# Chapter 02 How to Calculate Present Values

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1 Chapter 02 How to Calculate Present Values Multiple Choice Questions 1. The present value of \$100 expected in two years from today at a discount rate of 6% is: A. \$ B. \$ C. \$ D. \$ Present Value is defined as: A. Future cash flows discounted to the present at an appropriate discount rate B. Inverse of future cash flows C. Present cash flow compounded into the future 3. If the interest rate is 12%, what is the 2-year discount factor? A B C If the present value of the cash flow X is \$240, and the present value cash flow Y \$160, then the present value of the combined cash flow is: A. \$240 B. \$160 C. \$80 D. \$

2 5. The rate of return is also called: I) discount rate; II) hurdle rate; III) opportunity cost of capital A. I only B. I and II only C. I, II, and III D. None of the given ones 6. Present value of \$121,000 expected to be received one year from today at an interest rate (discount rate) of 10% per year is: A. \$121,000 B. \$100,000 C. \$110, One year discount factor at a discount rate of 25% per year is: A B. 1.0 C The one-year discount factor at an interest rate of 100% per year is: A. 1.5 B. 0.5 C Present Value of \$100,000 that is, expected, to be received at the end of one year at a discount rate of 25% per year is: A. \$80,000 B. \$125,000 C. \$100,

3 10. If the one-year discount factor is , what is the discount rate (interest rate) per year? A. 10% B. 20% C. 30% 11. If the present value of \$480 to be paid at the end of one year is \$400, what is the one-year discount factor? A B C If the present value of \$250 expected to be received one year from today is \$200, what is the discount rate? A. 10% B. 20% C. 25% 13. If the one-year discount factor is 0.90, what is the present value of \$120 to be received one year from today? A. \$100 B. \$96 C. \$ If the present value of \$600 expected to be received one year from today is \$400, what is the one-year discount rate? A. 15% B. 20% C. 25% D. 50% 2-3

4 15. The present value formula for one period cash flow is: A. PV = C 1 (1 + r) B. PV = C 1 /(1 + r) C. PV = C 1 /r 16. The net present value formula for one period is: I) NPV = C 0 + [C 1 /(1 + r)]; II) NPV = PV required investment; and III) NPV = C 0 /C 1 A. I only B. I and II only C. III only 17. An initial investment of \$400,000 will produce an end of year cash flow of \$480,000. What is the NPV of the project at a discount rate of 20%? A. \$176,000 B. \$80,000 C. \$0 (zero) 18. If the present value of a cash flow generated by an initial investment of \$200,000 is \$250,000, what is the NPV of the project? A. \$250,000 B. \$50,000 C. \$200,

5 19. What is the present value of the following cash flow at a discount rate of 9%? A. \$372, B. \$450,000 C. \$405, At an interest rate of 10%, which of the following cash flows should you prefer? A. Option A B. Option B C. Option C D. Option D 21. What is the net present value of the following cash flow at a discount rate of 11%? A. \$69, B. \$231, C. \$80, What is the present value of the following cash flow at a discount rate of 16% APR? A. \$136, B. \$122, C. \$158,

6 23. What is the net present value (NPV) of the following cash flows at a discount rate of 9%? A. \$122, B. \$200,000 C. \$155, The following statements regarding the NPV rule and the rate of return rule are true except: A. Accept a project if its NPV > 0 B. Reject a project if the NPV < 0 C. Accept a project if its rate of return > 0 D. Accept a project if its rate of return > opportunity cost of capital 25. An initial investment of \$500 produces a cash flow \$550 one year from today. Calculate the rate of return on the project A. 10% B. 15% C. 25% D. none of the above 26. According to the net present value rule, an investment in a project should be made if the: A. Net present value is greater than the cost of investment B. Net present value is greater than the present value of cash flows C. Net present value is positive D. Net present value is negative 2-6

7 27. Which of the following statements regarding the net present value rule and the rate of return rule is not true? A. Accept a project if NPV > cost of investment B. Accept a project if NPV is positive C. Accept a project if return on investment exceeds the rate of return on an equivalent investment in the financial market D. Reject a project if NPV is negative 28. The opportunity cost of capital for a risky project is A. The expected rate of return on a government security having the same maturity as the project B. The expected rate of return on a well-diversified portfolio of common stocks C. The expected rate of return on a portfolio of securities of similar risks as the project 29. A perpetuity is defined as: A. Equal cash flows at equal intervals of time for a specific number of periods B. Equal cash flows at equal intervals of time forever C. Unequal cash flows at equal intervals of time forever 30. Which of the following is generally considered an example of a perpetuity: A. Interest payments on a 10-year bond B. Interest payments on a 30-year bond C. Consols 2-7

