1. Chapter 5 The Time Value of Money
Laurence Booth, Sean Cleary and Pamela Peterson Drake

2. 5.2 Annuities and perpetuities 5.3 Nominal and effective rates
Outline of the chapter
5.1 Time is money
Simple versus compound interest
Future value of an amount
Present value of an amount
5.2 Annuities and perpetuities
Ordinary annuities
Annuity due
Deferred annuities
Perpetuities
5.3 Nominal and effective rates
APR
EAR
Solving for the rate
5.4 Applications
Savings plans
Loans and mortgages
Saving for retirement

3. 5.1 Time value of money

4. Simple interest
Simple interest is interest that is paid only on the principal amount.
Interest = rate × principal amount of loan
Note: Some credit unions offer simple interest loans
Booth Cleary Drake

5. Simple interest: example
A 2-year loan of $1,000 at 6% simple interest
At the end of the first year, interest = 6% × $1,000 = $60
At the end of the second year, interest = 6% × $1,000 = $60
and loan repayment of $1,000
Key: in the second year, only the principal earns interest
Booth Cleary Drake

6. Compound interest
Compound interest is interest paid on both the principal and any accumulated interest.
Interest = rate × principal + accumulated interest

7. Compounding
Compounding is translating a present value into a future value, using compound interest.
Future value = Present value × Future value interest factor
Future value interest factor is also referred to as the compound factor.

8. Terminology and notation
Meaning
Future value FV
Value at some specified future point in time
Present value PV
Value today
Interest i
Compensation for the use of funds
Number of periods n
Number of periods between the present value and the future value
Compound factor
(1 + i)n
Translates a present value into a future value
We subscript FV and PV at times when we are dealing with complex situations and need to distinguish the timing of these values.
Booth Cleary Drake

9. Compare: simple versus compound
Suppose you deposit $5,000 in an account that pays 5% interest per year. What is the balance in the account at the end of four years if interest is:
Simple interest?
Compound interest?

10. Simple interest Year Beginning Add interest Ending 1 $5,000.00
+ (5% × $5,000) =
$5,250.00 2
$5,500.00 3
$5,750.00 4
$6,000.00

11. Compound interest Year Beginning Compounding Ending 1 $5,000.00
× =
$5,250.00 2
$5,512.50 3
$5,788.13 4
$6,077.53

12. Interest on interest
How much interest on interest? Interest on interest = FVcompound – FVsimple Interest on interest = $6, – 6, = $77.53

13. Comparison Year End of year balance Simple interest Compound interest
$5,000.00 1
$5,250.00 2
$5,500.00
$5,512.50 3
$5,750.00
$5,788.13 4
$6,000.00
$6,077.53
Bottom line: The value grows faster with compound interest compared to simple interest.
Booth Cleary Drake

14. Try it: Simple v. compound
Suppose you are comparing two accounts:
The Bank A account pays 5.5% simple interest.
The Bank B account pays 5.4% compound interest.
If you were to deposit $10,000 in each, what balance would you have in each bank at the end of five years?

15. Try it: Answer Bank A: $12,750.00 Bank B: $13,007.78
Booth Cleary Drake

16. A note about interest
Because compound interest is so common, assume that interest is compounded unless otherwise indicated.

17. Short-cuts
Example:
Consider $1,000 deposited for three years at 6% per year.

18. The long way
FV1 = $1, × (1.06) = $1,060.00
FV2 = $1, × (1.06) = $1,123.60
FV3 = $1, × (1.06) = $1,191.02 or
FV3 = $1,000 × (1.06)3 = $1,191.02
FV3 = $1,000 × = $1,191.02
Future value factor

19. Short-cut: Calculator
Known values:
PV = 1,000
n = 3
i = 6%
Solve for: FV

20. Input three known values, solve for the one unknown
Known: PV, i , n
Unknown: FV
HP10B
BAIIPlus
HP12C
TI83/84
1000 +/- PV
3 N
6 I/YR FV
1000 CHS PV
3 n
6 i
[APPS] [Finance]
[TVM Solver]
N =3
I%=6
PV = -1000
FV [Alpha] [Solve}

21. Short-cut: spreadsheet Microsoft Excel or Google Docs
=FV(RATE,NPER,PMT,PV,TYPE)
TYPE default is 0, end of period
=FV(.06,3,0,-1000) or A 1 6% 2 3
-1000 4
=FV(A1,A2,0,A3)
Type:
0 = end of period
1 = beginning of period
Basically, this is not relevant until we get to annuities.
Booth Cleary Drake

22. Problems Set 1

23. Problem 1.1
Suppose you deposit $2,000 in an account that pays 3.5% interest annually.
How much will be in the account at the end of three years?
How much of the account balance is interest on interest?
Booth Cleary Drake

24. Problem 1.2
If you invest $100 today in an account that pays 7% each year for four years and 3% each year for five years, how much will you have in the account at the end of the nine years?

