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 years 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 Blandings 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 Roberts 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 doesnt 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 tablethat 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 Chous 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 sons 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 budgetthat 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.