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

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