8 31. You would like to have enough money saved to receive \$100,000 per year perpetuity after retirement so that you and your family can lead a good life. How much would you need to save in your retirement fund to achieve this goal (assume that the perpetuity payments start one year from the date of your retirement. The interest rate is 12.5%)? A. \$1,000,000 B. \$10,000,000 C. \$800, What is the present value of \$10,000 per year perpetuity at an interest rate of 10%? A. \$10,000 B. \$100,000 C. \$200, You would like to have enough money saved to receive \$80,000 per year perpetuity after retirement so that you and your family can lead a good life. How much would you need to save in your retirement fund to achieve this goal (assume that the perpetuity payments start one year from the date of your retirement. The interest rate is 8%)? A. \$7,500,000 B. \$750,000 C. \$1,000, You would like to have enough money saved to receive a \$50,000 per year perpetuity after retirement so that you and your family can lead a good life. How much would you need to save in your retirement fund to achieve this goal (assume that the perpetuity payments starts on the day of retirement. The interest rate is 8%)? A. \$1,000,000 B. \$675,000 C. \$625,

9 35. You would like to have enough money saved to receive an \$80,000 per year perpetuity after retirement so that you and your family can lead a good life. How much would you need to save in your retirement fund to achieve this goal (assume that the perpetuity payments starts on the day of retirement. The interest rate is 10%)? A. \$1,500,000 B. \$880,000 C. \$800, An annuity is defined as A. Equal cash flows at equal intervals of time for a specified period of time B. Equal cash flows at equal intervals of time forever C. Unequal cash flows at equal intervals of time forever 37. If you receive \$1,000 payment at the end each year for the next five years, what type of cash flow do you have? A. Uneven cash flow stream B. An annuity C. An annuity due 38. If the three-year present value annuity factor is and two-year present value annuity factor is 1.833, what is the present value of \$1 received at the end of the 3 years? A. \$ B. \$0.84 C. \$

10 39. If the five-year present value annuity factor is and four-year present value annuity factor is , what is the present value at the \$1 received at the end of five years? A. \$ B. \$ C. \$ What is the present value annuity factor at a discount rate of 11% for 8 years? A B C What is the present value annuity factor at an interest rate of 9% for 6 years? A B C What is the present value of \$1000 per year annuity for five years at an interest rate of 12%? A. \$6, B. \$3, C. \$ What is the present value of \$5000 per year annuity at a discount rate of 10% for 6 years? A. \$21, B. \$3, C. \$16,

11 44. After retirement, you expect to live for 25 years. You would like to have \$75,000 income each year. How much should you have saved in the retirement to receive this income, if the interest is 9% per year (assume that the payments start on the day of retirement)? A. \$736, B. \$802, C. \$2,043, After retirement, you expect to live for 25 years. You would like to have \$75,000 income each year. How much should you have saved in the retirement to receive this income, if the interest is 9% per year (assume that the payments start one year after the retirement)? A. \$736, B. \$6,352, C. \$1,875, For \$10,000 you can purchase a 5-year annuity that will pay \$ per year for five years. The payments are made at the end of each year. Calculate the effective annual interest rate implied by this arrangement: (approximately) A. 8% B. 9% C. 10% 47. If the present value annuity factor for 10 years at 10% interest rate is , what is the present value annuity factor for an equivalent annuity due? A B C

12 48. If the present annuity factor is , what is the present value annuity factor for an equivalent annuity due if the interest rate is 9%? A B C For \$10,000 you can purchase a 5-year annuity that will pay \$ per year for five years. The payments are made at the beginning of each year. Calculate the effective annual interest rate implied by this arrangement: (approximately) A. 8% B. 9% C. 10% D. none of the above 50. John House has taken a \$250,000 mortgage on his house at an interest rate of 6% per year. If the mortgage calls for twenty equal annual payments, what is the amount of each payment? A. \$21, B. \$10, C. \$16, John House has taken a 20-year, \$250,000 mortgage on his house at an interest rate of 6% per year. What is the value of the mortgage after the payment of the fifth annual installment? A. \$128, B. \$211, C. \$141,

13 52. If the present value of \$1.00 received n years from today at an interest rate of r is , then what is the future value of \$1.00 invested today at an interest rate of r% for n years? A. \$ B. \$2.594 C. \$1.70 D. Not enough information to solve the problem 53. If the present value of \$1.00 received n years from today at an interest rate of r is 0.621, then what is the future value of \$1.00 invested today at an interest rate of r% for n years? A. \$1.00 B. \$1.61 C. \$1.621 D. Not enough information to solve the problem 54. If the future value of \$1 invested today at an interest rate of r% for n years is , what is the present value of \$1 to be received in n years at r% interest rate? A. \$ B. \$1.00 C. \$ If the future value annuity factor at 10% and 5 years is , calculate the equivalent present value annuity factor A B C D. None of the given ones 56. If the present value annuity factor at 10% APR for 10 years is , what is the equivalent future value annuity factor? A B C

