Time Value Of Money Part 1

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The Time Value of Money

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Time Value Of Money Part 1

  1. 1. ALAN ANDERSON, Ph.D. ECI RISK TRAINING www.ecirisktraining.com
  2. 2. 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 2
  3. 3. The time value of money is one of the most fundamental concepts in finance; it is based on the notion that receiving a sum of money in the future is less valuable than receiving that sum today. This is because a sum received today can be invested and earn interest. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 3
  4. 4. The four basic time value of money concepts are:  future value of a sum  present value of a sum  future value of an annuity  present value of an annuity (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 4
  5. 5. If a sum is invested today, it will earn interest and increase in value over time. The value that the sum grows to is known as its future value. Computing the future value of a sum is known as compounding. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 5
  6. 6. The future value of a sum depends on the interest rate earned and the time horizon over which the sum is invested. This is shown with the following formula: FVN = PV(1+I)N (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 6
  7. 7. where: FVN = future value of a sum invested for N periods I = periodic rate of interest PV = the present or current value of the sum invested (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 7
  8. 8. Suppose that a sum of $1,000 is invested for four years at an annual rate of interest of 3%. What is the future value of this sum? (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 8
  9. 9. In this case, N=4 I=3 PV = $1,000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 9
  10. 10. Using the future value formula, FVN = PV(1+I)N FV4 = 1,000(1+.03)4 FV4 = 1,000(1.125509) FV4 = $1,125.51 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 10
  11. 11. The present value of a sum is the amount that would need to be invested today in order to be worth that sum in the future. Computing the present value of a sum is known as discounting. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 11
  12. 12. The formula for computing the present value of a sum is: FVN PV = (1 + I ) N (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 12
  13. 13. How much must be deposited in a bank account that pays 5% interest per year in order to be worth $1,000 in three years?
 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 13
  14. 14. In this case, N=3 I=5 FV3 = $1,000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 14
  15. 15. FVN 1, 000 PV = = (1 + I ) N (1.05) 3 1, 000 = = $863.84 1.1576 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 15
  16. 16. An annuity is a periodic stream of equally-sized payments. The two basic types of annuities are:  ordinary annuity  annuity due (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 16
  17. 17. With an ordinary annuity, the first payment takes place one period in the future. With an annuity due, the first payment takes place immediately. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 17
  18. 18. The formulas used to compute the future value and present value of a sum can be easily extended to the case of an annuity. (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 18
  19. 19. The formula for computing the future value of an ordinary annuity is: ⎡ (1 + I ) − 1 ⎤ N FVAN = PMT ⎢ ⎥ ⎣ I ⎦ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 19
  20. 20. where: FVAN = future value of an N-period ordinary annuity PMT = the value of the periodic payment (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 20
  21. 21. Suppose that a sum of $1,000 is invested at the end of each of the next four years at an annual rate of interest of 3%. What is the future value of this ordinary annuity? (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 21
  22. 22. In this case, N=4 I=3 PMT = $1,000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 22
  23. 23. Using the formula, ⎡ (1 + I ) − 1 ⎤ N FVAN = PMT ⎢ ⎥ ⎣ I ⎦ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 23
  24. 24. ⎡ (1 + .03) − 1 ⎤ 4 FVA4 = 1,000 ⎢ ⎥ = $4,183.63 ⎣ .03 ⎦ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 24
  25. 25. The future value of the annuity can also be obtained by computing the future value of each term and then combining the results: (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 25
  26. 26. 1,000(1.03)3 + 1,000(1.03)2 + 1,000(1.03)1 + 1,000(1.03)0 = 1,092.73 + 1,060.90 + 1,030.00 + 1,000.00 = $4,183.63 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 26
  27. 27. The future value of an annuity due is computed as follows: FVAdue = FVAordinary(1+I) (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 27
  28. 28. Referring to the previous example, the future value of an annuity due would be: 4,183.63(1+.03) = $4,309.14 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 28
  29. 29. The formula for computing the present value of an ordinary annuity is: ⎡ 1 ⎤ 1− ⎢ (1 + I )N ⎥ PVAN = PMT ⎢ ⎥ ⎢ I ⎥ ⎢ ⎣ ⎥ ⎦ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 29
  30. 30. where: PVAN = future value of an N-period ordinary annuity PMT = the value of the periodic payment (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 30
  31. 31. How much must be invested today in a bank account that pays 5% interest per year in order to generate a stream of payments of $1,000 at the end of each of the next three years?
 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 31
  32. 32. In this case, N=3 I=5 PMT = $1,000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 32
  33. 33. Using the formula, ⎡ 1 ⎤ 1− ⎢ (1 + I )N ⎥ PVAN = PMT ⎢ ⎥ ⎢ I ⎥ ⎢ ⎣ ⎥ ⎦ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 33
  34. 34. ⎡ 1 ⎤ 1− ⎢ (1 + .05)3 ⎥ PVA3 = 1, 000 ⎢ ⎥ = $2, 723.25 ⎢ .05 ⎥ ⎢ ⎣ ⎥ ⎦ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 34
  35. 35. The present value of the annuity can also be obtained by computing the present value of each term and then combining the results: (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 35
  36. 36. 1,000(1.05)-3 + 1,000(1.05)-2 + 1,000(1.05)-1 = 863.84 + 907.03 + 952.38 = $2723.25 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 36
  37. 37. The present value of an annuity due is computed as follows: PVAdue = PVAordinary(1+I) (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 37
  38. 38. Referring to the previous example, the present value of an annuity due would be: 2,723.25(1+.05) = $2,859.41 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 38

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