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# Berk Chapter 4: The Time Value Of Money

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### Berk Chapter 4: The Time Value Of Money

1. 1. Chapter 4 The Time Value of Money
2. 2. Chapter Outline <ul><li>4.1 The Timeline </li></ul><ul><li>4.2 The Three Rules of Time Travel </li></ul><ul><li>4.3 Valuing a Stream of Cash Flows </li></ul><ul><li>4.4 Calculating the Net Present Value </li></ul><ul><li>4.5 Perpetuities, Annuities, and Other Special Cases </li></ul>
3. 3. Chapter Outline (cont’d) <ul><li>4.6 Solving Problems with a Spreadsheet Program </li></ul><ul><li>4.7 Solving for Variables Other Than Present Value or Future Value </li></ul>
4. 4. Learning Objectives <ul><li>Draw a timeline illustrating a given set of cash flows. </li></ul><ul><li>List and describe the three rules of time travel. </li></ul><ul><li>Calculate the future value of: </li></ul><ul><li>A single sum. </li></ul><ul><li>An uneven stream of cash flows, starting either now or sometime in the future. </li></ul><ul><li>An annuity, starting either now or sometime in the future. </li></ul>
5. 5. Learning Objectives <ul><li>Several cash flows occurring at regular intervals that grow at a constant rate each period. </li></ul><ul><li>Calculate the present value of: </li></ul><ul><li>A single sum. </li></ul><ul><li>An uneven stream of cash flows, starting either now or sometime in the future. </li></ul><ul><li>An infinite stream of identical cash flows. </li></ul><ul><li>An annuity, starting either now or sometime in the future. </li></ul>
6. 6. Learning Objectives <ul><li>Given four out of the following five inputs for an annuity, compute the fifth: (a) present value, (b) future value, (c) number of periods, (d) periodic interest rate, (e) periodic payment. </li></ul><ul><li>Given three out of the following four inputs for a single sum, compute the fourth: (a) present value, (b) future value, (c) number of periods, (d) periodic interest rate. </li></ul><ul><li>Given cash flows and present or future value, compute the internal rate of return for a series of cash flows. </li></ul>
7. 7. 4.1 The Timeline <ul><li>A timeline is a linear representation of the timing of potential cash flows. </li></ul><ul><li>Drawing a timeline of the cash flows will help you visualize the financial problem. </li></ul>
8. 8. 4.1 The Timeline (cont’d) <ul><li>Assume that you made a loan to a friend. You will be repaid in two payments, one at the end of each year over the next two years. </li></ul>
9. 9. 4.1 The Timeline (cont’d) <ul><li>Differentiate between two types of cash flows </li></ul><ul><ul><li>Inflows are positive cash flows. </li></ul></ul><ul><ul><li>Outflows are negative cash flows, which are indicated with a – (minus) sign. </li></ul></ul>
10. 10. 4.1 The Timeline (cont’d) <ul><li>Assume that you are lending \$10,000 today and that the loan will be repaid in two annual \$6,000 payments. </li></ul><ul><li>The first cash flow at date 0 (today) is represented as a negative sum because it is an outflow. </li></ul><ul><li>Timelines can represent cash flows that take place at the end of any time period – a month, a week, a day, etc. </li></ul>
11. 11. Textbook Example 4.1
12. 12. Textbook Example 4.1 (cont’d)
13. 13. 4.2 Three Rules of Time Travel <ul><li>Financial decisions often require combining cash flows or comparing values. Three rules govern these processes. </li></ul><ul><li>Table 4.1 The Three Rules of Time Travel </li></ul>
14. 14. The 1st Rule of Time Travel <ul><li>A dollar today and a dollar in one year are not equivalent. </li></ul><ul><li>It is only possible to compare or combine values at the same point in time. </li></ul><ul><ul><li>Which would you prefer: A gift of \$1,000 today or \$1,210 at a later date? </li></ul></ul><ul><ul><li>To answer this, you will have to compare the alternatives to decide which is worth more. One factor to consider: How long is “later?” </li></ul></ul>