14 57. If the present value annuity factor at 12% APR for 5 years is , what is the equivalent future value annuity factor? A B C If the present value annuity factor at 8% APR for 10 years is 6.71, what is the equivalent future value annuity factor? A B C You are considering investing in a retirement fund that requires you to deposit \$5,000 per year, and you want to know how much the fund will be worth when you retire. What financial technique should you use to calculate this value? A. Future value of a single payment B. Future value of an annuity C. Present value of an annuity 60. Mr. Hopper is expected to retire in 25 years and he wishes accumulate \$750,000 in his retirement fund by that time. If the interest rate is 10% per year, how much should Mr. Hopper put into the retirement fund each year in order to achieve this goal? [Assume that the payments are made at the end of each year] A. \$4, B. \$2,500 C. \$7,

15 61. Mr. Hopper is expected to retire in 30 years and he wishes accumulate \$1,000,000 in his retirement fund by that time. If the interest rate is 12% per year, how much should Mr. Hopper put into the retirement fund each year in order to achieve this goal? A. \$4, B. \$8, C. \$4, You would like to have enough money saved to receive a growing annuity for 20 years, growing at a rate of 5% per year, the first payment being \$50,000 after retirement. That way, you hope that you and your family can lead a good life after retirement. How much would you need to save in your retirement fund to achieve this goal.(assume that the growing annuity payments start one year from the date of your retirement. The interest rate is 10%)? A. \$1,000,000 B. \$425, C. \$605, You would like to have enough money saved to receive a growing annuity for 25 years, growing at a rate of 4% per year, the first payment being \$60,000 after retirement, so that you and your family can lead a good life. How much would you need to save in your retirement fund to achieve this goal? (assume that the growing perpetuity payments start one year from the date of your retirement. The interest rate is 12%)? A. \$1,500,000 B. \$632,390 C. \$452, The discount rate is used for calculating the NPV is: A. Determined by the financial markets B. Found by the government C. Found by the CEO 2-15

16 65. The managers of a firm can maximize stockholder wealth by: A. Taking all projects with positive NPVs B. Taking all projects with NPVs greater than the cost of investment C. Taking all projects with NPVs greater than present value of cash flow D. All of the above 66. If you invest \$100 at 12% APR for three years, how much would you have at the end of 3 years using simple interest? A. \$136 B. \$ C. \$ If you invest \$100 at 12% APR for three years, how much would you have at the end of 3 years using compound interest? A. \$136 B. \$ C. \$ Which of the following statements is true? A. The process of discounting is the inverse of the process of compounding. B. Ending balances using simple interest is always greater than the ending balance using compound interest at positive interest rates. C. Present value of an annuity due is always less than the present value of an equivalent annuity at positive interest rates. D. All of the above are true. 69. The concept of compound interest is most appropriately described as: A. Interest earned on an investment B. The total amount of interest earned over the life of an investment C. Interest earned on interest 2-16

17 70. Ms. Colonial has just taken out a \$150,000 mortgage at an interest rate of 6% per year. If the mortgage calls for equal monthly payments for twenty years, what is the amount of each payment? (Assume monthly compounding or discounting.) A. \$ B. \$ C. \$ are true 71. An investment at 10.47% effective rate compounded monthly is equal to a nominal (annual) rate of: A % B. 9.57% C. 10% 72. An investment at 12% nominal rate compounded monthly is equal to an annual rate of: A % B % C. 12% 73. Mr. William expects to retire in 30 years and would like to accumulate \$1 million in the pension fund. If the annual interest rate is 12% per year, how much should Mr. Williams put into the pension fund each month in order to achieve his goal? Assume that Mr. Williams will deposit the same amount each month into his pension fund and also use monthly compounding. A. \$ B. \$ C. \$

18 74. An investment at 10% nominal rate compounded continuously is equal to an equivalent annual rate of: A % B % C % 75. The present value of a \$100 per year perpetuity at 10% per year interest rate is \$1000. What would be the present value if the payments were compounded continuously? A. \$ B. \$ C. \$ True / False Questions 76. The rate of return, discount rate, hurdle rate or opportunity cost of capital all means the same. True False 77. A dollar today is worth more than a dollar tomorrow if the interest rate is positive. True False 78. The present value of a future cash flow can be found by dividing it by an appropriate discount factor. True False 79. Net present value is found by subtracting the required investment from the present value of future cash flows. True False 2-18

