).
So far, Nathan and Stephanie have built up $9,000 in the college savings
account. Their longrun financial plan is to contribute $3,000 a year at
the beginning of each of the next five years (at t = 0, 1, 2, 3, and 4).
Then they plan to make 6 equal annual contributions at the end of each of
the following 6 years (t = 5, 6, 7, 8, 9, and 10).
Their investment
account is expected to earn 8%. How large must the annual payments be in
the subsequent 6 years (t = 5, 6, 7, 8, 9, and 10) to meet their
daughterโs anticipated college costs?
a. $2,513
b. $3,110
c. $5,758
d. $2,976
e. $4,349
Loan amortization: repayment of principal
63.
Answer: e
HARD
Your company has just taken out a 1year installment loan for $100,000.
Monthly payments on the loan will be $8,978, due at the end of each month.
What percentage of the 3rd monthly payment will go toward the repayment of
principal?
a. 91.70%
b. 94.81%
c. 86.79%
d. 88.54%
e. 89.06%
Loan amortization: interest paid
64.
Answer: b
HARD
A homeowner just obtained a 30year (360month) mortgage for $120,000.
The mortgage has a fixed nominal annual rate of 8%, with monthly payments.
What percentage of the total payments made during the first 3 months will
go toward payment of interest?
a. 94.81%
b. 90.79%
c. 86.79%
d. 88.54%
e. 91.70%
Chapter 2: Time Value of Money
Page 33
Multiple Choice: Conceptual
PV and discount rate
65.
Answer: a
EASY
You are analyzing the value of an investment by calculating the present
value of its expected cash flows.
Which of the following would cause
the investment to look better?
a. The discount rate decreases.
b. The cash flows are extended over a longer period of time, but the
total amount of the cash flows remains the same.
c. The discount rate increases.
d. The riskiness of the projectโs cash flows increases.
e. The total amount of cash flows remains the same, but more of the
cash flows are received in the later years and less are received in
the earlier years.
Time value concepts
66.
Answer: e
EASY
Which of the following statements is NOT CORRECT, assuming positive
interest rates?
a. A 5year $100 annuity due will have a higher present value than
similar ordinary annuity.
b. A 15year, $100,000 mortgage will have larger monthly payments than
an otherwise similar 30year mortgage.
c. If an investment pays 10% interest compounded annually, its
effective rate will also be 10%.
d. Securities A and B offer the same nominal rate of interest, but A
pays interest quarterly and B pays semiannually. Investment B will
have the higher present value.
e. An investment’s nominal interest rate will always be equal to or
greater than its effective annual rate.
Time value concepts
67.
Answer: c
EASY
A lump sum payment of $1,000 is due at the end of 5 years. The nominal
interest rate is 10%, semiannual compounding.
Which of the following
statements is CORRECT?
a. The present value of the $1,000 would be greater if interest were
compounded monthly rather than semiannually.
b. The periodic rate is greater than 5%.
c. The periodic interest rate is 5%.
d. The present value would be greater if the lump sum were discounted
back for more periods.
e. The PV if the $1,000 lump sum has a higher present value than the PV
of a 5year, $200 ordinary annuity.
Page 34
Chapter 2: Time Value of Money
Time value concepts
68.
Answer: e
EASY
Which of the following investments will have the highest future value
at the end of 5 years? Assume that the effective annual rate for all
investments is the same and greater than zero.
a. A pays $50 at the end of every 6month period for the next 5 years
(a total of 10 payments).
b. B pays $50 at the beginning of every 6month period for the next
5 years (a total of 10 payments).
c. C pays $500 at the end of 5 years (a total of one payment).
d. D pays $100 at the end of every year for the next 5 years (a total
of 5 payments).
e. E pays $100 at the beginning of every year for the next 5 years (a
total of 5 payments).
Effective annual rate
69.
Answer: b
EASY
Which of the following bank accounts has the highest effective annual
return?
a. An account that pays 10% nominal interest with monthly compounding.
b. An account that pays 10% nominal interest with daily compounding.
c. An account that pays 10% nominal interest with annual compounding.
d. An account that pays 9% nominal interest with daily compounding.
e. An account that pays 9% nominal interest with monthly compounding.
Effective annual rate
70.
Answer: d
EASY
You are interested in investing your money in a bank account. Which of
the following banks provides you with the highest effective rate of
interest?
a. Bank 1; 8.0% with monthly compounding.
b. Bank 2; 8.0% with annual compounding.
c. Bank 3; 8.0% with quarterly compounding.
d. Bank 4; 8.0% with daily (365day) compounding.
e. Bank 5; 8.2% with annual compounding.
Chapter 2: Time Value of Money
Page 35
Quarterly compounding
71.
EASY
Your bank account pays an 8% nominal rate of interest. The interest is
compounded quarterly. Which of the following statements is CORRECT?
a. The periodic rate of interest
interest is 4%.
b. The periodic rate of interest
interest is greater than 8%.
c. The periodic rate of interest
interest is 8%.
d. The periodic rate of interest
interest is 8%.
e. The periodic rate of interest
interest is greater than 8%.
Annuities
72.
Answer: e
is
2%
and
the
effective
rate
of
is
8%
and
the
effective
rate
of
is
4%
and
the
effective
rate
of
is
8%
and
the
effective
rate
of
is
2%
and
the
effective
rate
of
Answer: c
MEDIUM
Suppose someone offered you the choice of two equally risky annuities,
each paying $10,000 per year for five years.
One is an ordinary (or
deferred) annuity, while the other is an annuity due.
Which of the
following statements is CORRECT?
a. The present value of the ordinary annuity must exceed the present
value of the annuity due, but the future value of an ordinary
annuity may be less than the future value of the annuity due.
b. The present value of the annuity due exceeds the present value of
the ordinary annuity, while the future value of the annuity due is
less than the future value of the ordinary annuity.
c. The present value of the annuity due exceeds the present value of
the ordinary annuity, and the future value of the annuity due also
exceeds the future value of the ordinary annuity.
d. If interest rates increase, the difference between the present value
of the ordinary annuity and the present value of the annuity due
remains the same.
e. The present value of the ordinary annuity exceeds the present value
of the annuity due, and the future value of an ordinary annuity also
exceeds the future value of the annuity due.
Page 36
Chapter 2: Time Value of Money
Amortization
73.
Answer: e
MEDIUM
A $10,000 loan is to be amortized over 5 years, with annual endofyear
payments. Given these facts, which of these statements is CORRECT?
a. The annual payments would be larger if the interest rate were lower.
b. If the loan were amortized over 10 years rather than 5 years, and if
the interest rate were the same in either case, the first payment
would include more dollars of interest under the 5year amortization
plan.
c. The last payment would have a higher proportion of interest than the
first payment.
d. The proportion of interest versus principal repayment would be the
same for each of the 5 payments.
e. The proportion of each payment that represents interest as opposed
to repayment of principal would be higher if the interest rate were
higher.
Amortization
74.
Answer: b
MEDIUM
Which of the following statements regarding a 30year (360month)
$100,000 fixedrate mortgage is CORRECT?
(Ignore all taxes and
transactions costs.)
a. The remaining balance after three years will be $100,000 less the
total amount of interest paid during the first 36 months.
b. The proportion of the monthly payment that goes towards repayment of
principal will be higher 10 years from now than it will be this
year.
c. The monthly payment on the mortgage will steadily decline over time.
d. The outstanding balance gets paid off at a faster rate early in a
loanโs life, rather than later.
e. Because it is a fixed rate mortgage, the amount paid in interest per
payment is constant.
Amortization
75.
Answer: a
MEDIUM
Which of the following statements regarding a 30year, $100,000
mortgage with a nominal interest rate of 10%, compounded monthly, is
NOT CORRECT?
a. The monthly payments will decline over time.
b. The proportion of the monthly payment that represents interest will
be lower for the last payment than for the first payment on the
loan.
c. The total dollar amount of principal being paid off each month gets
larger as the loan approaches maturity.
d. The amount paid toward interest in the first payment would be lower
if the nominal interest rate were 8%.
e. Over 90% of the first payment goes toward interest.
Chapter 2: Time Value of Money
Page 37
Time value concepts
76.
Answer: d
MEDIUM
Which of the following is NOT CORRECT?
a. The present value of a 5year, $100 annuity due will exceed the
present value of a 5year, $100 ordinary annuity.
b. If a loan has a nominal rate of 10%, then the effective rate can
never be less than 10%.
c. If there is annual compounding, then the effective, periodic, and
nominal rates of interest are all the same.
d. An investment that compounds interest semiannually, and has a
nominal rate of 10%, will have an effective rate less than 10%.
e. The proportion of the payment of a fully amortized loan that goes
toward interest declines over time.
PART II โ Questions and Problems from Prior Test Bank not used in Part I
Multiple Choice: Problems
EASY (#77 through #86)
Growth rate
77.
Answer: d
In 1958 the average tuition for one year at an Ivy League school was
$1,800.
Thirty years later, in 1988, the average cost was $13,700.
What was the growth rate in tuition over the 30year period?
a. 12%
b. 9%
c. 6%
d. 7%
e. 8%
Interest rate
78.
Answer: b
South Penn Trucking is financing a new truck with a loan of $10,000 to
be repaid in 5 annual endofyear installments of $2,504.56.
What
annual interest rate is the company paying?
a. 7%
b. 8%
c. 9%
d. 10%
e. 11%
Page 38
Chapter 2: Time Value of Money
Effect of inflation
79.
Answer: c
At an inflation rate of 9%, the purchasing power of $1 would be cut in
half in 8.04 years.
How long to the nearest year would it take the
purchasing power of $1 to be cut in half if the inflation rate were
only 4%?
a. 12 years
b. 15 years
c. 18 years
d. 20 years
e. 23 years
Time for a sum to double
80.
Answer: d
You are currently investing your money in a bank account that has a
nominal annual rate of 7%, compounded monthly. How many years will it
take for you to double your money?
a. 8.67
b. 9.15
c. 9.50
d. 9.93
e. 10.25
Time for lump sum to grow
81.
Answer: e
Jill currently has $300,000 in a brokerage account that pays 10%
interest.
Assuming Jill makes no additional contributions to the
account, how many years will it take for her to have $1,000,000 in the
account?
a. 23.33 years
b. 3.03 years
c. 16.66 years
d. 33.33 years
e. 12.63 years
Monthly loan payments
82.
Answer: c
You are considering buying a new, $15,000 car, and you have $2,000 to
put toward a down payment.
If you can negotiate a nominal annual
interest rate of 10% and finance the car over 60 months, what are your
monthly car payments?
a. $216.67
b. $252.34
c. $276.21
d. $285.78
e. $318.71
Chapter 2: Time Value of Money
Page 39
FV of an annuity
83.
Answer: e
What is the future value of a 5year ordinary annuity with annual
payments of $200, evaluated at 15%?
a. $ 670.44
b. $ 842.91
c. $1,169.56
d. $1,522.64
e. $1,348.48
PV of an annuity
84.
Answer: a
What is the present value of a 5year, $200 ordinary annuity, evaluated
at 15%?
a. $ 670.43
b. $ 842.91
c. $1,169.56
d. $1,348.48
e. $1,522.64
PV of a perpetuity
85.
Answer: c
You can buy a perpetuity that pays $1,000 annually, and your required
rate of return on this investment is 15%. You should be indifferent to
buying or not buying the investment if it were offered at a price of
a. $5,000.00
b. $6,000.00
c. $6,666.67
d. $7,500.00
e. $8,728.50
Required annuity payments
86.
Answer: b
If a 5year ordinary annuity has a present value of $1,000, and if the
interest rate is 10%, what is the amount of each annuity payment?
a. $240.42
b. $263.80
c. $300.20
d. $315.38
e. $346.87
Page 40
Chapter 2: Time Value of Money
EASY/MEDIUM (#87 through #94)
Quarterly compounding and FV
87.
Answer: a
If $100 is placed in an account earning
quarterly, what will it be worth in 5 years?
a
nominal
4%,
compounded
a. $122.02
b. $105.10
c. $135.41
d. $120.90
e. $117.48
PV of an uneven CF stream
88.
Answer: b
A real estate investment has the following expected cash flows:
Year
1
2
3
4
Cash Flows
$10,000
25,000
50,000
35,000
If the discount rate is 8%, what is the investmentโs present value?
a. $103,799
b. $ 96,110
c. $ 95,353
d. $120,000
e. $ 77,592
Effective annual rate
89.
Answer: c
Bank A offers to lend a firm funds for an expansion, at a nominal rate
of 8%, compounded monthly. Bank B will charge 9%, with interest due at
the end of the year.
What is the difference in the effective annual
rates charged by the two banks?
a. 0.25%
b. 0.50%
c. 0.70%
d. 1.00%
e. 1.25%
Chapter 2: Time Value of Money
Page 41
Effective annual rate
90.
Answer: b
You recently received a no annual fee credit card offer that states
that the annual percentage rate (APR) is 18% on outstanding balances.
What is the effective annual interest rate?
(Hint:
Remember these
companies bill you monthly.)
a. 18.81%
b. 19.56%
c. 19.25%
d. 20.00%
e. 18.00%
Effective annual rate
91.
Answer: b
Which of the following investments has the highest effective annual
rate (EAR)? (Assume that all CDs are of equal risk.)
a. A bank CD that pays 10% quarterly.
b. A bank CD that pays 10% monthly.
c. A bank CD that pays 10.2% annually.
d. A bank CD that pays 10% semiannually.
e. A bank CD that pays 9.6% daily (on a 365day basis).
Effective annual rate
92.
Answer: b
Elizabeth has $35,000 in an investment account, but she wants the account
to grow to $100,000 in 10 years without making any additional
contributions to the account. What effective annual rate of interest does
she need to earn on the account to meet her goal?
a. 9.03%
b. 11.07%
c. 10.23%
d. 8.65%
e. 12.32%
Effective annual rate
93.
Answer: a
Which one of the following investments provides the highest effective
rate of return?
a. An investment that has a 9.9% nominal rate and quarterly annual
compounding.
b. An investment that has a 9.7% nominal rate and daily (365)
compounding.
c. An investment that has a 10.2% nominal rate and annual compounding.
d. An investment that has a 10% nominal rate and semiannual
compounding.
e. An investment that has a 9.6% nominal rate and monthly compounding.
Page 42
Chapter 2: Time Value of Money
Nominal and effective rates
94.
Answer: b
An investment pays you 9% interest compounded semiannually.
A second
investment of equal risk, pays interest compounded quarterly. What
nominal rate of interest would you have to receive on the second
investment in order to make you indifferent between the two
investments?
a. 8.71%
b. 8.90%
c. 9.00%
d. 9.20%
e. 9.31%
MEDIUM (#95 through #122)
FV of an annuity
95.
Answer: a
Today is your 23rd birthday, and you just received a gift of $1,000.
You have used the money to open up a brokerage account. Your plan is
to contribute an additional $2,000 to the account each year on your
birthday, up through and including your 65th birthday, starting next
year. The account has an annual expected return of 12%. How much do
you expect to have in the account right after you make the final $2,000
contribution on your 65th birthday?
a. $2,045,442
b. $1,811,996
c. $2,292,895
d. $1,824,502
e. $2,031,435
FV under monthly compounding
96.
Answer: a
Bill plans to deposit $200 into a bank account at the end of every
month. The bank account has a nominal interest rate of 8% and interest
is compounded monthly. How much will Bill have in the account at the
end of 2ยฝ years (30 months)?
a. $ 6,617.77
b. $
502.50
c. $ 6,594.88
d. $22,656.74
e. $ 5,232.43
Chapter 2: Time Value of Money
Page 43
PV of an uneven CF stream
97.
Answer: c
Assume that you will receive $2,000 a year in Years 1 through 5, $3,000
a year in Years 6 through 8, and $4,000 in Year 9, with all cash flows
to be received at the end of the year.
If you require a 14% rate of
return, what is the present value of these cash flows?
a. $ 9,851
b. $13,250
c. $11,714
d. $15,129
e. $17,353
FV of a sum
98.
Answer: b
You deposited $1,000 in a savings account that pays 8% interest,
compounded quarterly, planning to use it to finish your last year in
college.
Eighteen months later, you decide to go to the Rocky
Mountains to become a ski instructor rather than continue school, so
you close out your account. How much money will you receive?
a. $1,171
b. $1,126
c. $1,082
d. $1,163
e. $1,008
FV of annuity due
99.
Answer: d
Starting on her 23rd birthday, Janet plans to start saving for her
retirement. She will contribute $1,000 to a brokerage account each year
on her birthday, starting today.
Her 42nd and final contribution will
th
take place on her 64 birthday. Janetโs aunt gave her $10,000 today to
get the account started. If the account has an expected annual return of
10%, how much will Janet expect to have in her account on her 65th
birthday?
a. $ 985,704
b. $1,034,489
c. $1,085,274
d. $1,139,038
e. $1,254,041
Page 44
Chapter 2: Time Value of Money
Time value of money and retirement
100.
Answer: b
Today, Bruce and Brenda each have $150,000 in an investment account.
No other contributions will be made to their investment accounts. Both
have the same goal: They each want their account to reach $1 million,
at which time each will retire. Bruce has his money invested in riskfree securities with an expected annual return of 5%.
Brenda has her
money invested in a stock fund with an expected annual return of 10%.
How many years after Brenda retires will Bruce retire?
a. 12.6
b. 19.0
c. 19.9
d. 29.4
e. 38.9
FV of a sum
101.
Answer: d
Suppose you put $100 into a savings account today, the account pays a
nominal annual interest rate of 6%, compounded semiannually, and you
withdraw $100 after 6 months.
What would your ending balance be 20
years after the initial $100 deposit was made?
a. $226.20
b. $115.35
c. $ 62.91
d. $ 9.50
e. $ 3.00
FV of an annuity
102.
Answer: e
Your bank account pays a nominal interest rate of 6%, compounded daily.
Your plan is to deposit $500 in the account today, and deposit $1,000
in the account at the end of each of the next three years.
How much
will you have in the account at the end of three years, after making
your final deposit?
a. $2,591
b. $3,164
c. $3,500
d. $3,779
e. $3,788
Chapter 2: Time Value of Money
Page 45
FV of an annuity
103.
Answer: c
Terry Austin is 30 years old and is saving for her retirement.
She
plans to make 36 contributions to her retirement account at the
beginning of each of the next 36 years. The first contribution will be
made today (t = 0) and the final contribution will be made 35 years
from today (t = 35). The retirement account will earn a return of 10%
a year. If each contribution she makes is $3,000, how much will be in
the retirement account 35 years from now (t = 35)?
a. $894,380
b. $813,073
c. $897,380
d. $987,118
e. $978,688
FV of an annuity
104.
Answer: d
20th
Today is your
birthday, and your parents just gave you $5,000 that
you plan to use to open a stock brokerage account.
You plan to add
$500 to the account each year on your birthday.
Your first $500
contribution will come one year from now on your 21st birthday. Your 45th
and final $500 contribution will occur on your 65th birthday. You plan
to withdraw $5,000 from the account five years from now on your 25 th
birthday to take a trip to Europe. You also anticipate that you will
need to withdraw $10,000 from the account 10 years from now on your 30th
birthday to take a trip to Asia. You expect that the account will have
an average annual return of 12%. How much money do you anticipate that
you will have in the account on your 65th birthday, following your final
contribution?
a. $385,863
b. $413,028
c. $457,911
d. $505,803
e. $566,498
FV of annuity due
105.
Answer: d
You are saving money so that you can purchase a house in five years.
You plan to contribute six payments of $3,000 a year.
The first
payment will be made today (t = 0) and the final payment will be made
five years from now (t = 5).
If you earn 11% in your investment
account, how much money will you have in the account five years from
now (at t = 5)?
a. $19,412
b. $20,856
c. $21,683
d. $23,739
e. $26,350
Page 46
Chapter 2: Time Value of Money
FV of annuity due
106.
Answer: e
Today is your 21st birthday, and you are opening up an investment
account. You plan to contribute $2,000 per year on your birthday. The
first contribution will be made today, and the 45 th, and final,
contribution will be made on your 65th birthday. If you earn 10% a year
on your investments, how much money will you have in the account on
your 65th birthday, immediately after making your final contribution?
a. $1,581,590.64
b. $1,739,749.71
c. $1,579,590.64
d. $1,387,809.67
e. $1,437,809.67
FV under monthly compounding
107.
Answer: e
You just put $1,000 in a bank account that pays 6% nominal annual
interest, compounded monthly.
How much will you have in your account
after 3 years?
a. $1,006.00
b. $1,056.45
c. $1,180.32
d. $1,191.00
e. $1,196.68
FV under monthly compounding
108.
Answer: d
Steven just deposited $10,000 in a bank account that has a 12% nominal
interest rate, compounded monthly.
Steven also plans to contribute
another $10,000 to the account one year (12 months) from now and
another $20,000 to the account two years from now. How much will be in
the account three years (36 months) from now?
a. $57,231
b. $48,993
c. $50,971
d. $49,542
e. $49,130
FV under daily compounding
109.
Answer: a
You have $2,000 invested in a bank account that pays a 4% nominal
interest rate with daily compounding. How much money will you have in
the account in 132 days? (Assume there are 365 days in each year.)
a. $2,029.14
b. $2,028.93
c. $2,040.00
d. $2,023.44
e. $2,023.99
Chapter 2: Time Value of Money
Page 47
FV under nonannual compounding
110.
Answer: d
Josh and John (2 brothers) are each trying to save enough money to buy
their own cars.
Josh is planning to save $100 from every paycheck.
(He is paid every 2 weeks.)
John plans to put aside $150 each month
but has already saved $1,500.
Interest rates are currently quoted at
10%. Joshโs bank compounds interest every two weeks while Johnโs bank
compounds interest monthly. At the end of 2 years they will each spend
all their savings on a car.
What is the price of the most expensive
car purchased?
a. $5,744.29
b. $5,807.48
c. $5,703.02
d. $5,797.63
e. $5,898.50
FV under quarterly compounding
111.
Answer: c
An investment pays $100 every six months (semiannually) over the next
2.5 years.
Interest, however, is compounded quarterly, at a nominal
rate of 8%.
What is the future value of the investment after 2.5
years?
a. $520.61
b. $541.63
c. $542.07
d. $543.98
e. $547.49
FV under quarterly compounding
112.
Answer: d
Rachel wants to take a trip to England in 3 years, and saving to pay
for the trip.
Today (8/1/05) she made an initial deposit of $1,000.
Her plan is to add $2,000 to the account one year from now (8/1/06) and
another $3,000 to the account two years from now (8/1/07). The account
has a nominal interest rate of 7%, but the interest is compounded
quarterly. How much will Rachel have in the account three years from
today (8/1/08)?
a. $6,724.84
b. $6,701.54
c. $6,895.32
d. $6,744.78
e. $6,791.02
Page 48
Chapter 2: Time Value of Money
FV of an uneven CF stream
113.
Answer: e
You are saving money for your first house, and you plan to make regular
deposits into a brokerage account earning 14%.
Your first deposit of
$5,000 will be made today.
You also plan to make four additional
deposits at the beginning of each of the next four years. Your plan is
to increase your deposits by 10% a year. (That is, you plan to deposit
$5,500 at t = 1, and $6,050 at t = 2, etc.) How much money will be in
your account after five years?
a. $24,697.40
b. $30,525.00
c. $32,485.98
d. $39,362.57
e. $44,873.90
Present value
114.
Answer: c
Which of the following securities has the largest present value?
Assume in all cases that the annual interest rate is 8% and that there
are no taxes.
a. A fiveyear ordinary annuity that pays you $1,000 each year.
b. A fiveyear zero coupon bond that has a face value of $7,000.
c. A preferred stock issue that pays an $800 annual dividend in perpetuity.
(Assume that the first dividend is received one year from today.)
d. A sevenyear zero coupon bond that has a face value of $8,500.
e. A security that pays you $1,000 at the end of 1 year, $2,000 at the
end of 2 years, and $3,000 at the end of 3 years.
PV under monthly compounding
115.
Answer: b
You have just bought a 10year security that pays $500 every six
months. Another equally risky security also has a maturity of 10 years,
and pays 10%, compounded monthly (that is, the nominal rate is 10%).
What price should you have paid for the security that you just
purchased?
a. $6,108.46
b. $6,175.82
c. $6,231.11
d. $6,566.21
e. $7,314.86
PV under nonannual compounding
116.
Answer: c
An investment pays $500 at the end of every 6 months for the next 3
years.
The nominal interest rate is 12%; compounded quarterly.
What
is the present value of the investment?
a. $2,458.66
b. $2,444.67
c. $2,451.73
d. $2,463.33
Chapter 2: Time Value of Money
Page 49
e. $2,437.56
PV of an annuity
117.
Answer: a
A magazine subscription is running out and you can renew it by sending
$10 a year (the regular rate) or get a lifetime subscription to the
magazine for $100. Your cost of capital is 7%. How many years would
you have to live to make the lifetime subscription the better buy?
Payments for the regular subscription are made at the beginning of each
year. (Round up if necessary to obtain a whole number of years.)
a. 15 years
b. 10 years
c. 18 years
d. 7 years
e. 8 years
Value of a perpetuity
118.
Answer: c
You are willing to pay $15,625 to purchase a perpetuity that will pay
you and your heirs $1,250 each year, forever. If your required rate of
return does not change, how much would you be willing to pay if this
were a 20year annual payment, ordinary annuity instead of a
perpetuity?
a. $10,342
b. $11,931
c. $12,273
d. $13,922
e. $17,157
FV of an uneven CF stream
119.
Answer: d
After graduation, you plan to work for 10 years and then visit
Australia. You expect to save $1,000 a year for the first 5 years and
$2,000 annually for the next 5 years.
These savings cash flows will
start in one year.
In addition, your family has just given you a
$5,000 graduation gift. If your gift and all future contributions are
put into an account that pays 8% compounded annually, what will your
financial โstakeโ be when you leave for Australia 10 years from now?
a. $21,432
b. $28,393
c. $16,651
d. $31,148
e. $20,000
Page 50
Chapter 2: Time Value of Money
FV of an uneven CF stream
120.
Answer: c
Erika just put $10,000 into a new savings account, and she plans to
contribute another $20,000 one year from now, and $50,000 two years
from now. The savings account pays 6% annual interest. With no other
deposits or withdrawals, how much will she have in the account 10 years
from today?
a. $ 8,246.00
b. $116,937.04
c. $131,390.46
d. $164,592.62
e. $190,297.04
PV of an uneven CF stream
121.
Answer: a
What is the present value of the following cash flows, if the discount
rate is 12%?
0

