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# Session2 Time Value Of Money (2)

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### Session2 Time Value Of Money (2)

1. 1. <ul><li>Compounding and Discounting Single Sums </li></ul>The Time Value of Money
2. 2. <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
3. 3. <ul><li>Translate \$1 today into its equivalent in the future (COMPOUNDING) . </li></ul>? If we can MEASURE this opportunity cost, we can: Today Future
4. 4. <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>? ? If we can MEASURE this opportunity cost, we can: Today Future Today Future
5. 5. <ul><li>Future Value </li></ul>
6. 6. 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 =
7. 7. 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 =
8. 8. 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
9. 9. 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
10. 10. 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 =
11. 11. 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 =
12. 12. 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
13. 13. 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
14. 14. 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 =
15. 15. <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 =
16. 16. <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
17. 17. <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
18. 18. 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 =
19. 19. <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 =
20. 20. <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
21. 21. <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
22. 22. Future Value - continuous compounding What is the FV of \$1,000 earning 8% with continuous compounding , after 100 years? 0 ? PV = FV =
23. 23. <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 =
24. 24. Future Value - continuous compounding What is the FV of \$1,000 earning 8% with continuous compounding , after 100 years? <ul><li>\$2.98m </li></ul>0 100 PV = -1000 FV = Mathematical Solution: FV = PV (e in ) FV = 1000 (e .08x100 ) = 1000 (e 8 ) FV = \$2,980,957. 99
25. 25. <ul><li>Present Value </li></ul>
26. 26. Present Value - single sums If you will receive \$100 one year from now, what is the PV of that \$100 if your opportunity cost is 6%? 0 ? PV = FV =
27. 27. <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 will receive \$100 one year from now, what is the PV of that \$100 if your opportunity cost is 6%? 0 1 PV = FV = 100
28. 28. <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 will receive \$100 one year from now, what is the PV of that \$100 if your opportunity cost is 6%? 0 1 PV = -94. 34 FV = 100
29. 29. <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 will receive \$100 one year from now, what is the PV of that \$100 if your opportunity cost is 6%? 0 1 PV = -94. 34 FV = 100
30. 30. Present Value - single sums If you will receive \$100 5 years from now, what is the PV of that \$100 if your opportunity cost is 6%? 0 ? PV = FV =
31. 31. <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 will receive \$100 5 years from now, what is the PV of that \$100 if your opportunity cost is 6%? 0 5 PV = FV = 100
32. 32. Present Value - single sums If you will receive \$100 5 years from now, what is the PV of that \$100 if your opportunity cost is 6%? <ul><li>-74. 73 </li></ul>0 5 PV = FV = 100 Calculator Solution: P/Y = 1 I = 6 N = 5 FV = 100 PV = -74.73
33. 33. <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 will receive \$100 5 years from now, what is the PV of that \$100 if your opportunity cost is 6%? 0 5 PV = -74. 73 FV = 100
34. 34. 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 ? PV = FV =
35. 35. <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
36. 36. <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
37. 37. <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
38. 38. 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 ? PV = FV =
39. 39. <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 = -5,000 FV = 11,933
40. 40. <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?
41. 41. 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 =
42. 42. <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
43. 43. 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>
44. 44. <ul><li>In every single sum future value and present value problem, there are 4 variables: </li></ul><ul><li>FV , PV , i , and n </li></ul><ul><li>When doing problems, you will be given 3 of these variables and asked to solve for the 4th variable. </li></ul><ul><li>Keeping this in mind makes “time value” problems much easier! </li></ul>Hint for single sum problems:
45. 45. <ul><li>Compounding and Discounting </li></ul><ul><li>Cash Flow Streams </li></ul>The Time Value of Money 0 1 2 3 4
46. 46. <ul><li>Annuity: a sequence of equal cash flows , occurring at the end of each period. </li></ul>Annuities
47. 47. <ul><li>Annuity: a sequence of equal cash flows, occurring at the end of each period. </li></ul>Annuities 0 1 2 3 4
48. 48. <ul><li>If you buy a bond, you will receive equal 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:
49. 49. <ul><li>If you buy a bond, you will receive equal 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:
50. 50. Future Value - annuity If you invest \$1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years? 0 1 2 3
51. 51. <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 at the end of the next 3 years, at 8%, how much would you have after 3 years? 1000 1000 1000 0 1 2 3
52. 52. <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 at the end of the next 3 years, at 8%, how much would you have after 3 years? 1000 1000 1000 0 1 2 3
53. 53. <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 at the end of the next 3 years, at 8%, how much would you have after 3 years?
