Present Value Problems
1. At an effective annual interest rate of 20 percent, how many years will it take a given amount to triple in value? (Round to the closest year.)
a. 5 years
b. 8 years
c. 6 years
d. 10 years
e. 9 years
2. You deposited $1,000 in a savings account that pays 8 percent 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 in school, so you close out your account. How much money will you receive?
3. What is the future value of a 5-year ordinary annuity with annual payments of $200, evaluated at a 15 percent interest rate?
a. $ 670.44
b. $ 842.91
4. What is the present value of a 5-year ordinary annuity with annual payments of $200, evaluated at a 15 percent interest rate?
a. $ 670.43
b. $ 842.91
5. You have the opportunity to buy a perpetuity which pays $1,000 annually. Your required rate of return on this investment is 15 percent. You should be essentially indifferent to buying or not buying the investment if it were offered at a price of
6. An investor is considering the purchase of 20 acres of land. An analysis indicates that if the land is used for cattle grazing, it will produce a cash flow of $1,000 per year indefinitely. If the investor requires a return of 10 percent on investments of this type, what is the most he or she should be willing to pay for the land?
a. $ 1,000
b. $ 10,000
c. $ 100,000
d. $ 150,000
7. 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 percent rate of return, what is the present value of these cash flows?
a. $ 9,851
8. Suppose the present value of a 2-year ordinary annuity is $100. If the discount rate is 10 percent, what must be the annual cash flow?
9. If a 5-year regular annuity has a present value of $1,000, and if the interest rate is 10 percent, what is the amount of each annuity payment?
10. If $100 is placed in an account that earns a nominal 4 percent, compounded quarterly, what will it be worth in 5 years?
11. 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 30-year period?
12. At an inflation rate of 9 percent, 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 percent?
a. 12 years
b. 15 years
c. 18 years
d. 20 years
e. 23 years
13. South Penn Trucking is financing a new truck with a loan of $10,000 to be repaid in 5 annual end-of-year installments of $2,504.56. What annual interest rate is the company paying?
14. Gomez Electronics needs to arrange financing for its expansion program. Bank A offers to lend Gomez the required funds on a loan where interest must be paid monthly, and the quoted rate is 8 percent. Bank B will charge 9 percent, with interest due at the end of the year. What is the difference in the effective annual rates charged by the two banks?
15. You recently received a letter from Cut-to-the-Chase National Bank that offers you a new credit card that has no annual fee. It states that the annual percentage rate (APR) is 18 percent on outstanding balances. What is the effective annual interest rate? (Hint: Remember these companies bill you monthly.)
16. Which of the following investments has the highest effective return (EAR)? (Assume that all CDs are of equal risk.)
a. A bank CD which pays 10 percent interest quarterly.
b. A bank CD which pays 10 percent monthly.
d. A bank CD which pays 10.2 percent annually.
d. A bank CD which pays 10 percent semiannually.
e. A bank CD which pays 9.6 percent daily (on a 365-day basis).
17. Which one of the following investments provides the highest effective return?
a. An investment which has a 9.9 percent nominal rate and quarterly annual compounding.
b. An investment which has a 9.7 percent nominal rate and daily (365) compounding.
c. An investment which has a 10.2 percent nominal rate and annual compounding.
d. An investment which has a 10 percent nominal rate and semiannual compounding.
e. An investment which has a 9.6 percent nominal rate and monthly compounding.
18. Which of the following investments would provide an investor the highest effective annual return?
a. An investment which has a 9 percent nominal rate with semiannual compounding.
b. An investment which has a 9 percent nominal rate with quarterly compounding.
c. An investment which has a 9.2 percent nominal rate with annual compounding.
d. An investment which has an 8.9 percent nominal rate with monthly compounding.
e. An investment which has an 8.9 percent nominal rate with quarterly compounding.
19. An investment pays you 9 percent 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?
20. Your subscription to Jogger's World Monthly is about to run out and you have the choice of renewing it by sending in the $10 a year regular rate or of getting a lifetime subscription to the magazine by paying $100. Your cost of capital is 7 percent. 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
21. Assume you are to receive a 20-year annuity with annual payments of $50. The first payment will be received at the end of Year 1, and the last payment will be received at the end of Year 20. You will invest each payment in an account that pays 10 percent. What will be the value in your account at the end of Year 30?
