Fm10e ch05

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Fm10e ch05

  1. 1. Chapter 5 - The Time Value of Money  2005, Pearson Prentice Hall
  2. 2. The Time Value of Money <ul><li>Compounding and Discounting Single Sums </li></ul>
  3. 3. <ul><li>We know that receiving $1 today is worth more than $1 in the future. This is due to opportunity costs . </li></ul><ul><li>The opportunity cost of receiving $1 in the future is the interest we could have earned if we had received the $1 sooner. </li></ul>Today Future
  4. 4. If we can measure this opportunity cost, we can:
  5. 5. If we can measure this opportunity cost, we can: <ul><li>Translate $1 today into its equivalent in the future (compounding) . </li></ul>
  6. 6. If we can measure this opportunity cost, we can: <ul><li>Translate $1 today into its equivalent in the future (compounding) . </li></ul>Today ? Future
  7. 7. If we can measure this opportunity cost, we can: <ul><li>Translate $1 today into its equivalent in the future (compounding) . </li></ul><ul><li>Translate $1 in the future into its equivalent today (discounting) . </li></ul>Today ? Future
  8. 8. If we can measure this opportunity cost, we can: <ul><li>Translate $1 today into its equivalent in the future (compounding) . </li></ul><ul><li>Translate $1 in the future into its equivalent today (discounting) . </li></ul>? Today Future Today ? Future
  9. 9. <ul><li>Compound Interest and Future Value </li></ul>
  10. 10. Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year?
  11. 11. Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year? 0 1 PV = FV =
  12. 12. Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year? <ul><li>Calculator Solution: </li></ul><ul><li>P/Y = 1 I = 6 </li></ul><ul><li>N = 1 PV = -100 </li></ul><ul><li>FV = $106 </li></ul>0 1 PV = -100 FV =
  13. 13. Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year? <ul><li>Calculator Solution: </li></ul><ul><li>P/Y = 1 I = 6 </li></ul><ul><li>N = 1 PV = -100 </li></ul><ul><li>FV = $106 </li></ul>0 1 PV = -100 FV = 106
  14. 14. Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year? <ul><li>Mathematical Solution: </li></ul><ul><li>FV = PV (FVIF i, n ) </li></ul><ul><li>FV = 100 (FVIF .06, 1 ) (use FVIF table, or) </li></ul><ul><li>FV = PV (1 + i) n </li></ul><ul><li>FV = 100 (1.06) 1 = $106 </li></ul>0 1 PV = -100 FV = 106
  15. 15. Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years?
  16. 16. Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years? 0 5 PV = FV =
  17. 17. Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years? <ul><li>Calculator Solution: </li></ul><ul><li>P/Y = 1 I = 6 </li></ul><ul><li>N = 5 PV = -100 </li></ul><ul><li>FV = $133.82 </li></ul>0 5 PV = -100 FV =
  18. 18. Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years? <ul><li>Calculator Solution: </li></ul><ul><li>P/Y = 1 I = 6 </li></ul><ul><li>N = 5 PV = -100 </li></ul><ul><li>FV = $133.82 </li></ul>0 5 PV = -100 FV = 133. 82
  19. 19. Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years? <ul><li>Mathematical Solution: </li></ul><ul><li>FV = PV (FVIF i, n ) </li></ul><ul><li>FV = 100 (FVIF .06, 5 ) (use FVIF table, or) </li></ul><ul><li>FV = PV (1 + i) n </li></ul><ul><li>FV = 100 (1.06) 5 = $ 133.82 </li></ul>0 5 PV = -100 FV = 133. 82
  20. 20. Future Value - single sums If you deposit $100 in an account earning 6% with quarterly compounding , how much would you have in the account after 5 years?
  21. 21. Future Value - single sums If you deposit $100 in an account earning 6% with quarterly compounding , how much would you have in the account after 5 years? 0 ? PV = FV =
  22. 22. <ul><li>Calculator Solution: </li></ul><ul><li>P/Y = 4 I = 6 </li></ul><ul><li>N = 20 PV = -100 </li></ul><ul><li>FV = $134.68 </li></ul>Future Value - single sums If you deposit $100 in an account earning 6% with quarterly compounding , how much would you have in the account after 5 years? 0 20 PV = -100 FV =
  23. 23. <ul><li>Calculator Solution: </li></ul><ul><li>P/Y = 4 I = 6 </li></ul><ul><li>N = 20 PV = -100 </li></ul><ul><li>FV = $134.68 </li></ul>Future Value - single sums If you deposit $100 in an account earning 6% with quarterly compounding , how much would you have in the account after 5 years? 0 20 PV = -100 FV = 134. 68
  24. 24. <ul><li>Mathematical Solution: </li></ul><ul><li>FV = PV (FVIF i, n ) </li></ul><ul><li>FV = 100 (FVIF .015, 20 ) (can’t use FVIF table) </li></ul><ul><li>FV = PV (1 + i/m) m x n </li></ul><ul><li>FV = 100 (1.015) 20 = $134.68 </li></ul>Future Value - single sums If you deposit $100 in an account earning 6% with quarterly compounding , how much would you have in the account after 5 years? 0 20 PV = -100 FV = 134. 68
  25. 25. Future Value - single sums If you deposit $100 in an account earning 6% with monthly compounding , how much would you have in the account after 5 years?
