TIME VALUE OF MONEY
PROBLEMS
Exercise 1 (FV)
Sales of the P.J. Cramer Company were $500,000 this year, and they are expected to grow at a
compound rate of 20 percent for the next six years. What will be the sales figure at the end of each of
the next six years?
Exercise 2 (PV)
Suppose you were to receive $1,000 at the end of 10 years. If your opportunity rate is 10 percent, what
is the present value of this amount if interest is compounded (a) annually? (b) quarterly? (c)
continuously?
Exercise 3 (Interest rate)
Vernal Equinox wishes to borrow $10,000 for three years. A group of individuals agrees to lend him
this amount if he contracts to pay them $16,000 at the end of the three years. What is the implicit
compound annual interest rate implied by this contract (to the nearest whole percent)?
Exercise 4 (Interest rate)
In connection with the United States Bicentennial, the Treasury once contemplated offering a savings
bond for $1,000 that would be worth $1 million in 100 years. Approximately what compound annual
interest rate is implied by these terms?
Exercise 5 (EAR, APR)
Suppose that an investment promises to pay a nominal 9.6 percent annual rate of interest. What is the
effective annual interest rate on this investment assuming that interest is compounded (a) annually? (b)
semiannually? (c) quarterly? (d) monthly? (e) daily (365 days)? (f ) continuously? (Note: Report your
answers accurate to four decimal places – e.g., 0.0987 or 9.87%.)
Exercise 6 (FV, EAR, Different compounding)
The following are exercises in future (terminal) values:
a. At the end of three years, how much is an initial deposit of $100 worth, assuming a compound annual
interest rate of (i) 100 percent? (ii) 10 percent? (iii) 0 percent?
b. At the end of five years, how much is an initial $500 deposit followed by five year-end, annual $100
payments worth, assuming a compound annual interest rate of (i) 10 percent? (ii) 5 percent? (iii) 0
percent?
c. At the end of six years, how much is an initial $500 deposit followed by five year-end, annual $100
payments worth, assuming a compound annual interest rate of (i) 10 percent? (ii) 5 percent? (iii) 0
percent?
d. At the end of three years, how much is an initial $100 deposit worth, assuming a quarterly
compounded annual interest rate of (i) 100 percent? (ii) 10 percent?
e. At the end of 10 years, how much is a $100 initial deposit worth, assuming an annual interest rate of
10 percent compounded (i) annually? (ii) semiannually? (iii) quarterly? (iv) continuously?
Exercise 7 (PV, One CF, EAR, Different compounding)
The following are exercises in present values:
a. $100 at the end of three years is worth how much today, assuming a discount rate of (i) 100 percent?
(ii) 10 percent? (iii) 0 percent?
b. What is the aggregate present value of $500 received at the end of each of the next three years,
assuming a discount rate of (i) 4 percent? (ii) 25 percent?
c. $100 is received at the end of one year, $500 at the end of two years, and $1,000 at the end of three
years. What is the aggregate present value of these receipts, assuming a discount rate of (i) 4 percent?
(ii) 25 percent?
d. $1,000 is to be received at the end of one year, $500 at the end of two years, and $100 at the end of
three years. What is the aggregate present value of these receipts assuming a discount rate of (i) 4
percent? (ii) 25 percent?
Exercise 8 (EAR, One CF)
A bank offers you a seven-month certificate of deposit (CD) at a 7.06 percent annual rate that would
provide a 7.25 percent effective annual yield. For the seven-month CD, is interest being compounded
daily, weekly, monthly, or quarterly?
And, by the way, having invested $10,000 in this CD, how much money would you receive when your
CD matures in seven months?
Exercise 9 (PV)
Lost Dutchman Mines, Inc., is considering investing in Peru. It makes a bid to the government to
participate in the development of a mine, the profits of which will be realized at the end of five years.
The mine is expected to produce $5 million in cash to Lost Dutchman Mines at that time. Other than
the bid at the outset, no other cash flows will occur, as the government will reimburse the company for
all costs. If Lost Dutchman requires a nominal annual return of 20 percent (ignoring any tax
consequences), what is the maximum bid it should make for the participation right if interest is
compounded (a) annually? (b) semiannually? (c) quarterly? (d) continuously?
Exercise 10 (Uneven stream of CFs)
The following cash-flow streams need to be analyzed:
Cash Flow
Stream
W
X
Y
Z
1
$100
600
200
2
$200
-
End of Year
3
$200
500
4
$300
-
5
$300
1,200
300
a. Calculate the future (terminal) value of each stream at the end of year 5 with a compound annual
interest rate of 10 percent.
b. Compute the present value of each stream if the discount rate is 14 percent.
