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# The Time Value of Money

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1. The Time Value of Money Compounding and Discounting Single Sums

2. Today Future We know that receiving \$1 today is worth more than \$1 in the future. This is duetoOPPORTUNITY COSTS. The opportunity cost of receiving \$1 in the future is theinterestwe could have earned if we had received the \$1 sooner.

3. If we can MEASURE this opportunity cost, we can: Today Future • Translate \$1 today into its equivalent in the future(COMPOUNDING). ?

4. If we can MEASURE this opportunity cost, we can: Today Future Today Future • Translate \$1 today into its equivalent in the future(COMPOUNDING). • Translate \$1 in the future into its equivalent today(DISCOUNTING). ? ?

5. Future Value

6. Future Value - single sumsIf you deposit \$100 in an account earning 6%, how much would you have in the account after 1 year? PV = FV = 0 1

7. Future Value - single sumsIf you deposit \$100 in an account earning 6%, how much would you have in the account after 1 year? Calculator Solution: I = 6 N = 1 PV = 100 FV = \$106 PV = 100 FV = 0 1

8. Future Value - single sumsIf you deposit \$100 in an account earning 6%, how much would you have in the account after 1 year? Calculator Solution: I = 6 N = 1 PV = 100 FV = \$106 PV = 100 FV = 106 0 1

9. Future Value - single sumsIf you deposit \$100 in an account earning 6%, how much would you have in the account after 1 year? Mathematical Solution: FV = PV (FVIF i, n) FV = 100 (FVIF .06, 1) (use FVIF table, or) FV = PV (1 + i)n FV = 100 (1.06)1 = \$106 PV = 100 FV = 106 0 1

10. Future Value - single sumsIf you deposit \$100 in an account earning 6%, how much would you have in the account after 5 years? PV = FV = 0 5

11. Future Value - single sumsIf you deposit \$100 in an account earning 6%, how much would you have in the account after 5 years? Calculator Solution: I = 6 N = 5 PV = 100 FV = \$133.82 PV = 100 FV = 0 5

12. Future Value - single sumsIf you deposit \$100 in an account earning 6%, how much would you have in the account after 5 years? Calculator Solution: I = 6 N = 5 PV = 100 FV = \$133.82 PV = 100 FV = 133.82 0 5

13. Future Value - single sumsIf you deposit \$100 in an account earning 6%, how much would you have in the account after 5 years? Mathematical Solution: FV = PV (FVIF i, n) FV = 100 (FVIF .06, 5) (use FVIF table, or) FV = PV (1 + i)n FV = 100 (1.06)5 = \$133.82 PV = 100 FV = 133.82 0 5

14. Future Value - single sumsIf you deposit \$100 in an account earning 6% with quarterly compounding, how much would you have in the account after 5 years? PV = FV = 0 ?

15. Future Value - single sumsIf you deposit \$100 in an account earning 6% with quarterly compounding, how much would you have in the account after 5 years? Calculator Solution: P/Y = 4 I = 6 N = 20 PV = 100 FV = \$134.68 PV = 100 FV = 0 20

16. Future Value - single sumsIf you deposit \$100 in an account earning 6% with quarterly compounding, how much would you have in the account after 5 years? Calculator Solution: P/Y = 4 I = 6 N = 20 PV = 100 FV = \$134.68 PV = 100 FV = 134.68 0 20

17. Future Value - single sumsIf you deposit \$100 in an account earning 6% with quarterly compounding, how much would you have in the account after 5 years? Mathematical Solution: FV = PV (FVIF i, n) FV = 100 (FVIF .06, 20) (can’t use FVIF table) FV = PV (1 + i/m) m x n FV = 100 (1.015)20 = \$134.68 PV = 100 FV = 134.68 0 20

18. Future Value - single sumsIf you deposit \$100 in an account earning 6% with monthly compounding, how much would you have in the account after 5 years? PV = FV = 0 ?

19. Future Value - single sumsIf you deposit \$100 in an account earning 6% with monthly compounding, how much would you have in the account after 5 years? Calculator Solution: P/Y = 12 I = 6 N = 60 PV = 100 FV = \$134.89 PV = 100 FV = 0 60

20. Future Value - single sumsIf you deposit \$100 in an account earning 6% with monthly compounding, how much would you have in the account after 5 years? Calculator Solution: P/Y = 12 I = 6 N = 60 PV = 100 FV = \$134.89 PV = 100 FV = 134.89 0 60

21. Future Value - single sumsIf you deposit \$100 in an account earning 6% with monthly compounding, how much would you have in the account after 5 years? Mathematical Solution: FV = PV (FVIF i, n) FV = 100 (FVIF .06, 60) (can’t use FVIF table) FV = PV (1 + i/m) m x n FV = 100 (1.005)60 = \$134.89 PV = -100 FV = 134.89 0 60

22. Present Value

23. Present Value - single sumsIf you will receive \$100 one year from now, what is the PV of that \$100 if your opportunity cost is 6%? PV = FV = 0 ?

24. Present Value - single sumsIf you will receive \$100 one year from now, what is the PV of that \$100 if your opportunity cost is 6%? Calculator Solution: P/Y = 1 I = 6 N = 1 FV = 100 PV = 94.34 PV = FV = 100 0 1

25. Present Value - single sumsIf you will receive \$100 one year from now, what is the PV of that \$100 if your opportunity cost is 6%? Calculator Solution: P/Y = 1 I = 6 N = 1 FV = 100 PV = 94.34 PV = 94.34 FV = 100 0 1

26. Present Value - single sumsIf you will receive \$100 one year from now, what is the PV of that \$100 if your opportunity cost is 6%? Mathematical Solution: PV = FV (PVIF i, n) PV = 100 (PVIF .06, 1) (use PVIF table, or) PV = FV / (1 + i)n PV = 100 / (1.06)1 = \$94.34 PV = 94.34 FV = 100 0 1

27. Present Value - single sumsIf you will receive \$100 5 years from now, what is the PV of that \$100 if your opportunity cost is 6%? PV = FV = 0 ?

