CHAPTER 3 THE TIME VALUE OF MONEY

Future
Value of a Lump Sum
P3-1. You
have $1,500 to invest today at 7 percent interest compounded annually.
a. How much will you have accumulated in the
account at the end of the following number of years?
1. Three years
2. Six years
3. Nine years
b. Use your findings in part (a) to calculate
the amount of interest earned in
1. the first three years (years 1 to 3)
2. the second three years (years 4 to 6)
3. the third three years (years 7 to 9)
c. Compare and contrast your findings in part
(b). Explain why the amount of interest earned increases in each succeeding
3-year period.

Present Value
of a Lump Sum
P3-2. You
just won a lottery that promises to pay you $1 million exactly 10 years from
today. Because the $1 million payment is guaranteed by the state in which you
live, opportunities exist to sell the claim today for an immediate lump-sum
cash payment.
a. What is the least you will sell your claim
for if you could earn the following rates of return on similar-risk investments
during the 10-year period?
1. 6
percent
2. 9
percent
3. 12
percent
b. Rework part (a) under the assumption that the
$1 million payment will be received in 15 rather than 10 years.
c. Based on your findings in parts (a) and (b),
discuss the effect of both the size of the rate of return and the time until
receipt of payment on the present value of a future sum.

P3-3. An Indiana state savings
bond can be converted to $100 at maturity six years from purchase. If the state
bonds are to be competitive with U.S. savings bonds, which pay 8 percent
annual interest (compounded annually), at what price must the state sell its bonds?
Ignore taxes and assume no cash payments on savings bonds prior to redemption.

Future Value of Cash Flow Streams
P3-4. Kim Edwards and Chris Phillips are both
newly minted 30-year old MBAs. Kim plans to invest $1,000 per month into her
401(k) beginning next month, while Chris intends to invest $2,000 per month,
but he does not plan to begin investing until 10 years after Kim begins
investing. Both Kim and Chris will retire at age 67, and the 401(k) plan
averages a 12 percent annual return compounded monthly. Who will have more
401(k) money at retirement?

P3-5. Robert Williams is
considering an offer to sell his medical practice, allowing him to retire five
years early. He has been offered $500,000 for his practice and can invest this
amount in an account earning 10 percent per year, compounded annually. If the
practice is expected to generate the following cash flows, should Robert accept
this offer and retire now?

End of Year

Cash Flow

1

$150,000

2

150,000

3

125,000

4

125,000

5

100,000

.

P3-6. For
the following questions, assume an annual annuity of $1,000 and a required
return of 12
percent.

a. What is the future value of an ordinary
annuity for 10 years?
b. If you earned an additional year’s worth of
interest on this annuity, what would be the future value?
c. What is the future value of a 10-year annuity
due?
d. What is the relationship between your answers
in parts (b) and (c)? Explain.

P3-7. Robert
Blanding’s employer offers its workers a two-month paid sabbatical every seven
years. Robert, who just started working for the firm, plans to spend his
sabbatical touring Europe at an estimated cost
of $25,000. To finance his trip, Robert plans to make six annual end-of-year
deposits of $2,500 each, starting this year, into an investment account earning
8% interest.

a. Will Robert’s account balance at the end of
seven years be enough to pay for his trip?

b. Suppose Robert increases his annual
contribution to $3,150. How large will his account balance be at the end of
seven years?

P3-8. Gina
Coulson has just contracted to sell a small parcel of land that she inherited a
few years ago. The buyer is willing to pay $24,000 at closing of the
transaction or will pay the amounts shown in the following table at the beginning of each of the next five
years. Because Gina doesn’t really need the money today, she plans to let it
accumulate in an account that earns 7 percent annual interest. Given her desire
to buy a house at the end of five years after closing on the sale of the lot,
she decides to choose the payment alternative—$24,000 lump sum or mixed stream
of payments in the following table—that provides the highest future value at
the end of five years.

Mixed Stream

Beginning of Year (t)

Cash Flow (CFt)

1

$ 2,000

2

4,000

3

6,000

4

8,000

5

10,000

a. What is the future value of the lump sum at
the end of year 5?
b. What is the future value of the mixed stream
at the end of year 5?
c. Based on your findings in parts (a) and (b),
which alternative should Gina take?
d. If Gina could earn 10 percent rather than 7
percent on the funds, would your recommendation in part (c) change? Explain.

.

P3-9. Dixon
Shuttleworth has been offered the choice among three retirement-planning investments.
The first investment offers a 5 percent return for the first 5 years, a 10
percent return for the next 5 years, and a 20 percent return thereafter. The
second investment offers 10 percent for the first 10 years and 15 percent
thereafter. The third investment offers a constant 12 percent rate of return.
Determine, for each of the given number of years, which of these investments is
the best for Dixon
if he plans to make one payment today into one of these funds and plans to
retire in the following number of years:
a. 15 years
b. 20 years
c. 30 years

Present Value of Cash Flow Streams
P3-10. For the
following questions, assume an end-of-year cash flow of $250 and a 10 percent
discount rate.
a. What is the present value of a single cash
flow?
b. What is the present value of a 5-year
annuity?
c. What is the present value of a 10-year
annuity?
d. What is the present value of a 100-year
annuity?
e. What is the present value of a $250
perpetuity?
f. Do you detect a relationship between the
number of periods of an annuity and its resemblance to a perpetuity?