  26. 26. Future Value - single sums If you deposit $100 in an account earning 6% with monthly compounding , how much would you have in the account after 5 years? 0 ? PV = FV =
  27. 27. <ul><li>Calculator Solution: </li></ul><ul><li>P/Y = 12 I = 6 </li></ul><ul><li>N = 60 PV = -100 </li></ul><ul><li>FV = $134.89 </li></ul>Future Value - single sums If you deposit $100 in an account earning 6% with monthly compounding , how much would you have in the account after 5 years? 0 60 PV = -100 FV =
  28. 28. <ul><li>Calculator Solution: </li></ul><ul><li>P/Y = 12 I = 6 </li></ul><ul><li>N = 60 PV = -100 </li></ul><ul><li>FV = $134.89 </li></ul>Future Value - single sums If you deposit $100 in an account earning 6% with monthly compounding , how much would you have in the account after 5 years? 0 60 PV = -100 FV = 134. 89
  29. 29. <ul><li>Mathematical Solution: </li></ul><ul><li>FV = PV (FVIF i, n ) </li></ul><ul><li>FV = 100 (FVIF .005, 60 ) (can’t use FVIF table) </li></ul><ul><li>FV = PV (1 + i/m) m x n </li></ul><ul><li>FV = 100 (1.005) 60 = $134.89 </li></ul>Future Value - single sums If you deposit $100 in an account earning 6% with monthly compounding , how much would you have in the account after 5 years? 0 60 PV = -100 FV = 134. 89
  30. 30. Future Value - continuous compounding What is the FV of $1,000 earning 8% with continuous compounding , after 100 years?
  31. 31. Future Value - continuous compounding What is the FV of $1,000 earning 8% with continuous compounding , after 100 years? 0 ? PV = FV =
  32. 32. <ul><li>Mathematical Solution: </li></ul><ul><li>FV = PV (e in ) </li></ul><ul><li>FV = 1000 (e .08x100 ) = 1000 (e 8 ) </li></ul><ul><li>FV = $2,980,957. 99 </li></ul>Future Value - continuous compounding What is the FV of $1,000 earning 8% with continuous compounding , after 100 years? 0 100 PV = -1000 FV =
  33. 33. Future Value - continuous compounding What is the FV of $1,000 earning 8% with continuous compounding , after 100 years? <ul><li>Mathematical Solution: </li></ul><ul><li>FV = PV (e in ) </li></ul><ul><li>FV = 1000 (e .08x100 ) = 1000 (e 8 ) </li></ul><ul><li>FV = $2,980,957. 99 </li></ul>0 100 PV = -1000 FV = $2.98m
  34. 34. <ul><li>Present Value </li></ul>
  35. 35. Present Value - single sums If you receive $100 one year from now, what is the PV of that $100 if your opportunity cost is 6%?
  36. 36. Present Value - single sums If you receive $100 one year from now, what is the PV of that $100 if your opportunity cost is 6%? 0 ? PV = FV =
  37. 37. <ul><li>Calculator Solution: </li></ul><ul><li>P/Y = 1 I = 6 </li></ul><ul><li>N = 1 FV = 100 </li></ul><ul><li>PV = -94.34 </li></ul>Present Value - single sums If you receive $100 one year from now, what is the PV of that $100 if your opportunity cost is 6%? 0 1 PV = FV = 100
  38. 38. <ul><li>Calculator Solution: </li></ul><ul><li>P/Y = 1 I = 6 </li></ul><ul><li>N = 1 FV = 100 </li></ul><ul><li>PV = -94.34 </li></ul>Present Value - single sums If you receive $100 one year from now, what is the PV of that $100 if your opportunity cost is 6%? PV = -94. 34 FV = 100 0 1
  39. 39. <ul><li>Mathematical Solution: </li></ul><ul><li>PV = FV (PVIF i, n ) </li></ul><ul><li>PV = 100 (PVIF .06, 1 ) (use PVIF table, or) </li></ul><ul><li>PV = FV / (1 + i) n </li></ul><ul><li>PV = 100 / (1.06) 1 = $94.34 </li></ul>Present Value - single sums If you receive $100 one year from now, what is the PV of that $100 if your opportunity cost is 6%? PV = -94. 34 FV = 100 0 1
  40. 40. Present Value - single sums If you receive $100 five years from now, what is the PV of that $100 if your opportunity cost is 6%?
  41. 41. Present Value - single sums If you receive $100 five years from now, what is the PV of that $100 if your opportunity cost is 6%? 0 ? PV = FV =
  42. 42. <ul><li>Calculator Solution: </li></ul><ul><li>P/Y = 1 I = 6 </li></ul><ul><li>N = 5 FV = 100 </li></ul><ul><li>PV = -74.73 </li></ul>Present Value - single sums If you receive $100 five years from now, what is the PV of that $100 if your opportunity cost is 6%? 0 5 PV = FV = 100
  43. 43. <ul><li>Calculator Solution: </li></ul><ul><li>P/Y = 1 I = 6 </li></ul><ul><li>N = 5 FV = 100 </li></ul><ul><li>PV = -74.73 </li></ul>Present Value - single sums If you receive $100 five years from now, what is the PV of that $100 if your opportunity cost is 6%? 0 5 PV = -74. 73 FV = 100
  44. 44. <ul><li>Mathematical Solution: </li></ul><ul><li>PV = FV (PVIF i, n ) </li></ul><ul><li>PV = 100 (PVIF .06, 5 ) (use PVIF table, or) </li></ul><ul><li>PV = FV / (1 + i) n </li></ul><ul><li>PV = 100 / (1.06) 5 = $74.73 </li></ul>Present Value - single sums If you receive $100 five years from now, what is the PV of that $100 if your opportunity cost is 6%? 0 5 PV = -74. 73 FV = 100
  45. 45. Present Value - single sums What is the PV of $1,000 to be received 15 years from now if your opportunity cost is 7%?
