Chapter 3—Present ValueMULTIPLE CHOICE 1. Which of the following cannot be calculated? Future value of a perpetuity. 2. You have the choice between two investments that have the same maturity and the same nominal return. Investment A pays simple interest, investment B pays compounded interest. Which one should you pick? B, because it has higher effective annual return. 3. For a positive r, future value will always exceed present value. 4. Which of the following statements is true? In an ordinary annuity payments occur at the end of the period. 5. The Springfield Crusaders just signed their quarterback to a 10 year $50 million contract. Is this contract really worth $50 million? (assume r >0) No, it would only be worth $50 million if it were all paid out today. 6. Last national bank offers a CD paying 7% interest (compounded annually). If you invest $1,000 how much will you have at the end of year 5. $1,402.55 7. You want to buy a house in 4 years and expect to need $25,000 for a down payment. If you have $15,000 to invest, how much interest do you have to earn (compounded annually) to reach your goal? 13.62% 8. You want to buy your dream car, but you are $5,000 short. If you could invest your entire savings of $2,350 at an annual interest of 12%, how long would you have to wait until you have accumulated enough money to buy the car? 6.66 years 9. How much do you have to invest today at an annual rate of 8%, if you need to have $5,000 6 years from today? $3,150.85 10. If you can earn 5% (compounded annually) on an investment, how long does it take for your money to triple? 22.52 years 11. As a result of an injury settlement with your insurance you have the choice between (1) receiving $5,000 today or (2) $6,500 in three years. If you could invest your money at 8% compounded annually, which option should you pick? (2), because it has a higher PV. NARRBEGIN: Multiple Cash Flows End of year Cash flow 1 $2,500 2 3,000 3 1,250 4 3,500 5 1,250 6 4,530 7 2,350 12. What is the future value of cash flows 1-5 at the end of year 5, assuming a 6% interest rate (compounded annually)? $13,093.74 13. What is the present value of these cash flows, if the discount rate is 10% annually? $12,620.90 14. You are planning your retirement and you come to the conclusion that you need to have saved $1,250,000 in 30 years. You can invest into an retirement account that guarantees you a 5% annual return. How much do you have to put into your account at the end of each year to reach your retirement goal? $18,814.30 15. You set up a college fund in which you pay $2,000 each year at the end of the year. How much money will you have accumulated in the fund after 18 years, if your fund earns 7% compounded annually? $67,998.07 16. You set up a college fund in which you pay $2,000 each year at the beginning of the year. How much money will you have accumulated in the fund after 18 years, if your fund earns 7% compounded annually? $72,757.93 17. When you retire you expect to live for another 30 years. During those 30 years you want to be able to withdraw $45,000 at the beginning of each year for living expenses. How much money do you have to have in your retirement account to make this happen. Assume that you can earn 8% on your investments. $547,128.27 18. You are offered a security that will pay you $2,500 at the end of the year forever. If your discount rate is 8%, what is the most you are willing to pay for this security? $31,250 19. What is the effective annual rate of 12% compounded monthly? 12.68% 20. If you invested $2,000 in an account that pays 12% interest, compounded continuously, how much would be in the account in 5 years? $3,644.24 21. You want to buy a new plasma television in 3 years, when you think prices will have gone down to a more reasonable level. You anticipate that the television will cost you $2,500. If you can invest your money at 8% compounded monthly, how much do you need to put aside today? $1,968.14 22. You found your dream house. It will cost you $175,000 and you will put down $35,000 as a down payment. For the rest you get a 30 year 6.25% mortgage. What will be your monthly mortgage payment (assume no early repayment)? $862 23. You want to buy a new car. The car you picked will cost you $32,000 and you decide to go with the dealer’s financing offer of 5.9% compounded monthly for 60 months. Unfortunately, you can only afford monthly loan payments of $300. However, the dealer allows you to pay off the rest of the loan in a one time lump sum payment at the end of the loan. How much do you have to pay to the dealer when the lump sum is due? $22,071.75 24. You are planning your retirement and you come to the conclusion that you need to have saved $1,250,000 in 30 years. You can invest into an retirement account that guarantees you a 5% return. How much do you have to put into your