The Time Value of Money

1. The Time Value of Money Chapter 9

2. The Time Value of Money • Which would you rather have ? • $100 today - or • $100 one year from today • Sooner is better !

3. The Time Value of Money • How about $100 today or $105 one year from today? • We revalue current dollars and future dollars using the time value of money • Cash flow time line graphically shows the timing of cash flows

4. 0 1 2 3 4 5 Cash Flow Time Lines • Time 0 is today; Time 1 is one period from today Interest rate Time 5% Cash Flows -100 105 Outflow Inflow

5. Future Value • Compounding • the process of determining the value of a cash flow or series of cash flows some time in the future when compound interest is applied

6. Future Value • PV = present value or starting amount, say, $100 • i = interest rate, say, 5% per year would be shown as 0.05 • INT = dollars of interest you earn during the year $100  0.05 = $5 • FVn = future value after n periods or $100 + $5 = $105 after one year • = $100 (1 + 0.05) = $100(1.05) = $105

7. Future Value

8. Future Value • The amount to which a cash flow or series of cash flows will grow over a given period of time when compounded at a given interest rate

9. Compounded Interest • Interest earned on interest

10. 0 1 2 3 4 5 Cash Flow Time Lines 5% Time -100 Interest 5.00 5.25 5.51 5.79 6.08 Total Value 105.00 110.25 115.76 121.55 127.63

11. Future Value Interest Factor for i and n (FVIFi,n) • The future value of $1 left on deposit for n periods at a rate of i percent per period • The multiple by which an initial investment grows because of the interest earned

12. Future Value Interest Factor for i and n (FVIFi,n) • FVn = PV(1 + i)n = PV(FVIFi,n) For $100 at i = 5% and n = 5 periods

13. Future Value Interest Factor for i and n (FVIFi,n) • FVn = PV(1 + i)n = PV(FVIFi,n) For $100 at i = 5% and n = 5 periods $100 (1.2763) = $127.63

14. Financial Calculator Solution • Five keys for variable input • N = the number of periods • I = interest rate per periodmay be I, INT, or I/Y • PV = present value • PMT = annuity payment • FV = future value

15. 0 1 2 3 4 5 Time 5% Cash Flows -100 5.00 Total Value 105.00 Two Solutions • Find the future value of $100 at 5% interest per year for five years • 1. Numerical Solution: 5.25 5.51 5.79 6.08 110.25 115.76 121.55 127.63 FV5 = $100(1.05)5 = $100(1.2763) = $127.63

16. Two Solutions 2. Financial Calculator Solution: Inputs: N = 5 I = 5 PV = -100 PMT = 0 FV = ? Output: = 127.63

17. Graphic View of the Compounding Process: Growth • Relationship among Future Value, Growth or Interest Rates, and Time Future Value of $1 i= 15% i= 10% i= 5% i= 0% 2 4 6 8 10 0 Periods

18. Present Value • Opportunity cost • the rate of return on the best available alternative investment of equal risk • If you can have $100 today or $127.63 at the end of five years, your choice will depend on your opportunity cost

19. Present Value • The present value is the value today of a future cash flow or series of cash flows • The process of finding the present value is discounting, and is the reverse of compounding • Opportunity cost becomes a factor in discounting

20. 0 1 2 3 4 5 Cash Flow Time Lines 5% PV = ? 127.63

21. Present Value • Start with future value: • FVn = PV(1 + i)n

22. 0 1 2 3 4 5 5% PV = ? 127.63 Two Solutions • Find the present value of $127.63 in five years when the opportunity cost rate is 5% • 1. Numerical Solution: ÷ 1.05 ÷ 1.05 ÷ 1.05 ÷ 1.05 -100.00 105.00 110.25 115.76 121.55

23. Two Solutions • Find the present value of $127.63 in five years when the opportunity cost rate is 5% • 2. Financial Calculator Solution: Inputs: N = 5 I = 5 PMT = 0 FV = 127.63 PV = ? Output: = -100

24. Graphic View of the Discounting Process Present Value of $1 • Relationship among Present Value, Interest Rates, and Time i= 0% i= 5% i= 10% i= 15% 2 4 6 8 10 12 14 16 18 20 Periods