22. You expect to receive $1,000 at the end of each of the next 3 years. You will deposit these payments into an account which pays 10 percent compounded semiannually. What is the future value of these payments, that is, the value at the end of the third year?
23. You are contributing money to an investment account 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 percent in your investment account, how much money will you have in the account five years from now (at t = 5)?
24. Your uncle has agreed to deposit $3,000 in your brokerage account at the beginning of each of the next five years (t = 0, t = 1, t = 2, t = 3 and t = 4). You estimate that you can earn 9 percent a year on your investments. How much will you have in your account four years from now (at t = 4)? (Assume that no money is withdrawn from the account until t = 4.)
25. You just put $1,000 in a bank account which pays 6 percent nominal annual interest, compounded monthly. How much will you have in your account after 3 years?
26. Assume that you can invest to earn a stated annual rate of return of 12 percent, but where interest is compounded semiannually. If you make 20 consecutive semiannual deposits of $500 each, with the first deposit being made today, what will your balance be at the end of Year 20?
27. You have $2,000 invested in a bank account that pays a 4 percent nominal annual interest with daily compounding. How much money will you have in the account at the end of July (i.e., in 132 days)? (Assume there are 365 days in each year.)
28. You are interested in saving money for your first house. Your plan is to make regular deposits into a brokerage account which will earn 14 percent. 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 percent 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?
29. Assume that your required rate of return is 12 percent and you are given the following stream of cash flows:
Year Cash Flow
If payments are made at the end of each period, what is the present value of the cash flow stream?
30. You are given the following cash flows. What is the present value (t = 0) if the discount rate is 12 percent?
0 12% 1 2 3 4 5 6 Periods
0 1 2,000 2,000 2,000 0 -2,000
31. You are given the following cash flow information. The appropriate discount rate is 12 percent for Years 1-5 and 10 percent for Years 6-10. Payments are received at the end of the year.
What should you be willing to pay right now to receive the income stream above?
32. A project with a 3-year life has the following probability distributions for possible end-of-year cash flows in each of the next three years:
Year 1 Year 2 Year 3
Prob Cash Flow Prob Cash Flow Prob Cash Flow
0.30 $300 0.15 $100 0.25 $200
0.40 500 0.35 200 0.75 800
0.30 700 0.35 600 0.15 900
Using an interest rate of 8 percent, find the expected present value of these uncertain cash flows. (Hint: Find the expected cash flow in each year, then evaluate those cash flows.)
b. $ 835.42
33. You just graduated, and you plan to work for 10 years and then to leave for the Australian "Outback" bush country. You figure you can save $1,000 a year for the first 5 years and $2,000 a year for the next 5 years. These savings cash flows will start one year from now. In addition, your family has just given you a $5,000 graduation gift. If you put the gift now, and your future savings when they start, into an account which pays 8 percent compounded annually, what will your financial "stake" be when you leave for Australia 10 years from now?
34. Foster Industries has a project which has the following cash flows:
t Cash Flow
What cash flow will the project have to generate in the fourth year in order for the project to have an internal rate of return = 15%?
a. $ 15.55
b. $ 58.95
35. You recently purchased a 20-year investment which 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. The investment cost you $5,544.87. Alternative investments of equal risk have a required return of 9 percent. What is the annual cash flow received at the end of each of the final 17 years, that is, what is X?
36. John Keene recently invested $2,566.70 in a project that is promising to return 12 percent per year. The cash flows are expected to be as follows:
End of Cash
What is the cash flow at the end of the 4th year?
b. $ 600
d. $ 655
37. An investment costs $3,000 today and provides cash flows at the end of each year for 20 years. The relevant cost of capital is 10 percent. The projected cash flows for years 1, 2, and 3 are $100, $200, and $300 respectively. What is the annual cash flow received for each of the years 4 through 20 (17 years)? (Assume the same payment for each of these years.)