account at the end of every month to reach your retirement goal? $1,501.94 25. When you retire you expect to live for another 30 years. During those 30 years you want to be able to withdraw $4,000 at the beginning of every month for living expenses. How much money do you have to have in your retirement account to make this happen. Assume that you can earn 8% on your investments. $548,768.20 26. If you were to invest $120 for two years, while earning 8% compound interest, what is the total amount of interest that you will earn? $19.97 27. If you were to invest $120 for two years, while earning 8% simple interest, what is the total amount of interest that you will earn? $19.20 28. If the rate of interest that investors can earn on a 2 year investment is zero then all of the above. 29. In the equation below, the exponent “3” represents $133.10 = $100 (1 + .1)3 the number of periods that the present value is left on deposit. 30. You are asked to choose between a 4 year investment that pays 10% compound interest and a similar investment that pays 11.5% simple interest. Which investment will you choose? the 10% compound interest investment 31. The amount that someone is willing to pay today, for a single cash flow in the future is the present value of the cash flow. 32. Pam is in need of cash right now and wants to sell the rights to a $1,000 cash flow that she will receive 5 years from today. If the discount rate for such a cash flow is 9.5%, then what is the fair price that someone should be willing to pay Pam today for rights to that future cash flow? $635.23 33. Your father’s pension recently vested and he is told that if he never works another day in his life, he will recieve a lump sum of $1,500,000 on his 65th birtday (exactly 15 years from today). Assume that your father needs to permanently retire today. What could he sell the rights to his lump sum for, today, if the correct discount rate for such a calcuation is 6%? $625,897.59 34. Your parents set up a trust for you that you will not have access to until your 30th birthday, which is exactly 9 years from today. By prior arrangement, the trust will be worth exactly $200,000 on your 30th birthday. You need cash today and are willing to sell the rights to that trust today for a set amount. If the discount rate for such a cash flow is 12%, what is the maximum amount that someone should be willing to pay you today for the rights to the trust on your 30th birthday? $72,122.01 35. In the equation below, the number “100” represents $75.13 = $100 / (1 + .1)3 the future value a cash flow to be received at a later date. 36. You will recieve a stream of payments beginning at the end of year 1 and the amount will increase by $10 each year until the final payment at the end of year 5. If the first payment is $50, what amount will you have at the end of year 5 if you can invest all amounts at a 7% interest rate ? $394.79 37. You will recieve a stream of $50 payments beginning at the end of year 1 until the final payment at the end of year 5. What amount will you have at the end of year 5 if you can invest all amounts at a 9% interest rate? $299.24 38. You will recieve a stream of annual $70 payments beginning at year 0 until the final payment at the beginning of year 5. What amount will you have at the end of year 5 if you can invest all amounts at a 11% interest rate? $553.90 39. You will recieve a stream of annual $70 payments at the beginning of year 0 until the final payment at the beginning of year 5. What amount will you have at the end of year 5 if you can invest all amounts at an 11% interest rate? $483.90 40. You are trying to prepare a budget based upon the amount of cash flow that you will have available 5 years from now. You are initially promised a regular annuity of $50 with the first payment to be made 1 year from now and the last payment 5 years from now. However, you are actually going to receive an annuity due with the same number of payments but where the first payment is to begin immediately. How much (or less) cash will you have 5 years from now based upon that error if the rate to invest funds is 10%? $30.52 41. An annuity can best be described as an even stream of payments to be recieved at a common interval over the life of the payments. 42. Which of the following should have the greatest value if the discount rate applying to the cash flows is a positive value? the future value of a stream of $5 payments to be received at the end of the next two years. 