ch06
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ch06

Course Number: FINANCE 302, Fall 2010

College/University: CSU LA

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Chapter 6 Discounted Cash Flows and Valuation LEARNING OBJECTIVES 1. Explain why cash flows occurring at different times must be discounted to a common date before they can be compared, and be able to compute the present value and future value for multiple cash flows. When making decisions involving cash flows over time, we should first identify the magnitude and timing of the cash flows and then discount each...

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6 Discounted Chapter Cash Flows and Valuation LEARNING OBJECTIVES 1. Explain why cash flows occurring at different times must be discounted to a common date before they can be compared, and be able to compute the present value and future value for multiple cash flows. When making decisions involving cash flows over time, we should first identify the magnitude and timing of the cash flows and then discount each individual cash flow to its present value. The process of discounting the cash flows adjusts them for the time value of money, because todays dollars are not equal in value to dollars in the future. Once all of the cash flows are in present value terms, the cash flows can be compared to make decisions. Section 6.1 discusses the computation of present values and future values of multiple cash flows. 2. Describe how to calculate the present value of an ordinary annuity and how an ordinary annuity differs from an annuity due. An ordinary annuity is a series of equally spaced level cash flows over time. The cash flows for an ordinary annuity are assumed to take place at the ends of the periods. To find the value of an ordinary annuity, we start by calculating the annuity factor, which is equal to (1 present value factor)/i. Then, we multiply this factor by the constant future payment. An annuity due is an annuity in which the cash flows occur at the beginnings of the periods. A lease is an example of an annuity due. In this case, we are effectively prepaying for the service. To calculate the value of an annuity due, we multiply the ordinary annuity value times (1 + i). Section 6.2 discusses the computation of level cash flows (annuities and perpetuities). 1 3. Explain what a perpetuity is and how it is used in business, and be able to calculate the value of a perpetuity. A perpetuity is like an annuity except that the cash flows are perpetualthey never end. British Treasury Department bonds, called consols, were the first widely used securities of this kind. The most common example of perpetuity today is preferred stock. The issuer of preferred stock promises to pay fixed rate dividends forever. The preferred stockholders must be paid before common stockholders. To calculate the present value of a perpetuity, we simply divide the promised constant dividend payment (CF) by the interest rate (i). Learn by Doing Application 6.8 in Section 6.2 illustrates an application and calculation of a perpetuity problem found in business. 4. Discuss growing annuities and perpetuities, as well as their application in business, and be able to calculate their value. Financial managers often need to value cash-flow streams that increase at a constant rate over time. These cash flow streams are called growing annuities or growing perpetuities. An example of a growing annuity would be a 10-year lease contract with an annual adjustment for the expected rate of inflation over the life of the contract. If the cash flows continue to grow at a constant rate indefinitely, this cash flow stream is called a growing perpetuity. Application and calculation of cash flows that grow at a constant rate are discussed in Section 6.3. 5. Discuss why the effective annual interest rate (EAR) is the appropriate way to annualize interest rates, and be able to calculate EAR. 2 The EAR is the annual growth rate that takes compounding into account. Thus, the EAR is the true cost of borrowing or lending money. When we need to compare interest rates, we must make sure that the rates to be compared have the same time and compounding periods. If interest rates are not comparable, they must be converted into common terms. The easiest way to convert rates to common terms is to calculate the EAR for each interest rate. The use and calculations of EAR are discussed in Section 6.4. I. True or False Questions 1. Calculating the present and future values of multiple cash flows is relevant only for individual investors. a. b. 2. True False Calculating the present and future values of multiple cash flows is relevant for businesses only. a. b. 3. True False In computing the present and future value of multiple cash flows, each cash flow is discounted or compounded at a different rate. 3 a. b. 4. True False The present value of multiple cash flows is greater than the sum of those cash flows. a. b. 5. True False When you pay the same amount every month as your insurance premium for a term life policy for a period of five years, the stream of cash flows is called a perpetuity. a. b. 6. True False When you pay the same amount every month on your car loan for a period of three years, the stream of cash flows is called an annuity. a. b. 7. True False In todays financial markets, the best example of a perpetuity is the common stock issued by firms. 