37) You have been given a choice between two retirement policies as described below.
Policy A: You will receive equal annual payments of $10,000 beginning 35 years from now for
10 years.
Policy B: You will receive one lump-sum of $100,000 in 40 years from now.
Which policy would you choose? Assume rate of interest is 6 percent.
38) A charitable foundation has $500,000 invested in an account that earns 7%. The foundation
has promised to begin making annual payments to beneficiaries in one year, and the first
payment will be $25,000. The foundation has promised that future payments will grow at a
constant rate forever. At what rate can the foundation afford to increase payments assuming that
it makes no additional deposits into the account?
A) 0%; it can’t afford to increase payments forever without adding more money to the account.
B) 1%
C) 2%
D) 3%
39) A university has $16,000,000 invested in its endowment. The university wants to withdraw
$800,000 from this endowment starting next year and continuing at annual intervals forever, with
each subsequent payment growing at 4% per year. What rate of return does the endowment have
to earn to sustain the desired withdrawals?
40) A wealthy benefactor wants to make a donation to a charity that will provide the charity with
annual income of $250,000 forever, with the first payment to the charity made exactly 10 years
from today. Assume that money donated to the charity will be invested in an account that earns
6%. How large must the donation be to generate the desired income stream?
41) How much money would you have to deposit today to create an income stream that pays
$10,000 one year from today and continues to make annual payments forever, with payments
after the first $10,000 growing at 4% per year? Assume money that you invest today to fund this
income stream earns a 7% rate of return.
A) $142,857
B) $250,000
C) $1,250,000
D) $333,333
42) A certain investment promises to pay you $2,500 per year forever with the first payment
starting next year. If you can earn a 5% return on similar investments, what’s the most you would
pay for this investment today?
A) $50,000
B) $5,000
C) $41,667
D) $62,500
43) A perpetuity will pay you $100 starting next year and continuing at that level forever. The
appropriate discount rate for the perpetuity is 10%. Calculate the present value of (1) the infinite
stream of payments, i.e., the entire perpetuity, and (2) the first 20 payments. Compare these two
present values and comment on what they can tell you about the present value of the perpetuity’s
payments in the very distant future (i.e., payments made later than 20 years in the future).
44) You inherited an investment portfolio worth $1 million. The portfolio earns a return of 9%
per year. You want to withdraw money from this portfolio once per year starting one year from
today, and you want to continue making withdrawals forever. Furthermore, you want to increase
your withdrawals at 3% per year to keep up with inflation. How large can your first withdrawal
be?
45) A certain investment promises to pay you $2,500 per year forever with the first payment
starting 5 years from now. If you can earn a 5% return on similar investments, what’s the most
you would pay for this investment today?
A) $50,000
B) $39,176
C) $41,135
D) $37,311
46) Ashley is planning to attend college when she graduates from high school 7 years from now.
She anticipates that she will need $20,000 at the beginning of each of the four college years to
pay for tuition and fees, and have some spending money (i.e., she needs to be able to withdraw
$20,000 from savings four times, with the first withdrawal taking place 7 years from now).
Ashley’s father has promised to help her save for college by making 7 deposits of $7,000 each
into an investment accounting earning 8 percent interest. His first payment comes a year from
today. Will there be enough money in the account for Ashley to pay for her college expenses?
Assume the rate of interest stays at 8 percent during the college years.
5.4 Mixed streams
1) You receive $100 today, $200 in one year, and $300 in two years. If you deposit these cash
flows into an account earning 12 percent, the value in the account three years from now is
________.
A) less than $600
B) $649
C) $727
D) $815
2) You receive $1,200 today, $2,200 in one year, and $3,300 in two years. If you deposit these
cash flows in an account earning 12%, how much money is in the account three years from now?
A) $6,221
B) $5,554
C) $7,269
D) $8,142
3) Find the future value at the end of year 3 of the following stream of cash flows received at the
end of each year, assuming the firm can earn 17 percent on its investments.
A) $16,320
B) $20,127
C) $23,548
D) $27,551
4) Find the future value at the end of year 3 of the following stream of cash flows received at the
end of each year, assuming the firm can earn 8 percent on its investments.
A) $51,780
B) $39,248
C) $47,944
D) $40,981
5) You receive $1,000 in 1 year, $1,200 in 2 years, and $1,300 in 3 years. The present value
today of these future receipts is ________ if the opportunity cost is 7 percent.
A) $2,500
B) $3,044
C) $4,036
D) $3,257
6) You will receive $100 in 1 year, $200 in 2 years, and $300 in 3 years. If you can earn 13% on
your investments, the present value of these future receipts is ________.
A) $453
B) $512
C) $801
D) $600
7) Find the present value of the following stream of cash flows assuming an opportunity cost of
14 percent.
A) $59,169
B) $92,443
C) $81,090
D) $51,903
8) Find the present value of the following stream of cash flows assuming an opportunity cost of
25 percent.
A) $27,168
B) $33,960
C) $72,656
D) $41,674
9) Find the present value of the following stream of cash flows assuming an opportunity cost of 9
percent.
A) $85,791
B) $187,838
C) $65,213
D) $79,345
10) Find the present value of the following stream of a firm’s cash flows, assuming that the firm’s
opportunity cost is 14 percent.