43. What is the present value of $25 to be received at the end of each year for the next 6 years if the discount rate is 12%? $102.79 44. What is the present value of $25 to be received at the beginning of each year for the next 6 years if the discount rate is 12%? $115.12 45. Forever Insurance Company has offered to pay you or your heirs $100 per year at the end of each year forever. If the correct discount rate for such a cash flow is 13%, what the the amount that you would be willing to pay Forever Insurance for this set of cash flows? $769.23 46. You would like to have $1,000 one year (365 days) from now and you find that the bank is paying 7% compounded daily. How much will you have to deposit with the bank today to be able to have the $1,000? $932.40 47. By increasing the number of compounding periods in a year, while holding the annual percentage rate constant, you will increase the annual percentage yield. 48. The ratio of interest to principal repayment on an amortizing loan decreases as the loan gets older. 49. You are trying to accumulate $2,000 at the end of 5 years by contributing a fixed amount at the end of each year. You initially decide to contribute $300 per year but find that you are coming up short of the $2,000 goal. What could you do to increase the value of the investment at the end of year 5? invest in an investment that has a higher rate of return. 50. If you hold the annual percentage rate constant while increasing the number of compounding periods per year, then the effective interest rate will increase. 51. A young couple buys their dream house. After paying their down payment and closing costs, the couple has borrowed $400,000 from the bank. The terms of the mortgage are 30 years of monthly payments at an APR of 6% with monthly compounding. What is the monthly payment for the couple? $2,398.20 52. A young couple buys their dream house. After paying their down payment and closing costs, the couple has borrowed $400,000 from the bank. The terms of the mortgage are 30 years of monthly payments at an APR of 6% with monthly compounding. Suppose the couple wants to pay off their mortgage early, and will make extra payments to accomplish this goal. Specifically, the couple will pay an EXTRA $2,000 every 12 months (this extra amount is in ADDITION to the regular scheduled mortgage payment). The first extra $2,000 will be paid after month 12. What will be the balance of the loan after the first year of the mortgage? $393,087.95 53. Uncle Fester puts $50,000 into a bank account earning 6%. You can't withdraw the money until the balance has doubled. How long will you have to leave the money in the account? 12 years 54. Which of the following statements are TRUE? Statement I: As you increase the interest rate, the future value of an investment increases. Statement II: As you increase the length of the investment (to receive some lump sum), the present value of the investment increases. Statement III: The present value of an ordinary annuity is larger than the present value of an annuity due. (all else equal) a. Statement I only 55. Consider the following set of cashflows to be received over the next 3 years: Year 1 2 3 Cashflow $100 $225 $300 If the discount rate is 10%, how would we write the formula to find the Future Value of this set of cash flows at year 3? $100 (1.10)2 + $225 (1.10) + $300 56. Which is NOT correct regarding an ordinary annuity and annuity due? As the length of the annuity increases, the future value of the annuity decreases. 57. After graduating from college with a finance degree, you begin an ambitious plan to retire in 25 years. To build up your retirement fund, you will make quarterly payments into a mutual fund that on average will pay 12% APR compounded quarterly. To get you started, a relative gives you a graduation gift of $5,000. Once retired, you plan on moving your investment to a money market fund that will pay 6% APR with monthly compounding. As a young retiree, you believe you will live for 30 more years and will make monthly withdrawals of $10,000. To meet your retirement needs, what quarterly payment should you make? $2,588.27 58. A bank account has a rate of 12% APR with quarterly compounding. What is the EAR for the account? 12.55% 59. An investor puts $200 in a money market account TODAY that returns 3% per year with monthly compounding. The investor plans to keep his money in the account for 2 years. What is the future value of his investment when he closes the account two years from today? $212.35 60. Suppose you take out a loan from the local mob boss for $10,000. Being a generous banker, the mob boss offers you an APR of 60% with monthly compounding. The length of the loan is 3 years with monthly payments. However, you want to get out of this arrangement as quickly as possible. You decide to pay off whatever balance remains after the first year of payments. What is your remaining balance after one year? $8,339.13 61. Suppose you are ready to buy your first house. To buy the house, you will take out a $140,000 mortgage from the bank. The bank offers you the mortgage for 30 years at an APR of 6.0% with interest compounded monthly. For your tenth monthly payment, what is the reduction in principal? a. $145.77 62. What is the future value of a 5-year ordinary annuity with annual payments of $250, evaluated at a 15 percent interest rate? $1,685.60 63. 