15. 15. The 2nd Rule of Time Travel <ul><li>To move a cash flow forward in time, you must compound it. </li></ul><ul><ul><li>Suppose you have a choice between receiving \$1,000 today or \$1,210 in two years. You believe you can earn 10% on the \$1,000 today, but want to know what the \$1,000 will be worth in two years. The time line looks like this: </li></ul></ul>
16. 16. The 2nd Rule of Time Travel (cont’d) <ul><li>Future Value of a Cash Flow </li></ul>
17. 17. Using a Financial Calculator: The Basics <ul><li>HP 10BII </li></ul><ul><ul><li>Future Value </li></ul></ul><ul><ul><li>Present Value </li></ul></ul><ul><ul><li>I/YR </li></ul></ul><ul><ul><ul><li>Interest Rate per Year </li></ul></ul></ul><ul><ul><ul><li>Interest is entered as a percent, not a decimal </li></ul></ul></ul><ul><ul><ul><ul><li>For 10%, enter 10, NOT .10 </li></ul></ul></ul></ul>
18. 18. Using a Financial Calculator: The Basics (cont’d) <ul><li>HP 10BII </li></ul><ul><ul><li>Number of Periods </li></ul></ul><ul><ul><li>Periods per Year </li></ul></ul><ul><ul><li>Gold -> C All </li></ul></ul><ul><ul><ul><li>Clears out all TVM registers </li></ul></ul></ul><ul><ul><ul><li>Should do between all problems </li></ul></ul></ul>
19. 19. Using a Financial Calculator: Setting the keys <ul><li>HP 10BII </li></ul><ul><ul><li>Gold -> C All (Hold down [C] button) </li></ul></ul><ul><ul><ul><li>Check P/YR </li></ul></ul></ul><ul><ul><li># -> Gold -> P/YR </li></ul></ul><ul><ul><ul><li>Sets Periods per Year to # </li></ul></ul></ul><ul><ul><li>Gold -> DISP -> # </li></ul></ul><ul><ul><ul><li>Gold and [=] button </li></ul></ul></ul><ul><ul><ul><li>Sets display to # decimal places </li></ul></ul></ul>
20. 20. Using a Financial Calculator <ul><li>HP 10BII </li></ul><ul><ul><li>Cash flows moving in opposite directions must have opposite signs. </li></ul></ul>
21. 21. Financial Calculator Solution <ul><li>Inputs: </li></ul><ul><ul><li>N = 2 </li></ul></ul><ul><ul><li>I = 10 </li></ul></ul><ul><ul><li>PV = 1,000 </li></ul></ul><ul><li>Output: </li></ul><ul><ul><li>FV = –1,210 </li></ul></ul>
22. 22. Figure 4.1 The Composition of Interest Over Time
23. 23. Textbook Example 4.2
24. 24. Textbook Example 4.2 (cont’d)
25. 25. Textbook Example 4.2 Financial Calculator Solution for n=7 years <ul><li>Inputs: </li></ul><ul><ul><li>N = 7 </li></ul></ul><ul><ul><li>I = 10 </li></ul></ul><ul><ul><li>PV = 1,000 </li></ul></ul><ul><li>Output: </li></ul><ul><ul><li>FV = –1,948.72 </li></ul></ul>
26. 26. Alternative Example 4.2 <ul><li>Problem </li></ul><ul><ul><li>Suppose you have a choice between receiving \$5,000 today or \$10,000 in five years. You believe you can earn 10% on the \$5,000 today, but want to know what the \$5,000 will be worth in five years. </li></ul></ul>
27. 27. Alternative Example 4.2 (cont’d) <ul><li>Solution </li></ul><ul><ul><li>The time line looks like this: </li></ul></ul><ul><ul><li>In five years, the \$5,000 will grow to: </li></ul></ul><ul><ul><li>\$5,000 × (1.10) 5 = \$8,053 </li></ul></ul><ul><ul><li>The future value of \$5,000 at 10% for five years is \$8,053. </li></ul></ul><ul><ul><li>You would be better off forgoing the gift of \$5,000 today and taking the \$10,000 in five years. </li></ul></ul>
28. 28. Alternative Example 4.2 Financial Calculator Solution <ul><li>Inputs: </li></ul><ul><ul><li>N = 5 </li></ul></ul><ul><ul><li>I = 10 </li></ul></ul><ul><ul><li>PV = 5,000 </li></ul></ul><ul><li>Output: </li></ul><ul><ul><li>FV = –8,052.55 </li></ul></ul>
29. 29. The 3rd Rule of Time Travel <ul><li>To move a cash flow backward in time, we must discount it. </li></ul><ul><li>Present Value of a Cash Flow </li></ul>
30. 30. Textbook Example 4.3
31. 31. Textbook Example 4.3
32. 32. Textbook Example 4.3 Financial Calculator Solution <ul><li>Inputs: </li></ul><ul><ul><li>N = 10 </li></ul></ul><ul><ul><li>I = 6 </li></ul></ul><ul><ul><li>FV = 15,000 </li></ul></ul><ul><li>Output: </li></ul><ul><ul><li>PV = –8,375.92 </li></ul></ul>