4 a. b. 8. True False Since the issuers of preferred stock promise to pay investors a fixed dividend, usually quarterly, forever, these are the most important perpetuities in the financial markets. c. d. 9. True False The present value of a perpetuity is the promised constant cash payment divided by the interest rate (i). e. f. 10. True False In ordinary annuities, cash flows occur at the beginning of each period. g. h. 11. True False In an annuity due, cash flows occur at the beginning of each period. 5 i. j. 12. True False The lease payments by a business on a warehouse rental are an example of an annuity due. k. l. 13. True False The present value of an annuity due is less than the present value of an ordinary annuity. m. n. 14. True False The present value of an annuity due is equal to the present value of an ordinary annuity. o. p. 2. True False The future value of an annuity due is greater than the future value of an ordinary annuity. a. True b. False 6 15. The future value of an annuity due is equal to the future value of an ordinary annuity. c. d. 16. True False Cash flow streams that increase at a constant rate over time are called growing annuities or growing perpetuities. e. f. 17. True False A fertilizer manufacturing company enters into a contract with a county parks and recreation department that calls for the company to sell 10 percent more of its best lawn feed every year for the next five years. If they also agree to maintain the total price as constant over the contract period, this growth in revenue is an example of a growing perpetuity. g. True h. False 7 18. You have received news about an inheritance that will pay you $5,000 next year. Beginning the following year, your inheritance will increase by 5 percent every year forever. This is a growing perpetuity. i. j. 19. True False Trey Hughes opened a pizza place last year. He expects to increase his revenue from last year by 7 percent every year for the next 10 years. This is an example of a growing annuity. k. l. 20. True False The APR is the annualized interest rate using compound interest. m. n. 21. True False The APR is defined as the simple interest charged per period multiplied by the number of periods per year. o. True 8 p. 22. False The correct way to annualize an interest rate is to compute the effective annual interest rate. q. r. 23. True False The correct way to annualize an interest rate is to compute the annual percentage rate (APR). s. t. 24. True False The effective annual interest rate (EAR) is defined as the annual growth rate that takes compounding into account. u. v. 25. True False The EAR is the true cost of borrowing and lending. w. True 9 x. 26. False The quoted interest rate is by convention a simple annual interest rate, such as the APR. y. z. 27. True False The quoted interest rate is by definition a simple annual interest rate, such as the EAR. aa. bb. 28. True False The Truth-in-Lending Act and the Truth-in-Savings Act require by law that the APR be disclosed on all consumer loans and savings plans and that it be prominently displayed on advertising and contractual documents. cc. dd. 29. True False Only the APR or some other quoted rate should be used as the interest rate factor for present or future value calculations. ee. True 10 ff. False 11 II. Multiple-Choice Questions 30. To solve future value problems with multiple cash flows involves which of the following steps? a. First, draw a time line to make sure that each cash flow is placed in the correct time period. b. c. Third, add up the future values. d. 31. Second, calculate the future value of each cash flow for its time period. All of the above are necessary steps. Which one of the following steps is NOT involved in solving future value problems? a. First, draw a time line to make sure that each cash flow is placed in the correct time period. b. c. Third, add up the values. d. 32. Second, discount each cash flow for its time period. All of the above are necessary steps. To solve present value problems with multiple cash flows involves which of the following steps? 12 a. First, draw a time line to make sure that each cash flow is placed in the correct time period. b. c. Third, add up the present values. d. 33. Second, calculate the present value of each cash flow for its time period. All of the above are necessary steps. Which one of the following steps is NOT involved in solving present value problems? a. First, draw a time line to make sure that each cash flow is placed in the correct time period. b. c. Third, add up the values. d. 34. Second, compound each cash flow for its time period. All of the above are necessary steps. Calculating the present and future values of multiple cash flows is relevant a. for businesses only. b. for individuals only c. for both individuals and businesses. d. none of the above. 13 35. In computing the present and future value of multiple cash flows, a. b. each cash flow is discounted or compounded at a different rate. c. earlier cash flows are discounted at a higher rate. d. 36. each cash flow is discounted or compounded at the same rate. later cash flows are discounted at a higher rate. In computing the present and future value of multiple cash flows, a. b. each cash flow is discounted or compounded at the same rate. c. earlier cash flows are discounted at a higher rate. d. 37. earlier cash flows are discounted at a lower rate. none of the above. The present value of multiple cash flows is a. greater than the sum of the cash flows. b. equal to the sum of all the cash flows. c. less than the sum of the cash flows. 