Chapter 9 Time Value of Money Discussion Questions 9-1. How is the future value (Appendix A) related to the present value of a single sum (Appendix B)? The future value represents the expected worth of a single amount, whereas the present value represents the current worth. FV = PV (1 + i)n future value 9-2. How is the present value of a single sum (Appendix B) related to the present value of an annuity (Appendix D)? The present value of a single amount is the discounted value for one future payment, whereas the present value of an annuity represents the discounted value of a series of consecutive future payments of equal amount. 9-3. Why does money have a time value? Money has a time value because funds received today can be invested to reach a greater value in the future. A person would rather receive $1 today than $1 in 10 years, because a dollar received today, invested at 6 percent, is worth $1.791 after 10 years. 9-4. Does inflation have anything to do with making a dollar today worth more than a dollar tomorrow? Inflation makes a dollar today worth more than a dollar in the future. Because inflation tends to erode the purchasing power of money, funds received today will be worth more than the same amount received in the future. 9-5. Adjust the annual formula for a future value of a single amount at 12 percent for 10 years to a semiannual compounding formula. What are the interest factors (FVIF) before and after? Why are they different? The more frequent compounding under the semiannual compounding assumption increases the future value so that semiannual compounding is worth .101 more per dollar. 9-6. If, as an investor, you had a choice of daily, monthly, or quarterly compounding, which would you choose? Why? The greater the number of compounding periods, the larger the future value. The investor should choose daily compounding over monthly or quarterly. 9-7. What is a deferred annuity? A deferred annuity is an annuity in which the equal payments will begin at some future point in time. 9-8. List five different financial applications of the time value of money. Different financial applications of the time value of money: Equipment purchase or new product decision Present value of a contract providing future payments Future value of an investment Regular payment necessary to provide a future sum Regular payment necessary to amortize a loan Determination of return on an investment Determination of the value of a bond Chapter 9 Problems 1. You invest $3,000 for three years at 12 percent. a. What is the value of your investment after one year? Multiply $3,000 × 1.12. b. What is the value of your investment after two years? Multiply your answer to part a by 1.12. c. What is the value of your investment after three years? Multiply your answer to part b by 1.12. This gives your final answer. d. Confirm that your final answer is correct by going to Appendix A (future value of $1), and looking up the future value for n = 3, and i = 12 percent. Multiply this tabular value by $3,000 and compare your answer to the answer in part c. There may be a slight difference due to rounding. 9-1. Solution: a. b. c. d. Calculator Solution: (a) N I/Y PV PMT FV 1 12 3,000 0 CPT FV −3,360 Answer: $3,360 (b) N I/Y PV PMT FV 2 12 3,000 0 CPT FV −3,763.20 Answer: $3,763.20 (c) N I/Y PV PMT FV 3 12 3,000 0 CPT FV −4,214.78 Answer: $4,214.78 Solution using TVM Tables: a. $3,000 × 1.12 = $3,360.00 b. $3,360 × 1.12 = $3,763.20 c. $3,763.20 × 1.12 = $4,214.78 d. $3,000 × 1.405 = $4,215.00 (Appendix A) 2. Present value (LO9-3) What is the present value of: a. $7,900 in 10 years at 11 percent? b. $16,600 in 5 years at 9 percent? c. $26,000 in 14 years at 6 percent? 9-2. Solution: a. b. c. Calculator Solution: (a) N I/Y PV PMT FV 10 11 CPT PV −2,782.26 0 7,900 Answer: $2,782.26 (b) N I/Y PV PMT FV 5 9 CPT PV −10,788.86 0 16,600 Answer: $10,788.86 (c) N I/Y PV PMT FV 14 6 CPT PV −11,499.83 0 26,000 Answer: $11,499.83 Appendix B PV = FV × PVIF a. $ 7,900 × .352 = $2,781 b. $16,600 × .650 = $10,790 c. $26,000 × .442 = $11,492 3. Present Value (LO9-3) a. What is the present value of $140,000 to be received after 30 years with a 14 percent discount rate? b. Would the present value of the funds in part a be enough to buy a $2,900 concert ticket? 9-3. Solution: (a) (b) No, $2,747.78 isn’t enough. Calculator Solution: (a) N I/Y PV PMT FV 30 14 CPT PV 2747.78 0 140,000 Answer: $2747.78 (b) No. You only have $2747.78. Appendix B PV = FV × PVIF (14%, 30 periods) a. $140,000 × .02 = $2,800 b. NO. You only have $2,800 in present value. 4. Present Value (LO9-4) You will receive $6,800 three years from now. The discount rate is 10 percent. a. What is the value of your investment two years from now? Multiply $6,800 × .909 (one year’s discount rate at 10 percent). b. What is the value of your investment one year from now? Multiply your answer to part a by .909 (one year’s discount rate at 10 percent). c. What is the value of your investment today? Multiply your answer to part b by .909 (one year’s discount rate at 10 percent). d. Confirm that your answer to part c is correct by going to Appendix B (present value of $1) for n = 3 and i = 10%. Multiply this tabular value by $6,800 and compare your answer to part c. There may be a slight difference due to rounding. 9-4. Solution: a. b. c. Calculator Solution: (a) N I/Y PV PMT FV 1 10 CPT −6,181.82 0 FV 6,800 Answer: $6,181.82 (b) N I/Y PV PMT FV 2 10 CPT −5,619.83 0 FV 6,800 Answer: $5,619.83 (c) N I/Y PV PMT FV 3 10 CPT −5,108.94 0 FV 6,800 Answer: $5,108.94 a. $6,800 × .909 = $6,181.20 b. $6,181.20× .909 = $5,618.71 c. $5,618.71× .909 = $5,107.41 d. Appendix B (10%, 3 periods) FV = FV × PVIF $6,800 ×.751 = $5,106.80 5. If you invest $9,000 today, how much will you have: a. In 2 years at 9 percent? b. In 7 years at 12 percent? c. In 25 years at 14 percent? d. In 25 years at 14 percent (compounded semiannually)? 