  46. 46. Present Value - single sums What is the PV of $1,000 to be received 15 years from now if your opportunity cost is 7%? 0 15 PV = FV =
  47. 47. <ul><li>Calculator Solution: </li></ul><ul><li>P/Y = 1 I = 7 </li></ul><ul><li>N = 15 FV = 1,000 </li></ul><ul><li>PV = -362.45 </li></ul>Present Value - single sums What is the PV of $1,000 to be received 15 years from now if your opportunity cost is 7%? 0 15 PV = FV = 1000
  48. 48. <ul><li>Calculator Solution: </li></ul><ul><li>P/Y = 1 I = 7 </li></ul><ul><li>N = 15 FV = 1,000 </li></ul><ul><li>PV = -362.45 </li></ul>Present Value - single sums What is the PV of $1,000 to be received 15 years from now if your opportunity cost is 7%? 0 15 PV = -362. 45 FV = 1000
  49. 49. <ul><li>Mathematical Solution: </li></ul><ul><li>PV = FV (PVIF i, n ) </li></ul><ul><li>PV = 100 (PVIF .07, 15 ) (use PVIF table, or) </li></ul><ul><li>PV = FV / (1 + i) n </li></ul><ul><li>PV = 100 / (1.07) 15 = $362.45 </li></ul>Present Value - single sums What is the PV of $1,000 to be received 15 years from now if your opportunity cost is 7%? 0 15 PV = -362. 45 FV = 1000
  50. 50. Present Value - single sums If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?
  51. 51. Present Value - single sums If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return? 0 5 PV = FV =
  52. 52. <ul><li>Calculator Solution: </li></ul><ul><li>P/Y = 1 N = 5 </li></ul><ul><li>PV = -5,000 FV = 11,933 </li></ul><ul><li>I = 19% </li></ul>Present Value - single sums If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return? 0 5 PV = -5000 FV = 11,933
  53. 53. <ul><li>Mathematical Solution: </li></ul><ul><li>PV = FV (PVIF i, n ) </li></ul><ul><li>5,000 = 11,933 (PVIF ?, 5 ) </li></ul><ul><li>PV = FV / (1 + i) n </li></ul><ul><li>5,000 = 11,933 / (1+ i) 5 </li></ul><ul><li>.419 = ((1/ (1+i) 5 ) </li></ul><ul><li>2.3866 = (1+i) 5 </li></ul><ul><li>(2.3866) 1/5 = (1+i) i = .19 </li></ul>Present Value - single sums If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?
  54. 54. Present Value - single sums Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500? 0 PV = FV =
  55. 55. <ul><li>Calculator Solution: </li></ul><ul><li>P/Y = 12 FV = 500 </li></ul><ul><li>I = 9.6 PV = -100 </li></ul><ul><li>N = 202 months </li></ul>Present Value - single sums Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500? 0 ? PV = -100 FV = 500
  56. 56. Present Value - single sums Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500? <ul><li>Mathematical Solution: </li></ul><ul><li>PV = FV / (1 + i) n </li></ul><ul><li>100 = 500 / (1+ .008) N </li></ul><ul><li>5 = (1.008) N </li></ul><ul><li>ln 5 = ln (1.008) N </li></ul><ul><li>ln 5 = N ln (1.008) </li></ul><ul><li>1.60944 = .007968 N N = 202 months </li></ul>
  57. 57. Hint for single sum problems: <ul><li>In every single sum present value and future value problem, there are four variables: </li></ul><ul><li>FV , PV , i and n . </li></ul><ul><li>When doing problems, you will be given three variables and you will solve for the fourth variable. </li></ul><ul><li>Keeping this in mind makes solving time value problems much easier! </li></ul>
  58. 58. The Time Value of Money <ul><li>Compounding and Discounting </li></ul><ul><li>Cash Flow Streams </li></ul>0 1 2 3 4
  59. 59. Annuities <ul><li>Annuity: a sequence of equal cash flows , occurring at the end of each period. </li></ul>
  60. 60. <ul><li>Annuity: a sequence of equal cash flows, occurring at the end of each period. </li></ul>Annuities 0 1 2 3 4
  61. 61. Examples of Annuities: <ul><li>If you buy a bond, you will receive equal semi-annual coupon interest payments over the life of the bond. </li></ul><ul><li>If you borrow money to buy a house or a car, you will pay a stream of equal payments. </li></ul>
  62. 62. <ul><li>If you buy a bond, you will receive equal semi-annual coupon interest payments over the life of the bond. </li></ul><ul><li>If you borrow money to buy a house or a car, you will pay a stream of equal payments. </li></ul>Examples of Annuities:
  63. 63. Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?