64. The present value of an ordinary annuity is $2,000. The annuity features monthly payments from an account that pays 12% APR (with monthly compounding). If this was an annuity due, what would be the present value? (assume that same interest rate and same payments)$2,020.00 65. Suppose that Hoosier Farms offers an investment that will pay $10 per year forever. How much is this offer worth if you need a 8% return on your investment? $125 66. Suppose a professional sports team convinces a former player to come out of retirement and play for three seasons. They offer the player $2 million in year 1, $3 million in year 2, and $4 million in year 3. Assuming end of year payments of the salary, how would we find the value of his contract today if the player has a discount rate of 12%? c. PV d. PV 67. Which statement is FALSE concerning the time value of money The EAR is always greater than the APR. 68. Suppose you made a $10,000 investment ten years ago in a speculative stock fund. Your investment today is worth $100,000. What annual compounded return did you earn over the ten year period?25.89% 69. An athlete was offered the following contract for the next three years: Year 1 2 3 Cashflow $5 million $7 million $9 million The athlete would rather have his salary in equal amounts at the END of each of the three years. If the discount rate for the athlete is 10%, what yearly amount would she consider EQUIVALENT to the offered contract? $6.87 million per year 70. Which of the following investment opportunities has the highest present value if the discount rate is 10%? Investment A Investment B Investment C Year 0 $200 $300 $400 Year 1 $300 $350 $350 Year 2 $400 $400 $300 Year 3 $400 $350 $250 Year 4 $400 $300 $200 Investment B 71. A bank is offering a new savings account that pays 8% per year. Which formula below shows the calculation for determining how long it will take a $100 investment to double? a. n b. n = 1.08ln(2) c. n = 2ln(1.08) d. n 72. In five years, you plan on starting graduate school to earn your MBA. You know that graduate school can be expensive and you expect you will need $15,000 per year for tuition and other school expenses. These payments will be made at the BEGINNING of the school year. To have enough money to attend graduate school, you decide to start saving TODAY by investing in a money market fund that pays 4% APR with monthly compounding. You will make monthly deposits into the account starting TODAY for the next five years. How much will you need to deposit each month to have enough savings for graduate school? (Assume that money that is not withdrawn remains in the account during graduate school and the MBA will take two years to complete.) $442.16 73. As a young graduate, you have plans on buying your dream car in three years. You believe the car will cost $50,000. You have two sources of money to reach your goal of $50,000. First, you will save money for the next three years in a money market fund that will return 8% annually. You plan on making $5,000 annual payments to this fund. You will make yearly investments at the BEGINNING of the year. The second source of money will be a car loan that you will take out on the day you buy the car. You anticipate the car dealer to offer you a 6% APR loan with monthly compounding for a term of 60 months. To buy your dream car, what monthly car payment will you anticipate? $627.73 74. Which of the following investments would have the highest future value (in year 5) if the discount rate is 12%? $500 to be received TODAY (year 0) 75. Cozmo Costanza just took out a $24,000 bank loan to help purchase his dream car. The bank offered a 5-year loan at a 6% APR. The loan will feature monthly payments and monthly compounding of interest. What is the monthly payment for this car loan? $463.99 76. A young graduate invests $10,000 in a mutual fund that pays 8% interest per year. What is the future value of this investment in 12 years? $25,182 77. An electric company has offered the following perpetuity to investors to raise capital for the firm. The perpetuity will pay $1 next year, and it is promised to grow at 5% per year thereafter. If you can earn 10% on invested money, how much would you pay today for this perpetuity? $20 78. Cozmo Costanza just took out a $24,000 bank loan to help purchase his dream car. The bank offered a 5-year loan at a 6% APR. The loan will feature monthly payments and monthly compounding of interest. Suppose that Cozmo would like to pay off the remaining balance on his car loan at the end of the second year (24 payments). What is the remaining balance on the car loan after the second year? $15,252 79. A $100 investment yields $112.55 in one year. The interest on the investment was compounded quarterly. From this information, what was the stated rate or APR of the investment? 