14 d. 38. none of the above. The future value of multiple cash flows is a. b. equal to the sum of all the cash flows. c. less than the sum of the cash flows d. 39. greater than the sum of the cash flows. none of the above. If your investment pays the same amount at the end of each year for a period of six years, the cash flow stream is called a. b. an ordinary annuity. c. an annuity due. d. 40. a perpetuity. none of the above. If your investment pays the same amount at the beginning of each year for a period of 10 years, the cash flow stream is called 15 a. b. an ordinary annuity. c. an annuity due. d. 41. a perpetuity. none of the above. If your investment pays the same amount at the end of each year forever, the cash flow stream is called a. b. an ordinary annuity. c. an annuity due. d. 42. a perpetuity. none of the above. Cash flows associated with annuities are considered to be a. b. a cash flow stream of the same amount (a constant cash flow stream). c. a mix of constant and uneven cash flow streams. d. 43. an uneven cash flow stream. none of the above. Which ONE of the following statements is true about amortization? 16 a. Amortization refers to the way the borrowed amount (principal) is paid down over the life of the loan. b. With an amortized loan, each loan payment contains some payment of principal and an interest payment. c. A loan amortization schedule is just a table that shows the loan balance at the beginning and end of each period, the payment made during that period, and how much of that payment represents interest and how much represents repayment of principal. d. 45. All of the above are true. Which one of the following statements is NOT true about amortization? a. Amortization refers to the way the borrowed amount (principal) is paid down over the life of the loan. b. With an amortized loan, each loan payment contains some payment of principal and an interest payment. c. With an amortized loan, a smaller proportion of each months payment goes toward interest in the early periods. d. A loan amortization schedule is just a table that shows the loan balance at the beginning and end of each period, the payment made during that period, and how 17 much of that payment represents interest and how much represents repayment of principal. 46. Which one of the following statements is true about amortization? a. With an amortized loan, a bigger proportion of each months payment goes toward interest in the early periods. b. With an amortized loan, a bigger proportion of each months payment goes toward interest in the later periods. c. With an amortized loan, a smaller proportion of each months payment goes toward interest in the early periods. d. 47. None of the above. The annuity transformation method is used to transform a. a present value annuity to a future value annuity. b. a present value annuity to a future value annuity. c. an ordinary annuity to an annuity due. d. a perpetuity to an annuity. 18 48. A firm receives a cash flow from an investment that will increase by 10 percent annually for an infinite number of years. This cash flow stream is called a. b. a growing perpetuity. c. an ordinary annuity. d. 49. an annuity due. a growing annuity. Your investment in a small business venture will produce cash flows that increase by 15 percent every year for the next 25 years. This cash flow stream is called a. b. a growing perpetuity. c. an ordinary annuity. d. 50. an annuity due. a growing annuity. Which one of the following statements is TRUE about the effective annual rate (EAR)? a. The effective annual interest rate (EAR) is defined as the annual growth rate that takes compounding into account. 19 b. The EAR conversion formula accounts for the number of compounding periods and, thus, effectively adjusts the annualized interest rate for the time value of money. c. d. 51. The EAR is the true cost of borrowing and lending. All of the above are true. The true cost of borrowing is the a. b. effective annual rate. c. quoted interest rate. d. 52. annual percentage rate. periodic rate. The true cost of lending is the a. annual percentage rate. b. effective annual rate. c. quoted interest rate. d. none of the above. 20 53. Which one of the following statements is NOT true? a. b. The EAR is the appropriate rate to do present and future value calculations. c. The EAR is the true cost of borrowing and lending. d. 54. The APR is the appropriate rate to do present and future value calculations. The EAR takes compounding into account. Which one of the following statements is NOT true? a. The Truth-in-Lending Act was passed by Congress to ensure that the true cost of credit was disclosed to consumers. b. The Truth-in-Savings Act was passed to provide consumers an accurate estimate of the return they would earn on an investment. c. The above two pieces of legislation require by law that the APR be disclosed on all consumer loans and savings plans. d. 55. All of the above are true statements. Which one of the following statements is NOT true? a. The correct way to annualize an interest rate is to compute the effective annual interest rate (EAR). 