  64. 64. Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years? 0 1 2 3
  65. 65. <ul><li>Calculator Solution: </li></ul><ul><li>P/Y = 1 I = 8 N = 3 </li></ul><ul><li>PMT = -1,000 </li></ul><ul><li>FV = $3,246.40 </li></ul>Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years? 1000 1000 1000 0 1 2 3
  66. 66. <ul><li>Calculator Solution: </li></ul><ul><li>P/Y = 1 I = 8 N = 3 </li></ul><ul><li>PMT = -1,000 </li></ul><ul><li>FV = $3,246.40 </li></ul>Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years? 1000 1000 1000 0 1 2 3
  67. 67. Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?
  68. 68. <ul><li>Mathematical Solution: </li></ul>Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?
  69. 69. <ul><li>Mathematical Solution: </li></ul><ul><li>FV = PMT (FVIFA i, n ) </li></ul>Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?
  70. 70. <ul><li>Mathematical Solution: </li></ul><ul><li>FV = PMT (FVIFA i, n ) </li></ul><ul><li>FV = 1,000 (FVIFA .08, 3 ) (use FVIFA table, or) </li></ul>Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?
  71. 71. <ul><li>Mathematical Solution: </li></ul><ul><li>FV = PMT (FVIFA i, n ) </li></ul><ul><li>FV = 1,000 (FVIFA .08, 3 ) (use FVIFA table, or) </li></ul><ul><li>FV = PMT (1 + i) n - 1 </li></ul><ul><li> i </li></ul>Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?
  72. 72. <ul><li>Mathematical Solution: </li></ul><ul><li>FV = PMT (FVIFA i, n ) </li></ul><ul><li>FV = 1,000 (FVIFA .08, 3 ) (use FVIFA table, or) </li></ul><ul><li>FV = PMT (1 + i) n - 1 </li></ul><ul><li> i </li></ul><ul><li>FV = 1,000 (1.08) 3 - 1 = $3246.40 </li></ul><ul><li> .08 </li></ul>Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?
  73. 73. Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?
  74. 74. Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%? 0 1 2 3
  75. 75. <ul><li>Calculator Solution: </li></ul><ul><li>P/Y = 1 I = 8 N = 3 </li></ul><ul><li>PMT = -1,000 </li></ul><ul><li>PV = $2,577.10 </li></ul>Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%? 1000 1000 1000 0 1 2 3
  76. 76. <ul><li>Calculator Solution: </li></ul><ul><li>P/Y = 1 I = 8 N = 3 </li></ul><ul><li>PMT = -1,000 </li></ul><ul><li>PV = $2,577.10 </li></ul>Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%? 1000 1000 1000 0 1 2 3
  77. 77. Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?
  78. 78. <ul><li>Mathematical Solution: </li></ul>Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?
  79. 79. <ul><li>Mathematical Solution: </li></ul><ul><li>PV = PMT (PVIFA i, n ) </li></ul>Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?
  80. 80. <ul><li>Mathematical Solution: </li></ul><ul><li>PV = PMT (PVIFA i, n ) </li></ul><ul><li>PV = 1,000 (PVIFA .08, 3 ) (use PVIFA table, or) </li></ul>Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?
  81. 81. <ul><li>Mathematical Solution: </li></ul><ul><li>PV = PMT (PVIFA i, n ) </li></ul><ul><li>PV = 1,000 (PVIFA .08, 3 ) (use PVIFA table, or) </li></ul><ul><li> 1 </li></ul><ul><li>PV = PMT 1 - (1 + i) n </li></ul><ul><li> i </li></ul>Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?
  82. 82. <ul><li>Mathematical Solution: </li></ul><ul><li>PV = PMT (PVIFA i, n ) </li></ul><ul><li>PV = 1,000 (PVIFA .08, 3 ) (use PVIFA table, or) </li></ul><ul><li> 1 </li></ul><ul><li>PV = PMT 1 - (1 + i) n </li></ul><ul><li> i </li></ul><ul><li> 1 </li></ul><ul><li>PV = 1000 1 - (1.08 ) 3 = $2,577.10 </li></ul><ul><li> .08 </li></ul>Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?