0
12%
1

1
2

2,000
3

2,000
4

2,000
5

0
6 Periods

2,000
a. $3,277
b. $4,804
c. $5,302
d. $4,289
e. $2,804
PV of uncertain cash flows
122.
Answer: e
A 3year project has the following probability distributions
possible endofyear cash flows in each of the next three years:
Prob
0.30
0.40
0.30
Year 1
Cash Flow
$300
500
700
Prob
0.15
0.35
0.35
0.15
Year 2
Cash Flow
$100
200
600
900
Prob
0.25
0.75
for
Year 3
Cash Flow
$200
800
If the interest rate is 8%, what is the expected present value of the
project?
(Hint:
Find the expected cash flow in each year, then
evaluate those cash flows.)
a. $1,204.95
b. $ 835.42
c. $1,519.21
d. $1,580.00
e. $1,347.61
Chapter 2: Time Value of Money
Page 51
MEDIUM/HARD (#123 through #155)
Value of missing payments
123.
Answer: d
You recently purchased a 20year investment that pays you $100 at t =
1, $500 at t = 2, $750 at t = 3, and some fixed cash flow, X, at the
end of each of the remaining 17 years.
You purchased the investment
for $5,544.87.
Alternative investments of equal risk have a required
return of 9%. What is the annual cash flow received at the end of each
of the final 17 years, that is, what is X?
a. $600
b. $625
c. $650
d. $675
e. $700
Effective annual rate
124.
Answer: b
If it were evaluated with an interest rate of 0%, a 10year regular
annuity would have a present value of $3,755.50.
If the future
(compounded) value of this annuity, evaluated at Year 10, is $5,440.22,
what effective annual interest rate must the analyst be using to find
the future value?
a. 7%
b. 8%
c. 9%
d. 10%
e. 11%
Effective annual rate
125.
Answer: c
You want to borrow $1,000 from a friend for one year, and you propose
to pay her $1,120 at the end of the year. She agrees to lend you the
$1,000, but she wants you to pay her $10 of interest at the end of each
of the first 11 months plus $1,010 at the end of the 12th month. How
much higher is the effective annual rate under your friendโs proposal
than under your proposal?
a. 0.00%
b. 0.45%
c. 0.68%
d. 0.89%
e. 1.00%
Page 52
Chapter 2: Time Value of Money
EAR and FV of an annuity
126.
Answer: d
You plan to invest $5,000 at the end of each of the next 10 years in an
account that has a 9% nominal rate with interest compounded monthly.
How much will be in your account at the end of the 10 years?
a. $ 75,965
b. $967,571
c. $ 84,616
d. $ 77,359
e. $ 80,631
EAR and FV of an annuity
127.
Answer: b
An investment pays $5,000 at the end of each of the next five years.
You plan to invest the money in an account paying 8% interest,
compounded monthly.
How much will you have in the account after
receiving the final $5,000 payment in 5 years (60 months)?
a. $ 25,335.56
b. $ 29,508.98
c. $367,384.28
d. $304,969.90
e. $ 25,348.23
Remaining loan balance
128.
Answer: a
A bank recently loaned you $15,000 to buy a car.
The loan is for five
years (60 months) and is fully amortized. The nominal rate on the loan is
12%, and payments are made at the end of each month.
What will be the
remaining balance on the loan after you make the 30th payment?
a. $ 8,611.17
b. $ 8,363.62
c. $14,515.50
d. $ 8,637.38
e. $ 7,599.03
Remaining mortgage balance
129.
Answer: c
Jerry and Faith Hudson recently obtained a 30year (360month),
$250,000 mortgage with a 9% nominal interest rate.
What will be the
remaining balance on the mortgage after five years (60 months)?
a. $239,024
b. $249,307
c. $239,700
d. $237,056
e. $212,386
Chapter 2: Time Value of Money
Page 53
Remaining mortgage balance
130.
Answer: d
You just bought a house and have a $150,000 mortgage. The mortgage is
for 30 years and has a nominal rate of 8% (compounded monthly). After
36 payments (3 years) what will be the remaining balance on your
mortgage?
a. $110,376.71
b. $124,565.82
c. $144,953.86
d. $145,920.12
e. $148,746.95
Amortization
131.
Answer: c
The Howe family recently bought a house.
The house has a 30year,
$165,000 mortgage with monthly payments and a nominal interest rate of
8%.
What is the total dollar amount of interest the family will pay
during the first three years of their mortgage?
(Assume that all
payments are made at the end of the month.)
a. $ 3,297.78
b. $38,589.11
c. $39,097.86
d. $43,758.03
e. $44,589.11
Required annuity payments
132.
Answer: c
A baseball player is offered a 5year contract
following amounts at the end of each year:
Year 1:
Year 2:
Year 3:
Year 4:
Year 5:
that
pays
him
the
$1.2 million
1.6 million
2.0 million
2.4 million
2.8 million
Instead of accepting the contract, the baseball player asks his agent
to negotiate a contract that has a present value of $1 million more
than the present value of that which has been offered.
Moreover, the
player wants to receive his payments in the form of a 5year annuity
due. All cash flows are discounted at 10%. If the team were to agree
to the playerโs terms, what would be the playerโs annual salary (in
millions of dollars)?
a. $1.500
b. $1.659
c. $1.989
d. $2.343
e. $2.500
Page 54
Chapter 2: Time Value of Money
Required annuity payments
133.
Answer: b
Karen and her twin sister, Kathy, are celebrating their 30 th birthday
today. Karen has been saving for her retirement ever since their 25 th
birthday. On their 25th birthday, she made a $5,000 contribution to her
retirement account.
Every year thereafter on their birthday, she has
added another $5,000 to the account.
Her plan is to continue
contributing $5,000 every year on their birthday. Her 41st, and final,
$5,000 contribution will occur on their 65th birthday.
So far, Kathy has not saved anything for her retirement but she wants
to begin today.
Kathy plans to also contribute a fixed amount every
year. Her first contribution will occur today, and her 36 th, and final,
contribution will occur on their 65th birthday.
Assume that both
investment accounts earn an annual return of 10%.
How large does
Kathyโs annual contribution have to be for her to have the same amount
in her account at age 65, as Karen will have in her account at age 65?
a. $9,000.00
b. $8,154.60
c. $7,398.08
d. $8,567.20
e. $7,933.83
Required annuity payments
134.
Answer: c
Jim and Nancy just got married today.
They want to start saving so
they can buy an average house five years from today. The average house
in their town today sells for $120,000. Housing prices are expected to
increase 3% a year. When they buy their house five years from now, Jim
and Nancy expect to get a 30year (360month) mortgage with a 7%
nominal interest rate. They want the monthly payment on their mortgage
to be $500 a month.
They are starting to save today for a down payment on the house. The
down payment plus the mortgage will equal the expected price of the
house. Their plan is to deposit $2,000 in a brokerage account today and
then deposit a fixed amount at the end of each of the next five years.
Assuming that the brokerage account has an annual return of 10%, how
much do Jim and Nancy need to deposit at the end of each year in order
to accomplish their goal?
a. $10,634
b. $ 9,044
c. $ 9,949
d. $ 9,421
e. $34,569
Chapter 2: Time Value of Money
Page 55
Required annuity payments
135.
Answer: a
Today is your 25th birthday.
Your goal is to have $2 million by the
time you retire at age 65. So far you have nothing saved, but you plan
on making the first contribution to your retirement account today. You
plan on making three other contributions to the account, one at age 30,
age 35, and age 40.
Since you expect that your income will increase
rapidly over the next several years, the amount that you contribute at
age 30 will be double what you contribute today, the amount at age 35
will be three times what you contribute today, and the amount at age 40
will be four times what you contribute today.
Assume that your
investments will produce an average annual return of 10%.
Given your
goal and plan, what is the minimum amount you need to contribute to
your account today?
a. $10,145
b. $10,415
c. $10,700
d. $10,870
e. $11,160
Monthly vs. quarterly compounding
136.
Answer: c
The First National Bank offers a 5% nominal interest rate, compounded
monthly on its savings accounts, while the Second National Bank offers
the same effective annual return, but interest is compounded quarterly.
What nominal rate does the Second National Bank offer on its savings
accounts?
a. 5.12%
b. 5.00%
c. 5.02%
d. 1.28%
e. 5.22%
Effective annual rate
137.
Answer: d
Steaks Galore needs capital for its expansion program.
One bank will
lend the required $1,000,000 if Steaks Galore agrees to pay interest each
quarter and repay the principal at the end of the year. The quoted rate
is 10%.
A second lender offers 9%, daily compounding (365day year),
with interest and principal due at the end of the year.
What is the
difference in the effective annual rates (EFF%) charged by the two banks?
a. 0.31%
b. 0.53%
c. 0.75%
d. 0.96%
e. 1.25%
Page 56
Chapter 2: Time Value of Money
Amortization
138.
Answer: c
You are buying a factory for $250,000 by paying 20% as a down payment,
while the rest of the balance will be paid off over 30 years at a 12%
interest rate. What are the 30 equal annual payments?
a. $20,593
b. $31,036
c. $24,829
d. $50,212
e. $ 6,667
Amortization: repayment of principal
139.
Answer: a
You have just taken out an installment loan for $100,000. Assume that
the loan will be repaid in 12 equal monthly installments of $9,456 and
that the first payment will be due one month from today. How much of
your third monthly payment will go toward the repayment of principal?
a. $7,757.16
b. $6,359.12
c. $7,212.50
d. $7,925.88
e. $8,333.33
Amortization: interest paid
140.
Answer: c
A homeowner just obtained a $90,000 mortgage. The mortgage is for 30
years (360 months) and has a fixed nominal annual rate of 9%, with
monthly payments. What percentage of the total payments made the first
two years will go toward payment of interest?
a. 89.30%
b. 91.70%
c. 92.59%
d. 93.65%
e. 94.76%
Amortization: repayment of principal
141.
Answer: e
You recently obtained a $135,000, 30year mortgage with a nominal
interest rate of 7.25%.
Assume that payments are made at the end of
each month. What portion of the total payments made during the fourth
year will go towards the repayment of principal?
a. 9.70%
b. 15.86%
c. 13.75%
d. 12.85%
e. 14.69%
Chapter 2: Time Value of Money
Page 57
Amortization: remaining loan balance
142.
Answer: b
The Bunker Family recently entered into a 30year mortgage for
$300,000, with an 8% nominal interest rate.
Interest is compounded
monthly, and all payments are due at the end of the month. What will
be the remaining balance on the mortgage after five years?
a. $ 14,790.43
b. $285,209.57
c. $300,000.00
d. $366,177.71
e. $298,980.02
Amortization: remaining loan balance
143.
Answer: d
Recently, Jamie and Jake each bought new cars.
Both received a loan
from a local bank with a nominal interest rate of 12% where payments
are made at the end of each month, and they both pay the same monthly
payment. Jamieโs loan is for $15,000; however, his loan matures at the
end of 4 years (48 months), while Jakeโs loan matures in 5 years (60
months). After 48 months Jamieโs loan will be paid off, but what will
be the remaining balance on Jakeโs loan?
a. $ 1,998.63
b. $ 2,757.58
c. $ 3,138.52
d. $ 4,445.84
e. $11,198.55
NPV and nonannual discounting
144.
Answer: b
Your lease calls for payments of $500 at the end of each month for the
next 12 months. Now your landlord offers you a new 1year lease that
calls for zero rent for 3 months, then rental payments of $700 at the
end of each month for the next 9 months. You keep your money in a bank
time deposit that pays a nominal annual rate of 5%.
By what amount
would your net worth change if you accept the new lease? (Hint: Your
return per month is 5%/12 = 0.4166667%.)
a. $509.81
b. $253.62
c. +$125.30
d. +$253.62
e. +$509.81
Page 58
Chapter 2: Time Value of Money
FV of an annuity
145.
Answer: c
John and Julie Johnson are interested in saving for their retirement.
John and Julie have the same birthday–both are 50 years old today. They
started saving for their retirement on their 25th birthday, when they
received a $20,000 gift from Julieโs aunt and deposited the money in an
investment account.
Every year thereafter, the couple added another
$5,000 to the account.
(The first contribution was made on their 26th
birthday and the 25th contribution was made today on their 50th birthday.)
John and Julie estimate that they will need to withdraw $150,000 from the
account 3 years from now, to help meet college expenses for their 5
children.
The couple plans to retire on their 58th birthday, 8 years
from today. They will make a total of 8 more contributions, one on each
of their next 8 birthdays with the last payment made on their 58th
birthday. If the couple continues to contribute $5,000 to the account on
their birthday, how much money will be in the account when they retire?
Assume that the investment account earns 12% a year.
a. $1,891,521
b. $2,104,873
c. $2,289,627
d. $2,198,776
e. $2,345,546
FV of annuity due
146.
Answer: a
To save money for a new house, you begin contributing money to a
brokerage account.
You plan to make ten $1,500 contributions to the
brokerage account at the beginning of each of the next 10 years,
starting today (t = 0) and ending in nine years (t = 9). Assume that
the brokerage account pays a 9% return with quarterly compounding. How
much money do you expect to have in the brokerage account nine years
from now (t = 9)?
a. $23,127.49
b. $25,140.65
c. $25,280.27
d. $21,627.49
e. $19,785.76
Chapter 2: Time Value of Money
Page 59
Value of missing cash flow
147.
Answer: d
Foster Industries has a project that has the following cash flows:
Year
0
1
2
3
4
Cash Flow
$300.00
100.00
125.43
90.12
?
What cash flow will the project have to generate in the fourth year in
order for the project to have a 15% rate of return?
a. $ 15.55
b. $ 58.95
c. $100.25
d. $103.10
e. $150.75
EAR and FV of annuity
148.
Answer: c
Today you opened up a local bank account. Your plan is to make five
$1,000 contributions to this account.
The first $1,000 contribution
will occur today and then every six months you will contribute another
$1,000 to the account. (So your final $1,000 contribution will be made
two years from today).
The bank account pays a 6% nominal annual
interest, and interest is compounded monthly.
After two years, you
plan to leave the money in the account earning interest, but you will
not make any further contributions to the account.
How much will you
have in the account 8 years from today?
a. $7,092
b. $7,569
c. $7,609
d. $7,969
e. $8,070
Page 60
Chapter 2: Time Value of Money
FV of investment account
149.
Answer: b
Kelly and Brian Johnson are a recently married couple whose parents
advised them to start saving immediately in order to have enough money
down the road to pay for their retirement and their childrenโs college
expenses. Today (t = 0) is their 25th birthday (the couple shares the
same birthday).
The couple plan to have two children (Dick and Jane). Dick is expected
to enter college 20 years from now (t = 20); Jane is expected to enter
college 22 years from now (t = 22).
So in years t = 22 and t = 23
there will be two children in college. Each child will take 4 years to
complete college, and college costs are paid at the beginning of each
year of college.
College costs per child will be as follows:
Year
20
21
22
23
24
25
Cost per child
$58,045
62,108
66,456
71,108
76,086
81,411
Children in college
Dick
Dick
Dick and Jane
Dick and Jane
Jane
Jane
Kelly and Brian plan to retire 40 years from now at age 65 (at t = 40).
They plan to contribute $12,000 per year at the end of each year for
the next 40 years into an investment account that earns 10% per year.
This account will be used to pay for the college costs, and also to
provide a nest egg for Kelly and Brianโs retirement at age 65. How big
will Kelly and Brianโs nest egg (the balance of the investment account)
be when they retire at age 65 (t = 40)?
a. $1,854,642
b. $2,393,273
c. $2,658,531
d. $3,564,751
e. $4,758,333
PV of an uneven CF stream
150.
Answer: c
Find the present value of an income stream that has a negative flow of
$100 per year for 3 years, a positive flow of $200 in the 4 th year, and
a positive flow of $300 per year in Years 5 through 8. The appropriate
discount rate is 4% for each of the first 3 years and 5% for each of
the later years.
Thus, a cash flow accruing in Year 8 should be
discounted at 5% for some years and 4% in other years.
All payments
occur at yearend.
a. $ 528.21
b. $1,329.00
c. $ 792.49
d. $1,046.41
e. $ 875.18
Chapter 2: Time Value of Money
Page 61
PV of an uneven CF stream
151.
Answer: d
Hillary is trying to determine the cost of health care to college
students and parentsโ ability to cover those costs.
She assumes that
the cost of one year of health care for a college student is $1,000
today, that the average student is 18 when he or she enters college,
that inflation in health care cost is rising at the rate of 10% per
year, and that parents can save $100 per year to help cover their
childrenโs costs.
All payments occur at the end of the relevant
period, and the $100/year savings will stop the day the child enters
college (hence 18 payments will be made). Savings can be invested at a
nominal rate of 6%, annual compounding.
Hillary wants a health care
plan that covers the fully inflated cost of health care for a student
for 4 years, during Years 19 through 22 (with payments made at the end
of Years 19 through 22).
How much would the government have to set
aside now (when a child is born), to supplement the average parentโs
share of a childโs college health care cost?
The lump sum the
government sets aside will also be invested at 6%, annual compounding.
a. $1,082.76
b. $3,997.81
c. $5,674.23
d. $7,472.08
e. $8,554.84
Required annuity payments
152.
Answer: b
You are saving for the college education of your two children.
One
child will enter college in 5 years, while the other child will enter
college in 7 years.
College costs are currently $10,000 per year and
are expected to grow at a rate of 5% per year. All college costs are
paid at the beginning of the year. You assume that each child will be
in college for four years.
You currently have $50,000 in your educational fund.
Your plan is to
contribute a fixed amount to the fund over each of the next 5 years.
Your first contribution will come at the end of this year, and your
final contribution will come at the date when you make the first
tuition payment for your oldest child.
You expect to invest your
contributions into various investments, which are expected to earn 8%
per year. How much should you contribute each year in order to meet the
expected cost of your childrenโs education?
a. $2,894
b. $3,712
c. $4,125
d. $5,343
e. $6,750
Page 62
Chapter 2: Time Value of Money
Required annuity payments
153.
Answer: b
A young couple is planning for the education of their two children.
They plan to invest the same amount of money at the end of each of the
next 16 years. The first contribution will be made at the end of the
year and the final contribution will be made at the end of the year the
older child enters college.
The money will be invested in securities that are certain to earn a
return of 8% each year. The older child will begin college in 16 years
and the second child will begin college in 18 years.
The parents
anticipate college costs of $25,000 a year (per child).
These costs
must be paid at the end of each year. If each child takes four years
to complete their college degrees, then how much money must the couple
save each year?
a. $ 9,612.10
b. $ 5,477.36
c. $12,507.29
d. $ 5,329.45
e. $ 4,944.84
Required annuity payments
154.
Answer: c
Your father, who is 60, plans to retire in 2 years, and he expects to live
independently for 3 years. He wants a retirement income that has, in the
first year, the same purchasing power as $40,000 has today. However, his
retirement income will be a fixed amount, so his real income will decline
over time. His retirement income will start the day he retires, 2 years
from today, and he will receive a total of 3 retirement payments.
Inflation is expected to be constant at 5%. Your father has $100,000