33. 33. Alternative Example 4.3 <ul><li>Problem </li></ul><ul><ul><li>Suppose you are offered an investment that pays \$10,000 in five years. If you expect to earn a 10% return, what is the value of this investment today? </li></ul></ul>
34. 34. Alternative Example 4.3 (cont’d) <ul><li>Solution </li></ul><ul><ul><li>The \$10,000 is worth: </li></ul></ul><ul><ul><ul><li>\$10,000 ÷ (1.10) 5 = \$6,209 </li></ul></ul></ul>
35. 35. Alternative Example 4.3: Financial Calculator Solution <ul><li>Inputs: </li></ul><ul><ul><li>N = 5 </li></ul></ul><ul><ul><li>I = 10 </li></ul></ul><ul><ul><li>FV = 10,000 </li></ul></ul><ul><li>Output: </li></ul><ul><ul><li>PV = –6,209.21 </li></ul></ul>
36. 36. Applying the Rules of Time Travel <ul><li>Recall the 1st rule: It is only possible to compare or combine values at the same point in time. So far we’ve only looked at comparing. </li></ul><ul><ul><li>Suppose we plan to save \$1000 today, and \$1000 at the end of each of the next two years. If we can earn a fixed 10% interest rate on our savings, how much will we have three years from today? </li></ul></ul>
37. 37. Applying the Rules of Time Travel (cont'd) <ul><li>The time line would look like this: </li></ul>
38. 38. Applying the Rules of Time Travel (cont'd)
39. 39. Applying the Rules of Time Travel (cont'd)
40. 40. Applying the Rules of Time Travel (cont'd)
41. 41. Applying the Rules of Time Travel <ul><li>Table 4.1 The Three Rules of Time Travel </li></ul>
42. 42. Textbook Example 4.4
43. 43. Textbook Example 4.4 (cont’d)
44. 44. Textbook Example 4.4 Financial Calculator Solution
45. 45. Alternative Example 4.4 <ul><li>Problem </li></ul><ul><ul><li>Assume that an investment will pay you \$5,000 now and \$10,000 in five years. </li></ul></ul><ul><ul><li>The time line would like this: </li></ul></ul>
46. 46. Alternative Example 4.4 (cont'd) <ul><li>Solution </li></ul><ul><ul><li>You can calculate the present value of the combined cash flows by adding their values today. </li></ul></ul><ul><ul><li>The present value of both cash flows is \$11,209. </li></ul></ul>
47. 47. Alternative Example 4.4 (cont'd) <ul><li>Solution </li></ul><ul><ul><li>You can calculate the future value of the combined cash flows by adding their values in Year 5. </li></ul></ul><ul><ul><li>The future value of both cash flows is \$18,053. </li></ul></ul>
48. 48. Alternative Example 4.4 (cont'd)
49. 49. 4.3 Valuing a Stream of Cash Flows <ul><li>Based on the first rule of time travel we can derive a general formula for valuing a stream of cash flows: if we want to find the present value of a stream of cash flows, we simply add up the present values of each. </li></ul>
50. 50. 4.3 Valuing a Stream of Cash Flows (cont’d) <ul><li>Present Value of a Cash Flow Stream </li></ul>
51. 51. Textbook Example 4.5
52. 52. Textbook Example 4.5 (cont’d)
53. 53. Textbook Example 4.5 Financial Calculator Solution
54. 54. Future Value of Cash Flow Stream <ul><li>Future Value of a Cash Flow Stream with a Present Value of PV </li></ul>
55. 55. Alternative Example 4.5 <ul><li>Problem </li></ul><ul><ul><li>What is the future value in three years of the following cash flows if the compounding rate is 5%? </li></ul></ul>
56. 56. Alternative Example 4.5 (cont'd) <ul><li>Solution </li></ul><ul><li>Or </li></ul>
57. 57. 4.4 Calculating the Net Present Value <ul><li>Calculating the NPV of future cash flows allows us to evaluate an investment decision. </li></ul><ul><li>Net Present Value compares the present value of cash inflows (benefits) to the present value of cash outflows (costs). </li></ul>
58. 58. Textbook Example 4.6
59. 59. Textbook Example 4.6 (cont'd)
60. 60. Textbook Example 4.6 Financial Calculator Solution
61. 61. Alternative Example 4.6 <ul><li>Problem </li></ul><ul><ul><li>Would you be willing to pay \$5,000 for the following stream of cash flows if the discount rate is 7%? </li></ul></ul>