Exercise 11 (Annuity)
You have been offered a note with four years to maturity, which will pay $3,000 at the end of each of
the four years. The price of the note to you is $10,200. What is the implicit compound annual interest
rate you will receive (to the nearest whole percent)?
Exercise 12 (Annuity)
The Happy Hang Glide Company is purchasing a building and has obtained a $190,000 mortgage loan
for 20 years. The loan bears a compound annual interest rate of 17 percent and calls for equal annual
installment payments at the end of each of the 20 years. What is the amount of the annual payment?
Exercise 13 (Annuity)
You have borrowed $14,300 at a compound annual interest rate of 15 percent. You feel that you will be
able to make annual payments of $3,000 per year on your loan. (Payments include both principal and
interest). How long will it be before the loan is entirely paid off (to the nearest year)?
Exercise 14 (Uneven CFs)
The H & L Bark Company is considering the purchase of a debarking machine that is expected to
provide cash flows as follows:
Cash Flow
End of Year
1
2
3
4
5
6
7
1,200 2,000 2,400 1,900 1,600 1,400 1,400
8
1,400
9
10
1,400 1,400
If the appropriate annual discount rate is 14 percent, what is the present value of this cash-flow stream?
Exercise 15 (Annuity)
Emerson Cammack wishes to purchase an annuity contract that will pay him $7,000 a year for the rest
of his life. The Philo Life Insurance Company figures that his life expectancy is 20 years, based on its
actuary tables. The company imputes a compound annual interest rate of 6 percent in its annuity
contracts.
a. How much will Cammack have to pay for the annuity?
b. How much would he have to pay if the interest rate were 8 percent?
Exercise 16 (Annuity)
On a contract you have a choice of receiving $25,000 six years from now or $50,000 twelve years from
now. At what implied compound annual interest rate should you be indifferent between the two
contracts?
Exercise 16 (Annuity)
Joe Hernandez has inherited $25,000 and wishes to purchase an annuity that will provide him with a
steady income over the next 12 years. He has heard that the local savings and loan association is
currently paying 6 percent compound interest on an annual basis. If he were to deposit his funds, what
year-end equal-dollar amount (to the nearest dollar) would he be able to withdraw annually such that
he would have a zero balance after his last withdrawal 12 years from now?
Exercise 17 (Annuity)
You need to have $50,000 at the end of 10 years. To accumulate this sum, you have decided to save a
certain amount at the end of each of the next 10 years and deposit it in the bank. The bank pays 8 percent
interest compounded annually for long-term deposits. How much will you have to save each year (to
the nearest dollar)?
If you deposit a certain amount at the beginning of each of the next 10 years, how much will you have
to save each year (to the nearest dollar)?
Exercise 18 (Annuity)
Muffin Megabucks is considering two different savings plans. The first plan would have her deposit
$500 every six months, and she would receive interest at a 7 percent annual rate, compounded
semiannually. Under the second plan she would deposit $1,000 every year with a rate of interest of 7.5
percent, compounded annually. The initial deposit with Plan 1 would be made six months from now
and, with Plan 2, one year hence.
a. What is the future (terminal) value of the first plan at the end of 10 years?
b. What is the future (terminal) value of the second plan at the end of 10 years?
c. Which plan should Muffin use, assuming that her only concern is with the value of her savings at the
end of 10 years?
d. Would your answer change if the rate of interest on the second plan were 7 percent?
Exercise 19 (Annuity)
Selyn Cohen is 63 years old and recently retired. He wishes to provide retirement income for himself
and is considering an annuity contract with the Philo Life Insurance Company. Such a contract pays
him an equal-dollar amount each year that he lives. For this cash-flow stream, he must put up a specific
amount of money at the beginning.
According to actuary tables, his life expectancy is 15 years, and that is the duration on which the
insurance company bases its calculations regardless of how long he actually lives.
a. If Philo Life uses a compound annual interest rate of 5 percent in its calculations, what must Cohen
pay at the outset for an annuity to provide him with $10,000 per year? (Assume that the expected annual
payments are at the end of each of the 15 years.)
b. What would be the purchase price if the compound annual interest rate is 10 percent?
c. Cohen had $30,000 to put into an annuity. How much would he receive each year if the insurance
company uses a 5 percent compound annual interest rate in its calculations? A 10 percent compound
annual interest rate?