19 80. The opportunity cost of capital is higher for safe investments than for risky ones. True False 81. A safe dollar is always worth less than a risky dollar because the rate of return on a safe investment is generally low and the rate of return on a risky investment is generally high. True False 82. "Accept investments that have positive net present values" is called the net present value rule. True False 83. "Accept investments that offer rates of return in excess of opportunity cost of capital". True False 84. The rate of return on any perpetuity is equal to the cash flow multiplied by the price. True False 85. An annuity is an asset that pays a fixed sum each year for a specified number of years. True False 86. The value of a five-year annuity is equal to the sum of two perpetuities. One makes its first payment in year 1, and the other makes its first payment in year 6. True False 87. An equal-payment home mortgage is an example of an annuity. True False 2-19

20 88. In the amortization of a mortgage loan with equal payments, the fraction of each payment devoted to interest steadily increases over time and the fraction devoted to reducing the loan decreases steadily. True False 89. In the case of a growing perpetuity, the present value of the cash flow is given by: [C 1 /(r - g)] where r > g. True False 90. Compound interest assumes that you are reinvesting the interest payments at the rate of return. True False Short Answer Questions 91. Briefly explain the term "discount rate." 92. Intuitively explain the concept of the present value. 2-20

21 93. State the "net present value rule." 94. Briefly explain the concept of risk. 95. State the "rate of return rule." 96. Discuss why a dollar tomorrow cannot be worth less than a dollar the day after tomorrow. 2-21

22 97. Define the term "perpetuity." 98. Describe how you would go about finding the present value of any annuity given the formula for the present value of a perpetuity. 99. What is the difference between simple interest and compound interest? 100. Briefly explain, "continuous compounding." 2-22

23 Chapter 02 How to Calculate Present Values Answer Key Multiple Choice Questions 1. The present value of \$100 expected in two years from today at a discount rate of 6% is: A. \$ B. \$ C. \$ D. \$89.00 PV = 100/(1.06^2) = Type: Easy 2. Present Value is defined as: A. Future cash flows discounted to the present at an appropriate discount rate B. Inverse of future cash flows C. Present cash flow compounded into the future Type: Easy 3. If the interest rate is 12%, what is the 2-year discount factor? A B C DF2 = 1/(1.12^2) = Type: Easy 2-23

24 4. If the present value of the cash flow X is \$240, and the present value cash flow Y \$160, then the present value of the combined cash flow is: A. \$240 B. \$160 C. \$80 D. \$400 PV (x + y) = PV (x) + PV (y) = = 400 Type: Easy 5. The rate of return is also called: I) discount rate; II) hurdle rate; III) opportunity cost of capital A. I only B. I and II only C. I, II, and III D. None of the given ones Type: Easy 6. Present value of \$121,000 expected to be received one year from today at an interest rate (discount rate) of 10% per year is: A. \$121,000 B. \$100,000 C. \$110,000 PV = (121,000)/(1.1) = 110,000 Type: Easy 2-24

25 7. One year discount factor at a discount rate of 25% per year is: A B. 1.0 C. 0.8 Discount Factor = 1/1.25 = 0.8 Type: Easy 8. The one-year discount factor at an interest rate of 100% per year is: A. 1.5 B. 0.5 C Discount factor = 1/(1 + 1) = 0.5 Type: Easy 9. Present Value of \$100,000 that is, expected, to be received at the end of one year at a discount rate of 25% per year is: A. \$80,000 B. \$125,000 C. \$100,000 PV = (100,000)/( ) = 80,000 Type: Easy 2-25

26 10. If the one-year discount factor is , what is the discount rate (interest rate) per year? A. 10% B. 20% C. 30% 1 + r = 1/(0.8333) = 1.20; r = 20% 11. If the present value of \$480 to be paid at the end of one year is \$400, what is the one-year discount factor? A B C Discount factor is = 400/480 = If the present value of \$250 expected to be received one year from today is \$200, what is the discount rate? A. 10% B. 20% C. 25% 1 + r = 250 /200 = 1.25; r = 25% 2-26

27 13. If the one-year discount factor is 0.90, what is the present value of \$120 to be received one year from today? A. \$100 B. \$96 C. \$108 PV = (120)(0.90) = If the present value of \$600 expected to be received one year from today is \$400, what is the one-year discount rate? A. 15% B. 20% C. 25% D. 50% 1 + r = (600)/(400) = 1.5; r = 50% 15. The present value formula for one period cash flow is: A. PV = C 1 (1 + r) B. PV = C 1 /(1 + r) C. PV = C 1 /r 2-27

28 16. The net present value formula for one period is: I) NPV = C 0 + [C 1 /(1 + r)]; II) NPV = PV required investment; and III) NPV = C 0 /C 1 A. I only B. I and II only C. III only 17. An initial investment of \$400,000 will produce an end of year cash flow of \$480,000. What is the NPV of the project at a discount rate of 20%? A. \$176,000 B. \$80,000 C. \$0 (zero) NPV = -400,000 + (480,000/1.2) = If the present value of a cash flow generated by an initial investment of \$200,000 is \$250,000, what is the NPV of the project? A. \$250,000 B. \$50,000 C. \$200,000 NPV = -200, ,000 = 50,000 Type: Easy 2-28