9-5. Solution: a. b. c. d. Calculator Solution: (a) N I/Y PV PMT FV 2 9 9,000 0 CPT FV −10,692.90 Answer: $10,692.90 (b) N I/Y PV PMT FV 7 12 9,000 0 CPT FV −19,896.13 Answer: $19,896.13 (c) N I/Y PV PMT FV 25 14 9,000 0 CPT FV −238,157.24 Answer: $238,157.24 (d) N I/Y PV PMT FV 50 7 9,000 0 CPT FV −265,113.23 Answer: $265,113.23 Appendix A FV = PV × FVIF a. $9,000 × 1.188 = $ 10,692 b. $9,000 × 2.211 = $ 19,899 c. $9,000 × 26.462 = $238,158 d. $9,000 × 29.457 = $265,113 (7%, 50 periods) 6. Present value (LO9-3) Your aunt offers you a choice of $20,100 in 20 years or $870 today. If money is discounted at 17 percent, which should you choose? 9-6. Solution: Take the $870 today instead of $20,100 in 20 years. Calculator Solution: N I/Y PV PMT FV 20 17 −869.92 0 20,100 Answer: $869.92 Appendix B PV = FV × PVIF (17%, 20 periods) PV = $20,100 × .043 = $864 Choose $870 today. 7. Present Value (LO9-3) Your uncle offers you a choice of $105,000 in 10 years or $47,000 today. If money is discounted at 9 percent, which should you choose? 9-7. Solution: Take the $47,000 today instead of $105,000 in 10 years. Calculator Solution: N I/Y PV PMT FV 10 9 CPT PV −44,353.13 0 105,000 Answer: $44,353.13 Appendix B PV = FV × PVIF (9%, 10 periods) PV = $105,000 × .422 = $44,310 Choose $47,000 today. 8. Present Value (LO9-3) Your father offers you a choice of $105,000 in 12 years or $47,000 today. a. If money is discounted at 8 percent, which should you choose? b. If money is still discounted at 8 percent, but your choice is between $105,000 in 9 years or $47,000 today, which should you choose? 9-8. Solution: a. Take $47,000 today instead of $105,000 in 12 years. b. Take the $105,000 in 9 years because it is worth $52,526.14, which is more than $47,000 today. Calculator Solution: (a) & (b) N I/Y PV PMT FV 12 8 CPT PV −41,696.94 0 105,000 Answer: $41,696.94 (c) & (d) N I/Y PV PMT FV 9 8 CPT PV −52,526.14 0 105,000 Answer: $52,526.14 a. Appendix B PV = FV × PVIF (8%, 12 periods) FV = $105,000 × .397 = $41,685 Choose $47,000 today. b. Appendix B PV = FV × PVIF (8%, 9 periods) FV = $105,000 × .500 = $52,500 Choose $105,000 in 9 years. 9. Present Value (LO9-3) You are going to receive $205,000 in 18 years. What is the difference in present value between using a discount rate of 12 percent versus 9 percent? 9-9. Solution: 9% Rate 12% Rate The Difference $43,458.12 –26,658.12 $16,800.60 Calculator Solution: At 12% N I/Y PV PMT FV 18 12 CPT PV −26,658.12 0 205,000 Answer: $26,658.12 At 9% N I/Y PV PMT FV 18 9 CPT PV −43,458.72 0 205,000 Answer: $43,458.72 The difference is 43,458.72 – 26,658.12 = $16,800.60 Appendix B The difference is $16,810 10. How much would you have to invest today to receive: a. $15,000 in 8 years at 10 percent? b. $20,000 in 12 years at 13 percent? c. $6,000 each year for 10 years at 9 percent? d. $50,000 each year for 50 years at 7 percent? 9-10. Solution: a. b. c. d. Calculator Solution: (a) N I/Y PV PMT FV 8 10 CPT PV −6,997.61 0 15,000 Answer: $6,997.61 (b) N I/Y PV PMT FV 12 13 CPT PV −4,614.12 0 20,000 Answer: $4,614.12 (c) N I/Y PV PMT FV 10 9 CPT PV −38,505.95 6,000 0 Answer: $38,505.95 (d) N I/Y PV PMT FV 50 7 CPT PV −690,037.31 50,000 0 Answer: $690,037.31 Appendix B (a and b) PV = FV × PVIF a. $15,000 × .467 = $7,005 b. $20,000 × .231 = $4,620 Appendix D (c and d) c. $ 6,000 × 6.418 = $38,508 d. $50,000 × 13.801 = $690,050 11. Future value (LO9-2) If you invest $8,500 per period for the following number of periods, how much would you have? a. 12 years at 10 percent. b. 50 years at 9 percent. 9-11. Solution: a. b. Calculator Solution: (a) N I/Y PV PMT FV 12 10 0 8,500 CPT FV −181,766.41 Answer: $181,766.41 (b) N I/Y PV PMT FV 50 9 0 8,500 CPT FV −6,928,210.23 Answer: $6,928,210.23 Appendix C FVA = A × FV IFA a. $8,500 × 21.384 = $ 181,764 b. $8,500 × 815.08 = $ 6,928,180 12. You invest a single amount of $10,000 for 5 years at 10 percent. At the end of 5 years you take the proceeds and invest them for 12 years at 15 percent. How much will you have after 17 years? 9-12. Solution: After 5 Years After 17 Years Calculator Solution: First step: N I/Y PV PMT FV 5 10 10,000 0 CPT FV −16,105.10 Answer: $16,105.10 Second step: N I/Y PV PMT FV 12 15 16,105.10 0 CPT FV −86,166.31 Answer: $86,166.31 Appendix A FV = PV × FVIF $10,000 × 1.611 = $16,110 Appendix A FV = PV × FVIF $16,110 × 5.350 = $86,188 13. Present value (LO9-3) Mrs. Crawford will receive $7,600 a year for the next 19 years from her trust. If a 14 percent interest rate is applied, what is the current value of the future payments? 9-13. Solution: Calculator Solution: N I/Y PV PMT FV 19 14 CPT PV −49,782.80 7,600 0 Answer: $49,782.80 Appendix D PVA = A × PVIFA (14%, 19 periods) = $7,600 × 6.550 = $49,780 14. Phil Goode will receive $175,000 in 50 years. His friends are very jealous of him. If the funds are discounted back at a rate of 14 percent, what is the present value of his future “pot of gold”? 9-14. Solution: Calculator Solution: N I/Y PV PMT FV 50 14 CPT PV −249.92 0 175,000 Answer: $249.92 Appendix B PV = FV × PVIF (14%, 50 periods) = $175,000 × .001 = $175 15. Present value (LO9-3) Sherwin Williams will receive $18,500 a year for the next 25 years as a result of a picture he has painted. If a discount rate of 12 percent is applied, should he be willing to sell out his future rights now for $165,000? 9-15. Solution: Sherwin Williams should take the $165,000 for his future rights now. Calculator Solution: N I/Y PV PMT FV 25 12 CPT PV −145,098.07 18,500 0 Answer: $145,098.07 Appendix D PVA = A × PVIFA (12%, 25 periods) PVA = $18,500 × 7.843 = $145,096 Yes, the present value of the annuity is worth less than $165,000. 16. Carrie Tune will receive $19,500 for the next 20 years as a payment for a new song she has written. If a 10 percent rate is applied, should she be willing to sell out her future rights now for $160,000? 