Exercise 20 (Annuity)
Earl E. Bird has decided to start saving for his retirement. Beginning on his twenty-first birthday, Earl
plans to invest $2,000 each birthday into a savings investment earning a 7 percent compound annual
rate of interest. He will continue this savings program for a total of 10 years and then stop making
payments. But his savings will continue to compound at 7 percent for 35 more years, until Earl retires
at age 65. Ivana Waite also plans to invest $2,000 a year, on each birthday, at 7 percent, and will do so
for a total of 35 years. However, she will not begin her contributions until her thirty-first birthday. How
much will Earl’s and Ivana’s savings programs be worth at the retirement age of 65? Who is better off
financially at retirement, and by how much?
Exercise 21 (Annuity)
When you were born, your dear old Aunt Minnie promised to deposit $1,000 in a savings account for
you on each and every one of your birthdays, beginning with your first. The savings account bears a 5
percent compound annual rate of interest. You have just turned 25 and want all the cash. However, it
turns out that dear old (forgetful) Aunt Minnie made no deposits on your fifth, seventh, and eleventh
birthdays. How much is in the account now – on your twenty-fifth birthday?
Exercise 22 (Annuity)
Assume that you will be opening a savings account today by depositing $100,000. The savings account
pays 5 percent compound annual interest, and this rate is assumed to remain in effect for all future
periods. Four years from today you will withdraw R dollars. You will continue to make additional
annual withdrawals of R dollars for a while longer – making your last withdrawal at the end of year 9
– to achieve the following pattern of cash flows over time. (Note: Today is time period zero; one year
from today is the end of time period 1; etc.)
How large must R be to leave you with exactly a zero balance after your final R withdrawal is made at
the end of year 9? (Tip: Making use of an annuity table or formula will make your work a lot easier)
Exercise 23 (Rule 72, PV, FV)
It took roughly 14 years for the Dow Jones Average of 30 Industrial Stocks to go from 1,000 to 2,000.
To double from 2,000 to 4,000 took only 8 years, and to go from 4,000 to 8,000 required roughly 2
years. To the nearest whole percent, what compound annual growth rates are implicit in these three
index-doubling milestones?
Exercise 24 (Mortgage)
Xu Lin recently obtained a 10-year, $50,000 loan. The loan carries an 8 percent compound annual
interest rate and calls for annual installment payments of $7,451.47 at the end of each of the next 10
years.
a. How much (in dollars) of the first year’s payment is principal?
b. How much total interest will be paid over the life of the loan? (Hint: You do not need to construct a
loan amortization table to answer this question. Some simple math is all you need.)
Exercise 25 (Mortgage)
You borrow $10,000 at 14 percent compound annual interest for four years. The loan is repayable in
four equal annual installments payable at the end of each year.
a. What is the annual payment that will completely amortize the loan over four years?
(You may wish to round to the nearest dollar.)
b. Of each equal payment, what is the amount of interest? The amount of loan principal?
(Hint: In early years, the payment is composed largely of interest, whereas at the end it is mainly
principal.)
MULTIPLE CHOICE QUESTIONS
1.
a.
b.
c.
d.
2.
$5,000 invested at 6%, compounded quarterly, will be worth how much after 5 years?
$6,691
$16,036
$6,734
$5,386
Tom deposits $10,000 in a savings deposit paying 4%, compounded monthly. What amount
would he have at the end of seven years?
a. $13,225
b. $13,159
c. $13,179
d. $13,325
3.
Judy would like to accumulate $70,000 by the time her son starts college in ten years. What
amount would she need to deposit now in a deposit account earning 6%, compounded yearly,
to accumulate her savings goal?
a. $4,200
b. $39,513
c. $39,088
d. $125,359
4.
What is the monthly payment on a $200,000 conventional fixed-rate mortgage, 7 percent,
financed for 15 years?
a. $1830
b. $1797
c. $1679
d. $1721
5.
With reference to the question above, what is the loan balance after 10 years if paid as agreed?
a. $92,721
b. $83,581
c. $85,492
d. $90.785
6.
What is the monthly payment on a $100,000 fixed rate loan with a 6.5% rate with a term of
30 years?
a. $657
b. $632
c. $638
d. $612
7.
With reference to the question above, if you added $100 to the monthly payment, how soon
would your loan be paid off?
a. 249 months
b. 227 months
c. 185 months
d. 278 months
8.
You have just purchased a home and borrowed $50,000, 7 percent for 25 years, payable
monthly. What is your monthly payment?
a. $338
b. $339
c. $353
d. $369
9.
a. $305
b. $265
c. $257
d. $297
With reference to the question above, what was the amount of interest paid in the first month
of the loan? (Assume a January payment in a 365 day year)
10.
How long does it take to repay one-half of the principal on a $70,000, 7 percent, 15 year
mortgage loan?
a. 112 months
b. 95 months
c. 123 months
d. 131 months