25. Solving for Time and Interest Rates • Compounding and discounting are reciprocals • FVn = PV(1 + i)n Four variables: PV, FV, i and n If you know any three, you can solve for the fourth

26. 0 1 2 3 4 5 i = ? -78.35 100 Solving for i • For $78.35 you can buy a security that will pay you $100 after five years • We know PV, FV, and n, but we do not know i FVn = PV(1 + i)n $100 = $78.35(1 + i)5 Solve for i

27. Numerical Solution • FVn = PV(1 + i)n • $100 = $78.35(1 + i)5

28. Financial Calculator Solution • Inputs: N = 5 PV = -78.35 PMT = 0 FV = 100 I = ? • Output: = 5 • This procedure can be used for any rate or value of n, including fractions

29. Solving for n • Suppose you know that the security will provide a return of 10 percent per year, that it will cost $68.30, and that you will receive $100 at maturity, but you do not know when the security matures. You know PV, FV, and i, but you do not know n - the number of periods.

30. 10% -68.30 100 Solving for n • FVn = PV(1 + i)n • $100 = $68.30(1.10)n • By trial and error you could substitute for n and find that n = 4 0 1 2 n-1 n=?

31. Financial Calculator Solution • Inputs: I = 10 PV = -68.30 PMT = 0 FV = 100 N = ? • Output: = 4.0

32. Annuity • An annuity is a series of payments of an equal amount at fixed intervals for a specified number of periods • Ordinary (deferred) annuity has payments at the end of each period • Annuity due has payments at the beginning of each period • FVAn is the future value of an annuity over n periods

33. Future Value of an Annuity • The future value of an annuity is the amount received over time plus the interest earned on the payments from the time received until the future date being valued • The future value of each payment can be calculated separately and then the total summed

34. 0 5% 1 2 3 100 315.25 Future Value of an Annuity • If you deposit $100 at the end of each year for three years in a savings account that pays 5% interest per year, how much will you have at the end of three years? 100 100.00 = 100 (1.05)0 105.00 = 100 (1.05)1 110.25 = 100 (1.05)2

35. Future Value of an Annuity

36. Future Value of an Annuity • Financial calculator solution: • Inputs: N = 3 I = 5 PV = 0 PMT = -100 FV = ? • Output: = 315.25 • To solve the same problem, but for the present value instead of the future value, change the final input from FV to PV

37. Annuities Due • If the three $100 payments had been made at the beginning of each year, the annuity would have been an annuity due. • Each payment would shift to the left one year and each payment would earn interest for an additional year (period).

38. 0 5% 1 2 3 100 315.25 Future Value of an Annuity • $100 at the end of each year 100 100.00 = 100 (1.05)0 105.00 = 100 (1.05)1 110.25 = 100 (1.05)2

39. 100 100 105.00 = 100 (1.05)1 0 5% 1 2 3 110.25 = 100 (1.05)2 115.7625 = 100 (1.05)3 100 331.0125 Future Value of an Annuity Due • $100 at the start of each year

40. Future Value of an Annuity Due • Numerical solution:

41. Future Value of an Annuity Due • Numerical solution:

42. Future Value of an Annuity Due • Financial calculator solution: • Inputs: N = 3 I = 5 PV = 0 PMT = -100 FV = ? • Output: = 331.0125

43. Present Value of an Annuity • If you were offered a three-year annuity with payments of $100 at the end of each year • Or a lump sum payment today that you could put in a savings account paying 5% interest per year • How large must the lump sum payment be to make it equivalent to the annuity?

44. 100 100 0 5% 1 2 3 100 272.325 Present Value of an Annuity

45. Present Value of an Annuity • Numerical solution:

46. Present Value of an Annuity

47. Present Value of an Annuity • Financial calculator solution: • Inputs: N = 3 I = 5 PMT = -100 FV = 0 PV = ? • Output: = 272.325

48. Present Value of an Annuity Due • Payments at the beginning of each year • Payments all come one year sooner • Each payment would be discounted for one less year • Present value of annuity due will exceed the value of the ordinary annuity by one year’s interest on the present value of the ordinary annuity

49. 285.941 Present Value of an Annuity Due 0 5% 1 2 3 100 100

50. Present Value of an Annuity Due • Numerical solution: