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# Chapter 4 The Time Value of Money

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1. Chapter 4 The Time Value of Money

2. Chapter Outline 4.1 The Timeline 4.2 The Three Rules of Time Travel 4.3 Valuing a Stream of Cash Flows 4.4 Calculating the Net Present Value 4.5 Perpetuities and Annuities

3. Chapter Outline (cont’d) 4.6 Solving Problems with a Spreadsheet or Calculator 4.7 Non-Annual Cash Flows 4.8 Solving for the Cash Payments 4.9 _The Internal Rate of Return

4. 4.1 The Timeline • A timeline is a linear representation of the timing of potential cash flows. • Drawing a timeline of the cash flows will help you visualize the financial problem.

5. 4.1 The Timeline (cont’d) • 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.

6. 4.1 The Timeline (cont’d) • Differentiate between two types of cash flows • Inflows are positive cash flows. • Outflows are negative cash flows, which are indicated with a – (minus) sign.

7. 4.1 The Timeline (cont’d) • Assume that you are lending \$10,000 today and that the loan will be repaid in two annual \$6,000 payments. • The first cash flow at date 0 (today) is represented as a negative sum because it is an outflow. • Timelines can represent cash flows that take place at the end of any time period – a month, a week, a day, etc.

8. Textbook Example 4.1

9. Textbook Example 4.1 (cont’d)

10. 4.2 Three Rules of Time Travel • Financial decisions often require combining cash flows or comparing values. • Three rules govern these processes. Table 4.1The Three Rules of Time Travel

11. The 1st Rule of Time Travel • A dollar today and a dollar in one year are not equivalent. • It is only possible to compare or combine values at the same point in time. • Which would you prefer: A gift of \$1,000 today or \$1,210 at a later date? • To answer this, you will have to compare the alternatives to decide which is worth more. One factor to consider: How long is “later?”

12. The 2nd Rule of Time Travel • To move a cash flow forward in time, you must compound it. • 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:

13. The 2nd Rule of Time Travel (cont’d) • Future Value of a Cash Flow

14. The 2nd Rule of Time Travel (cont’d) • The following graph shows the account balance and the composition of interest over time when an investor starts with an initial deposit of \$1000, shown in red, in an account earning 10% interest over a 20-year period. • Note that the turquoise area representing interest on interest grows, and by year 15 has become larger than the interest on the original deposit, shown in green. • In year 20, the interest on interest the investor earned is \$3727.50, while the total interest earned on the original \$1000 is \$2000.

15. Figure 4.1 The Composition of Interest Over Time

16. Textbook Example 4.2

17. Textbook Example 4.2 (cont’d)

18. Alternative Example 4.2 • Problem • 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.

19. Solution The time line looks like this: In five years, the \$5,000 will grow to: \$5,000 × (1.10)5 = \$8,053 The future value of \$5,000 at 10% for five years is \$8,053. You would be better off forgoing the gift of \$5,000 today and taking the \$10,000 in five years. Alternative Example 4.2 (cont’d)

20. The 3rd Rule of Time Travel • To move a cash flow backward in time, we must discount it. • Present Value of a Cash Flow

21. Textbook Example 4.3

22. Textbook Example 4.3

23. Alternative Example 4.3 • Problem • 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?

24. Alternative Example 4.3 (cont’d) • Solution • The \$10,000 is worth: • \$10,000 ÷ (1.10)5 = \$6,209

25. Applying the Rules of Time Travel • 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. • 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?

26. Applying the Rules of Time Travel (cont'd) • The time line would look like this:

27. Applying the Rules of Time Travel (cont'd)

28. Applying the Rules of Time Travel (cont'd)

29. Applying the Rules of Time Travel (cont'd)

30. Table 4.1 The Three Rules of Time Travel

31. Textbook Example 4.4

32. Textbook Example 4.4 (cont’d)

33. 4.3 Valuing a Stream of Cash Flows • 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.

34. Present Value of a Cash Flow Stream 4.3 Valuing a Stream of Cash Flows (cont’d)

35. Textbook Example 4.5

36. Future Value of Cash Flow Stream • Future Value of a Cash Flow Stream with a Present Value of PV

37. 4.4 Calculating the Net Present Value • Calculating the NPV of future cash flows allows us to evaluate an investment decision. • Net Present Value compares the present value of cash inflows (benefits) to the present value of cash outflows (costs).

38. Textbook Example 4.6

39. 4.5 Perpetuities and Annuities • Perpetuities • When a constant cash flow will occur at regular intervals forever it is called a perpetuity.

40. 4.5 Perpetuities and Annuities (cont’d) • The value of a perpetuity is simply the cash flow divided by the interest rate. • Present Value of a Perpetuity

41. Textbook Example 4.7

42. Textbook Example 4.7 (cont’d)

43. 4.5 Perpetuities and Annuities (cont’d) • Annuities • When a constant cash flow will occur at regular intervals for a finite number of N periods, it is called an annuity. • Present Value of an Annuity

44. Present Value of an Annuity • 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.

45. Present Value of an Annuity (cont’d) • You have created a 20-year annuity of \$5 per year, plus you will receive your \$100 back in 20 years. So: • Re-arranging terms:

46. Present Value of an Annuity • For the general formula, substitute P for the principal value and:

47. Future Value of an Annuity • Future Value of an Annuity

48. Textbook Example 4.9