in savings now, and he can earn 8% on savings now and in the future.
How much must he save each year, starting today and ending next year,
to meet his retirement goals?
a. $1,863
b. $2,034
c. $2,716
d. $5,350
e. $6,102
Chapter 2: Time Value of Money
Page 63
Required annuity payments
155.
Answer: c
You are considering an investment in a 40year security. The security
will pay $25 a year at the end of each of the first three years. The
security will then pay $30 a year at the end of each of the next 20
years. The nominal interest rate is assumed to be 8%, and the current
price (present value) of the security is $360.39.
Given this
information, what is the equal annual payment to be received from Year
24 through Year 40 (for 17 years)?
a. $35
b. $38
c. $40
d. $45
e. $50
HARD (#156 through #169)
Required annuity payments
156.
Answer: a
John and Jessica are saving for their childโs education.
Their
daughter is currently eight years old and will be entering college 10
years from now (t = 10).
College costs are currently $15,000 a year
and are expected to increase at a rate of 5% a year. They expect their
daughter to graduate in four years, and that all annual payments will
be due at the beginning of each year (t = 10, 11, 12, and 13).
Right now, John and Jessica have $5,000 in their college savings
account. Starting today, they plan to contribute $3,000 a year at the
beginning of each of the next five years (t = 0, 1, 2, 3, and 4). Then
their plan is to make six equal annual contributions at the end of each
of the following six years (t = 5, 6, 7, 8, 9, and 10).
Their
investment account is expected to have an annual return of 12%. How
large of an annual payment do they have to make in the subsequent six
years (t = 5, 6, 7, 8, 9, and 10) in order to meet their childโs
anticipated college costs?
a. $4,411
b. $7,643
c. $2,925
d. $8,015
e. $6,798
Page 64
Chapter 2: Time Value of Money
Required annuity payments
Answer: a
157. Today is Rachelโs 30th birthday.
Five years ago, Rachel opened a
brokerage account when her grandmother gave her $25,000 for her 25 th
birthday.
Rachel added $2,000 to this account on her 26 th birthday,
$3,000 on her 27th birthday, $4,000 on her 28th birthday, and $5,000 on
her 29th birthday. Rachelโs goal is to have $400,000 in the account by
her 40th birthday.
Starting today, she plans to contribute a fixed amount to the account
each year on her birthday. She will make 11 contributions, the first
one will occur today, and the final contribution will occur on her 40 th
birthday.
Complicating things somewhat is the fact that Rachel plans
to withdraw $20,000 from the account on her 35th birthday to finance the
down payment on a home. How large does each of these 11 contributions
have to be for Rachel to reach her goal? Assume that the account has
earned (and will continue to earn) an effective return of 12% a year.
a. $11,743.95
b. $10,037.46
c. $11,950.22
d. $14,783.64
e. $ 9,485.67
Required annuity payments
158.
Answer: c
John is saving for his retirement.
Today is his 40 th birthday.
John
first started saving when he was 25 years old.
On his 25th birthday,
John made the first contribution to his retirement account; he
deposited $2,000 into an account that paid 9% interest, compounded
monthly. Each year on his birthday, John contributes another $2,000 to
the account. The 15th (and last) contribution was made last year on his
39th birthday.
John wants to close the account today and move the money to a stock
fund that is expected to earn an effective return of 12% a year. Johnโs
plan is to continue making contributions to this new account each year
on his birthday.
His next contribution will come today (age 40) and
his final planned contribution will be on his 65 th birthday.
If John
wants to accumulate $3,000,000 in his account by age 65, how much must
he contribute each year until age 65 (26 contributions in all) to
achieve his goal?
a. $11,892
b. $13,214
c. $12,471
d. $10,388
e. $15,572
Chapter 2: Time Value of Money
Page 65
Required annuity payments
159.
Answer: a
Joe and June Green are planning for their childrenโs college education.
Joe would like his kids to attend his alma mater where tuition is
currently $25,000 per year. Tuition costs are expected to increase by
5% each year.
Their children, David and Daniel, just turned
2 and 3 years old today, September 1, 2002. They are expected to begin
college the year in which they turn 18 years old and each will complete
his schooling in four years.
College tuition must be paid at the
beginning of each school year.
Grandma Green invested $10,000 in a mutual fund the day each child was
born.
This was to begin the boysโ college fund (a combined fund for
both children). The investment has earned and is expected to continue
to earn 12% per year. Joe and June will now begin adding to this fund
every August 31st (beginning with August 31, 2003) to ensure that there
is enough money to send the kids to college.
How much money must Joe and June put into the college fund each of the
next 15 years if their goal is to have all of the money in the
investment account by the time Daniel (the oldest son) begins college?
a. $5,928.67
b. $7,248.60
c. $4,822.66
d. $7,114.88
e. $5,538.86
Required annuity payments
160.
Answer: b
Bob is 20 years old today and is starting to save money, so that he can
get his MBA.
He is interested in a 1year MBA program.
Tuition and
expenses are currently $20,000 per year, and they are expected to
increase by 5% per year.
Bob plans to begin his MBA when he is 26
years old, and since all tuition and expenses are due at the beginning
of the school year, Bob will make his one single payment six years from
today. Right now, Bob has $25,000 in a brokerage account, and he plans
to contribute a fixed amount to the account at the end of each of the
next six years (t = 1, 2, 3, 4, 5, and 6). The account is expected to
earn an annual return of 10% each year. Bob plans to withdraw $15,000
from the account two years from today (t = 2) to purchase a used car,
but he plans to make no other withdrawals from the account until he
starts the MBA program. How much does Bob need to put in the account
at the end of each of the next six years to have enough money to pay
for his MBA?
a. $1,494
b. $ 580
c. $4,494
d. $2,266
e. $3,994
Page 66
Chapter 2: Time Value of Money
Required annuity payments
161.
Answer: e
Suppose you are deciding whether to buy or lease a car. If you buy the
car, it will cost $17,000 today (t = 0). You expect to sell the car four
years (48 months) from now for $6,000 (at t = 48). As an alternative to
buying the car, you can lease the car for 48 months. All lease payments
would be made at the end of the month.
The first lease payment would
occur next month (t = 1) and the final lease payment would occur 48
months from now (t = 48). If you buy the car, you would do so with cash,
so there is no need to consider financing.
If you lease the car, there
is no option to buy it at the end of the contract. Assume that there are
no taxes, and that the operating costs are the same regardless of whether
you buy or lease the car. Assume that all cash flows are discounted at a
nominal annual rate of 12%, so the monthly periodic rate is
1%.
What is the breakeven lease payment?
(That is, at what monthly
payment would you be indifferent between buying and leasing the car?)
a. $333.00
b. $336.62
c. $339.22
d. $343.51
e. $349.67
Required annuity payments
162.
Answer: c
24th
Today is Craigโs
birthday, and he wants to begin saving for
retirement. To get started, his plan is to open a brokerage account, and
to put $1,000 into the account today. Craig intends to deposit $X into
the account each year on his subsequent birthdays until the age of 64.
In other words, Craig plans to make 40 contributions of $X. The first
contribution will be made one year from now on his 25th birthday, and the
40th (and final) contribution will occur on his 64th birthday.
Craig
plans to retire at age 65 and he expects to live until age 85. Once he
retires, Craig estimates that he will need to withdraw $100,000 from the
account each year on his birthday in order to meet his expenses. (That
is, Craig plans to make 20 withdrawals of
$100,000 eachโthe first
withdrawal will occur on his 65th birthday and the final one will occur
on his 84th birthday.) Craig expects to earn 9% a year in his brokerage
account.
Given his plans, how much does he need to deposit into the
account for each of the next 40 years, in order to reach his goal?
(That is, what is $X?)
a. $2,379.20
b. $2,555.92
c. $2,608.73
d. $2,657.18
e. $2,786.98
Chapter 2: Time Value of Money
Page 67
Annuity due vs. ordinary annuity
163.
Answer: e
Bill and Bob are both 25 years old today. Each wants to begin saving for
his retirement. Both plan on contributing a fixed amount each year into
brokerage accounts that have annual returns of 12%. Both plan on retiring
at age 65, 40 years from today, and both want to have $3 million saved by
age 65.
The only difference is that Bill wants to begin saving today,
whereas Bob wants to begin saving one year from today. In other words,
Bill plans to make 41 total contributions (t = 0, 1, 2, … 40), while Bob
plans to make 40 total contributions (t = 1, 2, … 40). How much more
than Bill will Bob need to save each year in order to accumulate the same
amount as Bill does by age 65?
a. $796.77
b. $892.39
c. $473.85
d. $414.48
e. $423.09
Amortization
164.
Answer: b
The Florida Boosters Association has decided to build new bleachers for
the football field.
Total costs are estimated to be $1 million, and
financing will be through a bond issue of the same amount.
The bond
will have a maturity of 20 years, a coupon rate of 8%, and has annual
payments.
In addition, the Association must set up a reserve to pay
off the loan by making 20 equal annual payments into an account that
pays 8%, annual compounding.
The interestaccumulated amount in the
reserve will be used to retire the entire issue at its maturity
20 years hence. The Association plans to meet the payment requirements
by selling season tickets at a $10 net profit per ticket.
How many
tickets must be sold each year to service the debt (to meet the
interest and principal repayment requirements)?
a. 5,372
b. 10,186
c. 15,000
d. 20,459
e. 25,000
Page 68
Chapter 2: Time Value of Money
Effective annual rate
165.
Answer: c
You have some money on deposit in a bank account that pays a nominal
(or quoted) rate of 8.0944%, but with interest compounded daily (using
a 365day year).
Your friend owns a security that calls for the
payment of $10,000 after 27 months.
The security is just as safe as
your bank deposit, and your friend offers to sell it to you for $8,000.
If you buy the security, by how much will the effective annual rate of
return on your investment change?
a. 1.87%
b. 1.53%
c. 2.00%
d. 0.96%
e. 0.44%
PMT and quarterly compounding
166.
Answer: b
Your employer has agreed to make 80 quarterly payments of $400 each
into a trust account to fund your early retirement. The first payment
will be made 3 months from now. At the end of 20 years (80 payments),
you will be paid 10 equal annual payments, with the first payment to be
made at the beginning of Year 21 (or the end of Year 20).
The funds
will be invested at a nominal rate of 8%, quarterly compounding, during
both the accumulation and the distribution periods.
How large will
each of your 10 receipts be? (Hint: You must find the EAR and use it
in one of your calculations.)
a. $ 7,561
b. $10,789
c. $11,678
d. $12,342
e. $13,119
Value of unknown withdrawal
167.
Answer: d
Steve and Robert were college roommates, and each is celebrating their
30th birthday today.
When they graduated from college nine years ago
(on their 21st birthday), they each received $5,000 from family members
for establishing investment accounts.
Steve and Robert have added
$5,000 to their separate accounts on each of their following birthdays
(22nd through 30th birthdays).
Steve has withdrawn nothing from the
account, but Robert made one withdrawal on his 27th birthday. Steve has
invested the money in Treasury bills that have earned a 6% annual
return, while Robert has invested his money in stocks that have earned
a 12% annual return.
Both Steve and Robert have the same amount in
their accounts today.
How much did Robert withdraw on his 27 th
birthday?
a. $ 7,832.22
b. $ 8,879.52
c. $10,865.11
d. $15,545.07
Chapter 2: Time Value of Money
Page 69
e. $13,879.52
Nonannual compounding
168.
A financial planner
receiving cash flows.
present value.
Answer: a
has offered you three possible options for
You must choose the option that has the highest
(1) $1,000 now and another $1,000 at the beginning of each of the 11
subsequent months during the remainder of the year, to be deposited
in an account paying a 12% nominal annual rate, but compounded
monthly (to be left on deposit for the year).
(2) $12,750 at the end of the year (assume a 12% nominal interest rate
with semiannual compounding).
(3) A payment scheme of 8 quarterly payments made over the next two
years. The first payment of $800 is to be made at the end of the
current quarter. Payments will increase by 20% each quarter. The
money is to be deposited in an account paying a 12% nominal annual
rate, but compounded quarterly (to be left on deposit for the
entire 2year period).
Which one would you choose?
a. Choice 1
b. Choice 2
c. Choice 3
d. Either one, since they all have the same present value.
e. Neither one, since they all have negative present values.
Breakeven annuity payment
169.
Answer: a
Linda needs a new car and is deciding whether to buy or lease the car.
She estimates that if she buys the car, it will cost her $17,000 today (t
= 0) and that she would sell the car four years from now for $7,000 (at t
= 4). If she were to lease the car she would make a fixed lease payment
at the end of each of the next 48 months (4 years).
Assume that the
operating costs are the same regardless of whether she buys or leases the
car.
Assume that if she leases, there are no upfront costs and that
there is no option to buy the car after four years. Linda estimates that
she should use a 6% nominal interest rate to discount the cash flows.
What is the breakeven lease payment?
(That is, at what monthly lease
payment would she be indifferent between buying and leasing the car?)
a. $269.85
b. $271.59
c. $275.60
d. $277.39
e. $279.83
Page 70
Chapter 2: Time Value of Money
Multiple Part:
(The following information applies to the next two problems.)
A 30year, $115,000 mortgage has a nominal annual rate of 7%.
are made at the end of each month.
Required mortgage payment
170.
All payments
Answer: b
What is the monthly payment on the mortgage?
a. $760.66
b. $765.10
c. $772.29
d. $774.10
e. $776.89
Remaining mortgage balance
171.
Answer: e
What is the remaining balance on the mortgage after 5 years?
a. $106,545.45
b. $106,919.83
c. $107,623.52
d. $107,988.84
e. $108,251.33
(The following information applies to the next two problems.)
Today is your 21st birthday and your parents gave you a gift of $2,000. You
just put this money in a brokerage account, and your plan is to add $1,000 to
the account each year on your birthday, starting on your 22nd birthday.
Time to accumulate a lump sum
172.
Answer: d
If you earn 10% a year in the brokerage account, what is the minimum
number of whole years it will take for you to have at least $1,000,000
in the account?
a. 41
b. 43
c. 45
d. 47
e. 48
Required annual rate of return
173.
Answer: c
Assume that you want to have $1,000,000 in the account by age 60 (39
years from today). What annual rate of return will you need to earn on
your investments in order to reach this goal?
a. 12.15%
b. 12.41%
c. 12.57%
d. 12.66%
Chapter 2: Time Value of Money
Page 71
e. 12.91%
(The following information applies to the next two problems.)
Your family recently bought a house.
You have a $100,000, 30year mortgage
with a 7.2% nominal annual interest rate. Interest is compounded monthly and
all payments are made at the end of the month.
Monthly mortgage payments
174.
Answer: c
What is the monthly payment on the mortgage?
a. $639.08
b. $674.74
c. $678.79
d. $685.10
e. $691.32
Amortization
175.
Answer: d
What percentage of the total payments during the first three years is
going towards the principal?
a. 9.6%
b. 10.3%
c. 11.7%
d. 12.9%
e. 13.4%
(The following information applies to the next two problems.)
The Jordan family recently purchased their first home.
The house has a 15year (180month), $165,000 mortgage.
The mortgage has a nominal annual
interest rate of 7.75%.
All mortgage payments are made at the end of the
month.
Monthly mortgage payments
Answer: d
176. What is the monthly payment on the mortgage?
a. $1,065.63
b. $1,283.61
c. $1,322.78
d. $1,553.10
e. $1,581.97
Remaining mortgage balance
177.
Answer: c
What will be the remaining balance on the mortgage after one year (right
after the 12th payment has been made)?
a. $152,879.31
b. $155,362.50
c. $158,937.91
Page 72
Chapter 2: Time Value of Money
d. $160,245.39
e. $160,856.84
(The following information applies to the next two problems.)
Victoria and David have a 30year, $75,000 mortgage with an 8% nominal annual
interest rate. All payments are due at the end of the month.
Amortization
178.
What percentage of their
towards interest payments?
Answer: d
monthly
payments
the
first
year
will
go
a. 7.76%
b. 9.49%
c. 82.17%
d. 90.51%
e. 91.31%
Amortization
179.
Answer: a
If Victoria and David were able to refinance their mortgage and replace
it with a 7% nominal annual interest rate, how much (in dollars) would
their monthly payment decline?
a. $ 51.35
b. $ 59.78
c. $ 72.61
d. $ 88.37
e. $104.49
(The following information applies to the next two problems.)
Karen and Keith have a $300,000, 30year (360month) mortgage. The mortgage
has a 7.2% nominal annual interest rate. Mortgage payments are made at the
end of each month.
Monthly mortgage payment
180.
Answer: c
What is the monthly payment on the mortgage?
a. $1,759.41
b. $1,833.33
c. $2,036.36
d. $2,055.29
e. $3,105.25
Amortization
181.
Answer: b
What percentage of the total payments the first year (the first twelve
months) will go towards repayment of principal?
a. 11.88%
b. 12.00%
Chapter 2: Time Value of Money
Page 73
c. 13.21%
d. 13.55%
e. 14.16%
Page 74
Chapter 2: Time Value of Money
Web Appendix 2A
Multiple Choice: Problems
PV continuous compounding
2A1.
Answer: b
EASY/MEDIUM
In six yearsโ time, you are scheduled to receive money from a trust
established by your grandparents.
When the trust matures there will
be $100,000 in the account.
If the account earns 9% compounded
continuously, how much is in the account today?
a. $ 23,456
b. $ 58,275
c. $171,600
d. $ 59,627
e. $ 61,385
FV continuous compounding
2A2.
Answer: a
MEDIUM
Assume one bank offers you a nominal annual interest rate of 6%
compounded daily while another bank offers you continuous compounding
at a 5.9% nominal annual rate. You decide to deposit $1,000 with each
bank.
Exactly two years later you withdraw your funds from both
banks. What is the difference in your withdrawal amounts between the
two banks?
a. $ 2.25
b. $ 0.09
c. $ 1.12
d. $ 1.58
e. $12.58
Continuous compounding
2A3.
Answer: b
MEDIUM
You have $5,438 in an account that pays 10% interest, compounded
continuously. If you deposited some funds 10 years ago, how much was
your original deposit?
a. $1,000
b. $2,000
c. $3,000
d. $4,000
e. $5,000
Chapter 2: Time Value of Money
Page 75
Continuous compounded interest rate
2A4.
Answer: a
MEDIUM/HARD
In order to purchase your first home you need a down payment of
$19,000 four years from today. You currently have $14,014 to invest.
In order to achieve your goal, what nominal interest rate, compounded
continuously, must you earn on this investment?
a. 7.61%
b. 7.26%
c. 6.54%
d. 30.56%
e. 19.78%
Payment and continuous compounding
2A5.
Answer: d
MEDIUM/HARD
You place $1,000 in an account that pays 7% interest compounded
continuously.
You plan to hold the account exactly three years.
Simultaneously, in another account you deposit money that earns
8% compounded semiannually.
If the accounts are to have the same
amount at the end of the three years, how much of an initial deposit