P3-11. Log on to Hugh
Chou’s financial calculator web page
(http://www.interest.com/hugh/calc/simple.org), and look over the various
calculator links available. Refer back to some of the earlier time value
problems, and rework them with these calculators. Now, run through several
numerical scenarios to determine the impact of changing variables on your
results.

P3-12. Assume
that you just won the state lottery. Your prize can be taken either in the form
of $40,000 at the end of each of the next 25 years (i.e., $1 million over 25
years) or as a lump sum of $500,000 paid immediately.

a. If you expect to be able to earn 5 percent
annually on your investments over the next 25 years, ignoring taxes and other
considerations, which alternative should you take? Why?
b. Would your decision in part (a) be altered if
you could earn 7 percent rather than 5 percent on your investments over the
next 25 years? Why?
c. On a strict economic basis, at approximately
what earnings rate would you be indifferent when choosing between the two
plans?

P3-13. Matt Sedgwick,
facilities and operations manager for the Birmingham Buffalo professional
football team, has come up with an idea for generating income. Matt wants to
expand the stadium by building skyboxes sold with lifetime (perpetual) season
tickets. Each skybox will be guaranteed 10 season tickets at a cost of $200 per
ticket per year for life. If each skybox costs $100,000 to build, what is the
minimum selling price that Matt will have to charge for the skyboxes to break
even, if the required return is 10 percent?

P3-14. Melissa
Gould wants to invest today in order to assure adequate funds for her son’s
college education. She estimates that her son will need $20,000 at the end of
18 years; $25,000 at the end of 19 years; $30,000 at the end of 20 years; and
$40,000 at the end of 21 years. How much will Melissa have to invest in a fund
today if the fund earns the following interest rate?
a. 6 percent per year with annual compounding
b. 6 percent per year with quarterly compounding
c. 6 percent per year with monthly compounding

P3-15. Joan
Wallace, corporate finance specialist for Big Blazer Bumpers, has been charged
with the responsibility of funding an account to cover anticipated future
warranty costs. Warranty costs are expected to be $5 million per year for three
years, with the first costs expected to occur four years from today. How much
will Joan have to place into an account today earning 10 percent per year to
cover these expenses?

P3-16. Ruth Nail
has just received two offers for her seaside home. The first offer is for $1
million today. The second offer is for an owner-financed sale with a payment
schedule as follows:

End of Year

Payment

0 (Today)

$200,000

1

200,000

2

200,000

3

200,000

4

200,000

5

300,000

Assuming
no differential tax treatment between the two options and that Ruth earns a
rate of 8 percent on her investments, which offer should she take?

P3-17. Landon
Lowman, star quarterback of the university football team, has been approached
about forgoing his last two years of eligibility and making himself available
for the professional football draft. Talent scouts estimate that Landon could
receive a signing bonus of $1 million today along with a 5-year contract for $3
million per year (payable at the end of the year). They further estimate that
he could negotiate a contract for $5 million per year for the remaining seven
years of his career. The scouts believe, however, that Landon will be a much
higher draft pick if he improves by playing out his eligibility. If he stays at
the university, he is expected to receive a $2 million signing bonus in two
years along with a 5-year contract for $5 million per year. After that, the
scouts expect Landon to obtain a 5-year contract for $6 million per year to
take him into retirement. Assume that Landon can earn a 10 percent return over
this time. Should Landon stay or go?

P3-18. As part
of your personal budgeting process, you have determined that in each of the
next five years you will have budget shortfalls. In other words, you will need
the amounts shown in the following table at the end of the given year to
balance your budget—that is, inflows equal outflows. You expect to be able to
earn 8 percent on your investments during the next five years and wish to fund
the budget shortfalls over these years with a single initial deposit.

End
of Year Budget
Shortfall
1
$ 5,000
2
4,000
3
6,000
4
10,000
5
3,000

a. How large must the lump-sum deposit be today
into an account paying 8 percent annual interest to provide for full coverage
of the anticipated budget shortfall?
b. What effect would an increase in your
earnings rate have on the amount calculated in part a? Explain.

P3-19. Use the
following table of cash flows to answer parts a – c. Assume an 8 percent
discount rate.

End of Year

Cash Flow

1

$10,000

2

10,000

3

10,000

4

12,000

5

12,000

6

12,000

7

12,000

8

15,000

9

15,000

10

15,000

a. Solve for the present value of the cash flow
stream by summing the present value of each individual cash flow.
b. Now, solve for the present value by summing
the present value of the three separate annuities (one current and two
deferred).
c. Which method is better for a long series of
cash flows with embedded annuities?

P3-20. Given the
mixed streams of cash flows shown in the following table, answer parts (a) and
(b):

Cash Flow Stream

Year

A

B

1

$
50,000

$
10,000

2

40,000

20,000

3

30,000

30,000

4

20,000

40,000

5

10,000

50,000

Totals

$150,000

$150,000

a. Find the
present value of each stream, using a 15 percent discount rate.
b. Compare the
calculated present values, and discuss them in light of the fact that the undiscounted
total cash flows amount to $150,000 in each case.