12.00% 80. What is the future value at year 3 of the following set of cash flows if the discount rate is 11%? Year 0 1 2 3 Cash flow $100 $125 $200 $225 $738 81. A $200 investment in an account that pays 7% continuous interest would be worth how much in twenty years?$811 82. If you invest $5,000 in a mutual fund with a total annual return (interest rate) of 8% and you re-invest the proceeds each year, what will be the value of your investment after five years? $7,346.64 83. You buy a house for $220,000 in a neighborhood where home prices have risen 5% annually on average. You suspect that growth in home prices will slow to an average of 3.5% per year over the next five years. If your growth estimate of 3.5% growth is correct, how much less will your house be worth in five years compared with 5% growth? $19,490.95 84. You inherit $15,000 from your aunt. You decide to invest the money in a three-year CD that pays 4% interest to use as a down payment on a house. How much money will you have when the CD matures? $16,872.96 85. If you need $35,000 for a down payment on a house in six years, how much money must you invest today at 7% interest compounded annually to achieve your goal? $23,321.98 86. Your firm is evaluating a project that should generate revenue of $4,600 in year 1, $5,200 in year two, $5,900 in year three, and $5,700 in year four. The firm receives each cash flow at the end of each year. If your firm's required return is 12%, what is the future value of these cash flows at the end of year four? $25,293.55 87. If you deposit $9,000 at the end of each year in an account earning 8% interest, what will be the value of the account in 25 years? $657,953.46 88. You would like to retire with $1 million on your 60th birthday. If you start saving equal annual amounts on your 26th birthday, make your last deposit on your 60th birthday, and earn 10% interest on your money, how much must you invest each year to achieve your goal? $3,689.71 89. If you deposit $9,000 at the beginning of each year in an account earning 8% interest, what will be the value of the account in 25 years? $710,589.74 90. A report from the marketing department indicates that a new product will generate the following revenue stream: $62,500 in the first year, $89,400 in year two, $136,200 in year three, $128,300 in year four, and $112,000 in year five. If your firm's discount rate is 11% and the cash flows are received at the end of each year, what is the present value of this cash flow stream? a. $379,435.35 91. Your firm rents office space for $250,000 per year, due at the beginning of each year. If your firm's hurdle rate is 10%, what is the present value of five years' worth of rent? $1,042,466.36 92. Great Lakes Christmas Tree Co. expects to pay an annual dividend of $2 per share in perpetuity on its preferred shares starting one year from now. The firm is committed solely to its steady North American Christmas tree business (as opposed to, say, diversifying into landscape shrubbery). This profile warrants a required return of 6%. What is the present value of this dividend stream for investors? $33.33 93. Having acquired great fortune based on your mastery of finance, you decide to set up a charity. You'd like to give the finance department of your alma mater $100,000 next year, and you want to make an annual contribution in perpetuity, with each year’s contribution growing by 4%. The university can generate an 8% return on invested capital. What is the value of a lump-sum donation needed today to accomplish this? $2,500,000 94. If you invest $2,500 in a bank account that pays 6% interest compounded quarterly, how much will you have in five years? $3,367.14 95. Your credit card carries a 9.9% annual percentage rate, compounded daily. What is the effective annual rate, or annual percentage yield? 10.41% 96. Calculate the annual payment for a 20-year mortgage on a $3.5 million building at a 7.5% interest rate. Assume that the entire building is financed and that payments are made at the end of each year, starting at the end of the first year and ending at the end of the 20th year. $343,322.67 97. Calculate the monthly payment for a 20-year mortgage on a $3.5 million building at a 7.5% interest rate. Assume that the entire building is financed and that payments are made at the end of each month, starting at the end of the first month and ending at the end of the last month. $28,195.76 98. You decide that your family would be comfortable living on an annual income of $150,000, growing at 4% per year. You’d also like to continue generating this cash flow for your descendents, forever. With investment returns of 8%, how much wealth would you need today