A) $131,068
B) $149,417
C) $485,897
D) $104,322
11) During her four years at college, Hayley received the following amounts of money at the end
of each year from her grandmother. She deposited her money in a savings account paying 6
percent rate of interest. How much money will Hayley have on graduation day?
12) You have provided your friend with a service worth $8,500. Your friend offers you the
following sequence of end-of-year cash flows instead of paying $8,500 today. Should you accept
his offer if your opportunity cost is 8 percent?
13) Calculate the present value of $5,800 received at the end of year 1, $6,400 received at the
end of year 2, and $8,700 at the end of year 3, assuming an opportunity cost of 13 percent.
14) Calculate the present value of $800 received at the beginning of year 1, $400 received at the
beginning of year 2, and $700 received at the beginning of year 3, assuming an opportunity cost
of 9 percent.
15) Calculate the combined future value at the end of year 3 of $1,000 received at the end of year
1, $3,000 received at the end of year 2, and $5,000 received at the end of year 3, all sums
deposited at 5 percent.
16) You are considering the purchase of new equipment for your company and you have
narrowed down the possibilities to two models which perform equally well. However, the
method of paying for the two models is different. Model A requires $5,000 per year payment for
the next five years. Model B requires the following payment schedule. Which model should you
buy if your opportunity cost is 8 percent?
17) Last Christmas, Danny received an annual bonus of $1,500. These annual bonuses are
expected to grow by 5 percent for the next 5 years. How much will Danny have at the end of the
fifth year if he invests his Christmas bonuses (including the most recent bonus) in an account
paying 8 percent per year?
18) Calculate the present value of the following stream of cash flows, assuming that the firm’s
opportunity cost is 15 percent.
5.5 Compounding interest more frequently than annually
1) The nominal (stated) annual rate is the rate of interest actually paid or earned.
2) The nominal and effective rates are equivalent for annual compounding.
3) The effective annual rate increases with increasing compounding frequency.
4) The annual percentage rate (APR) is the nominal rate of interest, found by multiplying the
periodic rate by the number of periods in one year.
5) The annual percentage yield (APY) is the effective rate of interest that must be disclosed to
customers by banks on their savings products as a result of “truth in savings laws.”
6) The effective rate of interest is the contractual rate of interest charged by a lender or promised
by a borrower.
7) The effective rate of interest differs from the nominal rate of interest in that it reflects the
impact of compounding frequency.
8) For any interest rate and for any period of time, the more frequently interest is compounded,
the greater the amount of money that has to be invested today in order to accumulate a given
future amount.
9) The effective rate of interest and compounding frequency are inversely related.
10) The rate of interest agreed upon contractually charged by a lender or promised by a borrower
is the ________ interest rate.
A) effective
B) nominal
C) discounted
D) continuous
11) The rate of interest actually paid or earned, also called the annual percentage rate (APR), is
the ________ interest rate.
A) effective
B) nominal
C) discounted
D) continuous
12) The future value of $200 received today and deposited at 8 percent compounded
semiannually for three years is ________.
A) $380
B) $158
C) $253
D) $252
13) The future value of $100 received today and deposited in an account for four years paying
semiannual interest of 6 percent is ________.
A) $450
B) $127
C) $889
D) $134
14) The future value of $200 received today and deposited for three years in an account which
pays semiannual interest of 8 percent is ________.
A) $253.00
B) $252.00
C) $158.00
D) $134.66
15) The future value of an annuity of $1,000 each quarter for 10 years, deposited at 12 percent
compounded quarterly is ________.
A) $17,549
B) $75,401
C) $93,049
D) $11,200
16) What is the highest effective rate attainable with a 12 percent nominal rate?
A) 12.00%
B) 12.55%
C) 12.75%
D) 12.95%
17) Gina has planned to start her college education four years from now. To pay for her college
education, she has decided to save $1,000 a quarter for the next four years in an investment
account paying 12 percent interest. How much will she have at the end of the fourth year?
A) $1,574
B) $19,116
C) $20,157
D) $16,000
18) How much would Sophie have in her account at the end of 10 years if she deposit $2,000
into the account today if she earned 8 percent interest and interest is compounded continuously?
A) $4,317
B) $4,134
C) $4,451
D) $4,521
19) Assume Julian has a choice between two deposit accounts. Account A has an annual
percentage rate of 7.55 percent but with interest compounded monthly. Account B has an annual
percentage rate of 7.45 percent with interest compounded continuously. Which account provides
the highest effective annual return?
A) Account A
B) Account B
C) Both provide the same effective annual return.
D) We don’t have sufficient information to make a choice.
20) Calculate the future value of $6,490 received today and deposited for five years in an
account which pays interest of 14 percent compounded semiannually.
21) Calculate the future value of $10,000 received today and deposited for six years in an
account which pays interest of 12 percent compounded quarterly.
22) Jeanne has just graduated from high school and has received an award for $5,000. She would
like to deposit the money in an interest earning account until she graduates from college (i.e.,
four years from now). In her search for the highest interest earning account, she has narrowed the
list down to the following two accounts: 1) bank A pays 9 percent interest compounded annually,
and 2) bank B pays 8 percent interest compounded semiannually. Which is the better offer, and
how much will Jeanne have upon graduation from college?