  83. 83. Other Cash Flow Patterns The Time Value of Money 0 1 2 3
  84. 84. Perpetuities <ul><li>Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity. </li></ul><ul><li>You can think of a perpetuity as an annuity that goes on forever . </li></ul>
  85. 85. Present Value of a Perpetuity <ul><li>When we find the PV of an annuity , we think of the following relationship: </li></ul>
  86. 86. Present Value of a Perpetuity <ul><li>When we find the PV of an annuity , we think of the following relationship: </li></ul>PV = PMT (PVIFA i, n )
  87. 87. <ul><li>Mathematically, </li></ul>
  88. 88. <ul><li>Mathematically, </li></ul><ul><li>(PVIFA i, n ) = </li></ul>
  89. 89. <ul><li>Mathematically, </li></ul><ul><li>(PVIFA i, n ) = </li></ul>1 - 1 (1 + i) n i
  90. 90. <ul><li>Mathematically, </li></ul><ul><li>(PVIFA i, n ) = </li></ul><ul><li>We said that a perpetuity is an annuity where n = infinity. What happens to this formula when n gets very, very large? </li></ul>1 - 1 (1 + i) n i
  91. 91. <ul><li>When n gets very large, </li></ul>
  92. 92. <ul><li>When n gets very large, </li></ul>1 - 1 (1 + i) n i
  93. 93. <ul><li>When n gets very large, </li></ul><ul><li>this becomes zero. </li></ul>1 - 1 (1 + i) n i
  94. 94. <ul><li>When n gets very large, </li></ul><ul><li>this becomes zero. </li></ul><ul><li>So we’re left with PVIFA = </li></ul>1 i 1 - 1 (1 + i) n i
  95. 95. <ul><li>So, the PV of a perpetuity is very simple to find: </li></ul>Present Value of a Perpetuity
  96. 96. <ul><li>So, the PV of a perpetuity is very simple to find: </li></ul>Present Value of a Perpetuity PMT i PV =
  97. 97. <ul><li>What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment? </li></ul>
  98. 98. <ul><li>What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment? </li></ul>PMT $10,000 i .08 PV = =
  99. 99. <ul><li>What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment? </li></ul>PMT $10,000 i .08 = $125,000 PV = =
  100. 100. Ordinary Annuity vs. Annuity Due $1000 $1000 $1000 4 5 6 7 8
  101. 101. Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8
  102. 102. Begin Mode vs. End Mode ordinary annuity year year year 5 6 7 1000 1000 1000 4 5 6 7 8
  103. 103. Begin Mode vs. End Mode ordinary annuity year year year 5 6 7 1000 1000 1000 4 5 6 7 8 PV in END Mode
  104. 104. Begin Mode vs. End Mode ordinary annuity year year year 5 6 7 1000 1000 1000 4 5 6 7 8 PV in END Mode FV in END Mode
  105. 105. Begin Mode vs. End Mode annuity due year year year 6 7 8 1000 1000 1000 4 5 6 7 8
  106. 106. Begin Mode vs. End Mode annuity due year year year 6 7 8 1000 1000 1000 4 5 6 7 8 PV in BEGIN Mode
  107. 107. Begin Mode vs. End Mode annuity due year year year 6 7 8 1000 1000 1000 4 5 6 7 8 PV in BEGIN Mode FV in BEGIN Mode
  108. 108. Earlier, we examined this “ordinary” annuity:
  109. 109. Earlier, we examined this “ordinary” annuity: 1000 1000 1000 0 1 2 3
  110. 110. Earlier, we examined this “ordinary” annuity: <ul><li>Using an interest rate of 8%, we find that: </li></ul> 1000 1000 1000 0 1 2 3
  111. 111. Earlier, we examined this “ordinary” annuity: <ul><li>Using an interest rate of 8%, we find that: </li></ul><ul><li>The Future Value (at 3) is $3,246.40 . </li></ul> 1000 1000 1000 0 1 2 3
  112. 112. Earlier, we examined this “ordinary” annuity: <ul><li>Using an interest rate of 8%, we find that: </li></ul><ul><li>The Future Value (at 3) is $3,246.40 . </li></ul><ul><li>The Present Value (at 0) is $2,577.10 . </li></ul> 1000 1000 1000 0 1 2 3
  113. 113. What about this annuity? <ul><li>Same 3-year time line, </li></ul><ul><li>Same 3 $1000 cash flows, but </li></ul><ul><li>The cash flows occur at the beginning of each year, rather than at the end of each year. </li></ul><ul><li>This is an “annuity due.” </li></ul>1000 1000 1000 0 1 2 3
  114. 114. Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? 0 1 2 3
  115. 115. <ul><li>Calculator Solution: </li></ul><ul><li>Mode = BEGIN P/Y = 1 I = 8 </li></ul><ul><li>N = 3 PMT = -1,000 </li></ul><ul><li>FV = $3,506.11 </li></ul>Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? 0 1 2 3 -1000 -1000 -1000
  116. 116. Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? <ul><li>Calculator Solution: </li></ul><ul><li>Mode = BEGIN P/Y = 1 I = 8 </li></ul><ul><li>N = 3 PMT = -1,000 </li></ul><ul><li>FV = $3,506.11 </li></ul>0 1 2 3 -1000 -1000 -1000
  117. 117. Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? <ul><li>Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: </li></ul>
  118. 118. Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? <ul><li>Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: </li></ul><ul><li>FV = PMT (FVIFA i, n ) (1 + i) </li></ul>