to provide this income starting with $150,000 one year from now? $3,750,000 99. Atlas Map Co. has purchased a new building for $45 million. If the value of the building increases at a rate of 5% per year, how much will the building be worth in 20 years? $119,398,397 100. A stainless steel products manufacturer with an 8.5% cost of capital receives a $3,000,000 order, payable at the end of three years. What is the annual payment amount made at the end of each year with the equivalent present value? $919,618 101. Hamilton Industries needs a bulldozer. The purchasing manager has her eye on a new model that will be available in three years at a price of $75,000. If Hamilton's discount rate is 11%, how much money does she need now to pay for the bulldozer when it’s available? $54,839 102. If you deposit $10,000 today in an account that pays 5% interest compounded annually for five years, how much interest will you earn? $2,762.82 103. Mendelson Implements records the following cash flows at the end of each year for a project. If the firm's discount rate is 11%, what is the value of the project at the end of the last year? Year Cash flow 1 $794,633 2 $542,149 3 $836,200 4 $716,080 5 $520,354 $4,293,253 104. Herbilux Botanicals forecasts the following cash flows at the end of each year for a project. If the firm's discount rate is 9%, what is the present value of the project? Year Cash flow 1 $ 697,000 2 $ 631,000 3 $ 574,000 4 $ 898,000 5 $9,981,000 $8,736,914 105. Mendez Implements records the following cash flows at the beginning of each year for a project. If the firm's discount rate is 11%, what is the value of the project at the end of the last year? Year Cash flow 1 $794,633 2 $542,149 3 $836,200 4 $716,080 5 $520,354 $4,765,511 106. Mayfield Development, LLC forecasts the following cash flows at the beginning of each year for a project. If the firm's discount rate is 9%, what is the present value of the project? Year Cash flow 1 $ 697,000 2 $ 631,000 3 $ 574,000 4 $ 898,000 5 $9,981,000 $9,523,236 107. You’ve just won $1 million dollars in a lottery. For your prize, you may except a $1 million lump sum paid immediately, a constant perpetuity of $80,000 per year (with the first payment arriving in one year), or a stream of cash flows that starts at $45,000 next year and grows at 3.5% per year in perpetuity. If the interest rate is 8%, wish of these choices has a higher present value? all three choices have the same present value 108. A financial advisor recommends saving $1,000,000 for a comfortable retirement. With investment returns of 8%, what is the annual year-end cash flow generated by the $1 million for 25 years, assuming you spend all of the principal and interest? $93,679 109. If you invest $2,500 in a bank account that pays 6% interest compounded monthly, how much will you have in five years? $3,372.13 110. A few years after graduating from college, you decide to get an MBA. This endeavor sets you back $100,000 in loans. Luckily, you have the option to consolidate these loans at 5%. You opt for a 30year payback period with monthly installments due at the end of each month.. What is the monthly payment on the consolidated loan? $536.82 111. Your aunt is evaluating her retirement pension. She can retire at age 65 and collect $1,000 per month for the rest of her life. Assume that payments begin one month after her 65th birthday. If your aunt lives to be exactly 80 years old and can earn 7% interest (compounded monthly), what is the equivalent lump sum she would need at retirement to equal the value of the pension? $111,256 112. Prudent Policy Life Insurance Co. offers a 10-year term life insurance policy with a $250,000 benefit and annual premiums of $200, paid at the beginning of each year. If Prudent can earn 8% on invested capital, what is the present value to the firm of the premiums from one policy, assuming the policy holder outlives the policy term? $1,449 113. Prudent Policy Life Insurance Co. offers a 10-year term life insurance policy with a $250,000 benefit and annual premiums of $200, paid at the beginning of each year. If Prudent can earn 8% on invested capital, what is the future value to the firm of the premiums from one policy, assuming the policy holder outlives the policy term? a. $3,129 114. You are evaluating a perpetuity. The first payment is $100, and it arrives in one year. Each subsequent annual payment will increase by 10%. If the discount rate is 8%, what is the present value of this perpetuity? The present value is infinite 115. You invest $10,000 in August 2004. In August 2009, the investment is worth $12,000. What was your compound annual rate of return over the period? 3.71% 116. If a bank lends you $10,000 and requires that you make payments of $2,500 at the end of each of the next five years, what interest rate is the bank charging? 7.93%