38. If you buy a factory for $250,000 and the terms are 20 percent down, the balance to be paid off over 30 years at a 12 percent rate of interest on the unpaid balance, what are the 30 equal annual payments?
e. $ 6,667
39. Drexel Corporation has been enjoying a phenomenal rate of growth since its inception one year ago. Currently, its assets total $100,000. If growth continues at the current rate of 12 percent compounded quarterly, what will total assets be at the end of 10 quarters?
40. If it were evaluated with an interest rate of 0 percent, a 10-year 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?
41. Steaks Galore needs to arrange financing for its expansion program. One bank offers to lend the required $1,000,000 on a loan which requires interest to be paid at the end of each quarter. The quoted rate is 10 percent, and the principal must be repaid at the end of the year. A second lender offers 9 percent, daily compounding (365-day 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?
42. On January 1, 1993, a graduate student developed a 5-year financial plan which would provide enough money at the end of her graduate work (January 1, 1998) to open a business of her own. Her plan was to deposit $8,000 per year for 5 years, starting immediately, into an account paying 10 percent compounded annually. Her activities proceeded according to plan except that at the end of her third year (1/1/96) she withdrew $5,000 to take a Caribbean cruise, at the end of the fourth year (1/1/97) she withdrew $5,000 to buy a used Prelude, and at the end of the fifth year (1/1/98) she had to withdraw $5,000 to pay to have her dissertation typed. Her account, at the end of the fifth year, was less than the amount she had originally planned on by how much?
43. You are willing to pay $15,625 to purchase a perpetuity which 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 20-year, annual payment, ordinary annuity instead of a perpetuity?
44. You are currently at time period 0, and you will receive the first payment on an annual payment annuity of $100 in perpetuity at the end of this year. Six full years from now you will receive the first payment on an additional $150 in perpetuity, and at the end of time period 10 you will receive the first payment on an additional $200 in perpetuity. If you require a 10 percent rate of return, what is the combined present value of these three perpetuities?
45. Find the present value of an income stream which has a negative flow of $100 per year for 3 years, a positive flow of $200 in the 4th year, and a positive flow of $300 per year in Years 5 through 8. The appropriate discount rate is 4 percent for each of the first 3 years and 5 percent for each of the later years. Thus, a cash flow accruing in Year 8 should be discounted at 5 percent for some years and 4 percent in other years. All payments occur at year-end.
a. $ 528.21
c. $ 792.49
e. $ 875.18
46. Your parents start saving for your sister's college education. She will begin college when she turns age 18 and will need $4,000 at that time and at the end of each of the following 3 years. They will make a deposit at the end of this year in an account which pays 6 percent compounded annually, and an identical deposit at the end of each year with the last deposit occurring when she turns age 18. If an annual deposit of $1,484 will allow them to reach their goal, how old is your sister now?
47. 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 percent 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 at which 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 percent per year. How much should you contribute each year in order to meet the expected cost of your children's education?
48. Joe and Jane are interested in saving money to put their two children, John and Susy through college. John is currently 12 years old and will enter college in six years. Susy is 10 years old and will enter college in 8 years. Both children plan to finish college in four years.
College costs are currently $15,000 a year (per child), and are expected to increase at 5 percent a year for the foreseeable future. All college costs are paid at the beginning of the school year. Up until now, Joe and Jane have saved nothing but they expect to receive $25,000 from a favorite uncle in three years.
To provide for the additional funds that are needed, they expect to make 12 equal payments at the beginning of each of the next twelve years--the first payment will be made today and the final payment will be made on Susy's 21st birthday (which is also the day that the last payment must be made to the college). If all funds are invested in a stock fund which is expected to earn 12 percent, how large should each of the annual contributions be?
a. $ 7,475.60
b. $ 7,798.76
c. $ 8,372.67
d. $ 9,675.98
49. John and Barbara Roberts are starting to save for their daughter's college education.
* Assume that today's date is September 1, 1997.
* College costs are currently $10,000 a year and are expected to increase at a rate equal to 6 percent per year for the foreseeable future. All college payments are due at the beginning of the year. (So for example, college will cost $10,600 for the year beginning September 1, 1998).
* Their daughter will enter college 15 years from now (September 1, 2012). She will be enrolled for four years. Therefore the Roberts will need to make four tuition payments. The first payment will be made on September 1, 2012, the final payment will be made on September 1, 2015. Notice that because of rising tuition costs, the tuition payments will increase each year.
* The Roberts would also like to give their daughter a lump-sum payment of $50,000 on September 1, 2016, in order to help with a down payment on a home, or to assist with graduate school tuition.
* The Roberts currently have $10,000 in their college account. They anticipate making 15 equal contributions to the account at the end of each of the next 15 years. (The first contribution would be made on September 1, 1998, the final contribution will be made on September 1, 2012).
* All current and future investments are assumed to earn an 8 percent return. (Ignore taxes.)
How much should the Roberts contribute each year in order to reach their goal?
50. Jim and Nancy are interested in saving money for their son's education. Today is their son's 8th birthday. Their son will enter college ten years from now on his 18th birthday, and will attend for four years. All college costs are due at the beginning of the year, so Jim and Nancy will have to make payments on their son's 18th, 19th, 20th and 21st birthdays (t = 10, 11, 12, 13). They estimate that the college their son wants to attend will cost $35,000 the first year (t = 10) and that the costs will increase 7 percent each year (the final college payment will be made 13 years from now).
Currently, Jim and Nancy have $20,000 in an investment account. They also plan to contribute a fixed amount at the end of each of the next ten years (t = 1, 2, 3, ... 10). Their invested money will be in an account which pays 9 percent interest compounded annually. How much money do Jim and Nancy need to contribute to the account in each of the next ten years?
51. Joe and June Green are planning for their children's 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 percent each year. Their children, David and Daniel, just turned 2 and 3 years old today, September 1, 1997. 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 boys' 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, 1998) 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?
52. 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, i.e., the first contribution will be made at the end of the year and the final contribution will be made at the time the oldest child enters college.
The money will be invested in securities that are certain to earn a return of 8 percent each year. The oldest 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. $ 5,071.63
d. $ 5,329.45
e. $ 4,944.84
53. 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?
54. 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 percent, 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 which pays 8 percent, annual compounding. The interest-accumulated 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, i.e., to meet the interest and principal repayment requirements?
55. A financial planner has offered you three possible options for receiving cash flows. You must choose the option that has the highest present value.
(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 percent 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 percent 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 percent each quarter. The money is to be deposited in an account paying a 12 percent nominal annual rate, but compounded quarterly (to be left on deposit for the entire 2-year 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. Choice 1, if the payments were made at the end of each month.
1. c. 6 years
2. b. $1,126
3. e. $1,348.48
4. a. $ 670.43
5. c. $6,666.67
6. b. $ 10,000
7. c. $11,714
8. c. $57.62
9. b. $263.80
10. a. $122.02
11. d. 7%
12. c. 18 years
13. b. 8%
14. c. 0.70%
15. b. 19.56%
16. b. A bank CD which pays 10 percent monthly.
17. a. An investment which has a 9.9 percent nominal rate and quarterly annual compounding.
18. b. An investment which has a 9 percent nominal rate with quarterly compounding.
19. b. 8.90%
20. a. 15 years
21. b. $7,427.83
22. c. $3,318
23. d. $23,739
24. b. $17,954.13
25. e. $1,196.68
26. d. $62,527.47
27. a. $2,029.14
28. e. $44,873.90
29. a. $66,909
30. a. $3,277
31. d. $125,870
32. e. $1,347.61
33. d. $31,148
34. d. $103.10
35. d. $675
36. c. $1,157
37. d. $417.87
38. c. $24,829
39. e. $134,392
40. b. 8%
41. d. 0.96%
42. b. $16,550
43. c. $12,273
44. d. $2,779.58
45. c. $ 792.49
46. e. 10
47. b. $3,712
48. a. $ 7,475.60
49. b. $3,618.95
50. b. $5,848
51. a. $5,928.67
52. b. $ 5,071.63
53. a. $7,757.22
54. b. 10,186
55. a. Choice 1