25. Discounting

26. Discounting
Discounting is translating a future value into a present value.
The discount factor is the inverse of the compound factor: 𝑖 𝑛
To translate a future value into a present value, PV= FV 1+𝑖 𝑛

27. Example
Suppose you have a goal of saving $100,000 three years from today. If your funds earn 4% per year, what lump-sum would you have to deposit today to meet your goal?

28. Example, continued Known values: Unknown: PV FV = $100,000 n = 3

29. Example, continued
PV = $100, =$100,000 × PV = $100,000 × PV = $88, Check: FV3 = $88, × ( )3 = $100,000

30. Short-cut: Calculator
HP10B
BAIIPlus
HP12C
TI83/84
/- PV
3 N
4 I/YR PV
CHS PV 3n 4i
[APPS] [Finance] [TVM Solver]
N =3
I%=4
FV =
PV [Alpha] [Solve]

31. Short-cut: spreadsheet Microsoft Excel or Google Docs
=PV(RATE,NPER,PMT,PV,TYPE)
TYPE default: end of period
=PV(.06,3,0,-1000) or A 1 6% 2 3
100000 4
=PV(A1,A2,0,A3)

32. Try it: Present value
What is the today’s value of $10,000 promised ten years from now if the discount rate is 3.5%?

33. Try it: Answer Given: PV = $10,000 1+0.035 10 = $7,089.19 FV = $10,000
Solve for PV
PV = $10, = $7,089.19
Booth Cleary Drake

34. Frequency of compounding
If interest is compounded more than once per year, we need to make an adjustment in our calculation.
The stated rate or nominal rate of interest is the annual percentage rate (APR).
The rate per period depends on the frequency of compounding.

35. Discrete compounding: Adjustments
Adjust the number of periods and the rate per period.
Suppose the nominal rate is 10% and compounding is quarterly:
The rate per period is 10%  4 = 2.5%
The number of periods is number of years × 4

36. Continuous compounding: Adjustments
The compound factor is eAPR x n.
The discount factor is 1 eAPR x n .
Suppose the nominal rate is 10%.
For five years, the continuous compounding factor is e0.10 x 5 =
The continuous compounding discount factor for five years is 1 ÷ e0.10 x 5 =

37. Try it: Frequency of compounding
If you invest $1,000 in an investment that pays a nominal 5% per year, with interest compounded semi-annually, how much will you have at the end of 5 years?

38. Try it: Answer Given: FV = $1,000 × (1 + 0.025)10 = $1,280.08
PV = $1,000
n = 5 × 2 = 10
i = 0.05  2 = 0.25
Solve for FV
FV = $1,000 × ( )10 = $1,280.08

39. Problem Set 2

40. Problem 2.1
Suppose you set aside an amount today in an account that pays 5% interest per year, for five years. If your goal is to have $1,000 at the end of five years, what would you need to set aside today?

41. Problem 2.2
Suppose you set aside an amount today in an account that pays 5% interest per year, compounded quarterly, for five years. If your goal is to have $1,000 at the end of five years, what would you need to set aside today?

42. Problem 2.3
Suppose you set aside an amount today in an account that pays 5% interest per year, compounded continuously, for five years. If your goal is to have $1,000 at the end of five years, what would you need to set aside today?

43. 5.2 Annuities and Perpetuities1 2 3 4 5 | CF
PV?
FV?
5.2 Annuities and Perpetuities

44. What is an annuity? An annuity is a periodic cash flow.
Same amount each period
Regular intervals of time
The different types depend on the timing of the first cash flow.

45. Type of annuities Type First cash flow Examples Ordinary
One period from today
Mortgage
Annuity due
Immediately
Lottery payments
Rent
Deferred annuity
Beyond one period from today
Retirement savings

46. Time lines: 4-payment annuity1 2 3 4 5 |
Ordinary PV CF FV
Annuity due
Deferred annuity

47. Key to valuing annuities
The key to valuing annuities is to get the timing of the cash flows correct.
When in doubt, draw a time line.