9-16. Solution: Carrie Tune should not accept $160,000 for the future rights because they are worth more than that. Calculator Solution: N I/Y PV PMT FV 20 10 CPT PV −166,014.49 19,500 0 Answer: $166,014.49 Appendix D PVA = A × PVIFA (10%, 20 periods) PVA = $19,500 × 8.514 = $166,023 No, the present value of the annuity is worth more than $160,000. 17. The Clearinghouse Sweepstakes has just informed you that you have won $1 million. The amount is to be paid out at the rate of $20,000 a year for the next 50 years. With a discount rate of 10 percent, what is the present value of your winnings? 9-17. Solution: Calculator Solution: N I/Y PV PMT FV 50 10 CPT PV −198,296.29 20,000 0 Answer: $198,296.29 Appendix D PVA = A × PVIFA (10%, 50 periods) PVA = $20,000 × 9.915 = $198,300 18. Present value (LO9-3) Rita Gonzales won the $41 million lottery. She is to receive $1.5 million a year for the next 19 years plus an additional lump sum payment of $12.5 million after 19 years. The discount rate is 14 percent. What is the current value of her winnings? 9-18. Solution: Annuity Part Lump Sum Part Total Value $ 9,825,553.24 Present value of annuity + 1,036,854.55 Present value of lump sum $10,862,407.79 Total value Calculator Solution: First part: N I/Y PV PMT FV 19 14 CPT PV −9,825,553.24 1,500,000 0 Answer: $9,825,553.24 Second part: N I/Y PV PMT FV 19 14 CPT PV −1,036,854.55 0 12,500,000 Answer: $1,036,854.55 Total = 9,825,553.24 + 1,036,854.55 = $10,862,407.79 Appendix D PVA = A × PVIFA (14%, 19 periods) PVA = $1,500,000 × 6.550 = $9,825,000 Appendix B PV = FV × PVIF (14%, 19 periods) PV = $12,500,000 × .083 = $1,037,500 $ 9,825,000 1,037,500 $10,862,500 19. Al Rosen invests $25,000 in a mint condition 1952 Mickey Mantle Topps baseball card. He expects the card to increase in value 12 percent per year for the next 10 years. How much will his card be worth after 10 years? 9-19. Solution: Calculator Solution: Part one: N I/Y PV PMT FV 10 12 25,000 0 CPT FV −77,646.21 Answer: $77,646.21 Appendix A FV = PV × FVIF (12%, 10 periods) = $25,000 × 3.106 = $77,650 20. Future value (LO9-2) Christy Reed has been depositing $2,000 in her savings account every December since 2001. Her account earns 7 percent compounded annually. How much will she have in December 2012? (Assume that a deposit is made in December of 2012. Make sure to count the years carefully.) 9-20. Solution: Calculator Solution: N I/Y PV PMT FV 12 7 0 2,000 CPT FV −35,776.90 Answer: $35,776.90 Appendix C FVA = A × FVIFA (7%, n = 12) FVA = $2,000 × 17.888 = $35,776.00 21. Future value (LO9-2) At a growth (interest) rate of 10 percent annually, how long will it take for a sum to double? To triple? Select the year that is closest to the correct answer. 9-21. Solution: Tip To solving this problem The key to solving this problem algebraically is to know that To Double To Triple Calculator Solution: To double: N I/Y PV PMT FV CPT N 7.2725 10 1.00 0 −2.00 Answer: 7.27 years To triple: N I/Y PV PMT FV CPT N 11.5267 10 1.00 0 −3.00 Answer: 11.53 years Appendix A If the sum is doubling, then the tabular value must equal 2. In Appendix A, looking down the 10% column, we find the factor closest to 2 (1.949) in the 7-year row. The factor closest to 3 (3.138) is in the 12-year row. 22. Present value (LO9-3) If you owe $35,000 payable at the end of eight years, what amount should your creditor accept in payment immediately if she could earn 13 percent on her money? 9-22. Solution: Calculator Solution: N I/Y PV PMT FV 8 13 CPT PV −13,165.60 0 35,000 Answer: $13,165.60 Appendix B PV = FV × PVIF (13%, 8 periods) PV = $35,000 × .376 = $13,160 23. Jack Hammer invests in a stock that will pay dividends of $2.00 at the end of the first year; $2.20 at the end of the second year; and $2.40 at the end of the third year. Also, he believes that at the end of the third year he will be able to sell the stock for $33. What is the present value of all future benefits if a discount rate of 11 percent is applied? (Round all values to two places to the right of the decimal point.) 9-23. Solution: First Dividend Second Dividend Third Dividend Selling Price Present Value Total $24.13 Selling price + 1.80 First dividend + 1.79 Second dividend + 1.75 Third dividend $29.47 Present total value First dividend: N I/Y PV PMT FV 1 11 CPT PV −1.80 0 2.00 Answer: $1.80 Second dividend: N I/Y PV PMT FV 2 11 CPT PV −1.79 0 2.20 Answer: $1.79 Third dividend: N I/Y PV PMT FV 3 11 CPT PV −1.75 0 2.40 Answer: $1.75 Selling price: N I/Y PV PMT FV 3 11 CPT PV −24.13 0 33.00 Answer: $39.15 Total = 1.80 + 1.79 + 1.75 + 24.13 = $29.47 Appendix B PV = FVIF Discount rate = 11 percent $ 2.00 × .901 = $ 1.80 2.20 × .802 = 1.79 2.40 × .731 = 1.75 33.00 × .731 = 24.12 $29.46 24. Les Moore retired as president of Goodman Snack Foods Company but is currently on a consulting contract for $35,000 per year for the next 10 years. a. If Mr. Moore’s opportunity cost (potential return) is 10 percent, what is the present value of his consulting contract? b. Assuming Mr. Moore will not retire for two more years and will not start to receive his 10 payments until the end of the third year, what would be the value of his deferred annuity? 9-24. Solution: Present Value of the Annuity Discount two years off Calculator Solution: (a) N I/Y PV PMT FV 10 10 CPT PV −215,059.85 35,000 0 Answer: $215,059.85 (b) First part: Find the PV of the 10 payments of annuity: N I/Y PV PMT FV 10 10 CPT PV −215,059.85 35,000 0 Answer: $215,059.85 Second part: Find the PV of the above FV lump sum: N I/Y PV PMT FV 2 10 CPT PV −177,735.41 0 215,059.85 Answer: $177,735.41 Appendix D a. PVA = A × PVIFA (10%, 10 periods) PVA = $35,000 × 6.145 = $215,075 b. Deferred annuity—Appendix D PVA = A × PVIFA (i = 10%, 10 periods) PVA = $35,000 × 6.145 = $215,075 Now, discount back this value for two periods. PV = FV × PVIF (i = 10%, 2 periods) Appendix B $215,075 × .826 = $177,652 OR Appendix D PVA = $35,000 (6.814 – 1.7360, where n = 12, n = 2, and i = 10%) = $35,000(5.078) = $177,730 The answer is slightly different from the preceding answer due to rounding in the tables. 