62. 62. Alternative Example 4.6 (cont’d) <ul><li>Solution </li></ul><ul><ul><li>The present value of the benefits is: </li></ul></ul><ul><ul><li>3000 / (1.05) + 2000 / (1.05) 2 + 1000 / (1.05) 3 = 5366.91 </li></ul></ul><ul><ul><li>The present value of the cost is \$5,000, because it occurs now. </li></ul></ul><ul><ul><li>The NPV = PV (benefits) – PV (cost) </li></ul></ul><ul><ul><li> = 5366.91 – 5000 = 366.91 </li></ul></ul>
63. 63. Alternative Example 4.6 Financial Calculator Solution <ul><li>On a present value basis, the benefits exceed the costs by \$366.91. </li></ul>
64. 64. 4.5 Perpetuities, Annuities, and Other Special Cases <ul><li>When a constant cash flow will occur at regular intervals forever it is called a perpetuity. </li></ul>
65. 65. 4.5 Perpetuities, Annuities, and Other Special Cases (cont’d) <ul><li>The value of a perpetuity is simply the cash flow divided by the interest rate. </li></ul><ul><li>Present Value of a Perpetuity </li></ul>
66. 66. Textbook Example 4.7
67. 67. Textbook Example 4.7 (cont’d)
68. 68. Alternative Example 4.7 <ul><li>Problem </li></ul><ul><ul><li>You want to endow a chair for a female professor of finance at your alma mater. You’d like to attract a prestigious faculty member, so you’d like the endowment to add \$100,000 per year to the faculty member’s resources (salary, travel, databases, etc.) If you expect to earn a rate of return of 4% annually on the endowment, how much will you need to donate to fund the chair? </li></ul></ul>
69. 69. Alternative Example 4.7 (cont’d) <ul><li>Solution </li></ul><ul><ul><li>The timeline of the cash flows looks like this: </li></ul></ul><ul><ul><li>This is a perpetuity of \$100,000 per year. The funding you would need to give is the present value of that perpetuity. From the formula: </li></ul></ul><ul><ul><li>You would need to donate \$2.5 million to endow the chair. </li></ul></ul>
70. 70. Annuities <ul><li>When a constant cash flow will occur at regular intervals for a finite number of N periods, it is called an annuity. </li></ul><ul><li>Present Value of an Annuity </li></ul>
71. 71. Present Value of an Annuity <ul><li>To find a simpler formula, suppose you invest \$100 in a bank account paying 5% interest. As with the perpetuity, suppose you withdraw the interest each year. Instead of leaving the \$100 in forever, you close the account and withdraw the principal in 20 years. </li></ul>
72. 72. Present Value of an Annuity (cont’d) <ul><li>You have created a 20-year annuity of \$5 per year, plus you will receive your \$100 back in 20 years. So: </li></ul><ul><li>Re-arranging terms: </li></ul>
73. 73. Present Value of an Annuity <ul><li>For the general formula, substitute P for the principal value and: </li></ul>
74. 74. Textbook Example 4.8
75. 75. Textbook Example 4.8
76. 76. Textbook Example 4.8 Financial Calculator Solution <ul><li>Since the payments begin today, this is an Annuity Due. </li></ul><ul><ul><li>First: </li></ul></ul>
77. 77. Textbook Example 4.8 Financial Calculator Solution (cont’d) <ul><ul><li>Then: </li></ul></ul><ul><ul><ul><li>\$15 million > \$12.16 million, so take the lump sum. </li></ul></ul></ul>
78. 78. Future Value of an Annuity <ul><li>Future Value of an Annuity </li></ul>
79. 79. Textbook Example 4.9
80. 80. Textbook Example 4.9 (cont’d)
81. 81. Textbook Example 4.9 Financial Calculator Solution <ul><li>Since the payments begin in one year, this is an Ordinary Annuity. </li></ul><ul><ul><li>Be sure to put the calculator back on “End” mode: </li></ul></ul>
82. 82. Textbook Example 4.9 Financial Calculator Solution (cont’d) <ul><ul><li>Then: </li></ul></ul>
83. 83. Growing Perpetuities <ul><li>Assume you expect the amount of your perpetual payment to increase at a constant rate, g . </li></ul><ul><li>Present Value of a Growing Perpetuity </li></ul>
