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Time Value Of Money Part 2

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The Time Value of Money

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Time Value Of Money Part 2

1. 1. ALAN ANDERSON, Ph.D. ECI RISK TRAINING www.ecirisktraining.com
2. 2. The time value of money formulas can be used to solve for the appropriate rate of interest or time horizon given the present and future value of a sum. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 40
3. 3. The present and future value formulas can be used to solve for the rate of interest. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 41
4. 4. Suppose that an investor deposits \$10,000 in a bank account. The investor plans to keep these funds in the bank for ten years, with a goal of having \$20,000 at the end of that time. What rate of interest would he have to earn to double his money in ten years? (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 42
5. 5. This can be determined algebraically as follows: FVN = PV(1 + I)N FVN = (1 + I ) N PV (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 43
6. 6. FVN N = (1 + I ) PV FVN N −1= I PV (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 44
7. 7. In this example, 20, 000 10 − 1 = 0.07177 = 7.177% 10, 000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 45
8. 8. The present and future value formulas can also be used to solve for the time horizon. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 46
9. 9. Suppose that an investor deposits \$5,000 in a bank account that pays 6% interest per year. The investor wants to know how long it will take for these funds to be worth \$10,000. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 47
10. 10. This can be determined algebraically as follows: FVN = PV(1 + I)N FVN = (1 + I ) N PV (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 48
11. 11. ⎛ FVN ⎞ ln ⎜ ⎝ PV ⎠⎟ = N ln(1 + I ) ⎛ FVN ⎞ ln ⎜ ⎟ ⎝ PV ⎠ N= ln(1 + I ) (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 49
12. 12. In this example, ⎛ 10, 000 ⎞ ln ⎜ ⎟ ⎝ 5, 000 ⎠ N= = 11.896 ln(1 + .06) (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 50
13. 13. The Rule of 72 is a quick method for estimating the time horizon or the interest rate needed to double the value of an investment. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 51
14. 14. Dividing the interest rate into 72 gives the approximate number of years that it would take to double the value of an investment. For the example of the investor who needs to know how many years it would take to double his money at an interest rate of 6%, dividing 72 by 6 gives a result of 12, which is very close to the actual value of 11.896 years. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 52
15. 15. Dividing the number of years into 72 gives the approximate interest rate that would be required to double the value of an investment. For the example of the investor who needs to know what rate of interest is required to double his money in ten years, dividing 72 by 10 gives a result of 7.2%, which is very close to the actual value of 7.177%. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 53
16. 16. In the case of a stream of cash flows that are not equal, computing the future and present value of the cash flows is a more complex process. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 54
17. 17. The two basic types of uneven cash flows of interest in finance are: 1) an annuity with an additional payment during the final period 2) a cash flow stream with no pattern, known as an irregular stream of cash flows (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 55
18. 18. The cash flows from most bonds take the form of an annuity with an additional payment during the final period. Investment projects often generate irregular streams of cash flows to firms. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 56
19. 19. Suppose that a bond offers investors cash flows of \$100 each year for the next three years, with an additional payment of \$1,000 at the end of the third year. If the periodic rate of interest is 5%, what is the present value of this stream of cash flows? (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 57
20. 20. In this case, N=3 I=5 PMT = \$100 FV3 = \$1,000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 58
21. 21. ⎡ 1 ⎤ 1− ⎢ (1 + I )N ⎥ PVAN = PMT ⎢ ⎥ ⎢ I ⎥ ⎢ ⎣ ⎥ ⎦ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 59
22. 22. ⎡ 1 ⎤ 1− ⎢ (1 + .05)3 ⎥ PVA3 = 100 ⎢ ⎥ = \$272.32 ⎢ .05 ⎥ ⎢ ⎣ ⎥ ⎦ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 60
23. 23. FVN 1, 000 PV = = (1 + I ) N (1.05) 3 1, 000 = = \$863.84 1.1576 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 61
24. 24. Combining these results gives the present value of the cash flow stream: PVA3 + PV = 272.32 + 863.84 = \$1,136.16 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 62
25. 25. Suppose that an investment project produces cash flows of \$200 at the end of the next two years, and \$300 at the end of the following three years. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 63
26. 26. If the periodic rate of interest is 4%, what is the present value of these cash flows? In this case, the present value of each cash flow is computed using the PV formula; these results are combined to give the present value of the stream of irregular cash flows. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 64
27. 27. In this case, the present value is: 200 200 300 300 300 1 + 2 + 3 + 4 + 5 (1.04) (1.04) (1.04) (1.04) (1.04) = 192.31 + 184.91 + 266.67 + 256.44 + 246.58 = \$1,146.91 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 65
