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The Time Value of Money

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- 1. ALAN ANDERSON, Ph.D. ECI RISK TRAINING www.ecirisktraining.com
- 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. 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. 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. 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. 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. 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. 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. In this case, N=4 I=3 PV = $1,000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 9
- 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. 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. 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. 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. In this case, N=3 I=5 FV3 = $1,000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 14
- 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. 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. 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. 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. 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. 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. 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. In this case, N=4 I=3 PMT = $1,000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 22
- 23. Using the formula, ⎡ (1 + I ) − 1 ⎤ N FVAN = PMT ⎢ ⎥ ⎣ I ⎦ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 23
- 24. ⎡ (1 + .03) − 1 ⎤ 4 FVA4 = 1,000 ⎢ ⎥ = $4,183.63 ⎣ .03 ⎦ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 24
- 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. 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. 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. 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. 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. 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. 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. In this case, N=3 I=5 PMT = $1,000 (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 32
- 33. Using the formula, ⎡ 1 ⎤ 1− ⎢ (1 + I )N ⎥ PVAN = PMT ⎢ ⎥ ⎢ I ⎥ ⎢ ⎣ ⎥ ⎦ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 33
- 34. ⎡ 1 ⎤ 1− ⎢ (1 + .05)3 ⎥ PVA3 = 1, 000 ⎢ ⎥ = $2, 723.25 ⎢ .05 ⎥ ⎢ ⎣ ⎥ ⎦ (c) ECI RISK TRAINING 2009 www.ecirisktraining.com 34
- 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. 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. 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. 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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