54. 54. <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 at the end of the next 3 years, at 8%, how much would you have after 3 years?
55. 55. <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 at the end of the next 3 years, at 8%, how much would you have after 3 years?
56. 56. 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
57. 57. <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
58. 58. <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
59. 59. <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%?
60. 60. <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%?
61. 61. <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%?
62. 62. Other Cash Flow Patterns 0 1 2 3
63. 63. <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>Perpetuities
64. 64. <ul><li>When we find the PV of an annuity, we think of the following relationship: </li></ul>Present Value of a Perpetuity
65. 65. <ul><li>When we find the PV of an annuity , we think of the following relationship: </li></ul>PV = PMT (PVIFA i, n ) Present Value of a Perpetuity
66. 66. <ul><li>Mathematically, </li></ul><ul><li>(PVIFA i, n ) = </li></ul>
67. 67. <ul><li>Mathematically, </li></ul><ul><li>(PVIFA i, n ) = </li></ul>1 - 1 (1 + i) n i
68. 68. <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
69. 69. <ul><li>When n gets very large, </li></ul>
70. 70. <ul><li>When n gets very large, </li></ul><ul><li>this becomes zero. </li></ul>1 - 1 (1 + i) n i
71. 71. When n gets very large, this becomes zero. So we’re left with PVIFA = 1 - 1 (1 + i) n i 1 i
72. 72. <ul><li>So, the PV of a perpetuity is very simple to find: </li></ul>Present Value of a Perpetuity
73. 73. <ul><li>So, the PV of a perpetuity is very simple to find: </li></ul>PMT i PV = Present Value of a Perpetuity
74. 74. <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>
75. 75. <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>= \$125,000 PMT i PV = = \$10,000 .08
76. 76. Ordinary Annuity vs. Annuity Due \$1000 \$1000 \$1000 4 5 6 7 8
77. 77. Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8
78. 78. year year year 5 6 7 Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8
79. 79. year year year 5 6 7 Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8 PV in END Mode
80. 80. year year year 5 6 7 Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8 PV in END Mode FV in END Mode
81. 81. year year year 6 7 8 Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8
82. 82. year year year 6 7 8 Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8 PV in BEGIN Mode
83. 83. year year year 6 7 8 Begin Mode vs. End Mode 1000 1000 1000 4 5 6 7 8 PV in BEGIN Mode FV in BEGIN Mode
84. 84. <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 Earlier, we examined this “ordinary” annuity: 0 1 2 3
85. 85. <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 What about this annuity? 0 1 2 3
86. 86. 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
87. 87. <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
88. 88. 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
89. 89. 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>
90. 90. 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)
91. 91. 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)
92. 92. 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
93. 93. <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
94. 94. <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
95. 95. 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>
96. 96. 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)
97. 97. 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)
98. 98. <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
99. 99. <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
100. 100. <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
101. 101. <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
102. 102. <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
103. 103. <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
104. 104. <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
105. 105. 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>
106. 106. 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
107. 107. <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
108. 108. <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
109. 109. <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
110. 110. 119,624 0 1 2 3 4 5 6 7 8 0 0 0 0 40 40 40 40 40
111. 111. <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
112. 112. 119,624 69,226 0 1 2 3 4 5 6 7 8 0 0 0 0 40 40 40 40 40
113. 113. <ul><li>The PV of the cash flow stream is \$69,226. </li></ul>119,624 69,226 0 1 2 3 4 5 6 7 8 0 0 0 0 40 40 40 40 40
114. 114. 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>
115. 115. 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
116. 116. 0 1 2 3 . . . 360 400 400 400 400
117. 117. <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
118. 118. 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>
119. 119. 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>
120. 120. 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>
121. 121. <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
122. 122. 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>
123. 123. 0 1 2 3 . . . 360 ? ? ? ?
124. 124. <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 ? ? ? ?
125. 125. 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>
126. 126. 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>
127. 127. 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>
128. 128. <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? </li></ul><ul><li>The funds in your retirement account will compound at 10% annually. </li></ul>Team Assignment
129. 129. <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
130. 130. <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
131. 131. <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>27 28 29 30 31 32 33 34 35 250 250 250 250 250 1,042,466
132. 132. <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>27 28 29 30 31 32 33 34 35 250 250 250 250 250 1,042,466
133. 133. <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>