29 19. What is the present value of the following cash flow at a discount rate of 9%? A. \$372, B. \$450,000 C. \$405, PV = (100,000/1.09) + (150,000/(1.09^2)) + 200,000/(1.09^3) = 372, At an interest rate of 10%, which of the following cash flows should you prefer? A. Option A B. Option B C. Option C D. Option D PV(A) = ; PV(B) = ; PV(C) = ; A is preferred 2-29

30 21. What is the net present value of the following cash flow at a discount rate of 11%? A. \$69, B. \$231, C. \$80,000 NPV = -120,000 + (300,000/1.11) -(100,000/(1.11^2)) = 69, What is the present value of the following cash flow at a discount rate of 16% APR? A. \$136, B. \$122, C. \$158, PV = (-100,000/1.16) + (300,000/(1.16^2)) = 136, What is the net present value (NPV) of the following cash flows at a discount rate of 9%? A. \$122, B. \$200,000 C. \$155, NPV = -250,000 + (100,000/1.09) + (150,000/(1.09^2)) + (200,000/(1.09^3)) NPV = 122,

31 24. The following statements regarding the NPV rule and the rate of return rule are true except: A. Accept a project if its NPV > 0 B. Reject a project if the NPV < 0 C. Accept a project if its rate of return > 0 D. Accept a project if its rate of return > opportunity cost of capital 25. An initial investment of \$500 produces a cash flow \$550 one year from today. Calculate the rate of return on the project A. 10% B. 15% C. 25% D. none of the above Rate of return = ( )/500 = 10% Type: Easy 26. According to the net present value rule, an investment in a project should be made if the: A. Net present value is greater than the cost of investment B. Net present value is greater than the present value of cash flows C. Net present value is positive D. Net present value is negative 2-31

32 27. Which of the following statements regarding the net present value rule and the rate of return rule is not true? A. Accept a project if NPV > cost of investment B. Accept a project if NPV is positive C. Accept a project if return on investment exceeds the rate of return on an equivalent investment in the financial market D. Reject a project if NPV is negative 28. The opportunity cost of capital for a risky project is A. The expected rate of return on a government security having the same maturity as the project B. The expected rate of return on a well-diversified portfolio of common stocks C. The expected rate of return on a portfolio of securities of similar risks as the project 29. A perpetuity is defined as: A. Equal cash flows at equal intervals of time for a specific number of periods B. Equal cash flows at equal intervals of time forever C. Unequal cash flows at equal intervals of time forever Type: Easy 30. Which of the following is generally considered an example of a perpetuity: A. Interest payments on a 10-year bond B. Interest payments on a 30-year bond C. Consols Type: Easy 2-32

33 31. You would like to have enough money saved to receive \$100,000 per year perpetuity after retirement so that you and your family can lead a good life. How much would you need to save in your retirement fund to achieve this goal (assume that the perpetuity payments start one year from the date of your retirement. The interest rate is 12.5%)? A. \$1,000,000 B. \$10,000,000 C. \$800,000 PV = (100,000/0.125) = 800, What is the present value of \$10,000 per year perpetuity at an interest rate of 10%? A. \$10,000 B. \$100,000 C. \$200,000 PV = (10,000/0.1) = 100,000 Type: Easy 33. You would like to have enough money saved to receive \$80,000 per year perpetuity after retirement so that you and your family can lead a good life. How much would you need to save in your retirement fund to achieve this goal (assume that the perpetuity payments start one year from the date of your retirement. The interest rate is 8%)? A. \$7,500,000 B. \$750,000 C. \$1,000,000 PV = (80,000/0.08) = 1,000,

34 34. You would like to have enough money saved to receive a \$50,000 per year perpetuity after retirement so that you and your family can lead a good life. How much would you need to save in your retirement fund to achieve this goal (assume that the perpetuity payments starts on the day of retirement. The interest rate is 8%)? A. \$1,000,000 B. \$675,000 C. \$625,000 PV = [(50,000/0.08)](1.08) = 675, You would like to have enough money saved to receive an \$80,000 per year perpetuity after retirement so that you and your family can lead a good life. How much would you need to save in your retirement fund to achieve this goal (assume that the perpetuity payments starts on the day of retirement. The interest rate is 10%)? A. \$1,500,000 B. \$880,000 C. \$800,000 PV = [(80,000/0.1)] * (1.1) = 880, An annuity is defined as A. Equal cash flows at equal intervals of time for a specified period of time B. Equal cash flows at equal intervals of time forever C. Unequal cash flows at equal intervals of time forever Type: Easy 2-34