25. Juan Garza invested $20,000 10 years ago at 12 percent, compounded quarterly. How much has he accumulated? 9-25. Solution: Ten years with quarterly compounding means Calculator Solution: N I/Y PV PMT FV 40 3 20,000 0 CPT FV −65,240.76 Answer: $65,240.76 Appendix A FV = PV × FVIF (3%, 40 periods) FV = $20,000 × 3.262 = $65,240 26. Special compounding (LO9-5) Determine the amount of money in a savings account at the end of 10 years, given an initial deposit of $5,500 and a 12 percent annual interest rate when interest is compounded (a) annually, (b) semiannually, and (c) quarterly. 9-26. Solution: Annually Semiannually Quarterly Calculator Solution: (a) N I/Y PV PMT FV 10 12 5,500 0 CPT FV −17,082.17 Answer: $17,082.17 (b) N I/Y PV PMT FV 20 6 5,500 0 CPT FV −17,639.25 Answer: $17,639.25 (c) N I/Y PV PMT FV 40 3 5,500 0 CPT FV −17,941.21 Answer: $17,941.21 Appendix A FV = PV × FVIF a. $5,500 × 3.106 = $17,083 (n = 10; i = 12%) b. $5,500 × 3.207 = $17,639 (n = 20; i = 6%) c. $5,500 × 3.262 = $17,941 (n = 40; i = 3%) 27. Annuity due (LO9-4) As stated in the chapter, annuity payments are assumed to come at the end of each payment period (termed an ordinary annuity). However, an exception occurs when the annuity payments come at the beginning of each period (termed an annuity due). To find the present value of an annuity due, subtract 1 from n and add 1 to the tabular value. To find the future value of an annuity, add 1 to n and subtract 1 from the tabular value. For example, to find the future value of a $100 payment at the beginning of each period for five periods at 10 percent, go to Appendix C for n = 6 and i = 10 percent. Look up the value of 7.716 and subtract 1 from it for an answer of 6.716 or $671.60 ($100 × 6.716). What is the future value of a 15-year annuity of $1,800 per period where payments come at the beginning of each period? The interest rate is 12 percent. 9-27. Solution: Calculator Solution: Set calculator beginning N I/Y PV PMT FV 15 12 0 1,800 CPT FV 75,155.90 Answer: $75,155.90 Appendix C FVA = A × FVIFA n = 16, i = 12% 42.753 – 1 = 41.753 FVA = $1,800 × 41.753 = $75,155 28. Annuity due (LO9-4) What is the present value of a 10-year annuity of $3,000 per period in which payments come at the beginning of each period? The interest rate is 12 percent. (Hint: Adjust the number of periods and the resulting value from the present value of an annuity table.) 9-28. Solution: Calculator Solution: Set calculator to beginning N I/Y PV PMT FV 10 12 CPT PV −18,984.75 3,000 0 Answer: $18,984.75 Appendix D PVA = A × PVIFA n = 9, i = 12% 5.328 + 1 = 6.328 PVA = $3,000 × 6.328 = $18,984 29. Present value alternative (LO9-3) Your grandfather has offered you a choice of one of the three following alternatives: $7,500 now; $2,200 a year for nine years; or $31,000 at the end of nine years. Assuming you could earn 10 percent annually, which alternative should you choose? If you could earn 11 percent annually, would you still choose the same alternative? 9-29. Solution: 10% annually Option 1 Option 2 Option 3 At a rate of 10 percent annually, the best deal is $31,000 in nine years because it has the highest present value. 11% annually Option 1 Option 2 Option 3 At a rate of 11 percent annually, the best option is Option 2 because $2,200 a year for nine years is the best deal. Calculator Solution: (a-1) (first alternative) Present value of $7,500 received now: $7,500 (second alternative) Present value of annuity of $2,200 for nine years: N I/Y PV PMT FV 9 10 CPT PV –12,669.85 2,200 0 Answer: $12,669.85 (third alternative) Present value of $31,000 received in nine years: N I/Y PV PMT FV 9 10 CPT PV –13,147.03 0 31,000 Answer: $13,147.03 (a-2) Select $31,000 received at end of nine years. (b-1) (first alternative) Present value of $7,500 received today: $7,500 (second alternative) Present value of annuity of $2,200 at 11 percent for nine years: N I/Y PV PMT FV 9 11 CPT PV –12,181.50 2,200 0 Answer: $12,181.50 (third alternative) Present value of $31,000 received in nine years at 11 percent: N I/Y PV PMT FV 9 11 CPT PV –12,118.67 0 31,000 Answer: $12,118.67 (b-2) Select $2,200 received for nine years. (first alternative) Present value of $7,500 received now: $7,500 (second alternative) Present value of annuity of $2,200 for nine years: Appendix D (third alternative) Present value of $31,000 received in nine years: Appendix B Select $31,000 to be received in nine years. 9-29. (Continued) Revised answers based on 11 percent. (first alternative) Present value of $7,500 received today: $7,500 (second alternative) Present value of annuity of $2,200 at 11 percent for nine years: Appendix D (third alternative) Present value of $31,000 received in nine years at 11 percent: Appendix B Select $2,200 a year for nine years. As the interest rate (discount rate) increases, the present value declines. 30. You need $28,974 at the end of 10 years, and your only investment outlet is an 8 percent long-term certificate of deposit (compounded annually). With the certificate of deposit, you make an initial investment at the beginning of the first year. a. What single payment could be made at the beginning of the first year to achieve this objective? b. What amount could you pay at the end of each year annually for 10 years to achieve this same objective? 9-30. Solution: Calculator Solution: (a) N I/Y PV PMT FV 10 8 CPT PV −13,420.57 0 28,974 Answer: $13,420.57 (b) N I/Y PV PMT FV 10 8 0 CPT PMT −2,000.06 28,974 Answer: $2,000.06 a. Appendix B PV = FV × PVIF (8%, 10 periods) = $28,974 × .463 = $13,415 b. Appendix C A = FVA/FVIFA (8%, 10 periods) = $28,974/14.487 = $2,000 31. Quarterly compounding (LO9-5) Beverly Hills started a paper route on January 1, 2009. Every three months, she deposits $550 in her bank account, which earns 8 percent annually but is compounded quarterly. On December 31, 2012, she used the entire balance in her bank account to invest in an investment at 7 percent annually. How much will she have on December 31, 2015? 