do you need to make now in the account that pays 8% interest
compounded semiannually?
a. $1,006.42
b. $ 986.73
c. $ 994.50
d. $ 975.01
e. $ 962.68
Continuous compounding
2A6.
Answer: d
MEDIUM/HARD
For a 10year deposit, what annual rate payable semiannually will
produce the same effective rate as 4% compounded continuously?
a. 2.02%
b. 2.06%
c. 3.95%
d. 4.04%
e. 4.12%
Continuous compounding
2A7.
Answer: b
MEDIUM/HARD
How much should you be willing to pay for an account today that will
have a value of $1,000 in 10 years under continuous compounding if the
nominal rate is 10%?
a. $354
b. $368
c. $385
d. $376
e. $370
Page 76
Chapter 2: Time Value of Money
Continuous compounding
2A8.
Answer: b
HARD
If you receive $15,000 today and can invest it at a 5% annual rate
compounded continuously, what will be your ending value after 20
years?
a. $35,821
b. $40,774
c. $75,000
d. $81,342
e. $86,750
Chapter 2: Time Value of Money
Page 77
CHAPTER 2
ANSWERS AND SOLUTIONS
1.
FV of a lump sum
N
I/YR
PV
PMT
FV
2.
3.
4.
5.
Page 78
Answer: e
EASY
Answer: b
EASY
Answer: d
EASY
100
5%
$1.00
$0.00
$131.50
PV of a lump sum
N
I/YR
PV
PMT
FV
EASY
10
8%
$100.00
$0.00
$215.89
FV of a lump sum
N
I/YR
PV
PMT
FV
Answer: c
3
4%
$2,000
$0
$2,249.73
FV of a lump sum
N
I/YR
PV
PMT
FV
EASY
5
10%
$100
$0
$161.05
FV of a lump sum
N
I/YR
PV
PMT
FV
Answer: a
3
6%
$1,888.92
$0
2,249.73
Chapter 2: Time Value of Money
6.
PV of a lump sum
N
I/YR
PV
PMT
FV
7.
8.
9.
10.
11.
EASY
Answer: b
EASY
Answer: e
EASY
Answer: a
EASY
14.21
5.00%
$100.00
$0
$200.00
Number of periods
N
I/YR
PV
PMT
FV
Answer: e
5
14.87%
$1.00
$0
$2.00
Number of periods
N
I/YR
PV
PMT
FV
EASY
10
5.00%
$613.81
$0
$1,000.00
Simple growth rate
N
I/YR
PV
PMT
FV
Answer: c
3
4%
$889.00
$0
$1,000.00
Interest rate on a simple lump sum investment
N
I/YR
PV
PMT
FV
EASY
100
10%
$0.73
$0
$10,000
PV of a lump sum
N
I/YR
PV
PMT
FV
Answer: a
6.64
11.00%
$2.00
$0
$4.00
Chapter 2: Time Value of Money
Page 79
12.
FV of an ordinary annuity
N
I/YR
PV
PMT
FV
13.
14.
15.
16.
17.
EASY
Answer: d
EASY
Answer: a
EASY
Answer: c
EASY
5
5.00%
$12,988.43
$3,000
$0.00
PV of an ordinary annuity
Page 80
Answer: b
3
9.00%
$0.00
$5,000
$17,865.65
PV of an ordinary annuity
N
I/YR
PV
PMT
FV
EASY
5
6.00%
$0.00
$3,000
$17,925.96
FV of an annuity due
BGN mode
N
I/YR
PV
PMT
FV
Answer: e
3
9.00%
$0.00
$5,000
$16,390.50
FV of an annuity due
BGN mode
N
I/YR
PV
PMT
FV
EASY
5
6.00%
$0.00
$3,000
$16,911.28
FV of an ordinary annuity
N
I/YR
PV
PMT
FV
Answer: c
Chapter 2: Time Value of Money
N
I/YR
PV
PMT
FV
18.
PV of an ordinary annuity
N
I/YR
PV
PMT
FV
19.
21.
22.
EASY
Answer: d
EASY
Answer: a
EASY
Answer: c
EASY
20
6.00%
$607,905.82
$50,000
$0.00
Payments on an ordinary annuity
N
I/YR
PV
PMT
FV
Answer: b
5
6.00%
$4,465.11
$1,000
$0.00
PV of an annuity due
BGN mode
N
I/YR
PV
PMT
FV
EASY
5
5.00%
$13,637.85
$3,000
$0.00
PV of an annuity due
BGN mode
N
I/YR
PV
PMT
FV
Answer: e
20
6.00%
$573,496.06
$50,000
$0.00
PV of an annuity due
BGN mode
N
I/YR
PV
PMT
FV
20.
5
6.00%
$4,212.36
$1,000
$0.00
15
6.00%
$200,000
$20,592.55
$0.00
Chapter 2: Time Value of Money
Page 81
23.
Payments on an ordinary annuity
N
I/YR
PV
PMT
FV
24.
25.
26.
27.
Page 82
Answer: d
EASY
Answer: a
EASY
Answer: c
EASY
20.91
8.00%
$500,000
$50,000.00
$0.00
Years to deplete an annuity due
BGN mode
N
I/YR
PV
PMT
FV
EASY
15
6.00%
$200,000
$19,426.94
$0.00
Years to deplete an ordinary annuity
N
I/YR
PV
PMT
FV
Answer: b
20
8.00%
$500,000
$47,153.80
$0.00
Payments on an annuity due
BGN mode
N
I/YR
PV
PMT
FV
EASY
20
8.00%
$500,000
$50,926.10
$0.00
Payments on an annuity due
BGN mode
N
I/YR
PV
PMT
FV
Answer: e
17.54
8.00%
$500,000
$50,000.00
$0.00
Chapter 2: Time Value of Money
28.
Interest rate implicit in an annuity
Answer: e
EASY
29.
N
20
I/YR
4.22%
PV
$1,000,000
PMT
$75,000
FV
$0.00
Interest rate implicit in an annuity
Answer: b
EASY
Answer: d
EASY
Answer: a
EASY
Answer: c
EASY
Answer: e
EASY
Answer: b
EASY
BGN mode
N
I/YR
PV
PMT
FV
30.
Interest rate implicit in an annuity
BGN mode
N
I/YR
PV
PMT
FV
31.
6
6.00%
$5,974.77
$1,000
$1,500
PV of a perpetuity
I/YR
PMT
PV
34.
8
7.00%
$159,732
$25,000
$0.00
PV of an ordinary annuity plus an ending payment
N
I/YR
PV
PMT
FV
33.
10
24.63%
$45,000
$10,000
$0.00
PV of an annuity due
BGN mode
N
I/YR
PV
PMT
FV
32.
30
3.08%
$10,000,000
$500,000
$0.00
6.00%
$100
$1,666.67 Divide PMT by I.
Rate of return on a perpetuity
Chapter 2: Time Value of Money
Page 83
Cost (PV)
PMT
I/YR
35.
Dollar payments on a perpetuity
Cost (PV)
I/YR
PMT
36.
CFs:
PV of CFs:
PV =
PV =
CFs:
PV of CFs:
PV =
PV =
38.
5%
0
$1,000
$1,000
$9,945.04
$9,945.04
Answer: a
EASY
Answer: c
EASY/MEDIUM
1
2
3
$2,000
$3,000
$5,000
$1,904.76
$2,721.09 $4,319.19
Found by summing individual PVs.
Found with a calculator or Excel to automate the process.
With a calculator, input the cash flows and I into the cash flow
register, then press the NPV key.
Answer: e
EASY/MEDIUM
N = Periods = years x periods/yr =
3
6
2
$0
I/Period = Nom. I/Periods per year =
8.00%
4.00%
$2,000
$2,530.64 Could be found using a calculator, the equation, or Excel. Note
that we must first convert to periods and rate per period.
PV of a lump sum, semiannually
Page 84
EASY
1
2
3
4
$100
$300
$0
$500
$92.59
$257.20
$0
$367.51
Find the individual PVs and sum them.
Automate the process using Excel or a calculator, by inputting
the data into the cash flow register and pressing the NPV key.
FV of a lump sum, semiannually
Years
Periods/Yr
PMT
Nom. I/YR
PV
FV
39.
8%
0
$0
$0
$717.31
$717.31
PV of an uneven cash flow stream
I/YR =
Answer: d
$1,500
8.00%
$120.00 Multiply Cost by I.
PV of an uneven cash flow stream
I/YR =
37.
$1,500
$105
7.00% Divide PMT by Cost.
Answer: b
EASY/MEDIUM
Chapter 2: Time Value of Money
Years
Periods/Yr
PMT
Nom. I/YR
FV
PV
40.
FV of an uneven cash flow stream
I/YR =
CFs:
FV of CFs:
FV =
FV =
FV =
41.
0
$1,000
5.0%
0
$500
10.0%
MEDIUM
Answer: a
MEDIUM
1
2
3
4
5
$50
$50
$50
$50
$1,050
I is the discount rate that causes the PV of the inflows to equal
the initial negative CF, and is found with Excel’s IRR function or
by inputting the CFs into a calculator and pressing the IRR key.
Answer: c
MEDIUM
1
2
3
4
$50
$60
$70
$516
I is the discount rate that causes the PV of the positive inflows to
equal the initial negative CF. I can be found using Excel’s IRR
function or by inputting the CFs into a calculator and pressing
the IRR key.
FV of a lump sum, monthly
Years
Periods/Yr
PMT
Nom. I/YR
PV
FV
Answer: d
1
2
3
4
$100
$300
$0
$500
$126
$350
$0
$500
Found by summing individual PVs.
Found with the NFV key in some calculators.
Found with a calculator by first finding the PV of the stream,
then finding the FV of that PV.
PV of the stream:
$717.31
FV of the PV:
$975.89
Interest rate built into an uneven cash flow stream
CFs:
I/YR
43.
8%
0
$0
$0
$975.89
$975.89
$975.89
Interest rate built into an uneven cash flow stream
CFs:
I/YR
42.
N = Periods = years x periods/yr =
3
6
2
$0
I/Period = Nom. I/Periods per year =
8.00%
4.00%
$2,000
$1,580.63 Could be found using a calculator, the equation, or Excel. Note
that we must first convert to periods and rate per period.
Answer: e
MEDIUM
N = Periods = years x periods/yr =
3
36
12
$0
I/Period = Nom. I/Periods per year =
8.00%
0.67%
$2,000
$2,540.47 Could be found using a calculator, the equation, or Excel. Note
that we must first convert to periods and rate per period.
Chapter 2: Time Value of Money
Page 85
44.
PV of a lump sum, monthly
Years
Periods/Yr
PMT
Nom. I/YR
FV
PV
45.
APR vs. effective annual rate
Nominal vs. effective annual rate
49.
6%
7
360
30
$5,000
$5,178.08
MEDIUM
Answer: c
MEDIUM
Days in month
Daily rate
Interest per day
30
0.0222%
$2.22
Rate/day
Days on deposit
MEDIUM
0.0167%
210
Answer: b
MEDIUM
8%
4
$25,000
$7,548.02 Found with a calculator, as the PMT.
Loan amortization: interest
Page 86
Answer: a
Answer: e
Loan amortization: payment
I/YR
Years
Amount borrowed
Payments
50.
8%
360
$10,000
$66.67
Fractional time periods
Nominal I/YR
Number of months
Days in year
Days in month
Amount deposited
Ending amount
MEDIUM
6%
12
6.17%
Interest charges, simple interest
Nominal I/YR
Days/yr
Amount borrowed
Interest per month
48.
Answer: d
15%
12
16.08%
Nominal I/YR
Periods/yr
EFF%
47.
MEDIUM
N = Periods = years x periods/yr =
3
36
12
$0
I/Period = Nom. I/Periods per year =
8.00%
0.67%
$2,000
$1,574.51 Could be found using a calculator, the equation, or Excel. Note
that we must first convert to periods and rate per period.
APR
Periods/yr
EFF%
46.
Answer: b
Answer: d
MEDIUM
Chapter 2: Time Value of Money
I/YR
Years
Amount borrowed
Interest in Year 1
51.
8%
4
$25,000
$2,000.00
Multiply the rate times the amt borrowed.
Comparing the effective cost of two bank loans
Answer: a
MEDIUM
This problem can be worked most easily using the interest conversion feature of a calculator.
It could also be worked using the conversion formula. We applied the formula using Excel.
Nominal rate, Bank A
7%
Nominal rate, Bank B
8%
Periods/yr, A
12
Periods/yr, B
1
EFF% A
7.23%
EFF% B
8.00%
Difference
0.77%
52.
Mortgage payments
N
I
PV
PMT
FV
53.
54.
Answer: c
MEDIUM
360
0.5833%
$190,000
$1,264
$0.00
Loan amortization: principal repayment
Answer: e
MEDIUM/HARD
Interest rate
8%
Years
4
Amount borrowed
$25,000
Step 1: Find the PMT
Step 2: Find the 1st year’s interest
Step 3: Subtract the interest from the payment; this is repayment of principal
$7,548
$2,000
$5,548
Loan amortization: ending balance
MEDIUM/HARD
Chapter 2: Time Value of Money
Answer: b
Page 87
55.
Interest rate
Years
Amount borrowed
Step 1: Find the PMT
Step 2: Find the 1st year’s interest
Step 3: Subtract the interest from the payment; this is repayment of principal
Step 4: Subtract the repayment of principal from the beginning amount owed
8%
4
$25,000
$7,548
$2,000
$5,548
$19,452
Retirement planning
MEDIUM/HARD
Answer: d
Interest rate
Years to retirement
Years in retirement
Amount saved per year
Step 1: Find the amount at age 65; use the FV function
Step 2: Find the PMT for a 30 year ordinary annuity using that FV as the PV
56.
Nonannual compounding
Interest rate
Periods/year
Monthly rate
1st deposit
2nd deposit
3rd deposit
57.
10%
35
30
$3,000
$813,073
$86,250
Answer: a
12%
12
1.00%
$5,000
$10,000
$15,000
Years on
deposit:
3
2
1
Lifetime subscription vs. annual payments
Months on
deposit:
36
24
12
MEDIUM/HARD
Ending
Amount
$7,153.84
$12,697.35
$16,902.38
$36,753.57
Answer: c
MEDIUM/HARD
Find N for an annuity due with the indicated terms to determine how long you must live to
make the lifetime subscription worthwhile.
Interest rate
7%
Annual cost
$100
Lifetime subscription cost
$1,000
Number of payments made
15.70 rounded to
16
Recall, we used BEGIN mode (because it is an annuity due), so it takes 16 payments before
the lifetime subscription is better. Since the 1st payment occurs today, the 16th payment
occurs at t = 15 (15 years from now). So, you must live for: 16 – 1 = 15 years.
58.
Comparing the effective cost of two bank loans
Page 88
Answer: e
MEDIUM/HARD
Chapter 2: Time Value of Money
Students must understand that “simple interest with interest paid monthly” means that the
bank gets the interest at the end of each month, hence it can invest it, presumably at the same
nominal rate. This results in the same effective rate as if it were stated as “6%, monthly
compounding.” The problem can be worked most easily using the interest conversion feature
of a calculator. It could also be worked using the conversion formula. We applied the formula
using Excel.
Nominal rate, Bank A
Nominal rate, Bank B
Periods/yr, A
Periods/yr, B
EFF% A
EFF% B
Difference
59.
6%
7%
12
1
6.17%
7.00%
0.83%
Retirement planning
Answer: b
Tom’s retirement account
N
46
I/YR
10%
PV
$0
PMT
$4,000
FV
$3,167,181 = on 70th bday, accounts must equal =
60.
MEDIUM/HARD
Jerry’s retirement account
N
41
I/YR
10%
PV
$0
PMT
$6,492
FV
$3,167,181
Saving to start a business
Answer: d
HARD
There are 3 cash flow streams: the gift and the two annuities. The gift will grow for 10 years.
Then there is a 5year annuity that will compound for an additional 5 years. Finally, there is a
second 5year annuity. The sum of the compounded values of those three sets of cash flows
is the final amount.
Nominal interest rate
Gift
1st annuity
2nd annuity
Total years
Annuity years
61.
8%
$20,000
$5,000
$10,000
10
5
Year 5
NA
$29,333
NA
Year 10
$43,178
$43,100
$58,666
Final amt:
Cash flow required to provide a given rate of return
Chapter 2: Time Value of Money
$144,944
Answer: a
HARD
Page 89
This is a very difficult problem. It would not generally be appropriate for a regular inclass
exam. It would be better for a takehome exam. We must find a value of X such that the PV
of the positive CFs, discounted at 9%, will equal the initial negative CF.
The problem is relatively easy with Excel but quite hard with a calculator because it’s hard to
conceptualize the required setup and steps.
Excel solution: Set the problem up as shown below. Put a guess–we initially guessed
$10,000–in the boxed cell under the first X. The IRR initially is less than 9%, so raise the
guess, and keep iterating until IRR = 9%. This value of X is the required payment for the
investment to provide the 9% rate of return. The problem can be worked faster if you know
how to use Goal Seek; then you tell Excel to change the X cell to the value that causes IRR =
$10,158.58
9%. It turns out to be exactly:
I/YR =
9%
0
1
2
3
4
5
6
7
$50,000
$5,000
$10,000 $15,000
X
X
X
X
$50,000
$5,000
$10,000 $15,000 $10,000 $10,000
$10,000
$10,000
IRR =
8.78%
Calculator
solution:
Calculator solution:
Step
CF register
to find
thethe
PVPV
ofofthe
at the
theend
endofof year
Year
$25,413.29
Step 1:
1. Use Use
CF register
to find
the4 4known
knowncash
cash flows
flows at
3:3. $25,413.29
Step2:
findfind
the the
FVFV
at the
endend
ofof
Period
PV for
for33years.
years.$32,910.95
$32,910.95
Step 2. NowNow
at the
period3,3,i.e.,
i.e.,compound
compound PV
Step
3:
Now
find
the
PMT
for
a
4year
annuity
with
the
PV.
$10,158.58$10,158.58
Step 3.
Now find the PMT for a 4year annuity with this PV.
62.
Saving for college
Page 90
Answer: c
HARD
Chapter 2: Time Value of Money
Current college costs
College cost inflation
Account return
First 5 payments
Current account balance
$16,000
4%
8%
$3,000
$9,000
First, determine each year of college’s costs.
Year 1 of college (t = 10)
=
Year 2 of college (t = 11)
=
Year 3 of college (t = 12)
=
Year 4 of college (t = 13)
=
$23,684
$24,631
$25,617
$26,641
The PV (at t = 10) of all of these college costs is:
$89,601 This is what they need at t = 10.
After the first 5 payments, the college account will have (at t=4):
$29,844.20
6 more contributions are left in order to get the required funds for college costs.
N
I
PV
PMT
FV
63.
6
8%
$29,844
$5,758
$89,601.29
Loan amortization: repayment of principal
N
I
PV
PMT
FV
12
1.17%
$100,000
$8,978
$0
% paid toward prin. =
Answer: e
HARD
Answer: b
HARD
89.06%
Amortization schedule(first 4 years)
Year
1
2
3
4
64.
Beg. Balance
Payment
$100,000.00 $8,978
$92,187.41 $8,978
$84,283.76 $8,978
$76,288.01 $8,978
Interest
Principal
$1,165.41
$1,074.36
$982.25
$889.07
$7,812.59
$7,903.64
$7,995.75
$8,088.93
Loan amortization: interest paid
Chapter 2: Time Value of Money
End.
Balance
$92,187.41
$84,283.76
$76,288.01
$68,199.08
Page 91
N
I
PV
PMT
FV
360
0.67%
$120,000
$881
$0
Paid toward interest
Total payments
% Paid toward interest
$2,398
$2,642
90.79%
65.
Amortization schedule(first 3 months)
Year
Beg. Balance
Payment
Interest
Principal
End. Balance
End.
Year
Beg. Balance Payment
Interest
Principal
1
$120,000.00
$880.52
$800.00
$80.52
$119,919.48
Balance
2
$119,919.48
$880.52
$799.46
$81.05
$119,838.43
$120,000.00
1
$880.52
$800.00
$80.52 #########
3
$119,838.43
$880.52
$798.92
$81.59
$119,756.84
$119,919.48
2
$880.52
$799.46
$81.05 #########
Total
payments:
$2,641.55
$2,398.39
$243.17
$119,838.43
3
$880.52
$798.92
$81.59 #########
Total payments:
$243.17
$2,641.55 $2,398.39
PV and discount rate
Answer: a EASY
66.
Time value concepts
Answer: e
EASY
67.
Time value concepts
Answer: c
EASY
Statement a is false, because the more compoundings per year, the less
the present value of a lump sum. Statement b is false and statement c
is true, because the periodic rate is 5%.
Statement d is false,
because the more periods a lump sum is discounted, the less its present
value is.
Statement e is false, because the annuity has cash flows
occurring sooner than the lump sum is due, so the sum of the PVs of its
cash flows will be greater.
68.
Time value concepts
Answer: e
EASY
Since the effective rates are all the same, the correct answer must be
the investment that has the largest amount of money compounding for the
longest time, which is statement e. This can be proven by assuming an
effective annual rate and performing the calculations.
69.
Effective annual rate
Answer: b
EASY
The bank account that pays the highest nominal rate with the most
frequent rate of compounding will have the highest EAR. Consequently,
statement b is the correct choice.
70.
Effective annual rate
Answer: d
EASY
The investment with the highest nominal rate and most frequent
compoundings will have the highest effective rate.
For this reason,
Bank 4 (statement d) dominates Banks 13 (statements a, b, and c).
Therefore, Bank 4โs effective rate (8.328%) must be calculated and
compared to Bank 5โs effective rate (8.2%). Therefore, statement d is
the correct choice.
Page 92
Chapter 2: Time Value of Money
71.
Quarterly compounding
Answer: e
EASY
If the nominal rate is 8% and there is quarterly compounding, the periodic
rate must be 8%/4 = 2%.
The effective rate will be greater than the
nominal rate; it will be 8.24%. So the correct answer is statement e.
72.
Annuities
Answer: c
MEDIUM
Statement a is false, because the present value of an annuity due (all
else equal) is greater than the present value of an ordinary annuity,
since cash flows are received sooner and reinvested longer. Statements
b and e are false, because the present and future values of an annuity
due (all else equal) are also greater, since the cash flows are
reinvested longer. Statement c is true for the same reasons. Statement
d is false, because the difference in PVs of the two annuities will
change as interest rates change.
73.
Amortization
Answer: e
MEDIUM
Statement a is false, because the interest rate and annual payments are
positively correlated (higher rates = higher payments, and vice versa).
Statement b is false, because interest during Year 1 would be the interest
rate times the beginning balance, which are identical in both cases.
Statement c is false, because early payments in a loanโs life have a lot
of interest, but later payments include very little interest since the
balance is low at this point.
Statement d is false, because this
composition changes with each payment, as more principal and less interest
is paid with each payment. Statement e is true, because if the interest
rate is higher, then a greater proportion of total payments (and hence,
each individual payment) goes toward interest.
74.
Amortization
Answer: b
MEDIUM
Statement a is false, because the remaining balance will be the loan
amount less the amount paid toward principal.
Statement b is true,
because the percentage paid toward principal increases throughout the
loanโs life.
Statement c is false, because the payment will be same
each month.
Statement d is false, because the balance gets paid off
quicker at the end of a bondโs life when the balance is lower and more
of the payment goes toward principal.
Statement e is false, because
fixedrate means that the interest rate applied to the outstanding
balance (which does change) will be fixed.
75.
Amortization
Answer: a
MEDIUM
Statement a is false, because monthly payments will not decline over time,
they will stay the same. Statement b is true, because the percentage paid
toward interest declines over time. Statement c is true, because interest
due for every payment gets progressively smaller, which means that the
portion toward principal gets larger. Statement d is true, because if the
interest rate is lower, less is paid toward interest.
Statement e is
Chapter 2: Time Value of Money
Page 93
76.
true. Therefore, answer choice a is the correct answer.
Time value concepts
77.
Growth rate
Time Line:
1958 i = ?

1,800
1959

1988

13,700
โข โข โข
Interest rate
2

2,504.56
3

2,504.56
4

2,504.56
Financial calculator solution:
Inputs: N = 5; PV = 10000; PMT = 2504.56; FV = 0.
8%.
Effect of inflation
Time Line:
0
4%

1.00
Output:
I/YR =
Answer: b
Time Line:
0
1
i = ?


10,000
2,504.56
79.
MEDIUM
Answer: d
Financial calculator solution:
Inputs:
N = 30; PV = 1800; PMT = 0; FV = 13700.
7.0%.
78.
Answer: d
1

5 Years

2,504.56
Output:
I/YR =
Answer: c
โข โข โข
n = ?

0.50
Years
Financial calculator solution:
Inputs: I/YR = 4; PV = 1; PMT = 0; FV = 0.50.
Output: N = 17.67 ๏ป 18 years.
80.
Time for a sum to double
Answer: d
I/YR = 7/12; PV = 1; PMT = 0; FV = 2; and solve for N = 119.17 months
= 9.93 years.
81.
Time for lump sum to grow
Answer: e
Financial calculator solution:
Inputs: I/YR = 10; PV = 300000; PMT = 0; FV = 1000000.
years.
82.
Monthly loan payments
Outputs: N = 12.63
Answer: c
First, find the monthly interest rate = 0.10/12 = 0.8333%/month. Now,
enter in your calculator N = 60; I/YR = 0.8333; PV = 13000; FV = 0;
and solve for PMT = $276.21.
Page 94
Chapter 2: Time Value of Money
83.
FV of an annuity
Time Line:
0 15%
1


200
Answer: e
2

200
3

200
4

200
5 Years

200
FV = ?
Financial calculator solution:
Inputs:
N = 5; I/YR = 15; PV = 0; PMT =
$1,348.48.
84.
Output:
PV of an annuity
Time Line:
0 15%
1


PV = ?
200
FV =
Answer: a
2

200
3

200
4

200
5 Years

200
Financial calculator solution:
Inputs: N = 5; I/YR = 15; PMT = 200; FV = 0.
85.
200.
Output:
PV = $670.43.
PV of a perpetuity
Answer: c
V = PMT/i = $1,000/0.15 = $6,666.67.
86.
Required annuity payments
Time line:
0
1
10%


PV = 1,000 PMT = ?
Answer: b
2

PMT
3

PMT
Financial calculator solution:
Inputs: N = 5; I/YR = 10; PV = 1000; FV = 0.
87.
4

PMT
Output:
5 Years

PMT
PMT = $263.80.
Quarterly compounding and FV
Answer: a
Time line:
01% 1
 
100
2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20 Qtrs
           
FV = ?
Financial calculator solution:
Inputs: N = 20; I/YR = 1; PV = 100; PMT = 0.
88.
PV of an uneven CF stream
Chapter 2: Time Value of Money
Output:
FV = $122.02.
Answer: b
Page 95
NPV = $10,000/1.08 + $25,000/(1.08)2 + $50,000/(1.08)3 + $35,000/(1.08)4
= $9,259.26 + $21,433.47 + $39,691.61 + $25,726.04
= $96,110.38 ๏ป $96,110.
89.
Financial calculator solution (using the cash flow register):
Inputs: CF0= 0; CF1 = 10000; CF2 = 25000; CF3 = 50000; CF4 = 35000; I/YR =
8.
Output: NPV = $96,110.39 ๏ป $96,110.
Effective annual rate
Answer: c
Bank A:
8%, monthly.
r
EARA = ๏ฆ๏ง1 + Nom ๏ถ๏ท
๏จ
m
m ๏ธ
0.08 ๏ถ
๏ฆ
= ๏ง1 +
๏ท
12 ๏ธ
๏จ
Bank B:
โ1
12
โ 1 = 8.30%.
9%, interest due at end of year
EARB = 9%.
9.00% – 8.30% = 0.70%.
90.
Effective annual rate
Answer: b
Use the formula for calculating effective rates from nominal rates as
follows:
EAR = (1 + 0.18/12)12 – l = 0.1956 or 19.56%.
91.
Effective annual rate
Answer: b
Convert each of the alternatives to an effective annual rate (EAR) for
comparison. This problem can be solved with either the EAR formula or
a financial calculator.
a. EAR = 10.38%.
b. EAR = 10.47%.
c. EAR = 10.20%.
d. EAR = 10.25%.
e. EAR = 10.07%.
Therefore, the highest effective return is choice b.
92.
Effective annual rate
Answer: b
Financial calculator solution:
Inputs: N = 10; PV = 35000; PMT = 0; FV = 100000.
11.07%.
93.
Effective annual rate
Outputs:
I/YR =
Answer: a
Convert each of the alternatives to an effective annual rate (EAR) for
comparison. This problem can be solved with either the EAR formula or
Page 96
Chapter 2: Time Value of Money
a financial calculator.
a. EAR = 10.2736%.
b. EAR = 10.1846%.
c. EAR = 10.2000%.
d. EAR = 10.2500%.
e. EAR = 10.0339%.
94.
95.
Therefore, the highest effective return is choice a.
Nominal and effective rates
Answer: b
1st investment:
Enter the following:
NOM% = 9; P/YR = 2; and solve for EFF% = 9.2025%.
2nd investment:
Enter the following:
EFF% = 9.2025; P/YR = 4; and solve for NOM% = 8.90%.
FV of an annuity
Answer: a
The payments start next year, so the calculator should be in END mode.
Enter the following data in your calculator:
N = 42; I/YR = 12; PV = 1000; PMT = 2000. Output: FV = $2,045,442.
96.
97.
FV under monthly compounding
Answer: a
Step 1:
The interest rate must match the payment period, and since the
payments are monthly, you need the monthly periodic rate.
Periodic rate = 8%/12 = 0.667%.
Step 2:
Enter the numbers given into your financial calculator:
N = 30; I/YR = 8/12 = 0.667; PV = 0; PMT = 200. Output:
= $6,617.77.
PV of an uneven CF stream
FV
Answer: c
Time Line:
0 14% 1
2
3
4
5
6
7
8
9 Years










PV = ? 2,000 2,000 2,000 2,000 2,000 3,000 3,000 3,000 4,000
Financial calculator solution (using the cash flow register):
Inputs: CF0 = 0; CF1 = 2000; Nj = 5; CF2 = 3000; Nj = 3; CF3 = 4000;
I/YR = 14.
Output: NPV = $11,714.
98.
FV of a sum
Time Line:
0 2%
1


1,000
Answer: b
2

Chapter 2: Time Value of Money
3

4

5

6 Qtrs

FV = ?
Page 97
Financial calculator solution:
Inputs: N = 6; I/YR = 2; PV = 1000; PMT = 0.
99.
Output:
FV = $1,126.
FV of annuity due
Answer: d
Since payments begin today and occur every year on Janetโs birthday, the
calculator must be set to BEGIN mode.
Now, find the future value of
these payments with your financial calculator:
BEG N = 42; I/YR = 10; PV = 10000; PMT = 1000. Output: FV = $1,139,038.
100.
101.
Time value of money and retirement
Step 1:
Find the number of years it will take for each $150,000
investment to grow to $1,000,000.
BRUCE:
I/YR = 5; PV = 150000; PMT = 0; FV = 1000000; and
solve for N = 38.88.
BRENDA: I/YR = 10; PV = 150000; PMT = 0; FV = 1000000; and
solve for N = 19.90.
Step 2:
Calculate the difference in the length of time for the
accounts to reach $1 million:
Bruce will be able to retire in 38.88 years, or 38.88 โ 19.90
= 19.0 years after Brenda does.
FV of a sum
Time Line:
0
1
3%