21 b. The APR is the annualized interest rate using simple interest. c. The correct way to annualize an interest rate is to compute the annual percentage rate (APR). d. You can find the interest rate per period by dividing the quoted annual rate by the number of compounding periods. 56. FV of multiple cash flows: Chandler Corp. is expecting a new project to start producing cash flows, beginning at the end of this year. They expect cash flows to be as follows: 1 $643,547 2 $678,214 3 $775,908 4 5 $778,326 $735,444 If they can reinvest these cash flows to earn a return of 8.2 percent, what is the future value of this cash flow stream at the end of five years? (Round to the nearest dollar.) a. $3,889,256 b. $4,227,118 c. $5,214,690 d. $4, 809,112 22 57. FV of multiple cash flows: Stiglitz, Inc., is expecting the following cash flows starting at the end of the year$113,245, $132,709, $141,554, and $180,760. If their opportunity cost is 9.6 percent, find the future value of these cash flows. (Round to the nearest dollar.) a. b. $732,114 c. $685,312 d. 58. $644,406 $900,810 FV of multiple cash flows: Tariq Aziz will receive from his investment cash flows of $3,125, $3,450, and $3, 800. If he can earn 7.5 percent on any investment that he makes, what is the future value of his investment cash flows at the end of three years? (Round to the nearest dollar.) a. b. $10,944 c. $10,812 d. 59. $11,120 $12,770 FV of multiple cash flows: Shane Matthews has invested in an investment that will pay him $6,200, $6,450, $7,225, and $7,500 over the next four years. If his opportunity cost 23 is 10 percent, what is the future value of the cash flows he will receive? (Round to the nearest dollar.) a. b. $29,900 c. $30,455 d. 60. $27,150 $31,504 FV of multiple cash flows: International Shippers, Inc., have forecast earnings of $1,233,400, $1,345,900, and $1,455,650 for the next three years. What is the future value of these earnings if the firms opportunity cost is 13 percent? (Round to the nearest dollar.) a. b. $4,551,446 c. $3,900,865 d. 61. $4,214,360 $4,875,212 PV of multiple cash flows: Jack Stuart has loaned money to his brother at an interest rate of 5.75 percent. He expects to receive $625, $650, $700, and $800 at the end of the next 24 four years as complete repayment of the loan with interest. How much did he loan out to his brother? (Round to the nearest dollar.) a. b. $2,250 c. $2,404 d. 62. $2,713 $2,545 PV of multiple cash flows: Ferris, Inc., has borrowed from their bank at a rate of 8 percent and will repay the loan with interest over the next five years. Their scheduled payments, starting at the end of the year are as follows$450,000, $560,000, $750,000, $875,000, and $1,000,000. What is the present value of these payments? (Round to the nearest dollar.) a. b. $2,615,432 c. $2431,224 d. 63. $2,735,200 $2,815,885 PV of multiple cash flows: Hassan Ali has made an investment that will pay him $11,455, $16,376, and $19,812 at the end of the next three years. His investment was to 25 fetch him a return of 14 percent. What is the present value of these cash flows? (Round to the nearest dollar.) a. b. $36,022 c. $41,675 d. 64. $33,124 $39,208 PV of multiple cash flows: Ajax Corp. is expecting the following cash flows$79,000, $112,000, $164,000, $84,000, and $242,000over the next five years. If the companys opportunity cost is 15 percent, what is the present value of these cash flows? (Round to the nearest dollar.) a. b. $414,322 c. $480,906 d. 65. $429,560 $477,235 PV of multiple cash flows: Pam Gregg is expecting cash flows of $50,000, $75,000, $125,000, and $250,000 from an inheritance over the next four years. If she can earn 11 26 percent on any investment that she makes, what is the present value of her inheritance? (Round to the nearest dollar.) a. b. $309,432 c. $412,372 d. 66. $361,998 $434,599 Present value of an annuity: Transit Insurance Company has made an investment in another company that will guarantee it a cash flow of $37,250 each year for the next five years. If the company uses a discount rate of 15 percent on its investments, what is the present value of this investment? (Round to the nearest dollar.) a. b. $124,868 c. $251,154 d. 67. $101,766 $186,250 Present value of an annuity: Herm Mueller has invested in a fund that will provide him a cash flow of $11,700 for the next 20 years. If his opportunity cost is 8.5 percent, what is the present value of this cash flow stream? (Round to the nearest dollar.) 27 a. b. $132,455 c. $110,721 d. 68. $234,000 $167,884 Present value of an annuity: Myers, Inc., will be making lease payments of $3,895.50 for a 10-year period, starting at the end of this year. If the firm uses a 9 percent discount rate, what is the present value of this annuity? (Round to the nearest dollar.) a. b. $29,000 c. $25,000 d. 69. $23,250 $20,000 Present value of an annuity: Lorraine Jackson won a lottery. She will have a choice of receiving $25,000 at the end of each year for the next 30 years, or a lump sum today. If she can earn a return of 10 percent on any investment she makes, what is the minimum amount she should be willing to accept today as a lump-sum payment? (Round to the nearest hundred dollars.) a. $750,000 28 b. c. $212,400 d. 70. $334,600 $235,700 Present value of an annuity: Craymore Tech is expecting cash flows of $67,000 at the end of each year for the next five years. If the firms discount rate is 17 percent, what is the present value of this annuity? (Round to the nearest dollar.) a. b. $241,653 c. $278,900 d. 71. $214,356 $197,776 Future value of an annuity: Carlos Menendez is planning to invest $3,500 every year for the next six years in an investment paying 12 percent annually. What will be the amount he will have at the end of the six years? (Round to the nearest dollar.) a. $21,000 b. $28,403 c. $24,670 d. $26,124 29 72. Future of value an annuity: Jayadev Athreya has started on his first job. He plans to start saving for retirement early. He will invest $5,000 at the end of each year for the next 45 years in a fund that will earn a return of 10 percent. How much will Jayadev have at the end of 45 years? (Round to the nearest dollar.) a. b. $3,594,524 c. $1,745,600 d. 73. $2,667,904 $5,233,442 Future value of an annuity: You plan to save $1,250 at the end of each of the next three years to pay for a vacation. If you can invest it at 7 percent, how much will you have at the end of three years? (Round to the nearest dollar.) a. $3,750 b. $3,918 c. $4,019 d. $4,589 30 74. Future value of an annuity: Zhijie Jiang is saving to buy a new car in four years. She will save $5,500 at the end of each of the next four years. If she invests her savings at 6.75 percent, how much will she have after four years? (Round to the nearest dollar.) a. b. $23,345 c. $27,556 d. 75. $22,000 $24,329 Future value of an annuity: Terri Garner will invest $3,000 in an IRA for the next 30 years. The investment will earn 13 percent annually. How much will she have at the end of 30 years? (Round to the nearest dollar.) a. b. $912,334 c. $748,212 d. 76. $897,598 $1,233,450 Computing annuity payment: Maricela Sanchez needs to have $25,000 in five years. If she can earn 8 percent on any investment, what is the amount that she will have to invest every year for the next five years? (Round to the nearest dollar.) 31 a. b. $4,261 c. $4,640 d. 77. $5,000 $4,445 Computing annuity payment: Jane Ogden wants to save for a trip to Australia. She will need $12,000 at the end of four years. She can invest a certain amount at the beginning of each of the next four years in a bank account that will pay her 6.8 percent annually. How much will she have to invest annually to reach her target? (Round to the nearest dollar.) a. b. $2,980 c. $2,538 d. 78. $3,000 $2,711 Computing annuity payment: Jackson Electricals has borrowed $27,850 from its bank at an annual rate of 8.5 percent. It plans to repay the loan in eight equal installments, beginning at the end of next year. What is its annual loan payment? (Round to the nearest dollar.) 32 a. b. $5,134 c. $4,939 d. 79. $4,708 $4,748 Computing annuity payment: John Harper has borrowed $17,400 to pay for his new truck. The annual interest rate on the loan is 9.4 percent, and the loan needs to be repaid in four years. What will be his annual payment if he begins his payment beginning now? (Round to the nearest dollar.) a. b. $5,450 c. $4,850 d. 80. $5,229 $4,953 Computing annuity payment: Trevor Smith wants to have a million dollars at retirement, which is 15 years away. He already has $200,000 in an IRA earning 8 percent annually. How much does he need to save each year, beginning at the end of this year to reach his target? Assume he could earn 8 percent on any investment he makes. (Round to the nearest dollar.) 33 a. b. $14,273 c. $10,900 d. 81. $13,464 $16,110 Perpetuity: Your father is 60 years old and wants to set up a cash flow stream that would be forever. He would like to receive $20,000 every year, beginning at the end of this year. If he could invest in account earning 9 percent, how much would he have to invest today to receive his perpetual cash flow? (Round to the nearest dollar.) a. b. $200,000 c. $189,000 d. 82. $222,222 $235,200 Perpetuity: A lottery winner was given a perpetual payment of $11, 444. She could invest the cash flows at 7 percent. What is the present value of this perpetuity? (Round to the nearest dollar.) a. $112,344 b. $163,486 34 c. d. 83. $191,708 $201,356 Perpetuity: Roger Barkley wants to set up a scholarship at his alma mater. He is willing to invest $500,000 in an account earning 10 percent. What will be the annual scholarship that can be given from this investment? (Round to the nearest dollar.) a. b. $500,000 c. $50,000 d. 84. $5,000 None of the above Perpetuity: Chris Collinge has funded a retirement investment with $250,000 earning a return of 5.75 percent. What is the value of the payment that he can receive in perpetuity? (Round to the nearest dollar.) a. $12,150 b. $15,250 c. $14,375 d. $14,900 35 85. Perpetuity: Jeff Conway wants to receive $25,000 in perpetuity and will invest his money in an investment that will earn a return of 13.5 percent annually. What is the value of the investment that he needs to make today to receive his perpetual cash flow stream? (Round to the nearest dollar.) a. b. $252,325 c. $144,350 d. 86. $640,225 $185,185 Annuity due: You plan to save $1,400 for the next four years, beginning now, to pay for a vacation. If you can invest it at 6 percent, how much will you have at the end of four years? Round to the nearest dollar. a. $6,124 b. $5,618 c. $4,019 d. $6,492 36 87. Annuity due: Mark Holcomb has a five-year loan on which he will make annual payments of $2,235, beginning now. If the interest rate on the loan is 8.3 percent, what is the present value of this annuity? (Round to the nearest dollar.) a. b. $8,854 c. $8,612 d. 88. $9,588 $9,122 Annuity due: Jenny Abel is investing $2,500 today and will do so at the beginning of another six years for a total of seven payments. If her investment can earn 12 percent, how much will she have at the end of seven years? (Round to the nearest dollar.) a. b. $28,249 c. $31,127 d. 89. $25,223 $29,460 Annuity due: Your inheritance will pay you $100,000 a year for five years beginning now. You can invest it in a CD that will pay 7.75 percent annually. What is the present value of your inheritance? (Round to the nearest dollar.) 37 a. b. $401,916 c. $433,064 d. 90. $399,356 $467,812 Growing perpetuity: Jack Benny is planning to invest in an insurance company product. The product will pay $10,000 at the end of this year. Thereafter, the payments will grow annually at a 3 percent rate forever. Jack will be able to invest his cash flows at a rate of 6.5 percent. What is the present value of this investment cash flow stream? (Round to the nearest dollar.) a. b. $312,766 c. $285,714 d. 91. $326,908 $258,133 Growing perpetuity: Norwood Investments is putting out a new product. The product will pay out $25,000 in the first year, and after that the payouts will grow by an annual rate of 2.5 percent forever. If you can invest the cash flows at 7.5 percent, how much will you be willing to pay for this perpetuity? (Round to the nearest dollar.) 38 a. b. $233,000 c. $250,000 d. 92. $312,000 $500,000 Growing annuity: Hill Enterprises is expecting tremendous growth from its newest boutique store. Next year the store is expected to bring in net cash flows of $675,000. The company expects its earnings to grow annually at a rate of 13 percent for the next 15 years. What is the present value of this growing annuity if the firm uses a discount rate of 18 percent on its investments? (Round to the nearest dollar.) a. b. $6,750,000 c. $7,115,449 d. 93. $6,448,519 $5,478,320 Growing annuity: Wilbon Corp. is evaluating whether it should take over the lease of an ethnic restaurant in Manhattan. The current owner had originally signed a 25-year lease, of which 16 years still remain. The restaurant has been growing steadily at a 7 percent growth for the last several years. Wilbon Corp. expects the restaurant to continue to grow 39 at the same rate for the remaining lease term. Last year, the restaurant brought in net cash flows of $310,000. If the firm evaluates similar investments at 15 percent, what is the present value of this investment? (Round to the nearest dollar.) a. b. $2,838,182 c. $3,109,460 d. 94. $2,966.350 $2,709,124 Effective annual rate: Desire Cosmetics borrowed $152,300 from a bank for three years. If the quoted rate (APR) is 11.75 percent, and the compounding is daily, what is the effective annual rate (EAR)? (Round to one decimal place.) a. b. 14.3% c. 12.5% d. 95. 11.75% 11.6% Effective annual rate: Largent Supplies Corp. has borrowed to invest in a project. The loan calls for a payment of $17,384 every month for three years. The lender quoted Largent a rate of 8.40 percent with monthly compounding. At what rate would you 40 discount the payments to find amount borrowed by Largent? (Round to two decimal places.) a. 8.40% b. 8.73% c. 8.95% d. None of the above. 41 III. Essay Questions 96. How is an annuity due different from the ordinary annuity? Answer: When constant cash flows are received or paid at the end of each period for a length of time, we have an ordinary annuity. If the same cash flows happen at the beginning of each period, we call it an annuity due. Cash flows received at the beginning of each period earn interest for an extra period compared to cash flows received at the end of each period for an investment of the same time frame. Thus, annuity dues have higher values than ordinary annuities. 97. The annual percentage rate (APR) is not the appropriate rate to do present or future value calculations. Explain this statement. Answer: The APR is the annualized interest rate using simple interest. In other words, the APR is the simple interest charged per period multiplied by the number of periods per year. However, the APR ignores the impact of compounding on cash flows. This makes it an inappropriate discount rate for doing present and future value calculations. An appropriate rate for such calculations is the effective annual rate (EAR). 98. What was the purpose behind the passage of the two consumer protection acts discussed in this chapter? 42 Answer: In 1968, Congress passed the Truth-in-Lending Act to ensure that all borrowers receive meaningful information about the cost of credit so they can make intelligent economic decisions. The act applies to all lenders that extend credit to consumers, and it covers credit card loans, auto loans, home mortgage loans, home equity loans, home improvement loans, and some small business loans. Similar legislation, the so-called Truth-in-Savings Act, applies to consumer savings vehicles such as consumer certificates of deposits (CDs). These two pieces of legislation require by law that the APR be disclosed on all consumer loans and savings plans and that it be prominently displayed on advertising and contractual documents. 