28. Present Value - single sumsIf you will receive \$100 5 years from now, what is the PV of that \$100 if your opportunity cost is 6%? Calculator Solution: P/Y = 1 I = 6 N = 5 FV = 100 PV = 74.73 PV = FV = 100 0 5

29. Present Value - single sumsIf you will receive \$100 5 years from now, what is the PV of that \$100 if your opportunity cost is 6%? 74.73 PV = FV = 100 0 5 Calculator Solution: P/Y = 1 I = 6 N = 5 FV = 100 PV = 74.73

30. Present Value - single sumsIf you will receive \$100 5 years from now, what is the PV of that \$100 if your opportunity cost is 6%? Mathematical Solution: PV = FV (PVIF i, n) PV = 100 (PVIF .06, 5) (use PVIF table, or) PV = FV / (1 + i)n PV = 100 / (1.06)5 = \$74.73 PV = 74.73 FV = 100 0 5

31. Present Value - single sumsWhat is the PV of \$1,000 to be received 15 years from now if your opportunity cost is 7%? PV = FV = 0 ?

32. Present Value - single sumsWhat is the PV of \$1,000 to be received 15 years from now if your opportunity cost is 7%? Calculator Solution: P/Y = 1 I = 7 N = 15 FV = 1,000 PV = 362.45 PV = FV = 1000 0 15

33. Present Value - single sumsWhat is the PV of \$1,000 to be received 15 years from now if your opportunity cost is 7%? Calculator Solution: P/Y = 1 I = 7 N = 15 FV = 1,000 PV = 362.45 PV = 362.45 FV = 1000 0 15

34. Present Value - single sumsWhat is the PV of \$1,000 to be received 15 years from now if your opportunity cost is 7%? Mathematical Solution: PV = FV (PVIF i, n) PV = 100 (PVIF .07, 15) (use PVIF table, or) PV = FV / (1 + i)n PV = 100 / (1.07)15 = \$362.45 PV = 362.45 FV = 1000 0 15

35. Present Value - single sumsIf you sold land for \$11,933 that you bought 5 years ago for \$5,000, what is your annual rate of return? PV = FV = 0 ?

36. PV = 5,000 FV = 11,933 0 5 Present Value - single sumsIf you sold land for \$11,933 that you bought 5 years ago for \$5,000, what is your annual rate of return? Calculator Solution: P/Y = 1 N = 5 PV = 5,000 FV = 11,933 I = 19%

37. Present Value - single sumsIf you sold land for \$11,933 that you bought 5 years ago for \$5,000, what is your annual rate of return? Mathematical Solution: PV = FV (PVIF i, n ) 5,000 = 11,933 (PVIF ?, 5 ) PV = FV / (1 + i)n 5,000 = 11,933 / (1+ i)5 .419 = ((1/ (1+i)5) 2.3866 = (1+i)5 (2.3866)1/5 = (1+i) i = .19

38. PV = FV = 0 Present Value - single sumsSuppose you placed \$100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to \$500?

39. PV = 100 FV = 500 0 ? Present Value - single sumsSuppose you placed \$100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to \$500? Calculator Solution: • P/Y = 12 FV = 500 • I = 9.6 PV = -100 • N = 202 months

40. Present Value - single sumsSuppose you placed \$100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to \$500? Mathematical Solution: PV = FV / (1 + i)n 100 = 500 / (1+ .008)N 5 = (1.008)N ln 5 = ln (1.008)N ln 5 = N ln (1.008) 1.60944 = .007968 N N = 202 months

41. Hint for single sum problems: • In every single sum future value and present value problem, there are 4 variables: • FV, PV, i, and n • When doing problems, you will be given 3 of these variables and asked to solve for the 4th variable. • Keeping this in mind makes “time value” problems much easier!

42. The Time Value of Money 0 1 2 3 4 Compounding and Discounting Cash Flow Streams

43. Annuities • Annuity: a sequence of equal cash flows, occurring at the end of each period.

44. Annuities 0 1 2 3 4 • Annuity: a sequence of equal cash flows, occurring at the end of each period.

45. Examples of Annuities: • If you buy a bond, you will receive equal coupon interest payments over the life of the bond. • If you borrow money to buy a house or a car, you will pay a stream of equal payments.

46. Examples of Annuities: • If you buy a bond, you will receive equal coupon interest payments over the life of the bond. • If you borrow money to buy a house or a car, you will pay a stream of equal payments.

47. 0 1 2 3 Future Value - annuityIf you invest \$1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years?

48. 0 1 2 3 Future Value - annuityIf you invest \$1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years? Calculator Solution: P/Y = 1 I = 8 N = 3 PMT = 1,000 FV = \$3,246.40 1000 1000 1000

49. 0 1 2 3 Future Value - annuityIf you invest \$1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years? Calculator Solution: P/Y = 1 I = 8 N = 3 PMT = 1,000 FV = \$3,246.40 1000 1000 1000

50. Future Value - annuityIf you invest \$1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years? Mathematical Solution: FV = PMT (FVIFA i, n) FV = 1,000 (FVIFA .08, 3) (use FVIFA table, or)