84. 84. Textbook Example 4.10
85. 85. Textbook Example 4.10 (cont’d)
86. 86. Alternative Example 4.10 <ul><li>Problem </li></ul><ul><ul><li>In Alternative Example 4.7, you planned to donate money to endow a chair at your alma mater to supplement the salary of a qualified individual by \$100,000 per year. Given an interest rate of 4% per year, the required donation was \$2.5 million. The University has asked you to increase the donation to account for the effect of inflation, which is expected to be 2% per year. How much will you need to donate to satisfy that request? </li></ul></ul>
87. 87. Alternative Example 4.10 (cont’d) The timeline of the cash flows looks like this: The cost of the endowment will start at \$100,000, and increase by 2% each year. This is a growing perpetuity. From the formula: You would need to donate \$5.0 million to endow the chair. <ul><li>Solution </li></ul>
88. 88. Growing Annuities <ul><li>The present value of a growing annuity with the initial cash flow c , growth rate g , and interest rate r is defined as: </li></ul><ul><ul><li>Present Value of a Growing Annuity </li></ul></ul>
89. 89. Textbook Example 4.11
90. 90. Textbook Example 4.11
91. 91. 4.6 Solving Problems with a Spreadsheet Program <ul><li>Spreadsheets simplify the calculations of TVM problems </li></ul><ul><ul><li>NPER </li></ul></ul><ul><ul><li>RATE </li></ul></ul><ul><ul><li>PV </li></ul></ul><ul><ul><li>PMT </li></ul></ul><ul><ul><li>FV </li></ul></ul><ul><li>These functions all solve the problem: </li></ul>
92. 92. Textbook Example 4.12
93. 93. Textbook Example 4.12 (cont’d)
94. 94. Textbook Example 4.13
95. 95. Textbook Example 4.13 (cont’d)
96. 96. 4.7 Solving for Variables Other Than Present Values or Future Values <ul><li>Sometimes we know the present value or future value, but do not know one of the variables we have previously been given as an input. For example, when you take out a loan you may know the amount you would like to borrow, but may not know the loan payments that will be required to repay it. </li></ul>
97. 97. Textbook Example 4.14
98. 98. Textbook Example 4.14 (cont’d)
99. 99. 4.7 Solving for Variables Other Than Present Values or Future Values (cont’d) <ul><li>In some situations, you know the present value and cash flows of an investment opportunity but you do not know the internal rate of return (IRR) , the interest rate that sets the net present value of the cash flows equal to zero. </li></ul>
100. 100. Textbook Example 4.15
101. 101. Textbook Example 4.15 (cont’d)
102. 102. Textbook Example 4.16
103. 103. Textbook Example 4.16 (cont’d)
104. 104. 4.7 Solving for Variables Other Than Present Values or Future Values (cont’d) <ul><li>In addition to solving for cash flows or the interest rate, we can solve for the amount of time it will take a sum of money to grow to a known value. </li></ul>
105. 105. Textbook Example 4.17
106. 106. Textbook Example 4.17
107. 107. Discussion of Data Case Key Topic <ul><li>Would your answer change if the certification job would require her to stay in Seattle but an MBA would allow her to move to New York City, where Natasha has always wanted to live? </li></ul><ul><ul><li>Hint: Consider the cost of living differences between Seattle and New York. Would her salary be different in Seattle versus New York? </li></ul></ul><ul><ul><li>http: //cgi .money. cnn . com/tools/costofliving/costofliving .html </li></ul></ul>
108. 108. Chapter Quiz <ul><li>Can you compare or combine cash flows at different times? </li></ul><ul><li>How do you calculate the present value of a cash flow stream? </li></ul><ul><li>What benefit does a firm receive when it accepts a project with a positive NPV? </li></ul><ul><li>How do you calculate the present value of a </li></ul><ul><ul><li>Perpetuity? </li></ul></ul><ul><ul><li>Annuity? </li></ul></ul><ul><ul><li>Growing perpetuity? </li></ul></ul><ul><ul><li>Growing annuity? </li></ul></ul>