100
100
102.
Answer: b
Answer: d
2

3

4

โข โข โข
40 6months
 Periods
FV = ?
Step 1:
Solve for amount on deposit at the end of 6 months:
0.06 ๏ถ
๏ฆ
$100๏ง1 +
๏ท โ $100 = $3.00.
2 ๏ธ
๏จ
Step 2:
Calculate the ending balance 20 years after
deposit of $100 was made:
Inputs: N = 39; I/YR = 3; PV = 3.00; PMT = 0.
$9.50.
FV of an annuity
the
initial
Output: FV =
Answer: e
Step 1:
Determine the effective annual rate:
The nominal rate is 6%, but we need the effective annual rate.
Using the calculator, input the following data:
NOM% = 6; P/YR = 365; and solve for EFF% = 6.1831%.
Step 2:
Determine the future value of the annuity:
N = 3; I/YR = 6.1831; PV = 500; PMT = 1000; and solve for FV
= $3,787.92 ๏ป $3,788.
Page 98
Chapter 2: Time Value of Money
103.
FV of an annuity
Answer: c
To calculate the solution to this problem, change your calculator to
BEGIN mode.
Then enter N = 35; I/YR = 10; PV = 0; PMT = 3000; and
solve for FV = $894,380.4160. Add the last payment of $3,000, and the
value at t = 35 is $897,380.4160 ๏ป $897,380.
104.
FV of an annuity
Answer: d
First, find the present values today of the two withdrawals to occur on
the 25th and 30th birthdays (in the 5th and 10th year of the problem,
respectively).
PV today of $5,000 withdrawal five years from now:
N = 5; I/YR = 12; PMT = 0; FV = 5000; and solve for PV = $2,837.13.
PV today of $10,000 withdrawal 10 years from now:
N = 10; I/YR = 12; PMT = 0; FV = 10000; and solve for PV = $3,219.73.
Now, we subtract the PV of these withdrawals from our initial investment:
$5,000.00 – $2,837.13 – $3,219.73 = $1,056.86.
Finally, we have our simple TVM setup with N, I/YR, PV, and PMT, solving
for FV:
N = 45; I/YR = 12; PV = 1056.86; PMT = 500; and solve for FV =
$505,803.08 ๏ป $505,803.
105.
FV of annuity due
Answer: d
There are a few ways to do this. One way is shown below.
To get the value at t = 5 of the first 5 payments:
BEGIN mode, N = 5; I/YR = 11; PV = 0; PMT = 3000; and solve for FV =
$20,738.58.
Now add on to this the last payment that occurs at t = 5.
$20,738.58 + $3,000 = $23,738.58 ๏ป $23,739.
106.
FV of annuity due
Answer: e
Step 1:
Calculate the value at t = 45 of the first 44 annuity
contributions:
Enter the following inputs in the calculator:
BEGIN mode, N = 44; I/YR = 10; PV = 0; PMT = 2000; and solve
for FV = $1,435,809.67.
Step 2:
Now add on to the FV (calculated in
contribution that occurs at t = 45:
$1,435,809.67 + $2,000.00 = $1,437,809.67.
Chapter 2: Time Value of Money
Step
1)
the
last
Page 99
107.
FV under monthly compounding
Answer: e
Financial calculator solution:
N = 3 ๏ด 12 = 36; I/YR = 6/12 = 0.5; PV = 1000; PMT = 0; and solve for
FV = $1,196.68.
108.
109.
FV under monthly compounding
Answer: d
Step 1:
Calculate the FV at t = 36 of the first deposit.
Enter N = 36; I/YR = 12/12 = 1; PV = 10000; PMT = 0; and
solve for FV = $14,308.
Step 2:
Calculate the FV at t = 36 of the second deposit.
Enter N = 24; I/YR = 12/12 = 1; PV = 10000; PMT = 0; and
solve for FV = $12,697.
Step 3:
Calculate the FV at t = 36 of the third deposit.
Enter N = 12; I/YR = 12/12 = 1; PV = 20000; PMT = 0; and
solve for FV = $22,537.
Step 4:
The sum of the future values gives you the answer, $49,542.
FV under daily compounding
Answer: a
Solve for FV as N = 132; I/YR = 4/365 = 0.0110; PV = 2000; PMT = 0;
and solve for FV = $2,029.14.
110.
FV under nonannual compounding
Answer: d
First, find the FV of Joshโs savings as: N = 2 ๏ด 26 = 52; I/YR = 10/26 =
0.3846; PV = 0; PMT = 100; and FV = $5,744.29.
Johnโs savings will have two components, a lump sum contribution of $1,500
and his monthly contributions. The FV of his regular savings is: N = 2 ๏ด
12 = 24; I/YR = 10/12 = 0.8333; PV = 0; PMT = 150; and FV = $3,967.04.
The FV of his previous savings is: N = 24; I/YR = 0.8333; PV = 1500; PMT
= 0; and FV = $1,830.59.
Summing the components of Johnโs savings yields $5,797.63, which is
greater than Joshโs total savings. Thus, the most expensive car purchased
costs $5,797.63.
111.
FV under quarterly compounding
Answer: c
The effective rate is given by:
NOM% = 8; P/YR = 4; and solve for EFF% = 8.2432%.
Page 100
Chapter 2: Time Value of Money
The nominal rate on a semiannual basis is given by:
EFF% = 8.2432; P/YR = 2; and solve for NOM% = 8.08%.
The future value is given by:
N = 2.5 ๏ด 2 = 5; I/YR = 8.08/2 = 4.04; PV = 0; PMT = 100; and solve for
FV = $542.07.
112.
FV under quarterly compounding
Answer: d
There are several ways of solving this. One way is:
First, find the periodic (quarterly) rate of 7%/4 = 1.75%.
Next, find the future value of each amount put in the account:
N = 12; I/YR = 1.75; PV = 1000; PMT = 0; and solve for FV =
$1,231.4393. N = 8; I/YR = 1.75; PV = 2000; PMT = 0; and solve for FV
= $2,297.7636. N = 4; I/YR = 1.75; PV = 3000; PMT = 0; and solve for
FV = $3,215.5771.
Add up the future values for the answer:
113.
FV of an uneven CF stream
$6,744.78.
Answer: e
First, calculate the payment amounts:
PMT0 = $5000, PMT1 = $5500, PMT2 = $6050, PMT3 = $6655, PMT4 = $7320.50.
Then, find the future value of each payment at t = 5: For PMT0, N = 5;
I/YR = 14; PV = 5000; PMT = 0; thus, FV = $9,627.0729. Similarly, for
PMT1, FV = $9,289.2809, for PMT2, FV = $8,963.3412, for PMT3, FV =
$8,648.8380, and for PMT4, FV = $8,345.3700. Finally, summing the future
values of the respective payments will give the balance in the account at
t = 5 or $44,873.90.
114.
Present value
Answer: c
Use your financial calculator to determine each securityโs
value, and then choose the one with the largest present value.
present
a. Enter the following inputs in your calculator:
N = 5; I/YR = 8; PMT = 1000; FV = 0; and solve for PV = $3,992.71.
b. Enter the following inputs in your calculator:
N = 5; I/YR = 8; PMT = 0; FV = 7000; and solve for PV = $4,764.08.
c. P = PMT/I = $800/0.08 = $10,000.
d. Enter the following inputs in your calculator:
N = 7; I/YR = 8; PMT = 0; FV = 8500; and solve for PV = $4,959.67.
Chapter 2: Time Value of Money
Page 101
e. Enter the following inputs in your calculator:
CF0 = 0; CF1 = 1000; CF2 = 2000; CF3 = 3000; I/YR = 8; and solve for
NPV = $5,022.10.
The preferred stock issue, statement c, has the largest present value
among these choices.
115.
PV under monthly compounding
Answer: b
Start by calculating the effective rate on the second security:
P/YR = 12; NOM% = 10; and solve for EFF% = 10.4713%.
Then, convert this effective rate to a semiannual rate:
EFF% = 10.4713; P/YR = 2; NOM% = 10.2107%.
Now, calculate the value of the first security as follows:
N = 10 ๏ด 2 = 20; I/YR = 10.2107/2 = 5.1054; PMT = 500; FV = 0; and solve
for PV = $6,175.82.
116.
PV under nonannual compounding
Answer: c
First, find the effective annual rate for a nominal rate of 12% with
quarterly compounding:
P/YR = 4; NOM% = 12; and EFF% = 12.55%.
In
order to discount the cash flows properly, it is necessary to find the
nominal rate with semiannual compounding that corresponds to the
effective rate calculated above.
Convert the effective rate to a
semiannual nominal rate as P/YR = 2; EFF% = 12.55; and NOM% = 12.18%.
Finally, find the PV as N = 2 ๏ด 3 = 6; I/YR = 12.18/2 = 6.09; PMT = 500;
FV = 0; and solve for PV = $2,451.73.
117.
PV of an annuity
Time Line:
0
7%

PVLifetime = 100
10
PVAnnual = 100
Answer: a
1

10
2

10
3

10
n = ?

10
Financial calculator solution:
Inputs: I/YR = 7; PV = 90; PMT = 10; FV = 0.
years.
118.
Value of a perpetuity
Years
Output:
N = 14.695 ๏ป 15
Answer: c
Time Line:
Page 102
Chapter 2: Time Value of Money
0 r = ? = 8% 1


PMT = 1,250 1,250
2

1,250
Solve for required return, r.
r =
20 Years

1,250
โข โข โข
We know Vp = PMT , thus,
r
$1,250
PMT
=
= 8%.
$15,625
Vp
Financial calculator solution:
Inputs: N = 20; I/YR = 8; PMT = 1250; FV = 0.
Output: PV = $12,272.68 ๏ป $12,273.
119.
FV of an uneven CF stream
Time Line:
0
1
8%


5,000
1,000
โข โข โข
Answer: d
5

1,000
6

2,000
โข โข โข
10 Years

2,000
FV = ?
Financial calculator solution:
Calculate PV of the cash flows, then bring them forward to FV using the
interest rate.
Inputs: CF0 = 5000; CF1 = 1000; Nj = 5; CF2 = 2000; Nj = 5; I/YR = 8.
Output: NPV = $14,427.45.
Inputs: N = 10; I/YR = 8; PV = 14427.45; PMT = 0.
Output: FV = $31,147.79 ๏ป $31,148.
120.
FV of an uneven CF stream
Answer: c
The easiest way to find the solution to this problem is to find the PV of
all her contributions today, and then find the FV of that PV 10 years from
now.
121.
Step 1:
Calculate the PV of all the deposits today:
CF0 = 10000; CF1 = 20000; CF2 = 50000; I/YR = 6; and solve for
NPV = $73,367.74653.
Step 2:
Calculate the FV 10 years from now of the PV of the deposits:
N = 10; I/YR = 6; PV = 73367.74653; PMT = 0; and solve for FV
= $131,390.46.
PV of an uneven CF stream
Answer: a
Time Line:
Chapter 2: Time Value of Money
Page 103
0 12%

0
PV = ?
1

1
2

2,000
3

2,000
4

2,000
5

0
6 Periods

2,000
Financial calculator solution:
Using cash flows
Inputs: CF0= 0; CF1= 1; CF2 = 2000; Nj = 3; CF3 = 0; CF4 = 2000; I/YR =
12.
Output: NPV = $3,276.615 ๏ป $3,277.
122.
PV of uncertain cash flows
Time Line:
0
8%

0
1

E(CF1)
Answer: e
2

E(CF2)
3 Years

E(CF3)
Calculate expected cash flows
E(CF1) = (0.30)($300) + (0.40)($500) + (0.30)($700) = $500.
E(CF2) = (0.15)($100) + (0.35)($200) + (0.35)($600) + (0.15)($900) = $430.
E(CF3) = (0.25)($200) + (0.75)($800) = $650.
Financial calculator solution:
Using cash flows
Inputs: CF0 = 0; CF1 = 500; CF2 = 430; CF3 = 650; I/YR = 8.
Output: NPV = $1,347.61.
123.
Value of missing payments
Answer: d
Find the FV of the price and the first three cash flows at t = 3.
To do this first find the present value of them.
CF0 = 5544.87; CF1 = 100; CF2 = 500; CF3 = 750; I/YR = 9; and solve for
NPV = $4,453.15.
Find the FV of this present value.
N = 3; I/YR = 9; PV = 4453.15; PMT = 0; FV = $5,766.96.
Now solve for X.
N = 17; I/YR = 9; PV = 5766.96; FV = 0; and solve for PMT = $675.
Page 104
Chapter 2: Time Value of Money
124.
Effective annual rate
Answer: b
Time Line:
0iiB == ?0%
1
A


PVA = 3,755.50 PMT
2
3


PMT
PMT
PMTB = PMTA = 375.55
4

PMT
โข โข โข
Financial calculator solution:
Calculate the PMT of the annuity
Inputs: N = 10; I/YR = 0; PV = 3755.50; FV = 0.
$375.55.
Calculate the effective annual interest rate
Inputs: N = 10; PV = 0; PMT = 375.55; FV = 5440.22.
Output: I/YR = 7.999 ๏ป 8.0%.
125.
10 Years

PMT
FVB = 5,440.22
Output: PMT =
Effective annual rate
Answer: c
Your proposal:
EAR1 = $120/$1,000
EAR1 = 12%.
Your friendโs proposal:
Interest is being paid each month ($10/$1,000 = 1% per month), so it
compounds, and the EAR is higher than rNom = 12%:
12
0.12 ๏ถ
๏ฆ
EAR2 = ๏ง1 +
๏ท – 1 = 12.68%.
12 ๏ธ
๏จ
Difference = 12.68% – 12.00% = 0.68%.
You could also visualize your friendโs proposal in a time line format:
0
1
2
11
12
 i = ?


โข โข โข


1,000
10
10
10
1,010
Insert those cash flows in the cash flow register of a calculator and
solve for IRR. The answer is 1%, but this is a monthly rate. The nominal
rate is 12(1%) = 12%, which converts to an EAR of 12.68% as follows:
Input into a financial calculator the following:
P/YR = 12; NOM% = 12; and solve for EFF% = 12.68%.
126.
EAR and FV of an annuity
Step 1:
Answer: d
Find the effective annual rate:
Enter the following input data in the calculator:
Chapter 2: Time Value of Money
Page 105
NOM% = 9; P/YR = 12; and solve for EFF% = 9.3807%.
Step 2:
127.
Calculate the FV of the $5,000 annuity at the end of 10 years:
Now, put the calculator in End mode, switch back to 1 P/YR,
and enter the following input data in the calculator:
N = 10; I/YR = 9.3807; PV = 0; PMT = 5000; and solve for FV =
$77,358.80 ๏ป $77,359.
EAR and FV of an annuity
0
12
 8.30% 
0
5,000
128.
24

5,000
Answer: b
36

5,000
48

5,000
60 Mos.

5,000
FV = ?
Step 1:
Because the interest is compounded monthly, but payments are
made annually, you need to find the interest rate for the
payment period (the effective rate for one year).
Enter the following input data in your calculator:
NOM% = 8; P/YR = 12; EFF% = 8.30%.
Now use this rate as the interest rate. Remember to switch back
P/YR = 1.
Step 2:
Find the FV of the annuity:
N = 5; I/YR = 8.30; PV = 0; PMT = 5000; and solve for FV =
$29,508.98.
Remaining loan balance
Answer: a
Step 1:
Solve for the monthly payment:
Enter the following input data in the calculator:
N = 60; I/YR = 12/12 = 1; PV = 15000; FV = 0; and solve for
PMT = $333.6667.
Step 2:
Determine the loan balance remaining after the 30th payment:
1 INPUT 30 โผ AMORT
Page 106
Chapter 2: Time Value of Money
= displays Int: $3,621.1746
= displays Prin: $6,388.8264
= displays Bal: $8,611.1736.
Therefore, the balance will be $8,611.17.
129.
Remaining mortgage balance
Answer: c
First, find the payment: Enter N = 360; I/YR = 9/12 = 0.75; PV =
250000; FV = 0; and solve for PMT = $2,011.56.
Use the calculatorโs amortization feature to find the remaining
mortgage balance:
5 years = 5 ๏ด 12 = 60 payments.
1 INPUT 60 โผ AMORT
= displays Int: $110,393.67
= displays Prin: $10,299.93
= displays Bal: $239,700.07.
130.
Remaining mortgage balance
Answer: d
Solve for the monthly payment as follows:
N = 30 ๏ด 12 = 360; I/YR = 8/12 = 0.667; PV = 150000; FV = 0; and solve
for PMT = $1,100.65/month.
Use the calculatorโs amortization
principal balance:
3 ๏ด 12 = 36 payments
1 INPUT 36 โผ AMORT
= displays Int: $35,543.52
= displays Prin: $4,079.88
= displays Bal: $145,920.12.
131.
132.
Amortization
feature
to
find
the
remaining
Answer: c
Step 1:
Determine the monthly payment of the mortgage:
Enter the following inputs in the calculator:
N = 360; I/YR = 8/12 = 0.6667; PV = 165000; FV = 0; and solve
for PMT = $1,210.7115.
Step 2:
Determine the amount of interest during the first 3 years of
the mortgage by using the calculatorโs amortization feature:
1 INPUT 36 โผ AMORT
= displays Int: $39,097.8616.
Required annuity payments
Chapter 2: Time Value of Money
Answer: c
Page 107
Enter CFs:
CF0 = 0; CF1 = 1.2; CF2 = 1.6; CF3 = 2.0; CF4 = 2.4; CF5 = 2.8.
I/YR = 10; NPV = $7.2937 million.
$1 + $7.2937 = $8.2937 million.
Now, calculate the annual payments:
BEGIN mode, N = 5; I/YR = 10; PV = 8.2937; FV = 0; and solve for PMT =
$1.989 million.
133.
134.
Required annuity payments
Answer: b
Step 1:
Work out how much Karen will have saved by age 65:
Enter the following inputs in the calculator:
N = 41; I/YR = 10; PV = 0; PMT = 5000; and solve for FV =
$2,439,259.
Step 2:
Figure the payments Kathy will need to make to have the same
amount saved as Karen:
Enter the following inputs in the calculator:
N = 36; I/YR = 10; PV = 0; FV = 2439259; and solve for PMT =
$8,154.60.
Required annuity payments
Step 1:
Answer: c
Figure out how much their house will cost when they buy it in
5 years:
Enter the following input data in the calculator:
N = 5; I/YR = 3; PV = 120000; PMT = 0; and solve for FV =
$139,112.89.
This is how much the house will cost.
Step 2:
Determine the maximum mortgage they can get, given that the
nominal interest rate will be 7%, it is a 360month mortgage,
and the payments will be $500:
N = 360; I/YR = 7/12 = 0.5833; PMT = 500; FV = 0; and solve
for PV = $75,153.78.
This is the PV of the mortgage (that is, the total amount they
can borrow).
Step 3:
Determine the down payment needed:
House prices are $139,112.89, and they can borrow
$75,153.78. This means the down payment will have to be:
only
Down payment = $139,112.89 – $75,153.78 = $63,959.11.
This is the amount they will have to save to buy their house.
Page 108
Chapter 2: Time Value of Money
Step 4:
Determine how much they need to deposit each year to reach
this goal:
N = 5; I/YR = 10; PV = 2000; FV = 63959.11; and solve for PMT
= $9,948.75 ๏ป $9,949.
135.
Required annuity payments
Answer: a
Hereโs a time line depicting the problem:
25

PMT
10%
30

2PMT
35

3PMT
40

4PMT
โข โข โข
65

FV = 2,000,000
$2,000,000 = PMT(1.10)40 + 2PMT(1.10)35 + 3PMT(1.10)30 + 4PMT(1.10)25
$2,000,000 = 45.259256PMT + 56.204874PMT + 52.348207PMT + 43.338824PMT
$2,000,000 = 197.15116PMT
$10,144.50 = PMT
PMT ๏ป $10,145.
136.
Monthly vs. quarterly compounding
Answer: c
There are several ways to do this, but the easiest is with the calculator:
137.
Step 1:
Find the effective rate on the account with monthly compounding:
NOM% = 5; P/YR = 12; and solve for EFF% = 5.1162%.
Step 2:
Translate the effective rate to a nominal rate based
quarterly compounding:
EFF% = 5.1162; P/YR = 4; and solve for NOM% = 5.0209% ๏ป 5.02%.
Effective annual rate
on
Answer: d
4
0.10 ๏ถ
๏ฆ
EARQtr = ๏ง1 +
๏ท – 1 = 10.38%.
4 ๏ธ
๏จ
365
0.09 ๏ถ
๏ฆ
EARDly = ๏ง1 +
– 1 = 9.42%.
๏ท
365 ๏ธ
๏จ
Difference = 10.38% – 9.42% = 0.96%.
138.
Amortization
Answer: c
Time Line:
Chapter 2: Time Value of Money
Page 109
0 12%
1