  119. 119. Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? <ul><li>Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: </li></ul><ul><li>FV = PMT (FVIFA i, n ) (1 + i) </li></ul><ul><li>FV = 1,000 (FVIFA .08, 3 ) (1.08) (use FVIFA table, or) </li></ul>
  120. 120. Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? <ul><li>Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: </li></ul><ul><li>FV = PMT (FVIFA i, n ) (1 + i) </li></ul><ul><li>FV = 1,000 (FVIFA .08, 3 ) (1.08) (use FVIFA table, or) </li></ul><ul><li>FV = PMT (1 + i) n - 1 </li></ul><ul><li> i </li></ul>(1 + i)
  121. 121. Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? <ul><li>Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: </li></ul><ul><li>FV = PMT (FVIFA i, n ) (1 + i) </li></ul><ul><li>FV = 1,000 (FVIFA .08, 3 ) (1.08) (use FVIFA table, or) </li></ul><ul><li>FV = PMT (1 + i) n - 1 </li></ul><ul><li> i </li></ul><ul><li>FV = 1,000 (1.08) 3 - 1 = $3,506.11 </li></ul><ul><li> .08 </li></ul>(1 + i) (1.08)
  122. 122. Present Value - annuity due What is the PV of $1,000 at the beginning of each of the next 3 years, if your opportunity cost is 8%? 0 1 2 3
  123. 123. <ul><li>Calculator Solution: </li></ul><ul><li>Mode = BEGIN P/Y = 1 I = 8 </li></ul><ul><li>N = 3 PMT = 1,000 </li></ul><ul><li>PV = $2,783.26 </li></ul>Present Value - annuity due What is the PV of $1,000 at the beginning of each of the next 3 years, if your opportunity cost is 8%? 0 1 2 3 1000 1000 1000
  124. 124. <ul><li> Calculator Solution: </li></ul><ul><li>Mode = BEGIN P/Y = 1 I = 8 </li></ul><ul><li>N = 3 PMT = 1,000 </li></ul><ul><li>PV = $2,783.26 </li></ul>Present Value - annuity due What is the PV of $1,000 at the beginning of each of the next 3 years, if your opportunity cost is 8%? 0 1 2 3 1000 1000 1000
  125. 125. Present Value - annuity due <ul><li>Mathematical Solution: </li></ul>
  126. 126. Present Value - annuity due <ul><li>Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: </li></ul>
  127. 127. Present Value - annuity due <ul><li>Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: </li></ul><ul><li>PV = PMT (PVIFA i, n ) (1 + i) </li></ul>
  128. 128. Present Value - annuity due <ul><li>Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: </li></ul><ul><li>PV = PMT (PVIFA i, n ) (1 + i) </li></ul><ul><li>PV = 1,000 (PVIFA .08, 3 ) (1.08) (use PVIFA table, or) </li></ul>
  129. 129. Present Value - annuity due <ul><li>Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: </li></ul><ul><li>PV = PMT (PVIFA i, n ) (1 + i) </li></ul><ul><li>PV = 1,000 (PVIFA .08, 3 ) (1.08) (use PVIFA table, or) </li></ul><ul><li> 1 </li></ul><ul><li>PV = PMT 1 - (1 + i) n </li></ul><ul><li> i </li></ul>(1 + i)
  130. 130. Present Value - annuity due <ul><li>Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: </li></ul><ul><li>PV = PMT (PVIFA i, n ) (1 + i) </li></ul><ul><li>PV = 1,000 (PVIFA .08, 3 ) (1.08) (use PVIFA table, or) </li></ul><ul><li> 1 </li></ul><ul><li>PV = PMT 1 - (1 + i) n </li></ul><ul><li> i </li></ul><ul><li> 1 </li></ul><ul><li>PV = 1000 1 - (1.08 ) 3 = $2,783.26 </li></ul><ul><li> .08 </li></ul>(1 + i) (1.08)
  131. 131. <ul><li>Is this an annuity ? </li></ul><ul><li>How do we find the PV of a cash flow stream when all of the cash flows are different? (Use a 10% discount rate.) </li></ul>Uneven Cash Flows 0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000
  132. 132. <ul><li>Sorry! There’s no quickie for this one. We have to discount each cash flow back separately. </li></ul>Uneven Cash Flows 0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000
  133. 133. <ul><li>Sorry! There’s no quickie for this one. We have to discount each cash flow back separately. </li></ul>Uneven Cash Flows 0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000
  134. 134. <ul><li>Sorry! There’s no quickie for this one. We have to discount each cash flow back separately. </li></ul>Uneven Cash Flows 0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000
  135. 135. <ul><li>Sorry! There’s no quickie for this one. We have to discount each cash flow back separately. </li></ul>Uneven Cash Flows 0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000
  136. 136. <ul><li>Sorry! There’s no quickie for this one. We have to discount each cash flow back separately. </li></ul>Uneven Cash Flows 0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000
  137. 137. <ul><li>period CF PV (CF) </li></ul><ul><li>0 -10,000 -10,000.00 </li></ul><ul><li>1 2,000 1,818.18 </li></ul><ul><li>2 4,000 3,305.79 </li></ul><ul><li>3 6,000 4,507.89 </li></ul><ul><li>4 7,000 4,781.09 </li></ul><ul><li>PV of Cash Flow Stream: $ 4,412.95 </li></ul>0 1 2 3 4 -10,000 2,000 4,000 6,000 7,000
  138. 138. Annual Percentage Yield (APY) <ul><li>Which is the better loan: </li></ul><ul><li>8% compounded annually , or </li></ul><ul><li>7.85% compounded quarterly ? </li></ul><ul><li>We can’t compare these nominal (quoted) interest rates, because they don’t include the same number of compounding periods per year! </li></ul><ul><li>We need to calculate the APY. </li></ul>
  139. 139. Annual Percentage Yield (APY)
  140. 140. Annual Percentage Yield (APY) APY = ( 1 + ) m - 1 quoted rate m
  141. 141. Annual Percentage Yield (APY) <ul><li>Find the APY for the quarterly loan: </li></ul>APY = ( 1 + ) m - 1 quoted rate m
  142. 142. Annual Percentage Yield (APY) <ul><li>Find the APY for the quarterly loan: </li></ul>APY = ( 1 + ) m - 1 quoted rate m APY = ( 1 + ) 4 - 1 .0785 4
  143. 143. Annual Percentage Yield (APY) <ul><li>Find the APY for the quarterly loan: </li></ul>APY = ( 1 + ) m - 1 quoted rate m APY = ( 1 + ) 4 - 1 APY = .0808, or 8.08% .0785 4
  144. 144. Annual Percentage Yield (APY) <ul><li>Find the APY for the quarterly loan: </li></ul><ul><li>The quarterly loan is more expensive than the 8% loan with annual compounding! </li></ul>APY = ( 1 + ) m - 1 quoted rate m APY = ( 1 + ) 4 - 1 APY = .0808, or 8.08% .0785 4
  145. 145. Practice Problems
  146. 146. Example <ul><li>Cash flows from an investment are expected to be $40,000 per year at the end of years 4, 5, 6, 7, and 8. If you require a 20% rate of return, what is the PV of these cash flows? </li></ul>
  147. 147. Example <ul><li>Cash flows from an investment are expected to be $40,000 per year at the end of years 4, 5, 6, 7, and 8. If you require a 20% rate of return, what is the PV of these cash flows? </li></ul>0 1 2 3 4 5 6 7 8 $0 0 0 0 40 40 40 40 40
  148. 148. <ul><li>This type of cash flow sequence is often called a “ deferred annuity .” </li></ul>0 1 2 3 4 5 6 7 8 $0 0 0 0 40 40 40 40 40
  149. 149. <ul><li>How to solve: </li></ul><ul><li>1) Discount each cash flow back to time 0 separately. </li></ul>0 1 2 3 4 5 6 7 8 $0 0 0 0 40 40 40 40 40
  150. 150. <ul><li>How to solve: </li></ul><ul><li>1) Discount each cash flow back to time 0 separately. </li></ul>0 1 2 3 4 5 6 7 8 $0 0 0 0 40 40 40 40 40
  151. 151. <ul><li>How to solve: </li></ul><ul><li>1) Discount each cash flow back to time 0 separately. </li></ul>0 1 2 3 4 5 6 7 8 $0 0 0 0 40 40 40 40 40
  152. 152. <ul><li>How to solve: </li></ul><ul><li>1) Discount each cash flow back to time 0 separately. </li></ul>0 1 2 3 4 5 6 7 8 $0 0 0 0 40 40 40 40 40
  153. 153. <ul><li>How to solve: </li></ul><ul><li>1) Discount each cash flow back to time 0 separately. </li></ul>0 1 2 3 4 5 6 7 8 $0 0 0 0 40 40 40 40 40
  154. 154. <ul><li>How to solve: </li></ul><ul><li>1) Discount each cash flow back to time 0 separately. </li></ul>0 1 2 3 4 5 6 7 8 $0 0 0 0 40 40 40 40 40
  155. 155. <ul><li>How to solve: </li></ul><ul><li>1) Discount each cash flow back to time 0 separately. </li></ul><ul><li>Or, </li></ul>0 1 2 3 4 5 6 7 8 $0 0 0 0 40 40 40 40 40
  156. 156. <ul><li>2) Find the PV of the annuity: </li></ul><ul><li>PV : End mode; P/YR = 1; I = 20; PMT = 40,000; N = 5 </li></ul><ul><li>PV = $119,624 </li></ul>0 1 2 3 4 5 6 7 8 $0 0 0 0 40 40 40 40 40
  157. 157. <ul><li>2) Find the PV of the annuity: </li></ul><ul><li>PV 3: End mode; P/YR = 1; I = 20; PMT = 40,000; N = 5 </li></ul><ul><li>PV 3 = $119,624 </li></ul>0 1 2 3 4 5 6 7 8 $0 0 0 0 40 40 40 40 40
  158. 158. 119,624 0 1 2 3 4 5 6 7 8 $0 0 0 0 40 40 40 40 40
  159. 159. <ul><li>Then discount this single sum back to time 0. </li></ul><ul><li>PV: End mode; P/YR = 1; I = 20; </li></ul><ul><li>N = 3; FV = 119,624; </li></ul><ul><li>Solve: PV = $69,226 </li></ul>119,624 0 1 2 3 4 5 6 7 8 $0 0 0 0 40 40 40 40 40
  160. 160. 69,226 0 1 2 3 4 5 6 7 8 $0 0 0 0 40 40 40 40 40 119,624
  161. 161. <ul><li>The PV of the cash flow stream is $69,226 . </li></ul>69,226 0 1 2 3 4 5 6 7 8 $0 0 0 0 40 40 40 40 40 119,624
  162. 162. Retirement Example <ul><li>After graduation, you plan to invest $400 per month in the stock market. If you earn 12% per year on your stocks, how much will you have accumulated when you retire in 30 years ? </li></ul>
  163. 163. Retirement Example <ul><li>After graduation, you plan to invest $400 per month in the stock market. If you earn 12% per year on your stocks, how much will you have accumulated when you retire in 30 years? </li></ul>0 1 2 3 . . . 360 400 400 400 400
  164. 164. 0 1 2 3 . . . 360 400 400 400 400
  165. 165. <ul><li>Using your calculator, </li></ul><ul><li>P/YR = 12 </li></ul><ul><li>N = 360 </li></ul><ul><li>PMT = -400 </li></ul><ul><li>I%YR = 12 </li></ul><ul><li>FV = $1,397,985.65 </li></ul>0 1 2 3 . . . 360 400 400 400 400
  166. 166. Retirement Example If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30?