9-31. Solution: Quarterly deposits for four years Investment growth over three years Calculator Solution: Step one: N I/Y PV PMT FV 16 2 0 550.00 CPT FV −10,251.61 Answer: $10,251.61 Step two: N I/Y PV PMT FV 3 7 10,251.61 0 CPT FV −12,558.66 Answer: $12,558.66 Appendix C FVA = A × FVIFA (2%, 16 periods) FVA = $550 × 18.639 = $10,251.45 after four years Appendix A FV = PV × FVIF (7%, 3 periods) FV = $10,251.45 × 1.225 FV = $12,558.03 after three more years 32. Yield (LO9-4) Franklin Templeton has just invested $9,260 for his son (age one). This money will be used for his son’s education 18 years from now. He calculates that he will need $71,231 by the time the boy goes to school. What rate of return will Mr. Templeton need in order to achieve this goal? 9-32. Solution: Franklin Templeton needs a 12 percent rate to achieve his goal of $71,231. Calculator Solution: N I/Y PV PMT FV 18 CPT I/Y 12.00 9,260.00 0 −71,231.00 Answer: 12.00 Appendix B Or Alternative solution Appendix A 33. Yield with interpolation (LO9-4) On January 1, 2011, Mr. Dow bought 100 shares of stock at $14 per share. On December 31, 2013, he sold the stock for $20 per share. What is his annual rate of return? Interpolate to find the answer. 9-33. Solution: Calculator Solution: N I/Y PV PMT FV 3 CPT I/Y 12.62 14.00 0 −20.00 Answer: 12.62% Appendix B Return is between 12–13 percent for three years. 12% + (.012/.019) (1%) 12% + .632 (1%) 12.63% 34. Yield with interpolation (LO9-4) C. D. Rom has just given an insurance company $35,000. In return, he will receive an annuity of $3,700 for 20 years. At what rate of return must the insurance company invest this $35,000 in order to make the annual payments? Interpolate. 9-34. Solution: Calculator Solution: N I/Y PV PMT FV 20 CPT I/Y 8.51 −35,000.00 3,700.00 0 Answer: 8.51% Appendix D 8% + (.359/.689) (1%) 8% + .521 (1%) = 8.52% 35. Betty Bronson has just retired after 25 years with the electric company. Her total pension funds have an accumulated value of $180,000, and her life expectancy is 15 more years. Her pension fund manager assumes he can earn a 9 percent return on her assets. What will be her yearly annuity for the next 15 years? 9-35. Solution: Calculator Solution: N I/Y PV PMT FV 15 9 180,000 CPT −22,330.60 0 Answer: $22,330.60 Appendix D 36. Morgan Jennings, a geography professor, invests $50,000 in a parcel of land that is expected to increase in value by 12 percent per year for the next five years. He will take the proceeds and provide himself with a 10-year annuity. Assuming a 12 percent interest rate, how much will this annuity be? 9-36. Solution: Part 1 Part 2 Calculator Solution: Step one: N I/Y PV PMT FV 5 12 50,000.00 0 CPT FV −88,117.08 Answer: $88,117.08 Step two: N I/Y PV PMT FV 10 12 88,117.08 CPT PMT −15,595.33 0 Answer: $15,595.33 Appendix A FV = PV × FVIF (12%, 5 periods) FV = $50,000 × 1.762 = $88,100 Appendix D A = PVA/PVIFA (12%, 10 periods) A = $88,100/5.650 = $15,593 37. Solving for an annuity (LO9-4) You wish to retire in 14 years, at which time you want to have accumulated enough money to receive an annual annuity of $17,000 for 19 years after retirement. During the period before retirement you can earn 8 percent annually, while after retirement you can earn 10 percent on your money. What annual contributions to the retirement fund will allow you to receive the $17,000 annuity? 9-37. Solution: Part 1 Part 2 Calculator Solution: Determine the present value of a 14-year annuity during retirement: N I/Y PV PMT FV 19 10 CPT PV −142,203.64 17,000 0 Answer: $142,203.64 To determine the annual deposit into an account earning 8% that is necessary to accumulate $142,203.64 after 14 years, solve for the annuity: N I/Y PV PMT FV 14 8 0 CPT PMT −5,872.56 142,203.64 Answer: $5,872.56 Annual contribution Determine the present value of an annuity during retirement: Appendix D To determine the annual deposit into an account earning 8 percent that is necessary to accumulate $142,205 after 14 years, use the future value of an annuity table. See Appendix C. 38. Del Monty will receive the following payments at the end of the next three years: $2,000, $3,500, and $4,500. Then, from the end of the 4th through the end of the 10th year, he will receive an annuity of $5,000 per year. At a discount rate of 9 percent, what is the present value of all three future benefits? 9-38. Solution: Payment #1 Payment #2 Payment #3 Annuity Value after three years Annuity Value at Time Zero Total Present Value $ 1,834.86 Payment #1 + 2,945.88 Payment #2 + 3,474.83 Payment #3 +19,431.81 Annuity value $27,687.38 Total present value Calculator Solution: First find the present value of the first three payments. N I/Y PV PMT FV 1 9 CPT PV −1,834.86 0 2,000 Answer: $1,834.86 N I/Y PV PMT FV 2 9 CPT PV −2,945.88 0 3,500 Answer: $2,945.88 N I/Y PV PMT FV 3 9 CPT PV −3,474.83 0 4,500 Answer: $3,474.83 Total = 1,834.86 + 2,945.88 + 3,474.83 = $8,255.57 as of now Then find the present value of the deferred annuity. N I/Y PV PMT FV 7 9 CPT PV −25,164.76 5,000 0 Answer: $25,164.76 as of the end of year 3. Then, find its PV as of now: N I/Y PV PMT FV 3 9 CPT PV −19,431.81 0 25,164.76 Answer: $19,431.81 as of now Finally, find the total present value of all future payments. 