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# Time Value of Money

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1. Time Value of Money FIL 404 Prepared by Keldon Bauer

2. Cash Flow Time Lines • You win a contest, and you have the option of taking \$1.4 million now or \$250,000 per year for five years. • Which should you take? • The answer comes through taking into consideration the time value of money.

3. 2 3 4 5 0 1 Cash Flow Time Lines • The first step is visualizing the cash flows by drawing a cash flow time line. • Time lines show when cash flows occur. • Time 0 is now.

4. 2 3 4 5 0 1 8% \$250K \$250K \$250K \$250K \$250K Cash Flow Time Lines • Outflows are listed as negatives. • Inflows are positive. • State the appropriate “interest rate,” which represents your opportunity costs

5. Future Value • Future value is higher than today, because if I had the money I would put it to work, it would earn interest. • The interest could then earn interest. • Compounding: allowing interest to earn interest on itself.

6. 2 3 4 5 0 1 8% Principal -1 Interest 0.08 0.0864 0.0933 0.1008 0.1088 Prev. Interest 0.00 0.0800 0.1664 0.2597 0.3605 Total 1.08 1.1664 1.2597 1.3605 1.4693 Future Value - Example • If you invest \$1,000 today at 8% interest per year, how much should you have in five years (in thousands).

7. Future Value • For one year, the future value can be defined as:

8. Future Value • The second year, the future value can be stated as follows:

9. Future Value • Therefore, the general solution to the future value problem is: • The Excel formula is: • =FV(Interest, Term, Payments, [Present Value], [Type])

10. Future Value - Excel

11. Future Value • Interest can be seen as the opportunity growth rate of money.

12. 2 3 4 5 0 1 8% PV=? \$500 Present Value • Present value is the value in today’s dollars of a future cash flow. • If we are interested in the present value of \$500 delivered in 5 years:

13. Present Value • The general solution to this problem follows from the solution to the future value problem:

14. Present Value - Excel • The Excel formula is: • =PV(Interest, Term, Payments, [Future Value], [Type])

15. Present Value - Excel

16. Present Value • Since the discount rate is the opportunity cost, the present value represents what I would have to give up now to get the future value specified.

17. 2 3 4 5 0 1 ?% \$100 \$500 Interest Rates • If we know the amount we need at time n and the amount we can invest at time zero, then we must only solve for the interest rate.

18. Interest Rates • Solving for interest rates algebraically:

19. 2 3 4 5 0 1 ?% \$100 \$500 Interest Rates - Example In Excel: =RATE(Term, Payment, Present Value, [Future Value], [Type], [Guess])

20. Interest Rate - Excel

21. Time Periods • If the present value, future value and interest rate are known, but the number of time periods is not. Then n can be found algebraically:

22. Time Periods - Example • If we use the last example of investing \$100, we want \$500 in future, and the current market interest is 8%, n can be found: In Excel: =NPER(Interest, Payment, Present Value, [Future Value], [Type])

23. Time Periods - Excel

24. Annuities • Definition: A series of equal payments at a fixed interval. • Two types: • Ordinary annuity: Payments occur at the end of each period. (Default in Excel) • Annuity due: Payments occur at the beginning of each period. (Set the type = 1 in Excel) • In Excel, use the same formulas introduced so far, just specify payment and type.

25. 2 3 4 5 0 1 8% \$100 \$100 \$100 \$100 \$100 FV=? What is the future value? Ordinary Annuity • Example is a regular payment of \$100 for five years earning 8% interest.

26. Ordinary Annuity – Future Value • The future value of an ordinary annuity can be found as follows:

27. 2 3 4 5 0 1 8% \$100 \$100 \$100 \$100 \$100 Ordinary Annuity - Example

28. Ordinary Annuity - Example

29. 2 3 4 5 0 1 8% \$100 \$100 \$100 \$100 \$100 FV=? What is the future value? Annuity Due • Example is a regular payment of \$100 for five years earning 8% interest.

30. Annuity Due – Future Value • The future value of an annuity due can be found by noticing that the annuity due is the same as an ordinary annuity, with one more compounding period:

31. 8% \$100 \$100 \$100 \$100 \$100 2 3 4 5 0 1 Annuity Due - Example

32. Annuity Due - Excel

33. 2 3 4 5 0 1 8% \$100 \$100 \$100 \$100 \$100 PV=? What is the present value? Ordinary Annuity - Present Value • Example is a regular payment of \$100 for five years earning 8% interest.

34. Ordinary Annuity - Present Value • The present value of an ordinary annuity can be found as follows:

35. 2 3 4 5 0 1 8% \$100 \$100 \$100 \$100 \$100 Ordinary Annuity - Example

36. Ordinary Annuity - Excel

37. 2 3 4 5 0 1 8% \$100 \$100 \$100 \$100 \$100 PV=? What is the present value? Annuity Due - Present Value • Example is a regular payment of \$100 for five years earning 8% interest.

38. Annuity Due - Present Value • The future value of an annuity due can be found as follows:

39. 8% \$100 \$100 \$100 \$100 \$100 2 3 4 5 0 1 Annuity Due - Example

40. Annuity Due - Excel

41. Annuities - Finding Interest Rate • Interest rates cannot be solved directly. • Calculators and computers search for the correct answer (there is only one correct answer). • It guesses and then iteratively goes higher or lower.

42. Perpetuities • What would you have to pay to be paid \$2,000 per year forever (given a market rate of 8%)?

43. Uneven Cash Flow Streams • If payments are irregular or come at irregular intervals, we can still find the PV (or FV). • Take the present value (or future value) of individual payments and sum them together.

44. \$ 92.59 2 3 4 5 0 1 8% \$171.47 \$238.15 \$294.01 \$100 \$200 \$300 \$400 \$500 \$340.29 Uneven Cash Flows - Example \$1,136.51 = Present Value

45. Uneven Cash Flows - Excel • Excel can do this in one argument: • =NPV(Interest, Array of Payments [starting with payment for time 1]). • If you want to include a payment in time zero, add it to the above argument separately.

46. Uneven Cash Flows - Excel

47. \$432.00 2 3 4 5 0 1 \$349.92 8% \$251.94 \$136.05 \$100 \$200 \$300 \$400 \$500 \$1,136.51 = Present Value Uneven Cash Flow - Example Future Value = \$1,669.91

48. Uneven Cash Flows - Excel

49. Finding Interest Rate • As with annuities, interest rates for uneven cash flow streams cannot be solved directly. • Calculators search for the correct answer, called an IRR (there may be more than one correct answer). • It guesses and then iteratively goes higher or lower.

50. 8% 2 3 4 5 0 1 -\$100 \$146.93 0 1 2 3 4 5 6 7 8 9 10 4% -\$100 \$148.02 Compounding • The more often one compounds interest, the faster it grows. Annual Semi-Annual