200,000 PMT = ?
2

PMT
3

PMT
โข โข โข
30 Years

PMT
Financial calculator solution:
Inputs: N = 30; I/YR = 12; PV = 200000; FV = 0.
Output: PMT = $24,828.73 ๏ป $24,829.
139.
Amortization: repayment of principal
Answer: a
Given:
Loan value = $100,000; Repayment period = 12 months; Monthly
payment = $9,456.
N = 12; PV = 100000; PMT = 9456; FV = 0; and solve for I/YR = 2.00% ๏ด
12 = 24.00%.
To find the amount of principal paid in the third month (or period),
use the calculatorโs amortization feature.
3 INPUT 3 โผ AMORT
= displays Int: $1,698.84
= displays Prin: $7,757.16
= displays Bal: $77,181.86.
140.
Amortization: interest paid
Answer: c
Enter the following inputs in the calculator:
N = 30 ๏ด 12 = 360; I/YR = 9/12 = 0.75; PV = 90000; FV = 0; PMT =
$724.16.
Total payments in the first 2 years are $724.16 ๏ด 24 = $17,379.85.
Use the calculatorโs amortization feature:
12 ๏ด 2 = 24 payments
1 INPUT 24 โผ AMORT
= displays Int: $16,092.44.
Percentage of first two years that is interest is:
$16,092.44/$17,379.85 = 0.9259 = 92.59%.
141.
Amortization: repayment of principal
Step 1:
Page 110
Answer: e
Calculate the monthly mortgage payment:
Enter the following inputs in the calculator:
N = 360; I/YR = 7.25/12 = 0.604167; PV = 135000; FV = 0; and
solve for PMT = $920.9380.
Chapter 2: Time Value of Money
142.
143.
Step 2:
Obtain the amortization schedule for the fourth year (months
3748) by using the calculatorโs amortization feature:
37 INPUT 48 โผ AMORT
= displays Int: $9,428.2512
= displays Prin: $1,623.0048.
Step 3:
Calculate the percentage of payments in the fourth year that
will go towards the repayment of principal:
$1,623.0048/($920.938 ๏ด 12) = 0.1469 = 14.69%.
Amortization: remaining mortgage balance
Answer: b
Step 1:
Find the monthly mortgage payment by entering the following
inputs in your calculator:
N = 360; I/YR = 8/12 = 0.667; PV = 300000; FV = 0; and solve
for PMT = $2,201.29.
Step 2:
Calculate the remaining principal balance after 5 years by
using your financial calculatorโs amortization feature.
60 INPUT โผ AMORT
= displays Int: $1,903.38
= displays Prin: $297.91
= displays Bal: $285,209.57.
Amortization: remaining loan balance
Answer: d
Step 1:
Calculate the common monthly payment using the information you
know about Jamieโs loan:
N = 48; I/YR = 12/12 = 1; PV = 15000; FV = 0; and solve for
PMT = $395.0075.
Step 2:
Calculate how much Jakeโs car cost using the information you
know about his loan and the monthly payment solved in Step 1:
N = 60; I/YR = 12/12 = 1; PMT = 395.0075; FV = 0; and solve
for PV = $17,757.5787.
Step 3:
Calculate the balance on Jakeโs loan at the end of 48 months
by using the calculatorโs amortization feature:
1 INPUT 48 โผ AMORT
= displays Int: $5,648.62
= displays Prin: $13,311.74
= displays Bal: $4,445.84.
Chapter 2: Time Value of Money
Page 111
144.
NPV and nonannual discounting
5%/12 =
Current 0 0.4167% 1
lease


0
500
Inputs
2

500
3

500
12
5/12 = 0.4167
N
I/YR
Output
New
lease
Answer: b
12

500
โข โข โข
PV
500
0
PMT
FV
= 5,840.61
5%/12 =
0 0.4167%

0
1

0
2

0
3

0
4

700
โข โข โข
12

700
CF0 = 0; CF13 = 0; CF412 = 700; I/YR = 0.4167; and solve for NPV =
$6,094.23.
Therefore, the PV of payments under the proposed lease would be greater
than the PV of payments under the old lease by $6,094.23 – $5,840.61 =
$253.62. Thus, your net worth would decrease by $253.62.
145.
FV of an annuity
Answer: c
Step 1:
The value of what they have saved so far is:
Enter the following input data in the calculator:
N = 25; I/YR = 12; PV = 20000; PMT = 5000; and solve for FV
= $1,006,670.638.
Step 2:
Deduct the amount to be paid out in 3 years:
Enter the following input data in the calculator:
N = 3; I/YR = 12; PMT = 0; FV = 150000; and solve for PV =
$106,767.037.
The value remaining is $1,006,670.638 โ $106,767.037 = $899,903.601.
Step 3:
Determine how much will be in the account on their 58th
birthday, after 8 more annual contributions:
Enter the following input data in the calculator:
N = 8; I/YR = 12; PV = 899903.601; PMT = 5000; and solve for
FV = $2,289,626.64 ๏ป $2,289,627.
Page 112
Chapter 2: Time Value of Money
146.
FV of annuity due
Answer: a
First, convert the 9% return with quarterly compounding to an effective
rate of 9.308332%.
With a financial calculator, NOM% = 9; P/YR = 4;
EFF% = 9.308332%. (Donโt forget to change P/YR = 4 back to P/YR = 1.)
Then calculate the FV of all but the final payment.
BEGIN MODE (1
P/YR) N = 9; I/YR = 9.308332; PV = 0; PMT = 1500; and solve for FV =
$21,627.49. You must then add the $1,500 at t = 9 to find the answer,
$23,127.49.
147.
Value of missing cash flow
Answer: d
Financial calculator solution:
Enter the first 4 cash flows, enter I/YR = 15, and solve for NPV =
$58.945. The future value of $58.945 will be the required cash flow.
N = 4; I/YR = 15; PV = 58.945; PMT = 0; and solve for FV = $103.10.
148.
EAR and FV of annuity
Answer: c
First, we must find the appropriate effective rate of interest.
your calculator enter the following data as inputs as follows:
NOM% = 6; P/YR = 12; and solve for EFF% = 6.167781%.
Using
Since the contributions are being made every 6 months, we need to
determine the nominal annual rate based on semiannual compounding. Enter
the following data in your calculator as follows:
EFF% = 6.167781%; P/YR = 2; and solve for NOM% = 6.0755%.
Now, use the periodic rate 6.0755%/2 = 3.037751% to calculate the FV of
the annuities due. Now, we must solve for the value of all contributions
as of the end of Year 2.
Enter the following data inputs in your
calculator:
N = 4; I/YR = 3.037751; PV = 1000; PMT = 1000; and solve for FV =
$5,313.14.
Chapter 2: Time Value of Money
Page 113
So, these contributions will be worth $5,313.14 as of the end of Year 2.
Now, we must find the value of this investment after the eighth year. For
this calculation, we can use annual periods and the effective annual rate
calculated earlier.
Enter the following data as inputs to your
calculator:
N = 6; I/YR = 6.167781; PV = 5313.14; PMT = 0; and solve for FV =
$7,608.65 ๏ป $7,609.
149.
FV of investment account
Answer: b
We need to figure out how much money they would have saved if they
didnโt pay for the college costs.
N = 40; I/YR = 10; PV = 0; PMT = 12000; and solve for FV = $5,311,110.67.
Now figure out how much they would use for college costs. First get the
college costs at one point in time, t = 20, using the cash flow register.
CF0 = 58045; CF1 = 62108; CF2 = 66,456 ๏ด 2 = 132912 (two kids in school);
CF3 = 71,108 ๏ด 2 = 142216; CF4 = 76086; CF5 = 81411; I/YR = 10; NPV =
$433,718.02.
The value of the college costs at year t = 20 is $433,718.02. What we
want is to know how much this is at t = 40.
N = 20; I/YR = 10; PV = 433718.02; PMT = 0; and solve for FV =
$2,917,837.96.
The amount in the nest egg at t = 40 is the amount saved less the
amount spent on college.
$5,311,110.67 – $2,917,837.96 = $2,393,272.71 ๏ป $2,393,273.
150.
PV of an uneven CF stream
Answer: c
Time Line:
i = 4%
0
Yrs

PV = ?
i = 5%
1
2
3
4
5
6
7
8

100

100

100

+200

+300

+300

+300

+300
277.51
1,070.00
1,203.60
792.49
Financial calculator solution:
Inputs: CF0 = 0; CF1 = 100; Nj = 3; I/YR = 4.
Output: NPV = 277.51.
Page 114
Chapter 2: Time Value of Money
Calculate the PV of CFs 48 as of time = 3 at i = 5%
Inputs: CF0 = 0; CF1 = 200; CF2 = 300; Nj = 4; I/YR = 5.
Output: NPV3 = $1,203.60.
Calculate PV of the FV of the positive CFs at time = 3
Inputs: N = 3; I/YR = 4; PMT = 0; FV = 1203.60.
Output: PV = $1,070.
Total PV = $1,070 – $277.51 = $792.49.
151.
PV of an uneven CF stream
Time Line:
0
1
i = 6%


+100
$8,554.84
1,082.76
$7,472.08
2

+100
โข โข โข
Answer: d
18

+100
19
20
21
22




6,115.91 6,727.50 7,400.25 8,140.27
PV of health care costs
PV of parentsโ savings
Lump sum government must set aside
Find the present value of parentโs savings:
100; FV = 0; and solve for PV = $1,082.76.
N = 18; I/YR = 6; PMT = –
Health care costs, Years 1922:
$1,000(1.1)19 = $6,115.91; 20
21
$1,000(1.1) = $6,727.50; $1,000(1.1) = $7,400.25; $1,000(1.1)22 = $8,140.27.
Find the present value of health care costs: CF 0 = 0; CF118 = 0; CF19 =
6115.91; CF20 = 6727.50; CF21 = 7400.25; CF22 = 8140.27; I/YR = 6; and
solve for NPV = 8,554.84 = PV of health care costs.
Consequently, the government must set aside $8,554.84 – $1,082.76 =
$7,472.08.
152.
Required annuity payments
Answer: b
College cost today = $10,000, Inflation = 5%.
CF0 = $10,000 ๏ด (1.05)5
$12,762.82 ๏ด 1 = $12,762.82; CF1 = $10,000 ๏ด (1.05)6 = $13,400.96 ๏ด 1
$13,400.96; CF2 = $10,000 ๏ด (1.05)7 = $14,071.00 ๏ด 2 = $28,142.00; CF3
$10,000 ๏ด (1.05)8 = $14,774.55 ๏ด 2 = $29,549.10; CF4 = $10,000 ๏ด (1.05)9
$15,513.28 ๏ด 1 = $15,513.28; CF5 = $10,000 ๏ด (1.05)10 = $16,288.95 ๏ด 1
$16,288.95.
=
=
=
=
=
Financial calculator solution:
Enter cash flows in CF register; I/YR = 8; solve for NPV = $95,244.08.
Calculate annuity:
N = 5; I/YR = 8; PV = 50000; FV = 95244.08; and solve for PMT = $3,712.15.
Chapter 2: Time Value of Money
Page 115
153.
154.
Required annuity payments
Answer: b
Step 1:
Calculate the present value of college costs at t = 16 (Treat
t = 16 as Year 0.):
Remember, costs are incurred at end of year.
CF0 = 25000; CF1 = 25000; CF2 = 50000; CF3 = 50000; CF4 = 25000;
CF5 = 25000; I/YR = 8; and solve for NPV = $166,097.03.
Step 2:
Calculate the annual required deposit:
N = 16; I/YR = 8; PV = 0; FV = 166097.03; then solve for PMT
= $5,477.36.
Required annuity payments
Infl. = 5%
0 i = 8%
1


40,000
100,000
PMT
Step 1:
Step 2:
Page 116
PMT
Retires
2

44,100
Answer: c
3

44,100
4

44,100
Goes on
Welfare
5

122,742
(116,640)
6,102
The retirement payments, which begin at t = 2, must be:
$40,000(1 + Infl.)2 = $40,000(1.05)2 = $44,100.
There will be 3 retirement payments of $44,100, made at t = 2, t =
Chapter 2: Time Value of Money
3, and t = 4. We find the PV of an annuity due at t = 2 as follows:
Set calculator to BEGIN mode. Then enter:
N = 3; I/YR = 8; PMT = 44100; FV = 0; and solve for PV =
$122,742. If he has this amount at t = 2, he can receive the
3 retirement payments.
The $100,000 now on hand will compound at 8% for 2 years:
$100,000(1.08)2 = $116,640.
So, he must save enough each year to accumulate an additional
$122,742 – $116,640 = $6,102:
Need at t = 2
$122,742
Will have
( 116,640)
Net additional needed
$ 6,102
He must make 2 payments, at t = 0 and at t = 1, such that they
will grow to a total of $6,102 at t = 2.
This is the FV of an annuity due found as follows:
Set calculator to BEGIN mode. Then enter:
N = 2; I/YR = 8; PV = 0; FV = 6102; and solve for PMT = $2,716.
Step 3:
Step 4:
Step 5:
155.
Required annuity payments
0 i = 8% 1

(360.39)

25
Answer: c
2
3
4

25

25

30
โข โข โข
298.25
62.14
23
24

30

PMT
40
โข โข โข

PMT
364.85
Calculate the NPV of payments in Years 123:
CF0 = 0; CF13 = 25; CF423 = 30; I/YR = 8; and solve for NPV = $298.25.
Difference between the securityโs price and PV of payments:
$360.39 – $298.25 = $62.14.
Calculate the FV of the difference between the purchase price and PV of
payments, Years 123:
N = 23; I/YR = 8; PV = 62.14; PMT = 0; and solve for FV = $364.85.
Calculate the value of the annuity payments in Years 2440:
N = 17; I/YR = 8; PV = 364.85; FV = 0; and solve for PMT = $40.
156.
Required annuity payments
0
1
12%


2

3

Answer: a
4

5

6

7

8

9

10

Savings: 5,000
Contrib. 3,000 3,000 3,000 3,000 3,000 PMT
PMT
PMT
PMT
PMT
PMT
11

12

13

College: 24,433 25,655 26,938 28,285
PV college costs = 88,947
Step 1:
Step 2:
Determine college costs:
College costs will be $15,000(1.05)10 = $24,433 at t = 10,
$15,000(1.05)11 = $25,655 at t = 11, $15,000(1.05)12 = $26,938
at t = 12, and $15,000(1.05)13 = $28,285 at t = 13.
Determine PV of college costs at t = 10:
Enter the cash flows into the cash flow register as follows:
CF0 = 24433; CF1 = 25655; CF2 = 26938; CF3 = 28285; I/YR = 12;
and solve for NPV = $88,947.
Chapter 2: Time Value of Money
Page 117
Step 3:
Step 4:
157.
Determine the value of their savings at t = 4 as follows:
N = 4; I/YR = 12; PV = 8000; PMT = 3000; and solve for FV =
$26,926.
Determine the value of the annual contributions from t = 5
through t = 10:
N = 6; I/YR = 12; PV = 26926; FV = 88947; and solve for PMT =
$4,411.
Required annuity payments
25

25,000
26

27

0
29

28

2,000 3,000 4,000
Answer: a
1
30

2
31

5,000 PMT
PMT
4,480.00
3,763.20
2,809.86
39,337.98
$55,391.04
10,132.62
$45,258.42
โข โข โข
6
35

7
36

PMT
PMT
โข โข โข
11
40

Years
Birthdays
PMT
20,000
FV = 400,000
Step 1:
158.
Compound cash flows from birthdays 25, 26, 27, and 28 to 29th
birthday:
$25,000(1.12)4 + $2,000(1.12)3 + $3,000(1.12)2 + 4,000(1.12) +
$5,000(1.12)0
= $39,337.98 + $2,809.86 + $3,763.20 + $4,480.00 + $5,000.00
= $55,391.04.
Step 2: Discount $20,000 withdrawal back to 29th birthday (6 years):
N = 6; I/YR = 12; PMT = 0; FV = 20000; and solve for PV =
$10,132.62. (Remember to add minus sign as this is a withdrawal.)
Step 3: Subtract the present value of the withdrawal from the compounded
values of the deposits to obtain the net amount on hand at
birthday 29 (after the $20,000 withdrawal is considered):
$55,391.04 – $10,132.62 = $45,258.42.
Step 4: Solve for the required annuity payment as follows:
N = 11; I/YR = 12; PV = 45258.42; FV = 400000; and solve for
PMT = $11,743.95.
Required annuity payments
Answer: c
Step 1:
Step 2:
Step 3:
Step 4:
Page 118
Convert the 9% monthly rate to an annual rate.
Enter NOM% = 9; P/YR = 12; and solve for EFF% = 9.3807%.
Compute the amount accumulated by age 40. Remember to change
P/YR from 12 to 1. BEGIN mode.
Then, enter N = 15; I/YR =
9.3807; PV = 0; PMT = 2000; and solve for FV = $66,184.35.
John needs $3 million in 25 years. Find the PV of this amount
today. Remember to change your calculator back from BEGIN to
END mode. Enter N = 25; I/YR = 12; FV = 3000000; PMT = 0; and
solve for PV = $176,469.92.
Find the shortfall today, the difference between the present value
of what he needs in 25 years and the present value of what heโs
accumulated today. $176,469.92 – $66,184.35 = $110,285.57.
Chapter 2: Time Value of Money
Step 5:
159.
Required annuity payments
Step 1
Step 2
Step 3
Step 4
Step 5
160.
Find the annuity needed to cover this shortfall.
Since the
contributions begin today this is an annuity due, so the
calculator must be set up in BEGIN mode. (Remember to change your
calculator back from BEGIN to END mode after working this
problem.)
BEGIN mode.
Then, enter N = 26; I/YR = 12; PV = 110285.57; FV = 0; and solve for PMT = $12,471.31 ๏ป $12,471.
Calculate the cost of tuition in each year:
$25,000(1.05)15 = $51,973.20; $25,000(1.05)16 = $54,571.86 ๏ด 2 =
$109,143.73; $25,000(1.05)17 = $57,300.46 ๏ด 2 = $114,600.92;
$25,000(1.05)18 = $60,165.48 ๏ด 2 = $120,330.96; $25,000(1.05)19 =
$63,173.75.
Find the present value of these costs at t = 15:
CF0 = 51973.20; CF1 = 109143.73; CF2 = 114600.92; CF3 =
120330.96; CF4 = 63173.75; I/YR = 12; and solve for NPV =
$366,579.37.
Calculate the FV of Grandmaโs deposits at t = 15:
Older son:
$10,000(1.12)18 = $ 76,899.66 (Deposit was made 3
years ago.)
Younger son: $10,000(1.12)17 = $ 68,660.41 (Deposit was made 2
years ago.)
Total = $145,560.07
Calculate net total amount needed at t = 15:
$366,579.37 – $145,560.07 = $221,019.30.
Calculate the annual required deposits:
N = 15; I/YR = 12; PV = 0; FV = 221019.30; and solve for PMT =
$5,928.67.
Required annuity payments
Step 1:
Step 2:
Step 3:
Answer: a
Answer: b
Find out what the cost of college will be in six years:
Enter the following input data in the calculator:
N = 6; I/YR = 5; PV = 20000; PMT = 0; and solve for FV =
$26,801.9128.
Calculate the present value of his college cost:
Enter the following input data in the calculator:
N = 6; I/YR = 10; PMT = 0; FV = 26801.9128; and solve for PV =
$15,128.98.
Find the present value today of the $15,000 that will be
withdrawn in two years for the purchase of a used car:
Enter the following input data in the calculator:
N = 2; I/YR = 10; PMT = 0; FV = 15000; and solve for PV =
Chapter 2: Time Value of Money
Page 119
Step 4:
161.
$12,396.69.
So in total, in todayโs dollars, he needs $15,128.98 +
$12,396.69 = $27,525.67, and his shortfall in todayโs dollars
is $25,000 – $27,525.67 = $2,525.67.
Find out how much Bob has to save at the end of each year to
make up the $2,525.67:
Enter the following input data in the calculator:
N = 6; I/YR = 10; PV = 2525.67; FV = 0; and solve for PMT =
$579.9125 ๏ป $580.
Required annuity payments
Answer: e
We must find the PV of the amount we can sell the car for in 4 years.
Enter the following data into your financial calculator:
N = 48; I/YR = 1; FV = 6000; PMT = 0; and solve for PV = $3,721.56.
This means that the total cost of the car, in present value terms is:
$17,000 โ $3,721.56 = $13,278.44.
Now, we need to find the lease payment that equates to this present
value. Enter the following data into your financial calculator:
N = 48; I/YR = 1; PV = 13278.44; FV = 0; and solve for PMT = $349.67.
162.
Required annuity payments
Here is the diagram of the problem:
24
25
64
65
0
1
40
41
9%