8,255.57 + 19,431.81 = $27,687.38 First find the present value of the first three payments. PV = FV × PVIF (Appendix B) i = 9% 1) $2,000 × .917 = $1,834 2) 3,500 × .842 = 2,947 3) 4,500 × .772 = 3,474 $8,255 Then find the present value of the deferred annuity. Appendix D will give a factor for a seven period annuity (4th year through the 10th year) at a discount rate of 9 percent. The value of the annuity at the beginning of the fourth year is: This value at the beginning of year 4 (end of year 3) must now be discounted back for three years to get the present value of the deferred annuity. Use Appendix B. Finally, find the total present value of all future payments. Present value of first three payments $ 8,225.00 Present value of the deferred annuity 19,427.38 $27,682.38 39. Bridget Jones has a contract in which she will receive the following payments for the next five years: $1,000, $2,000, $3,000, $4,000, and $5,000. She will then receive an annuity of $8,500 a year from the end of the 6th through the end of the 15th year. The appropriate discount rate is 14 percent. If she is offered $30,000 to cancel the contract, should she do it? 9-39. Solution: First Five Payments 1 : 2 : 3 : 4 : 5 : Present Value of the Annuity Discount five years for PV of deferred annuity $ 877.19 + 1,538.94 + 2,024.91 + 2,368.32 + 2,596.84 +23,027.24 $32,433.44 Since the present value of all future benefits under the contract is greater than $30,000, Bridget Jones should not accept this deal. Calculator Solution: First find the present value of the first five payments. N I/Y PV PMT FV 1 14 CPT PV −877.19 0 1,000 Answer: $877.19 N I/Y PV PMT FV 2 14 CPT PV −1,538.94 0 2,000 Answer: $1,538.94 N I/Y PV PMT FV 3 14 CPT PV −2,024.91 0 3,000 Answer: $2,024.91 N I/Y PV PMT FV 4 14 CPT PV −2,368.32 0 4,000 Answer: $2,368.32 N I/Y PV PMT FV 5 14 CPT PV −2,596.84 0 5,000 Answer: $2,596.84 Total = 877.19 + 1,538.94 + 2,024.91 + 2,368.32 + 2,596.84 = $9,406.20 Then find the present value of the deferred annuity. N I/Y PV PMT FV 10 14 CPT PV −44,336.98 8,500 0 Answer: $44,336.98 as of the end of year 5 Then find it PV as of now: N I/Y PV PMT FV 5 14 CPT PV −23,027.24 0 44,336.98 Answer: $23,027.2 4 as of now Finally, find the total present value of all future payments. 9,406.20 + 23,027.24 = $32,433.44 Since the present value of all future benefits under the contract is greater than $30,000, Bridget Jones should not accept this amount to cancel the contract. First find the present value of the first five payments. PV = FV × PVIF (Appendix B) i = 14% 1) $1,000 × .877 = $ 877 2) 2,000 × .769 = 1,538 3) 3,000 × .675 = 2,025 4) 4,000 × .592 = 2,368 5) 5,000 × .519 = 2,595 $9,403 Then, find the present value of the deferred annuity. Appendix D will give a factor for a 10-period annuity (6th year through the 15th year) at a discount rate of 14 percent. The value of the annuity at the beginning of the 6th year is: This value at the beginning of year 6 (end of year 5) must now be discounted back for five years to get the present value of the deferred annuity. Use Appendix B. Next, find the total present value of all future payments. Present value of first five payments $ 9,403.00 Present value of the deferred annuity 23,010.38 $32,413.38 40. Mark Ventura has just purchased an annuity to begin payment at the end of 2016 (that is the date of the first payment). Assume it is now the beginning of the year 2014. The annuity is for $8,000 per year and is designed to last 10 years. If the interest rate for this problem calculation is 13 percent, what is the most he should have paid for the annuity? 9-40. Solution: Annuity Discount off two years Calculator Solution: N I/Y PV PMT FV 10 13 CPT PV −43,409.95 8,000 0 Answer: $43,409.95 as of the end of year 2 Then, find its PV as of now: N I/Y PV PMT FV 2 13 CPT PV −33,996.36 0 43,409.95 Answer: $33,996.36 as of now The maximum that should be paid for the annuity is $33,996.36. Appendix D will give a factor for a 10-year annuity when the appropriate discount rate is 13 percent (5.426). The value of the annuity at the beginning of the year it starts (2011) is: The present value at the beginning of 2014 is found using Appendix B (two years at 13 percent). The factor is .783. Note we are discounting from the beginning of 2016 to the beginning of 2014. The maximum that should be paid for the annuity is $33,988. 41. Yield (LO9-4) If you borrow $9,441 and are required to pay back the loan in five equal annual installments of $2,750, what is the interest rate associated with the loan? 9-41. Solution: Calculator Solution: N I/Y PV PMT FV 5 CPT I/Y 14.00 −9,441.00 2,750.00 0 Answer: 14.00% Appendix D Interest rate = 14% Go across period 5 until you find 3.433. Go up to the percentage at the top of the column and find 14 percent. 42. Cal Lury owes $10,000 now. A lender will carry the debt for five more years at 10 percent interest. That is, in this particular case, the amount owed will go up by 10 percent per year for five years. The lender then will require that Cal pay off the loan over the next 12 years at 11 percent interest. What will his annual payment be? 9-42. Solution: Part 1 Part 2 Calculator Solution: Step one: N I/Y PV PMT FV 5 10 10,000 0 CPT FV −16,105.10 Answer: $16,105.10 is the amount owed after five years. Step two: N I/Y PV PMT FV 12 11 16,105.10 CPT PMT −2,480.62 0 Answer: $2,480.62 is the annual payment to retire the loan. Appendix A Appendix D 43. If your uncle borrows $60,000 from the bank at 10 percent interest over the seven-year life of the loan, what equal annual payments must be made to discharge the loan, plus pay the bank its required rate of interest (round to the nearest dollar)? How much of his first payment will be applied to interest? To principal? How much of his second payment will be applied to each? 9-43. Solution: Annual Payment Amount of first payment applied to interest and principal First year interest Applied to principal Amount of second payment applied to interest and principal After one year Second year interest Applied to principal Calculator Solution: N I/Y PV PMT FV 7 10 60,000.00 CPT PMT −12,324.33 0 Answer: $12,324.33 is the annual payment to retire the loan. First payment: $60,000 × .10 = $6,000 first year interest $12,324.33 – $6,000 = $6,324.33 applied to principal Second payment: First determine remaining principal $60,000 – $6,324.33 = $53,675.67 $53,675.67× .10 = $5,367.57 second year interest $12,324.33 – $5,367.57 = $6,956.76 applied to principal Appendix D First payment: $60,000 × .10 = $6,000 interest $12,325 – $6,000 = $6,325 applied to principal Second payment: First determine remaining principal and then the interest and principal payment. $60,000 – $6,325 = $53,675 remaining principal $53,675 × .10 = $ 5,368 interest $12,325 – $5,368 = $ 6,957 applied to principal 44. Larry Davis borrows $80,000 at 14 percent interest toward the purchase of a home. His mortgage is for 25 years. a. How much will his annual payments be? (Although home payments are usually on a monthly basis, we shall do our analysis on an annual basis for ease of computation. We will get a reasonably accurate answer.) b. How much interest will he pay over the life of the loan? c. How much should he be willing to pay to get out of a 14 percent mortgage and into a 10 percent mortgage with 25 years remaining on the mortgage? Assume current interest rates are 10 percent. Carefully consider the time value of money. Disregard taxes. 9-44. Solution: a. b. Total payments Total interest paid c. New annual payments Difference between 14 percent and 10 percent interest $11,639.87 – 8,813.45 $ 2,826.42 Amount that could be paid to refinance Calculator Solution: (a) N I/Y PV PMT FV 25 14 80,000 CPT PMT −11,639.87 0 Answer: Annual payment $11,639.87 (b) Total payments = 11,639.87 × 25 = $290,996.82 Total interest paid = 290,996.82 – 80,000 = 210,996.82 (c) N I/Y PV PMT FV 25 10 80,000 CPT PMT −8,813.45 0 Answer: Annual Payment $8,813.45 Difference between old and new payments = 11,639.87 – 8,813.45 = $2,826.42 P.V. of difference at 10 percent: N I/Y PV PMT FV 25 10 CPT PV −25,655.53 2,826.42 0 Answer: $25,655.53 is the amount that could be paid to refinance. Appendix D Appendix D c. New payments at 10 percent 9-44. (Continued) Difference between old and new payments PV of difference – Appendix D 45. Annuity with changing interest rates (LO9-4) You are chairperson of the investment fund for the Continental Soccer League. You are asked to set up a fund of semiannual payments to be compounded semiannually to accumulate a sum of $250,000 after nine years at a 10 percent annual rate (18 payments). The first payment into the fund is to take place six months from today, and the last payment is to take place at the end of the ninth year. a. Determine how much the semiannual payment should be. (Round to whole numbers.) On the day after the sixth payment is made (the beginning of the fourth year), the interest rate goes up to a 12 percent annual rate, and you can earn a 12 percent annual rate on funds that have been accumulated as well as all future payments into the fund. Interest is to be compounded semiannually on all funds. b. Determine how much the revised semiannual payments should be after this rate change (there are 12 payments and compounding dates). The next payment will be in the middle of the fourth year. (Round all values to whole numbers.) 9-45. Solution: a. b. Part 1: Value of first six payments at the beginning of year 4 Part 2 : FV of first six payments at the end of year 9 Part 3: Additional amount required $ 250,000.00 – 121,630.10 $ 128,369.90 Part 4 : New payment level Calculator Solution: (a) N I/Y PV PMT FV 18 5 0 CPT PV −8,886.68 250,000 Answer: $8,886.68 (b) First determine how much the old payments are equal to after 6 periods at 5%. N I/Y PV PMT FV 6 5 0 8,886.68 CPT FV −60,446.40 Answer: $60,446.40 Then, determine how much this value will grow to after 12 periods at 6 percent (semiannual rate). Then, determine how much this value will grow to after 12 periods at 6 percent. N I/Y PV PMT FV 12 6 60,446 0 CPT FV −121,629.23 Answer: $121,629.23 Subtract this value ($121,629.00) from $250,000 to determine how much you need to accumulate on the next 12 payments. 250,000 – 121,629 = $128,371 Determine the revised semiannual payment necessary to accumulate this sum after 12 periods at 6 percent. N I/Y PV PMT FV 12 6 0 CPT PV −7,609.45 128,371 Answer: $7,609.39 is the revised semiannual payment. Appendix C b. First determine how much the old payments are equal to after six periods at 5 percent. Use Appendix C. Then, determine how much this value will grow to after 12 periods at 6 percent (semiannual rate). Appendix A Subtract this value from $250,000 to determine how much you need to accumulate on the next 12 payments. Determine the revised semiannual payment necessary to accumulate this sum after 12 periods at 6 percent. Appendix C A = FVA/FVIFA A = $128,377 /16.870 A = $7,610 46. Your younger sister, Linda, will start college in five years. She has just informed your parents that she wants to go to Hampton University, which will cost $17,000 per year for four years (cost assumed to come at the end of each year). Anticipating Linda’s ambitions, your parents started investing $2,000 per year five years ago and will continue to do so for five more years. How much more will your parents have to invest each year for the next five years to have the necessary funds for Linda’s education? Use 10 percent as the appropriate interest rate throughout this problem (for discounting or compounding). 9-46. Solution: PV of college costs five years from today (Part 1) Accumulation of $2,000 per year for 10 years (Part 2) Part 1 minus Part 2 $53,887.71 –31,874.85 $22,012.86 Additional funds required in five years Additional contribution required for next five years Calculator Solution: Present value of college costs N I/Y PV PMT FV 4 10 CPT PV −53,887.71 17,000 0 Answer: $53,887.71 Accumulation based on investing $2,000 per year for 10 years. N I/Y PV PMT FV 10 10 0 2,000 CPT FV −31,874.85 Answer: $31,874.85 Additional funds required five years from now when Brittany starts college: PV of college costs − Accumulation based on $2,000 per year = 53,887.71 – 31,874.85 = $22,012.86 Added contribution for the next five years N I/Y PV PMT FV 5 10 0 CPT PV −3,605.65 22,012.86 Answer: $3,605.65 Present value of college costs Appendix D Accumulation based on investing $2,000 per year for 10 years. Appendix C Additional funds required five years from now. $53,890 PV of college costs 31,874 Accumulation based on $2,000 per year investment $22,016 Additional funds required Added contribution for the next five years Appendix C 47. Special consideration of annuities and time periods (LO9-4) Your parents have accumulated a $120,000 nest egg. They have been planning to use this money to pay college costs to be incurred by you and your sister, Courtney. However, Courtney has decided to forgo college and start a nail salon. Your parents are giving Courtney $15,000 to help her get started, and they have decided to take year-end vacations costing $10,000 per year for the next four years. How much money will your parents have at the end of four years to help you with graduate school, which you will start then? You plan to work on a master’s and perhaps a PhD. If graduate school costs $26,353 per year, approximately how long will you be able to stay in school based on these funds? Use 9 percent as the appropriate interest rate throughout this problem. Round all values to whole numbers. 9-47. Solution: Funds available after Nail Salon $120,000 – 15,000 $105,000 PV of vacations Funds available after vacations $105,000 – 32,397 $ 72,603 Funds four years later for graduate school Number of years of graduate school Calculator Solution: Funds available after the nail salon $ 120,000 Funding available before the nail salon –15,000 Nail salon ________________________________________ ________________________________________ $ 105,000 Funds available after the nail salon ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________ N I/Y PV PMT FV 4 9 CPT PV −32,397.20 10,000 0 Answer: $32,397.20 Less present value of vacation $ 105,000 –32,397 Less vacations ________________________________________ ________________________________________ $ 72,603 Remaining funds for graduate school ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________ Funds available four years later for graduate school: N I/Y PV PMT FV 4 9 72,603 0 CPT FV −102,485.05 Answer: $102,485 Number of years of graduate education N I/Y PV PMT FV CPT N 5.00 9 102,485 –26,353 0 Answer: Three years $120,000 Funds available Funds available after the nail salon $120,000 Funding available before the nail salon – 15,000 Nail salon 105,000 Funds available after the nail salon Less present value of vacation Appendix D $105,000 – 32,400 $72,600 Remaining funds for graduate school Available funds after four years. Appendix A Number of years of graduate education Appendix D with i = 9%, n = 5 for 3.890, the answer is five years. COMPREHENSIVE PROBLEM Medical Research Corporation (Comprehensive time value of money) Dr. Harold Wolf of Medical Research Corporation (MRC) was thrilled with the response he had received from drug companies for his latest discovery, a unique electronic stimulator that reduces the pain from arthritis. The process had yet to pass rigorous Federal Drug Administration (FDA) testing and was still in the early stages of development, but the interest was intense. He received the three offers described following this paragraph. (A 10 percent interest rate should be used throughout this analysis unless otherwise specified.) Offer I - $1,000,000 now plus $200,000 from year 6 through 15. Also, if the product did over $100 million in cumulative sales by the end of year 15, he would receive an additional $3,000,000. Dr. Wolf thought there was a 70 percent probability this would happen. Offer II - Thirty percent of the buyer’s gross profit on the product for the next four years. The buyer in this case was Zbay Pharmaceutical. Zbay’s gross profit margin was 60 percent. Sales in year one were projected to be $2 million and then expected to grow by 40 percent per year. Offer III - A trust fund would be set up for the next eight years. At the end of that period, Dr. Wolf would receive the proceeds (and discount them back to the present at 10 percent). The trust fund called for semiannual payments for the next eight years of $200,000 (a total of $400,000 per year). The payments would start immediately. Since the payments are coming at the beginning of each period instead of the end, this is an annuity due. To look up the future value of an annuity due in the tables, add 1 to n (16 + 1) and subtract 1 from the value in the table. Assume the annual interest rate on this annuity is 10 percent annually (5 percent semiannually). Determine the present value of the trust fund’s final value. Required: Find the present value of each of the three offers and indicate which one has the highest present value. CP 9-1. Solution: Medical Research Corporation Offer I $1,000,000 now plus: + $200,000 from year 6 through 15 (deferred annuity) Appendix D Appendix B + .70 × $3,000,000 = $2,100,000 Appendix B Total value of Offer I $1,000,000 Payment today 763,209 Present value of deferred annuity 501,900 Present value of $3 million bonus $2,265,109 Offer II Gross Profit Payment 30% Year Sales (60% of Sales) of Gross Profit 1 $2,000,000 $1,200,000 $360,000 2 2,800,000 1,680,000 504,000 3 3,920,000 2,352,000 705,600 4 5,488,000 3,292,800 987,600 Appendix B Year Payment PV Factor PV 1 $360,000 .909 $327,240 2 504,000 .826 416,304 3 705,600 .751 529,906 4 987,600 .683 674,531 Total value of Offer II $1,947,981 Offer III Future value of an annuity due (Appendix C) 8 years – semiannually n = 16 + 1 = 17 i = 10%/2 = 5% FVIFA = 25.840 – 1 = 24.840 (Appendix C) Present value of trust fund (Appendix B) CP 9-1. (Continued) Summary Value of Offer I $2,265,109 Value of Offer II $1,947,981 Value of Offer III $2,320,056 Select Offer III Solution Manual for Foundations of Financial Management Stanley B. Block, Geoffrey A. Hirt, Bartley R. Danielsen 9780077861612, 9781260013917, 9781259277160
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