28. 28. Each of the examples considered so far has been based on the assumption that interest is paid annually. When interest is paid more often than once per year, the present value and future value formulas must be adjusted. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 66
29. 29. Two adjustments must be made: 1) the periodic interest rate 2) the number of periods (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 67
30. 30.  The periodic interest rate equals: annual rate / number of periods per year  The number of periods equals: (number of years)(number of periods per year) (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 68
31. 31. Suppose that a sum of \$1,000 is invested for two years at an annual rate of interest of 4%. Compute the future value of this sum based on the assumption of: a) annual compounding b) semi-annual compounding (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 69
32. 32. With annual compounding, N=2 I=4 PV = \$1,000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 70
33. 33. Using the future value formula, FVN = PV(1+I)N FV2 = 1,000(1+.04)2 FV2 = 1,000(1.081600) FV2 = \$1081.60 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 71
34. 34. With semi-annual compounding, N=4 I=2 PV = \$1,000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 72
35. 35. Using the future value formula, FVN = PV(1+I)N FV4 = 1,000(1+.02)4 FV4 = 1,000(1.082432) FV4 = \$1082.43 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 73
36. 36. The more frequently interest is paid each year, the greater will be the future value of a sum or an annuity. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 74
37. 37. Compute the present value of \$1,000 to be received in four years using an annual interest rate of 6% with: a) annual compounding b) semi-annual compounding (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 75
38. 38. With annual compounding, N=4 I=6 FV4 = \$1,000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 76
39. 39. Using the present value formula, FVN 1000 PV = = = \$792.09 (1 + I ) N (1 + .06) 4 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 77
40. 40. With semi-annual compounding, N=8 I=3 FV8 = \$1,000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 78
41. 41. Using the present value formula, FVN 1000 PV = = = \$789.41 (1 + I ) N (1 + .03) 8 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 79
42. 42. The more frequently interest is paid each year, the smaller will be the present value of a sum or an annuity. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 80
43. 43. As the frequency of compounding increases, the present value of a sum or annuity decreases, while the future value of a sum or annuity increases. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 81
44. 44. The limiting compounding frequency is known as continuous compounding. In this case, interest is compounded at every instant in time. As a result, the number of compounding periods is infinite. The present and future value formulas with continuous compounding are: (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 82
45. 45. FVN = eIN FVN − IN PV = IN = FVN e e e = 2.7182818...... (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 83
46. 46. The present value of \$1,000 to be received in four years with an annual rate of interest of 5% compounded continuously is computed as follows: (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 84
47. 47. PV = 1,000e-(0.05)(4) = 1,000e-(0.20) = \$818.73 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 85
48. 48. The future value of \$1,000 invested for three years at an annual rate of interest of 4% compounded continuously is computed as follows: (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 86
49. 49. FV3 = 1,000e(0.04)(3) = 1,000e(0.12) = \$1,127.50 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 87
50. 50. In order to compare interest rates with different compounding frequencies, they can be converted into the effective annual rate (EAR). (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 88
51. 51. This is done with the following formula: M ⎛ APR ⎞ EAR = ⎜ 1 + ⎟ −1 ⎝ M ⎠ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 89
52. 52. where: APR = the annual percentage rate (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 90
53. 53. If a bank charges an APR of 6% per year, compounded quarterly for a loan, what is the effective annual rate? (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 91
54. 54. This can be determined with the formula, as follows: M ⎛ APR ⎞ EAR = ⎜ 1 + ⎟ −1 ⎝ M ⎠ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 92
55. 55. 4 ⎛ .06 ⎞ EAR = ⎜ 1 + ⎟ − 1 = 0.06136 ⎝ 4 ⎠ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 93
56. 56. This indicates that the borrower is actually paying 6.136% per year for this loan. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 94
57. 57. With continuous compounding, the EAR formula becomes: EAR = eAPR - 1 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 95
58. 58. If a bank charges an APR of 5% per year, continuously compounded, what is the effective annual rate? EAR = eAPR – 1 = e.05 – 1 = 0.051271 = 5.1271% (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 96
59. 59. For free problem sets based on this material along with worked-out solutions, write to info@ecirisktraining.com. To learn about training opportunities in finance and risk management, visit www.ecirisktraining.com (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 97