35 37. If you receive \$1,000 payment at the end each year for the next five years, what type of cash flow do you have? A. Uneven cash flow stream B. An annuity C. An annuity due Type: Easy 38. If the three-year present value annuity factor is and two-year present value annuity factor is 1.833, what is the present value of \$1 received at the end of the 3 years? A. \$ B. \$0.84 C. \$0.89 PV = ( ) * (1) = If the five-year present value annuity factor is and four-year present value annuity factor is , what is the present value at the \$1 received at the end of five years? A. \$ B. \$ C. \$ PV = ( ) * (1) =

36 40. What is the present value annuity factor at a discount rate of 11% for 8 years? A B C PV annuity factor = (1/0.11) - (1/((0.11)(1.11^8))) = What is the present value annuity factor at an interest rate of 9% for 6 years? A B C PV annuity factor = (1/0.09) - (1/((0.09)(1.09^6))) = What is the present value of \$1000 per year annuity for five years at an interest rate of 12%? A. \$6, B. \$3, C. \$ PV annuity factor = [(1/0.12) - (1/((0.12)(1.12^ 5)))] * 1000 = 3,

37 43. What is the present value of \$5000 per year annuity at a discount rate of 10% for 6 years? A. \$21, B. \$3, C. \$16, PV = [(1/0.10) - (1/((0.10)(1.10^6)))] * 5000 = 16, After retirement, you expect to live for 25 years. You would like to have \$75,000 income each year. How much should you have saved in the retirement to receive this income, if the interest is 9% per year (assume that the payments start on the day of retirement)? A. \$736, B. \$802, C. \$2,043,750 PV = [[(1/0.09) - (1/((0.09)(1.09^25)))] * 75,000] * (1.09) = 802, After retirement, you expect to live for 25 years. You would like to have \$75,000 income each year. How much should you have saved in the retirement to receive this income, if the interest is 9% per year (assume that the payments start one year after the retirement)? A. \$736, B. \$6,352, C. \$1,875,000 PV = [(1/0.09) - (1/((0.09)(1.09^25)))] * 75,000 = 736,

38 46. For \$10,000 you can purchase a 5-year annuity that will pay \$ per year for five years. The payments are made at the end of each year. Calculate the effective annual interest rate implied by this arrangement: (approximately) A. 8% B. 9% C. 10% Using a financial calculator: N = 5; PV = -10,000; PMT = ; FV = 0 Compute: I = 8.0% [calculator setting: END] 47. If the present value annuity factor for 10 years at 10% interest rate is , what is the present value annuity factor for an equivalent annuity due? A B C Annuity due: * 1.1 = If the present annuity factor is , what is the present value annuity factor for an equivalent annuity due if the interest rate is 9%? A B C annuity due factor = * 1.09 =

39 49. For \$10,000 you can purchase a 5-year annuity that will pay \$ per year for five years. The payments are made at the beginning of each year. Calculate the effective annual interest rate implied by this arrangement: (approximately) A. 8% B. 9% C. 10% D. none of the above Using a financial calculator: N = 5; PV = -10,000; PMT = ; FV = 0 Compute: I = 9.0% [Calculator setting: BEGIN (BGN)] 50. John House has taken a \$250,000 mortgage on his house at an interest rate of 6% per year. If the mortgage calls for twenty equal annual payments, what is the amount of each payment? A. \$21, B. \$10, C. \$16, (Use a financial calculator) PV = 250,000; I - = 6%; N = 20; FV = 0; Compute PMT = \$21, John House has taken a 20-year, \$250,000 mortgage on his house at an interest rate of 6% per year. What is the value of the mortgage after the payment of the fifth annual installment? A. \$128, B. \$211, C. \$141, Step 1: I = 6%; N = 20; PV = 250,000; FV = 0; Compute PMT = 21, Step 2: I = 6%; N = 15; PMT = 21,796.14; Compute PV = 211,

40 52. If the present value of \$1.00 received n years from today at an interest rate of r is , then what is the future value of \$1.00 invested today at an interest rate of r% for n years? A. \$ B. \$2.594 C. \$1.70 D. Not enough information to solve the problem FV = 1/(0.3855) = If the present value of \$1.00 received n years from today at an interest rate of r is 0.621, then what is the future value of \$1.00 invested today at an interest rate of r% for n years? A. \$1.00 B. \$1.61 C. \$1.621 D. Not enough information to solve the problem FV = 1/(0.621) = If the future value of \$1 invested today at an interest rate of r% for n years is , what is the present value of \$1 to be received in n years at r% interest rate? A. \$ B. \$1.00 C. \$ PV = 1/ =

41 55. If the future value annuity factor at 10% and 5 years is , calculate the equivalent present value annuity factor A B C D. None of the given ones PV = /(1.1)^5 = If the present value annuity factor at 10% APR for 10 years is , what is the equivalent future value annuity factor? A B C FV annuity factor = * (1.1^10) = If the present value annuity factor at 12% APR for 5 years is , what is the equivalent future value annuity factor? A B C FV annuity factor = * (1.12^5) =

42 58. If the present value annuity factor at 8% APR for 10 years is 6.71, what is the equivalent future value annuity factor? A B C FV annuity factor = 6.71 * (1.08^10) = You are considering investing in a retirement fund that requires you to deposit \$5,000 per year, and you want to know how much the fund will be worth when you retire. What financial technique should you use to calculate this value? A. Future value of a single payment B. Future value of an annuity C. Present value of an annuity Type: Easy 60. Mr. Hopper is expected to retire in 25 years and he wishes accumulate \$750,000 in his retirement fund by that time. If the interest rate is 10% per year, how much should Mr. Hopper put into the retirement fund each year in order to achieve this goal? [Assume that the payments are made at the end of each year] A. \$4, B. \$2,500 C. \$7, Future value annuity factor = [(1.1^25) - 1]/(0.1) = ; payment = 750,000/ =

43 61. Mr. Hopper is expected to retire in 30 years and he wishes accumulate \$1,000,000 in his retirement fund by that time. If the interest rate is 12% per year, how much should Mr. Hopper put into the retirement fund each year in order to achieve this goal? A. \$4, B. \$8, C. \$4,000 Future value annuity factor = [(1.12^30-1]/(0.12) = ; payment = 1,000,000/ = You would like to have enough money saved to receive a growing annuity for 20 years, growing at a rate of 5% per year, the first payment being \$50,000 after retirement. That way, you hope that you and your family can lead a good life after retirement. How much would you need to save in your retirement fund to achieve this goal.(assume that the growing annuity payments start one year from the date of your retirement. The interest rate is 10%)? A. \$1,000,000 B. \$425, C. \$605, PV = (50,000)[(1/( )) - {(1/( )}{(1.05^20)/(1.10^20)}] = 605,

44 63. You would like to have enough money saved to receive a growing annuity for 25 years, growing at a rate of 4% per year, the first payment being \$60,000 after retirement, so that you and your family can lead a good life. How much would you need to save in your retirement fund to achieve this goal? (assume that the growing perpetuity payments start one year from the date of your retirement. The interest rate is 12%)? A. \$1,500,000 B. \$632,390 C. \$452,165 PV = (60,000) [(1/( )) - {(1/( )}{(1.04^25)/(1.12^25)}] = 632, The discount rate is used for calculating the NPV is: A. Determined by the financial markets B. Found by the government C. Found by the CEO Type: Easy 65. The managers of a firm can maximize stockholder wealth by: A. Taking all projects with positive NPVs B. Taking all projects with NPVs greater than the cost of investment C. Taking all projects with NPVs greater than present value of cash flow D. All of the above 2-44

45 66. If you invest \$100 at 12% APR for three years, how much would you have at the end of 3 years using simple interest? A. \$136 B. \$ C. \$ FV = (100 * 0.12 * 3) = \$ If you invest \$100 at 12% APR for three years, how much would you have at the end of 3 years using compound interest? A. \$136 B. \$ C. \$ FV = 100 * (1.12^3) = \$ Which of the following statements is true? A. The process of discounting is the inverse of the process of compounding. B. Ending balances using simple interest is always greater than the ending balance using compound interest at positive interest rates. C. Present value of an annuity due is always less than the present value of an equivalent annuity at positive interest rates. D. All of the above are true. 2-45

46 69. The concept of compound interest is most appropriately described as: A. Interest earned on an investment B. The total amount of interest earned over the life of an investment C. Interest earned on interest 70. Ms. Colonial has just taken out a \$150,000 mortgage at an interest rate of 6% per year. If the mortgage calls for equal monthly payments for twenty years, what is the amount of each payment? (Assume monthly compounding or discounting.) A. \$ B. \$ C. \$ are true PMT = 150,000/[(1/0.005) - 1/((0.005 * (( )^240)))] = \$ An investment at 10.47% effective rate compounded monthly is equal to a nominal (annual) rate of: A % B. 9.57% C. 10% NOM = [(1.1047)^(1/12) - 1] * 12 = 0.1 = 10.00% 2-46

47 72. An investment at 12% nominal rate compounded monthly is equal to an annual rate of: A % B % C. 12% EAR = ((1.01)^12) - 1 = = 12.68% 73. Mr. William expects to retire in 30 years and would like to accumulate \$1 million in the pension fund. If the annual interest rate is 12% per year, how much should Mr. Williams put into the pension fund each month in order to achieve his goal? Assume that Mr. Williams will deposit the same amount each month into his pension fund and also use monthly compounding. A. \$ B. \$ C. \$ PMT = 1,000,000/{[(1/0.01) - (1/(0.01 * (1.01^360)))] * (1.01^360)} = \$ An investment at 10% nominal rate compounded continuously is equal to an equivalent annual rate of: A % B % C % (e^(0.1)) - 1 = = % 2-47

48 75. The present value of a \$100 per year perpetuity at 10% per year interest rate is \$1000. What would be the present value if the payments were compounded continuously? A. \$ B. \$ C. \$ (e^r) = 1.1 r = ln(1.1) = ; PV = 100/ = \$ True / False Questions 76. The rate of return, discount rate, hurdle rate or opportunity cost of capital all means the same. TRUE 77. A dollar today is worth more than a dollar tomorrow if the interest rate is positive. TRUE Type: Easy 78. The present value of a future cash flow can be found by dividing it by an appropriate discount factor. FALSE 2-48

49 79. Net present value is found by subtracting the required investment from the present value of future cash flows. TRUE 80. The opportunity cost of capital is higher for safe investments than for risky ones. FALSE 81. A safe dollar is always worth less than a risky dollar because the rate of return on a safe investment is generally low and the rate of return on a risky investment is generally high. FALSE 82. "Accept investments that have positive net present values" is called the net present value rule. TRUE 83. "Accept investments that offer rates of return in excess of opportunity cost of capital". TRUE 84. The rate of return on any perpetuity is equal to the cash flow multiplied by the price. FALSE 2-49

50 85. An annuity is an asset that pays a fixed sum each year for a specified number of years. TRUE Type: Easy 86. The value of a five-year annuity is equal to the sum of two perpetuities. One makes its first payment in year 1, and the other makes its first payment in year 6. FALSE 87. An equal-payment home mortgage is an example of an annuity. TRUE 88. In the amortization of a mortgage loan with equal payments, the fraction of each payment devoted to interest steadily increases over time and the fraction devoted to reducing the loan decreases steadily. TRUE 89. In the case of a growing perpetuity, the present value of the cash flow is given by: [C 1 /(r - g)] where r > g. TRUE 2-50

51 90. Compound interest assumes that you are reinvesting the interest payments at the rate of return. TRUE Short Answer Questions 91. Briefly explain the term "discount rate." Discount rate is the rate of return used for discounting future cash flows to obtain the present value. The discount rate can be obtained by looking at the rate of return, an equivalent investment opportunity in the capital market. 92. Intuitively explain the concept of the present value. If you have \$100 today, you can invest it and start earning interest on it. On the other hand, if you have to make a payment of \$100 one year from today, you do not need to invest \$100 today but a lesser amount. The lesser amount invested today plus the interest earned on it should add up to \$100. The present value of \$100 one year from today at an interest rate of 10% is \$ [PV = 100/1.1 = 90.91] 93. State the "net present value rule." Invest in projects with positive net present values. Net present value is the difference between the present value of future cash flows from the project and the initial investment. 2-51

52 94. Briefly explain the concept of risk. If the future cash flows from an investment are not certain then we call it a risky cash flow. That means there is an uncertainty about the future cash flows or future cash flows could be different from expected cash flows. The degree of uncertainty varies from investment to investment. Generally, uncertain cash flows are discounted using a higher discount rate than certain cash flows. This is only one method of dealing with risk. There are many ways to take risk into consideration while making financial decisions. 95. State the "rate of return rule." Invest as long as the rate of return on the investment exceeds the rate of return on equivalent investments in the capital market. 96. Discuss why a dollar tomorrow cannot be worth less than a dollar the day after tomorrow. If a dollar tomorrow is worth less than a dollar a day after tomorrow, it would be possible to earn a very large amount of money through "money machine" effect. This is only possible, if someone else is losing a very large amount of money. These conditions can only exist for a short period of time, and cannot exist in equilibrium as the source of money is quickly exhausted. Thus a dollar tomorrow cannot be worth less than a dollar the day after tomorrow. 97. Define the term "perpetuity." A perpetuity is defined as the same cash flow occurring each year forever. 2-52

53 98. Describe how you would go about finding the present value of any annuity given the formula for the present value of a perpetuity. The present value of any annuity can be thought of as the difference between two perpetuities one payment stating in year-1 (immediate) and one starting in year (n + 1) (delayed). By calculating difference between the present values of these two perpetuities today we can find the present value of an annuity. 99. What is the difference between simple interest and compound interest? When money is invested at compound interest, each interest payment is reinvested to earn more interest in subsequent periods. In the simple interest case, the interest is paid only on the initial investment Briefly explain, "continuous compounding." As frequency of compounding increases, the effective rate on an investment also increases. In case of continuous compounding the frequency of compounding is infinity. In this case, the nature of the function also changes. The effective interest rate is given by (e r - 1), where the value of e = e is the base for natural logarithms. 2-53

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