โข โข โข


1,000
X
X
100,000
Step 1:
Step 2:
Page 120
Answer: c
โข โข โข
84
60

100,000
Determine the PV at his 64th birthday of the cash outflows from
his 65th birthday to his 84th birthday.
Using a financial
calculator, enter the following input data:
N= 20; I/YR = 9; PMT = 100000; FV = 0; solve PV =
$912,854.57.
This is the amount needed in his account on his 64th birthday
in order to make 20 withdrawals of $100,000 from his account.
Determine the required annual payment (deposit) that will
Chapter 2: Time Value of Money
achieve this goal, given the $1,000 original deposit. Using a
financial calculator, enter the following input data:
N = 40; I/YR = 9; PV = 1000; FV = 912854.57; solve for PMT =
$2,608.73.
163.
Annuity due vs. ordinary annuity
Answer: e
There is more than one way to solve this problem.
Step 1:
Draw the time line:
Bill
25
26
0 r = 12% 1


PMT
PMT
27
2

PMT
Bob
PMT
PMT
โข โข โข
PMT
64
39

PMT
65
40

PMT
FV = $3M
PMT
FV = $3M
164.
Step 2:
Determine eachโs annual contribution:
Bill:
He starts investing today, so use the BEGIN mode of the
calculator.
Enter the following input data in the calculator:
N = 41; I/YR = 12; PV = 0; FV = 3,000,000 ๏ด 1.12 = 3360000; and
solve for PMT = $3,487.79.
(The FV is calculated as
$3,360,000 because the annuity will calculate the value to the
end of the year, until Bill is a second away from age 66.
Therefore, since he wants to have $3,000,000 by age 65, he
would have $3,000,000 ๏ด 1.12 one second before he turns 66.)
Bob: He starts investing at the end of this year, so use the
END mode of the calculator.
Enter the following input data in the calculator:
N = 40; I/YR = 12; PV = 0; FV = 3000000; and solve for PMT =
$3,910.88.
Step 3:
Determine the difference between the two payments:
The difference is $3,910.88 – $3,487.79 = $423.09.
Amortization
Time Line (in thousands):
0
1
2
3
20
i = 8%




โข โข โข

PMTC = 80
80
80
80
PMTR
PMTR
PMTR
FV = 1,000
Annual PMT Total = PMTCoupon + PMTReserve = $80,000 + PMTReserve.
Answer: b
Years
Financial calculator solution:
Long way Inputs: N = 20; I/YR = 8; PV = 0; FV = 1000000.
Output: PMT = $21,852.21.
Add coupon interest and reserve payment together
Annual PMTTotal = $80,000 + $21,852.21 = $101,852.21.
Total number of tickets = $101,852.21/$10.00 = 10,185.22 ๏ป 10,186.*
Short way Inputs: N = 20; I/YR = 8; PV = 1000000; FV = 0.
Chapter 2: Time Value of Money
Page 121
Output: PMT = $101,852.21.
Total number of tickets = $101,852.21/$10.00 ๏ป 10,186.*
*Rounded up to next whole ticket.
165.
Effective annual rate
Time Line:
0
0
i = ?

8,000
Answer: c
12
1

24
2

27 Months
2.25

10,000
Numerical solution:
Step 1: Find the effective annual rate (EAR) of interest on the bank
deposit:
EARDaily = (1 + 0.080944/365)365 – 1 = 8.43%.
Step 2: Find the EAR of the investment:
$8,000 = $10,000/(1 + i)2.25
(1 + i)2.25 = 1.25
1 + i = 1.25(1/2.25)
1 + i = 1.10426
i = 0.10426 ๏ป 10.43%
Step 3: Difference = 10.43% – 8.43% = 2.0%.
Financial calculator solution:
Calculate EARDaily using interest rate conversion feature
Inputs: P/YR = 365; NOM% = 8.0944. Output: EFF% = EAR = 8.43%.
Calculate EAR of the equal risk investment
Inputs: N = 2.25; PV = 8000; PMT = 0; FV = 10000.
Output: I/YR = 10.4259 ๏ป 10.43%.
Difference: 10.43% – 8.43% = 2.0%.
166.
PMT and quarterly compounding
0
1
i = 2%


+400
โข โข โข
80

+400
PMT
Answer: b
81

82

83

84

85

0
0
0
PMT
0
โข โข โข
115

116 Qtrs.

0
PMT
Find the FV at t = 80 of $400 quarterly payments:
N = 80; I/YR = 2; PV = 0; PMT = 400; and solve for FV = $77,508.78.
Find the EAR of 8%, compounded quarterly, so you can determine the
value of each of the receipts:
4
0.08 ๏ถ
๏ฆ
EAR = ๏ง1 +
๏ท – 1 = 8.2432%.
4 ๏ธ
๏จ
Now, determine the value of each of the receipts, remembering that this
is an annuity due.
Put the calculator in BEGIN mode and enter the following input data in
Page 122
Chapter 2: Time Value of Money
the calculator:
N = 10; I/YR = 8.2432; PV = 77508.78; FV = 0; and solve for PMT =
$10,788.78 ๏ป $10,789.
167.
Value of unknown withdrawal
Step 1:
Step 2:
Step 3:
Step 4:
168.
Answer: d
Find out how much Steve and Robert have in their accounts today:
You can get this from analyzing Steveโs account.
END mode: N = 9; I/YR = 6; PV = 5000; PMT = 5000; and solve
for FV = $65,903.9747.
Alternatively, BEGIN mode: N = 9; I/YR = 6; PV = 0; PMT =
5000; and solve for FV = $60,903.9747.
Then add the $5,000 for the last payment to get a total of
$65,903.9747.
This is also the value of Robertโs account today.
Find out how much Robert would have had if he had never
withdrawn anything:
END mode: N = 9; I/YR = 12; PV = 5000; PMT = 5000; and solve
for FV = $87,743.6753.
Alternatively, BEGIN mode: N = 9; I/YR = 12; PV = 0; PMT =
5000; and solve for FV = $82,743.6753.
Then add the $5,000 for the last payment to get a total of
$87,743.6753.
Find the difference in the value of Robertโs account due to
the withdrawal made:
However, since he took money out at age 27, he has only
$65,903.9747. The difference between what he has and what he
would have had is:
$87,743.6753 – $65,903.9747 = $21,839.7006.
Determine the amount of Robertโs withdrawal by discounting the
value found in Step 3:
N = 3; I/YR = 12; PMT = 0; FV = 21839.7006; then solve for PV
= $15,545.0675 ๏ป $15,545.07.
Nonannual compounding
Answer: a
To compare these alternatives, find the present value of each strategy
and select the option with the highest present value.
Option 1 can be valued as an annuity due.
Enter the following input data in the calculator:
BEGIN mode (to indicate payments will be received at the start of the
period) N = 12; I/YR = 12/12 = 1; PMT = 1000; FV = 0; and solve for PV
= $11,367.63.
Option 2 can be valued as a lump sum payment to be received in the future.
Enter the following input data in the calculator:
END mode (to indicate the lump sum will be received at the end of the year)
N = 2; I/YR = 12/2 = 6; PMT = 0; FV = 12750; and solve for PV =
$11,347.45.
Chapter 2: Time Value of Money
Page 123
Option 3 can be valued as a series of uneven cash flows.
The cash
flows at the end of each period are calculated as follows:
CF0 = $0.00; CF1 = $800.00; CF2 = $800.00(1.20) = $960.00; CF3 = $960.00
(1.20) = $1,152.00; CF4 = $1,152.00(1.20) = $1,382.40; CF5 = $1,382.40
(1.20) = $1,658.88; CF6 = $1,658.88(1.20) = $1,990.66; CF7 = $1,990.66
(1.20) = $2,388.79; CF8 = $2,388.79(1.20) = $2,866.54.
To find the present value of this cash flow stream using your financial
calculator enter:
END mode (to indicate the cash flows will occur at the end of each
period) 0 CFj; 800 CFj; 960 CFj; 1152 CFj; 1382.40 CFj; 1658.88 CFj;
1990.66 CFj; 2388.79 CFj; 2866.54 CFj (to enter the cash flows);I/YR =
12/4 = 3; solve for NPV = $11,267.37.
Choose the alternative with the highest present value, and hence select
Choice 1 (Answer a).
169.
Breakeven annuity payment
Step 1:
Answer: a
Calculate the NPV of purchasing the car
following data in your financial calculator:
by
entering
the
CF0 = 17000; CF147 = 0; CF48 = 7000; I/YR = 6/12 = 0.5; and
solve for NPV = $11,490.31.
Step 2:
Now, use the NPV calculated in Step 1 to determine the breakeven
lease payment that will cause the two NPVs to be equal. Enter
the following data in your financial calculator:
N = 48; I/YR = 0.5; PV = 11490.31; FV = 0; and solve for PMT =
$269.85.
170.
Required mortgage payment
Answer: b
Just enter the following data into your calculator and solve for the
monthly mortgage payment.
N = 360; I/YR = 7/12 = 0.583333; PV = 115000; FV = 0; and solve for PMT
= $765.0979 ๏ป $765.10.
171.
Remaining mortgage balance
Answer: e
With the data still input into your calculator, press 1 INPUT 60 โผ AMORT
= displays Interest: $39,157.2003
= displays Principal: $6,748.6737
= displays Balance: $108,251.3263
172.
Time to accumulate a lump sum
Answer: d
You must solve this time value of money problem for N (number of years)
by entering the following data in your calculator:
Page 124
Chapter 2: Time Value of Money
I/YR = 10; PV = 2000; PMT = 1000; FV = 1000000; and solve for N = 46.51.
Because there is a fraction of a year and the problem asks for whole
years, we must round up to the next year. Hence, the answer is 47 years.
173.
Required annual rate of return
Answer: c
Now, the time value of money problem has been modified to solve for
I/YR. Enter the following data in your calculator:
N = 39; PV = 2000; PMT = 1000; FV = 1000000; and solve for I/YR = 12.57%.
174.
Monthly mortgage payments
Answer: c
Enter the following data as inputs in your calculator:
N=30๏ด12 = 360; I/YR=7.2/12=0.60; PV=100000; FV=0; solve PMT = $678.79.
175.
Amortization
Answer: d
Determine the mortgage payment, then input:
1 INPUT 36 โผ AMORT
= Interest:
$21,280.8867
= Principal:
$3,155.4885
= Balance:
$96,844.5115.
The percentage that goes to principal =
176.
$3,155.49
$3,155.49
=
= 12.91%.
$24,436.44
36 ๏ด $678.79
Monthly mortgage payments
Answer: d
Using your financial calculator, enter the following data inputs:
N = 180; I/YR = 7.75/12 = 0.645833; PV = 165000; FV = 0; and solve for
PMT = $1,553.104993 ๏ป $1,553.10.
177.
Remaining mortgage balance
Answer: c
The complete solution looks like this:
Beginning
of Period
1
2
3
4
5
6
7
8
Mortgage
Balance
$165,000.00
164,512.52
164,021.89
163,528.09
163,031.11
162,530.91
162,027.49
161,520.81
Chapter 2: Time Value of Money
Payment
$1,553.10
1,553.10
1,553.10
1,553.10
1,553.10
1,553.10
1,553.10
1,553.10
Interest
$1,065.63
1,062.48
1,059.31
1,056.12
1,052.91
1,049.68
1,046.43
1,043.16
Ending
Mortgage Balance
$164,512.52
164,021.89
163,528.09
163,031.11
162,530.91
162,027.49
161,520.81
161,010.86
Page 125
9
10
11
12
161,010.86
160,497.62
159,981.06
159,461.16
1,553.10
1,553.10
1,553.10
1,553.10
1,039.86
1,036.55
1,033.21
1,029.85
160,497.62
159,981.06
159,461.16
158,937.91
Alternatively, using your financial calculator, do the following (with
the data still entered from the previous problem):
1 INPUT 12 โผ AMORT
= Interest: $12,575.172755
= Principal: $6,062.087161
= Balance: $158,937.912839
178.
Amortization
Step 1:
Find the monthly payment:
Step 2:
N=360;
I/YR=8/12=0.6667;
PV=75000;
FV=0;
solve
PMT
$550.3234.
Calculate value of monthly payments for the first year:
Step 3:
Total payments in the first year = $550.3234 ๏ด 12 = $6,603.8812.
Determine amount of interest during first year:
Step 4:
179.
Answer: d
1 INPUT 12 โผ AMORT
= Interest: $5,977.3581
= Principal: $626.5227
= Balance: $74,373.4773
Calculate the percentage that goes towards interest:
$5,977.3581/$6,603.8812 = 0.9051, or 90.51%.
Amortization
Step 1:
=
Answer: a
Calculate old monthly payment:
N = 360; I/YR = 8/12 = 0.6667; PV = 75000; FV = 0; and solve
for PMT = $550.3234.
Step 2:
Calculate new monthly payment:
N = 360; I/YR = 7/12 = 0.5833; PV = 75000; FV = 0; and solve
for PMT = $498.9769.
Step 3:
Calculate the difference between the 2 mortgage payments:
This represents a
$51.3465 ๏ป $51.35.
180.
savings
of
($550.3234
Monthly mortgage payment
Enter the following data in your calculator:
Page 126
โ
$498.9769)
=
Answer: c
Chapter 2: Time Value of Money
N = 360; I/YR = 7.2/12 = 0.60; PV = 300000; FV = 0; and solve for PMT =
$2,036.3646 ๏ป $2,036.36.
181.
Amortization
Using a financial calculator and the above information:
1 INPUT 12 โผ AMORT
= Interest: $21,504.5022
= Principal: $2,931.8730
= Balance: $297,068.1270
Answer: b
The percent paid toward principal = $2,931.87/($2,931.87 + $21,504.50)
= 12%.
WEB APPENDIX 2A SOLUTIONS
2A1.
PV continuous compounding
Answer: b
EASY/MEDIUM
PV = FVn/ein = $100,000/e0.09(6) = $100,000/1.7160 = $58,275.
2A2.
FV continuous compounding
Daily compounding:
FV2 = PV (1 + 0.06/365)365(2) = $1,000(1.12749) =
Continuous compounding:
FV2 = PVein = $1,000(e0.059(2)) = $1,000(1.12524) =
Difference between accounts
2A3.
Continuous compounding
Chapter 2: Time Value of Money
Answer: a
MEDIUM
$1,127.49
$1,125.24
$
2.25
Answer: b
MEDIUM
Page 127
0

1

i = e0.10
10

โข โข โข
PV = ?
Years
FV = 5,438
Numerical solution:
(Constant e = 2.7183 rounded.)
$5,438 = PVe0.10(10)
$5,438 = PVe1
PV = $5,438/e
= $5,438/2.7183 = $2,000.52 โ $2,000.
Financial calculator solution:
Use ex exponential key on calculator.
Calculate EAR with continuous
compounding.
Inputs: X = 0.10; press ex key.
Output: ex = 1.1052.
EAR = 1.1052 – 1.0 = 0.1052 = 10.52%.
Calculate PV of FV discounted continuously
Inputs: N=10; I/YR=10.52; PMT=0; FV=5438.
Output: PV=$2,000.
2A4.
Continuous compounded interest rate
Answer: a
MEDIUM/HARD
Calculate the growth factor using PV and FV which are given:
FVn = PV ein; $19,000 = $14,014 ei4
ei4 = 1.35579.
Take the natural logarithm of both sides:
i(4) ln e = ln 1.35579.
The natural log of e = 1.0.
Inputs: 1.35579. Press LN key. Output:
2A5.
i(4)ln e = ln 1.35579
i(4) = 0.30438
i = 0.0761 = 7.61%.
Payment and continuous compounding
0
Ic = e0.07
Is = 4%


1
2

Answer: d

2
4


3
6

MEDIUM/HARD
Years
6months
Periods
Account with
continuous
compounding
1,000
FVc = ? = 1,233.70
Account with
semiannual
compounding
PVs = ?
FVs = ? = 1,233.70
Step 1:
Page 128
LN = 0.30438.
Calculate the FV of the $1,000 deposit at 7% with continuous
compounding:
Using ex key:
Inputs: X = 0.21; press ex key. Output: ex = 1.2337.
Chapter 2: Time Value of Money
FVn = $1,000 e0.07(3) = $1,000(1.2337) = $1,233.70.
Step 2:
2A6.
Calculate the PV or initial deposit:
Inputs: N = 6; I/YR = 4; PMT = 0; FV = 1233.70.
Output: PV = $975.01.
Continuous compounding
Answer: d
MEDIUM/HARD
Answer: b
MEDIUM/HARD
Numerical solution:
20
i๏ถ
๏ฆ
e(0.04)(10)
= ๏ง1 + ๏ท
2๏ธ
๏จ
20
i๏ถ
๏ฆ
= ๏ง1 + ๏ท
2๏ธ
๏จ
i
e0.02 = 1 +
2
i
1.0202
= 1 +
2
i
= 0.0202
2
i
= 0.0404 = 4.04%.
e0.4
2A7.
Continuous compounding
Time Line:
0
i = 10.52%

PV = ?
1

2

โข โข โข
10
Years

FV = 1,000
Numerical solution:
$1,000 = PVe0.10(10) = PVe1.0
PV = $1,000/e = $1,000/2.7183 = $367.88 โ $368.
Financial calculator solution:
Use ex exponential key on calculator.
Calculate EAR with continuous
compounding.
Inputs: X = 0.10; press ex key. Output: ex = 1.1052.
Chapter 2: Time Value of Money
Page 129
EAR = 1.1052 – 1.0 = 0.1052 = 10.52%.
Calculate PV of FV discounting at the EAR:
Inputs: N = 10; I/YR = 10.52; PMT = 0; FV = 1000.
Output: PV = $367.78 โ $368.
2A8.
Continuous compounding
Time Line:
0
1
i = 5.127%


PV = 15,000
Answer: b
2

HARD
20
Years

FV = ?
โข โข โข
Numerical solution:
FV20 = $15,000e0.05(20) = $40,774.23 โ $40,774.
Financial calculator solution:
(Note: We carry the EAR to 5 decimal places for greater precision in
order to come closer to the correct exponential solution.)
Inputs: X = 0.05; press ex key. Output: ex = 1.05127.
EAR = 1.05127 – 1.0 = 0.05127 = 5.127%.
Calculate FV compounded continuously at EAR = 5.127%
Inputs: N = 20; I/YR = 5.127; PV = 15000; PMT = 0.
Output: FV = $40,773.38 โ $40,774.
Page 130
Chapter 2: Time Value of Money
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