1. Chapter 5 The Time Value of Money

2. Time Value The process of expressing –the present in the future (compounding) –the future in the present (discounting)

3. Time Value Payments are either –a single payment –a series of equal payments (an annuity)

4. Time Value Time value of money problems may be solved by using –Interest tables –Financial calculators –Software

5. Variables for Time Value of Money Problems PV = present value FV = future value PMT = annual payment N = number of time periods I = interest rate per period

6. Financial Calculators Express the cash inputs (PV, FV, and PMT) as cash inflows and cash outflows At least one of the cash variables must be –an inflow (+) –an outflow (-)

7. Future Value The future value of $1 takes a single payment in the present into the future The general equation for the future value of $1: P 0 (1 + i ) n = P n

8. Future Value Illustrated PV = -100 I = 8 N = 20 PMT = 0 FV = ? = 466.10

9. Greater Terminal Values Higher interest rates Longer time periods Result in greater terminal values

10. Greater Terminal Values

11. Present Value The present value of $1 brings a single payment in the future back to the present The general equation for the present value of $1: P 0 = P n (1+ i ) n

12. Present Value Illustrated FV = 100 I = 6 N = 5 PMT = 0 PV = ? = -74.73

13. Lower Present Values Higher interest rates Longer time periods Result in lower present values

14. Lower Present Values

15. Annuities - Future Sum The future sum of annuity takes a series of payments into the future Payments may be made –at the end of each time period (ordinary annuity) –at the beginning of each time period (annuity due)

16. FV Time Lines Ordinary annuity Year Payment - $100 100 100 0 1 2 3

17. FV Time Lines Annuity due Year Payment $100 100 100 - 0 1 2 3

18. Future Value of an Ordinary Annuity Illustrated PV = 0 PMT = -100 I = 5 N = 3 FV = ? = 315.25

19. Greater Terminal Values Higher interest rates Longer time periods Result in greater terminal values

20. Greater Terminal Values

21. Annuities - Present Value The present value of an annuity brings a series of payments in the future back to the present

22. Present Value of an Ordinary Annuity Illustrated FV = 0 PMT = 100 I = 6 N = 3 PV = ? = -267.30

23. Annuities - Present Value Higher interest rates result in lower present values But longer time periods increases the present value (because more payments are received)

24. Annuities - Present Value

25. Additional Time Value Illustrations The following is a series of problems or questions that use the time value of money.

26. Illustration 1 You deposit $1,000 in an account at the end of each year for twenty years. What is the total amount in the account if you earn 6 percent annually?

27. Future Value of an Ordinary Annuity The unknown: FV The givens: –PV = 0 –PMT = -1,000 –N = 20 –I = 6 The answer: $36,786

28. Interpretation For an annual cash payment of $1,000, you will have $36,786 after twenty years Of the $36,786 –$20,000 is the total cash outflow –$16,786 is the earned interest

29. Illustration 2 What is the present value of (or required cash outflow to purchase) an ordinary annuity of $1,000 for twenty years, if the rate of interest is 6 percent?

30. Present Value of an Annuity The unknown: PV The givens: –FV = 0 –PMT = 1,000 –N = 20 –I = 6 The answer: $11,470

31. Interpretation For a present payment of $11,470, the individual will annually receive $1,000 for the next twenty years The $11,470 is an immediate cash outflow The $1,000 annual payment to be received is a cash inflow

32. Illustration 3 What is the future value after ten years of a stock that cost $10 and appreciates at 9 percent annually?

33. Future Value of $1 The unknown: FV The givens: –PV = 10 –PMT = 0 –N = 10 –I = 9 The answer: $23.67

34. Interpretation A $10 stock will be worth $23.67 after 10 year if its price grows 9% annually.

35. Illustration 4 What is the cost of a stock that was sold for $23.67, held for 10 years and whose value appreciated 9 percent annually?

36. Present Value of $1 The unknown: PV The givens: –FV = 23.67 –PMT = 0 –N = 10 –I = 9 The answer: $10

37. Interpretation $23.67 received after ten years is worth $10 today if the return rate is 9 percent.

38. Interpretation of Future and Present Values These two problems are the same: In the first case the $10 is compounded into its future value ($23.67) In the second case the future value ($23.67) is discounted back to its present value ($10)

39. Illustration 5 A stock was purchased for $10 and sold for $23.67 after 10 years. What was the return?

40. Future Value of I The unknown: I The givens: –PV = 10 –PMT = 0 –N = 0 –FV = 23.67 The answer: 9%

41. Interpretation The yield on a $10 investment that was sold after 10 years for $23.67 is 9%.

42. Illustration 6 If an investment pay $50 a year for 10 years and repays $1,000 after 10 years, what is this investment worth today if you can earn 6 percent?

43. Determination of Present Value The unknown: PV The givens: –FV = 1,000 –PMT = 50 –I = 6 –N = 10 The answer: $926

44. Interpretation If you collect $50 a year for 10 years and receive $1,000 after 10 years, those cash inflows are currently worth $926 at 6 percent.

45. Illustration 7 Time value is used to determine a loan repayment schedule such as a mortgage.

46. Loan Repayment Schedule Amount borrowed (PV) = $80,000 Interest rate (I) = 8% Term of the loan (N) = 25 years No future value since loan is repaid Amount of the annual payment = $7,494.30

47. Loan Repayment Schedule PrincipalBalance PmntInterestRepaymentOwed 1$6,400.00$1,094.15$78,905.85 26,312.471,181.6877,724.17............ 25555.136,939.17.00

48. Illustration 8 You have $115,000 and spend $24,000 a year. If you earn 8% annually, how long will your funds last?

49. Determination of Number of Years The unknown: N The givens: –PV = 115,000 –I = 8 –FV = 0 –PMT = -24,000 The answer: 6.3 years

50. Interpretation If you have $115,000 and earn 8 percent annually, you can spend $24,000 per year for approximately 6 years and 4 months.

51. Non-annual Compounding More than one interest payment a year More frequent compounding

52. Non-annual Compounding Multiply number of years by frequency of compounding Divide interest rate by frequency of compounding

53. Periods less than One Year Same variables as in all time value problems except N < 1.

54. Illustration 9 What is the return on an ivestment that costs $98,543 and pays $100,000 after 45 days?

55. Determination of Return The unknown: I The givens: –PV = 98,543 –N = 0.1233 –FV = 100,000 –PMT = 0 The answer: 12.64%

56. Interpretation $98,543 invested for 45 days grows to $100,000 at 12.64 percent.