  167. 167. Retirement Example If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30? <ul><li>Mathematical Solution: </li></ul>
  168. 168. Retirement Example If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30? <ul><li>Mathematical Solution: </li></ul><ul><li>FV = PMT (FVIFA i, n ) </li></ul>
  169. 169. Retirement Example If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30? <ul><li>Mathematical Solution: </li></ul><ul><li>FV = PMT (FVIFA i, n ) </li></ul><ul><li>FV = 400 (FVIFA .01, 360 ) (can’t use FVIFA table) </li></ul>
  170. 170. Retirement Example If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30? <ul><li>Mathematical Solution: </li></ul><ul><li>FV = PMT (FVIFA i, n ) </li></ul><ul><li>FV = 400 (FVIFA .01, 360 ) (can’t use FVIFA table) </li></ul><ul><li>FV = PMT (1 + i) n - 1 </li></ul><ul><li> i </li></ul>
  171. 171. Retirement Example If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30? <ul><li>Mathematical Solution: </li></ul><ul><li>FV = PMT (FVIFA i, n ) </li></ul><ul><li>FV = 400 (FVIFA .01, 360 ) (can’t use FVIFA table) </li></ul><ul><li>FV = PMT (1 + i) n - 1 </li></ul><ul><li> i </li></ul><ul><li>FV = 400 (1.01) 360 - 1 = $1,397,985.65 </li></ul><ul><li> .01 </li></ul>
  172. 172. <ul><li>If you borrow $100,000 at 7% fixed interest for 30 years in order to buy a house, what will be your monthly house payment ? </li></ul>House Payment Example
  173. 173. House Payment Example <ul><li>If you borrow $100,000 at 7% fixed interest for 30 years in order to buy a house, what will be your monthly house payment? </li></ul>
  174. 174. 0 1 2 3 . . . 360 ? ? ? ?
  175. 175. <ul><li>Using your calculator, </li></ul><ul><li>P/YR = 12 </li></ul><ul><li>N = 360 </li></ul><ul><li>I%YR = 7 </li></ul><ul><li>PV = $100,000 </li></ul><ul><li>PMT = -$665.30 </li></ul>0 1 2 3 . . . 360 ? ? ? ?
  176. 176. House Payment Example <ul><li>Mathematical Solution: </li></ul>
  177. 177. House Payment Example <ul><li>Mathematical Solution: </li></ul><ul><li>PV = PMT (PVIFA i, n ) </li></ul>
  178. 178. House Payment Example <ul><li>Mathematical Solution: </li></ul><ul><li>PV = PMT (PVIFA i, n ) </li></ul><ul><li>100,000 = PMT (PVIFA .07, 360 ) (can’t use PVIFA table) </li></ul>
  179. 179. House Payment Example <ul><li>Mathematical Solution: </li></ul><ul><li>PV = PMT (PVIFA i, n ) </li></ul><ul><li>100,000 = PMT (PVIFA .07, 360 ) (can’t use PVIFA table) </li></ul><ul><li> 1 </li></ul><ul><li>PV = PMT 1 - (1 + i) n </li></ul><ul><li> i </li></ul>
  180. 180. House Payment Example <ul><li>Mathematical Solution: </li></ul><ul><li>PV = PMT (PVIFA i, n ) </li></ul><ul><li>100,000 = PMT (PVIFA .07, 360 ) (can’t use PVIFA table) </li></ul><ul><li> 1 </li></ul><ul><li>PV = PMT 1 - (1 + i) n </li></ul><ul><li> i </li></ul><ul><li> 1 </li></ul><ul><li>100,000 = PMT 1 - (1.005833 ) 360 PMT=$665.30 </li></ul><ul><li> .005833 </li></ul>
  181. 181. Team Assignment <ul><li>Upon retirement, your goal is to spend 5 years traveling around the world. To travel in style will require $250,000 per year at the beginning of each year. </li></ul><ul><li>If you plan to retire in 30 years , what are the equal monthly payments necessary to achieve this goal? The funds in your retirement account will compound at 10% annually. </li></ul>
  182. 182. <ul><li>How much do we need to have by the end of year 30 to finance the trip? </li></ul><ul><li>PV 30 = PMT (PVIFA .10, 5 ) (1.10) = </li></ul><ul><li>= 250,000 (3.7908) (1.10) = </li></ul><ul><li>= $1,042,470 </li></ul>27 28 29 30 31 32 33 34 35 250 250 250 250 250
  183. 183. <ul><li>Using your calculator, </li></ul><ul><li>Mode = BEGIN </li></ul><ul><li>PMT = -$250,000 </li></ul><ul><li>N = 5 </li></ul><ul><li>I%YR = 10 </li></ul><ul><li>P/YR = 1 </li></ul><ul><li>PV = $1,042,466 </li></ul>27 28 29 30 31 32 33 34 35 250 250 250 250 250
  184. 184. <ul><li>Now, assuming 10% annual compounding, what monthly payments will be required for you to have $1,042,466 at the end of year 30? </li></ul>1,042,466 27 28 29 30 31 32 33 34 35 250 250 250 250 250
  185. 185. <ul><li>Using your calculator, </li></ul><ul><li>Mode = END </li></ul><ul><li>N = 360 </li></ul><ul><li>I%YR = 10 </li></ul><ul><li>P/YR = 12 </li></ul><ul><li>FV = $1,042,466 </li></ul><ul><li>PMT = -$461.17 </li></ul>1,042,466 27 28 29 30 31 32 33 34 35 250 250 250 250 250
  186. 186. <ul><li>So, you would have to place $461.17 in your retirement account, which earns 10% annually, at the end of each of the next 360 months to finance the 5-year world tour. </li></ul>

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