1. Basic Finance The Time Value of Money7
An introduction to financial institutions, investments & Management
Eleventh Edition

2. Time Value
What is Time Value of Money?

3. Time Value Process of expressing
the present in the future (compounding)
the future in the present (discounting)

4. Time Value Compounding Future value of a dollar
Process by which interest is paid on interest that was previously earned.
Future value of a dollar
Amount to which a single payment will grow at some rate of interest.

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

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

7. 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

8. 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 (-)

9. Future Value P0(1 + i)n = Pn
Future value of $1 takes a single payment in the present into the future
General equation for the future value of $1:
P0(1 + i)n = Pn

10. Future Value Illustrated
PV = -100
I = 5
N = 20
PMT = 0
FV = ? =

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

12. Greater Terminal Values

13. Present Value
Present value of $1 brings a single payment in the future back to the present
General equation for the present value of $1: P0 = Pn (1+i)n

14. Present Value Present Value Discounting
Current value of a dollar to be received in the future.
Discounting
Process of determining present value.

15. Present Value Illustrated
FV = 100
I = 6
N = 5
PMT = 0
PV = ? =

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

17. Lower Present Values

18. Annuities
What is an Annuity?

19. Annuities - Future Sum Annuity: Series of equal, annual payments.
Future value of an annuity: Amount to which a series of equal payments will grow at some rate of interest.
Annuity Due: Annuity in which the payments are made at the beginning of the time period.
Ordinary Annuity: Annuity in which the payments are made at the end of the time period.

20. Annuities - Future Sum
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)

21. FV Time Lines
Ordinary annuity
Year
Payment $

22. FV Time Lines
Annuity due
Year
Payment $

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

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

25. Greater Terminal Values

26. Present Value of an Annuity
The present value of an annuity brings a series of payments in the future back to the present.

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

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

29. Annuities - Present Value

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

31. 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?

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

33. 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

34. 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?

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

36. 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

37. Illustration 3
You buy a stock for $10 and expect the price to increase 9 percent annually. After 10 years, what is the anticipated price of the stock?

38. Future Value of $1 The unknown: FV The givens: PV = -10 PMT = 0 N = 10
The answer: ($23.67)

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

40. 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?

41. Present Value of $1 The unknown: PV The givens: FV = 23.67 PMT = 0
The answer: -10($10)

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

43. 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)

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

45. Future Determination of the Interest Rate
The unknown: I
The givens:
PV = -10
PMT = 0
N = 0
FV = 23.67
The answer: 9(9%)

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

47. 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?

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

49. 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.

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

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

52. Loan Repayment Schedule
Principal
Balance
Pmnt
Interest
Repayment
Owed 1
$6,400.00
$1,094.15
$78,905.85 2
6,312.47
1,181.68
77,724.17 . 25
555.13
6,939.17
.00

53. Illustration 8
You have $470,000 and spend $96,000 a year. If you earn 8% annually, how long will your funds last?

54. Determination of Number of Years
The unknown: N
The givens:
PV = 470,000
I = 8
FV = 0
PMT = -96,000
The answer: (6.5 years)

55. Interpretation
If you have $470,000 and earn 8 percent annually, you can spend $96,000 per year for approximately 6 years and 6 months.

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

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

58. Non-annual Compounding Illustration
What is the future value of $100 that pays 8 percent compounded quarterly for five years?

59. Determination of Future Value
FV = $100 (1+0.08/4) 5x4 = $100 (1+0.02)20 = $148.59

60. Determination of Future Value
PV = -100 I = 8/4 = 2 N = 5x4 = 20 PMT = 0
FV =
The answer:
($148.59)

61. Interpretation
$100 grows to $ at 8 percent in five years with interest compounded quarterly.

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

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

64. Determination of Return
The unknown: I
The givens:
PV = -98,543
N =
FV = 100,000
PMT = 0
The answer: (12.64%)

65. $98,543 invested for 45 days grows to $100,000 at 12.64 percent.
Interpretation
$98,543 invested for 45 days grows to $100,000 at percent.