43 IV. Answers to True or False Questions 1. False 2. False 3. False 4. False 5. False 6. True 7. False 8. True 9. True 10. False 11. True 12. True 13. False 14. False 15. True 16. False 17. True 18. False 19. True 20. True 21. False 22. True 44 23. True 24. False 25. True 26. True 27. True 28. False 29. True 30. False 45 V. Answers to Multiple-Choice Questions 31. d 32. b 33. d 34. b 35. c 36. a 37. b 38. c 39. a 40. b 41. c 42. a 43. b 44. d 45. c 46. a 47. c 48. b 49. d 50. d 51. b 52. b 46 53. a 54. d 55. c 56. b 57. a 58. a 59. d 60. b 61. c 62. d 63. b 64. a 65. a 66. b 67. c 68. c 69. d 70. a 71. b 72. b 73. c 74. d 75. a 47 76. b 77. c 78. c 79. d 80. a 81. a 82. b 83. c 84. c 85. d 86. d 87. a 88. b 89. c 90. c 91. d 92. a 93. b 94. c 95. b 48 VI. Solutions to Multiple-Choice Questions 56. Solution: 0 1 2 3 4 5 $643,547 n = 5; $678,214 $775,908 $778,326 $735,444 i = 8.2% FV5 = $643,547(1.082) 4 + $678,214(1.082) 3 + $775,908(1.082) 2 + $778,326(1.082)1 + $735,444 = $882,042.10 + $859,109.52 + $908,374.12 + $842,148.73 + $735,444 = $4,227,118.47 57. Solution: 0 1 2 3 4 $113,245 n = 4; $132,709 $141,554 $180,760 i = 9.6% FV4 = $113,245(1.096) 3 + $132,709(1.096) 2 + $141,554(1.096)1 + $180,760 = $149,090.75 + $159,412.17 + $155,143.18 + $180,760 = $644,406.10 58. Solution: 0 1 2 3 49 $3,125 n = 3; $3,450 $3, 800 i = 7.5% FV3 = $3,125(1.075) 2 + $3,450(1.075)1 + $3,800 = $3,611.33 + $3,708.75 + $3,800 = $11,120.08 59. Solution: 0 1 2 3 4 $6,200 n = 4; $6,450 $7,225 $7,500 i = 10% FV4 = $6,200(1.10) 3 + $6,450(1.10) 2 + $7,225(1.10)1 + $7,500 = $8,252.20 + $7,804.50 + $7,947.50 + $7,500 = $31,504.20 60. Solution: 0 1 2 3 $1, 233,400 n = 3; $1,345,900 $1,455,650 i = 13% 50 FV3 = $1,233,400(1.13) 2 + $1,345,900(1.13)1 + $1,455,650 = $1,574,928.46 + $1,520,867 + $1,455,650 = $4,551,445.46 61. Solution: 0 1 2 3 4 $1625 n = 4; $650 $700 $800 i = 5.75% $625 $650 $700 $800 + + + 2 3 (1.0575) (1.0575) (1.0575) (1.0575) 4 = $591.02 + $581.24 + $591.91 + $639.69 PV = = $2,403.85 62. Solution: 0 1 2 3 4 5 $450,000 n = 5; $560,000 $750,000 $875,000 $1,000,000 i = 8% $450,000 $560,000 $750,000 $875,000 $1,000,000 + + + + (1.08) (1.08) 2 (1.08) 3 (1.08) 4 (1.08) 5 = $416,666.67 + $480,109.74 + $595,374.18 + $643,151.12 + $680,583.20 = $2,815,884.91 PV = 51 63. Solution: 0 1 2 3 $11,455 $16,376 n = 3; $19,812 i = 14% $11,455 $16,376 $19,812 + + (1.14) (1.14) 2 (1.14) 3 = $10,048.25 + $12,600.80 + $13,372.54 = $36,021.58 PV = 64. Solution: 0 1 2 3 4 5 $79,000 n = 5; $112,000 $164,000 $84,000 $242,000 i = 15% $79,000 $112,000 $164,000 $84,000 $242,000 + + + + (1.15) (1.15) 2 (1.15) 3 (1.15) 4 (1.15) 5 = $68,695.65 + $84,688.09 + $107,832.66 + $48,027.27 + $120,316.77 PV = = $429,560.45 65. Solution: 0 1 2 3 4 $50,000 n = 4; $75,000 $125,000 i = 11% 52 $250,000 $50,000 $75,000 $125,000 $250,000 + + + (1.11) (1.11) 2 (1.11) 3 (1.11) 4 = $45,045.05 + 60,871.68 + $91,398.92 + $164,682.74 = $361,998.39 PV = 66. Solution: 0 1 2 3 4 5 $37,250 n = 5; $37,250 $37,250 i = 15% Annual payment = PMT = $37,250 No. of payments = n = 5 Required rate of return = 15% Present value of investment = PVA5 1 1 (1 + i ) n PVA n = PMT i 1 1 (1.15) 5 = $37,250 = $37,250 3.3522 0.15 = $124,867.78 53 $37,250 $37,250 67. Solution: 0 1 2 3 19 20 $11,700 $11,700 n = 20; $11,700 $11,700 $11,700 9 10 i = 8.5% 1 1 (1 + i ) n PVA n = PMT i 1 1 (1.085) 20 = $11,700 = $11,700 9.4633 0.085 = $110,721.04 68. Solution: 0 1 2 3 $3,895.50 n = 10; $3,895.50 $3,895.50 i = 9% 54 $3,895.50 $3,895.50 1 1 (1 + i ) n PVA n = PMT i 1 1 (1.09)10 = $3,895.50 = $3,895.50 6.4177 0.09 = $24,999.99 55 69. Solution: 0 1 2 3 29 30 $25,000 $25,000 n = 30; $25,000 $25,000 4 5 i = 10% 1 1 (1 + i ) n PVA n = PMT i 1 1 (1.10) 30 = $25,000 0.10 = $235,672.86 70. $25,000 = $25,000 9.4269 Solution: 0 1 2 3 $67,000 n = 5; $67,000 $67,000 i = 17% 56 $67,000 $67,000 1 1 (1 + i ) n PVA n = PMT i 1 1 (1.17) 5 = $67,000 = $67,000 3.1993 0.17 = $214,356.19 57 71. Solution: 0 1 2 3 4 5 6 $3,500 n = 6; $3,500 $3,500 $3,500 $3,500 $3,500 i = 12% (1 + i ) n 1 FVA n = PMT i 6 (1.12) 1 = $3,500 = $3,500 8.1152 0.12 = $28,403.16 72. Solution: 0 1 2 3 44 45 $5,000 n = 45; $5,000 $5,000 i = 10% (1 + i ) n 1 FVA n = PMT i 45 (1.10) 1 = $5,000 = $5,000 718.9048 0.10 = $3,594.524.18 73. Solution: 0 1 2 3 58 $5,000 $5,000 $1,250 n = 3; $1,250 $1,250 i = 7% (1 + i ) n 1 FVA n = PMT i 3 (1.07) 1 = $1,250 = $1,250 3.2149 0.07 = $4,018.63 74. Solution: 0 1 2 3 4 $5,500 $5,500 n = 4; $5,500 $5,500 i = 6.75% (1 + i ) n 1 FVA n = PMT i (1.0675) 4 1 = $5,500 = $5,500 4.4235 0.0675 = $24,329.43 75. Solution: 0 1 2 3 29 30 $3,000 $3,000 $3,000 59 $3,000 $3,000 n = 30; i = 13% (1 + i ) n 1 FVA n = PMT i 30 (1.13) 1 = $3,000 = $3,000 293.1992 0.13 = $879,597.65 60 76. Solution: 0 1 2 3 4 5 PMT n = 5; PMT PMT i = 8% PMT PMT FVA = $25,000 (1 + i ) n 1 FVA n = PMT i 5 (1.08) 1 $25,000 = PMT 0.08 $25,000 $25,000 PMT = = 5 (1.08) 1 5.8666 0.08 = $4,261.41 77. Solution: 0 1 2 3 4 PMT PMT n = 4; PMT PMT i = 6.8% FVA = $12,000 61 (1 + i ) n 1 FVA n = PMT (1 + i ) i $12,000 PMT = (1.068) 4 1 0.068 (1.068) = $2,538.16 62 78. Solution: 0 1 2 3 7 8 PMT PVAn = $27,850 PMT PMT n = 8; PMT PMT i = 8.5% Present value of annuity = PVA = $27,850 Return on investment = i = 8.5% Payment required to meet target = PMT Using the PVA equation: 1 1 (1 + i ) n PVA n = PMT i $27,850 $27,850 PMT = = 1 5.6392 1 (1.085) 8 0.085 = $4,938.66 Each payment made by Jackson Electricals will be $4,938.66, starting at the end of next year. 79. Solution: 0 1 2 3 4 63 PMT PMT PVAn = $17,400 PMT PMT n = 4; i = 9.4% Present value of annuity = PVA = $17,400 Return on investment = i = 9.4% Payment required to meet target = PMT Type of annuity = Annuity due Using the PVA equation: 1 1 (1 + i ) n PVA n = PMT (1 + i ) i $17,400 $17,400 PMT = = 1 3.2115 1.094 1 (1.094) 4 (1.094) 0.094 = $4,952.49 Each payment made by John Harper will be $5,927.36, starting at the end of next year. 80. Solution: Retirement investment target in 15 years = $1,000,000 Amount invested in IRA account now = PV = $200,000 Return earned by investment = i = 8% Value of current investment in 15 years = FV15 FV15 = PV(1 + i )15 = $200,000(1.08)15 = $634,433.82 64 Balance of money needed to buy car = $1,000,000 -$634,433.82 =$365,566.18 = FVA Payment needed to reach target = PMT (1 + i ) n 1 FVA = PMT i FVA $365,566.18 $365,566.18 PMT = = = n 27.1521 1 (1 + i ) (1.08)15 1 0.08 i = $13,463.64 81. Solution: Annual payment needed = PMT = $20,000 Investment rate of return = i = 9% Term of payment = Perpetuity Present value of investment needed = PV PMT $20,000 = i 0.09 = $222,222.22 PV of Perpetuity = 82. Solution: Annual payment needed = PMT = $11,444 Investment rate of return = i = 7% Term of payment = Perpetuity Present value of investment needed = PV PMT $11,444 = i 0.07 = $163,485.71 PV of Perpetuity = 65 83. Solution: Annual payment needed = PMT Present value of investment = PVA = $500,000 Investment rate of return = i = 10% Term of payment = Perpetuity PMT i PMT = PV of Perpetuity i = $500,000 0.10 = $5,000 PV of Perpetuity = 84. Solution: Annual payment needed = PMT Present value of investment = PVA = $250,000 Investment rate of return = i = 5.75% Term of payment = Perpetuity PMT i PMT = PV of Perpetuity i = $250,000 0.0575 = $14,375 PV of Perpetuity = 85. Solution: Annual Payment needed = PMT = $25,000 Investment rate of return = i = 13.5% Term of payment = Perpetuity 66 Present value of investment needed = PV PMT $25,000 = i 0.135 = $185,185.19 PV of Perpetuity = 86. Solution: 0 1 2 3 4 $1,400 $1,400 n = 4; $1,400 $1,400 i = 6% (1 + i ) n FVA = PMT (1 + i ) i (1.06) 4 1 = $1,400 (1.06) 0.06 = $1,400 4.3746 1.06 = $6,491.93 67 87. Solution: 0 1 2 3 4 5 $2,235 $2,235 n = 5; $2,235 $2,235 $2,235 3 6 i = 8.3% Annual payment = PMT = $2,235 No. of payments = n = 5 Required rate of return = 8.3% Present value of investment = PVA5 1 1 (1 + i ) n PVA = PMT i (1 + i ) 1 1 (1.083) 5 = $2,235 (1.083) 0.083 = $2,235 3.9613 1.083 = $9,588.44 88. Solution: 0 1 2 7 PMT n = 7; PMT PMT PMT i = 12% 68 PMT Present value of annuity = PVA Return on investment = i = 9.4% Payment required to meet target = $2,500 Type of annuity = Annuity due (1 + i ) n FVA = PMT (1 + i ) i (1.12) 7 1 = $2,500 (1.12) 0.12 = $2,500 10.0890 1.12 = $28,249.23 89. Solution: 0 1 2 3 4 5 $100,000 n = 5; $100,000 $100,000 $100,000 i = 7.75% Annual payment = PMT = $100,000 No. of payments = n = 5 Required rate of return = 7.75% Present value of investment = PVA5 69 $100,000 1 1 (1 + i ) n PVA = PMT i (1 + i ) 1 1 (1.0775) 5 = $100,000 (1.0775) 0.0775 = $100,000 4.0192 1.0775 = $433,064.19 90. Solution: Cash flow at t=1 = CF1 = $10,000 Annual growth rate = g = 3% Discount rate = i = 6.5% Present value of growing perpetuity = PVA CF1 $10,000 = (i g) (0.065 0.03) = $285,714.29 PVA = 91. Solution: Cash flow at t=1 = CF1 = $25,000 Annual growth rate = g = 2.5% Discount rate = i = 7.5% Present value of growing perpetuity = PVA CF1 $25,000 = (i g) (0.075 0.025) = $500,000 PVA = 70 92. Solution: Time of growth = n = 15 years Next years expected net cash flow = CF1 = $675,000 Expected annual growth rate = g = 13% Firms required rate of return = i = 18% Present value of growing annuity = PVAn 1 + g n 1.13 15 CF1 $675,000 PVA n = 1 1 = (i g ) 1 + i (0.18 0.13) 1.18 = $13,500,000 477668 = $6,448,519.47 93. Solution: Time for lease to expire = n = 16 years Last years net cash flow = CF0 = $310,000 Expected annual growth rate = g = 7% Firms required rate of return = i = 15% Expected cash flow next year = CF1 = $310,000(1 + g) = $310,000(1.07) = $331,700 Present value of growing annuity = PVAn PVA n = 1 + g n 1.07 16 CF1 $331,700 1 = 1 (i g ) 1 + i (0.15 0.07) 1.15 = $4,146,250 0.684518 = $2,838,181.52 94. Solution: 71 Loan amount = PV = $152,300 Interest rate on loan = i = 11.75% Frequency of compounding = m = 365 Effective annual rate = EAR m1 i 0.1175 EAR = 1 + 1 = 1 + 365 m = 1.12455 1 = 12.46% 95. 365 1 Solution: Loan amount = PV Interest rate on loan = i = 8.4% Frequency of compounding = m = 12 Effective annual rate = EAR m1 12 i 0.084 EAR = 1 + 1 = 1 + 1 12 m = 1.0873 1 = 8.73% To discount present or future value of cash flows, the most appropriate rate is the EAR, that is, 8.73 percent. 72
MOST POPULAR MATERIALS FROM FINANCE 302
MOST POPULAR MATERIALS FROM FINANCE
MOST POPULAR MATERIALS FROM CSU LA