48. Example: PV of an annuity
What is the present value of a series of three cash flows of $4,000 each if the discount rate is 6%, with the first cash flow one year from today? 1 2 3 4 |
$4,000

49. Example: PV of an annuity The long way1 2 3 4 |
$4,000
$3,773.58 
3,559.99
3,358.48
$10,692.05

50. Example: PV of an annuity In table form
Year
Cash flow
Discount factor
Present value 1
$4,000.00
$3,773.58 2
3,559.99 3
3,358.48
$10,692.05
PV = $4, × = $10,692.05

51. Example: PV of an annuity Formula short-cuts
PV = $4,000 × 1 − PV = $4,000 × PV = $10,692.05

52. Example: PV of an annuity Calculator short cuts
Given:
PMT = $4,000
i = 6%
N = 3
Solve for PV

53. Example: PV of an annuity Spreadsheet short-cuts
=PV(RATE,NPER,PMT,FV,TYPE) =PV(.06,3,4000,0)
Note: Type is important for annuities
If Type is left out, it is assumed a 0
0 is for an ordinary annuity
1 is for an annuity due
Note: the default type is ordinary annuity, so this last term can be left off.
Booth Cleary Drake

54. Example: FV of an annuity
What is the future value of a series of three cash flows of $4,000 each if the discount rate is 6%, with the first cash flow one year from today? 1 2 3 4 |
$4,000

55. Example: FV of an annuity The long way1 2 3 4 |
$4,000.00 
4,240.00
4,494.40
$12,734.40

56. Example: FV of an annuity In table form
Year
Cash flow
Compound factor
Future value 1
$4,000.00
1.1236
$4,494.40 2
1.0600
4,240.00 3
1.0000
4,000.00
3.1836
$12,734.40
PV = $4, × = $12,734.40

57. Example: FV of an annuity Calculator short cuts
Given:
PMT = $4,000
i = 6%
N = 3
Solve for FV

58. Example: FV of an annuity Spreadsheet short-cuts
=FV(RATE,NPER,PMT,PV,type) =FV(.06,3,4000,0)

59. Annuity due
Consider a series of three cash flows of $4,000 each if the discount rate is 6%, with the first cash flow today.
What is the present value of this annuity?
What is the future value of this annuity?

60. The time line 1 2 3 |
$4,000.00
PV?
FV?
This is an annuity due

61. Present value of cash flow
Valuing an annuity due
Present value
Future value
Year
End of year cash flow
Compound factor
Present value of cash flow
$4,000.00 1
3,773.58 2
3,559.99
$11,333.57
Year
End of year cash flow
Factor
Future value
$4,000.00
$4,764.06 1
4,494.40 2
4,240.00
$13,498.46

62. Valuing an annuity due: Using calculators
Present value
Future value
PMT = 4000 N = 3 I = 6% BEG mode Solve for PV
PMT = 4000 N = 3 I = 6% BEG mode Solve for FV

63. Valuing an annuity due: Using spreadsheets
Present value =PV(RATE,NPER,PMT,FV,TYPE) =PV(0.06,3,4000,0,1) Future value

64. Any other way?
There is one period difference between an ordinary annuity and an annuity due. Therefore: PVannuity due = PVordinary annuity × (1 + i) and FVannuity due = FVordinary annuity × (1 + i)

65. Valuing a deferred annuity
A deferred annuity is an annuity that begins beyond one year from today.
That means that it could begin 2, 3, 4, … years from today, so each problem is unique.

66. Valuing a deferred annuity1 2 3 4 5 | CF
4-payment ordinary annuity, then discount value one period
PV0
←PV1
4-payment annuity due, then discount value two periods
←PV2

67. Example: Deferred annuity
What is the value today of a series of five cash flows of $6,000 each, with the first cash flow received four years from today, if the discount rate is 8%? 1 2 3 4 5 6 7 8 9 10 |
PV? CF

68. Example, cont.
Using an ordinary annuity: PV3 = $23, PV0 = $19, Using an annuity due: PV4 = $25,872.76
Discount 3 periods at 8%
Discount 4 periods at 8%

69. Example: Deferred annuity Calculator solutions
HP10B
BAIIPlus
TI83/84
0 CF
6000 CF
8 i
NPV
0 CF ↑ 1
0 CF ↑ 1 F1
0 CF ↑ 1 F2
0 CF ↑ 1 F3
6000 CF ↑ 1 F4
6000 CF ↑ 1 F5
6000 CF ↑ 1 F6
6000 CF ↑ 1 F7
6000 CF ↑ 1 F8
[2nd] { }
STO [2nd] L1
[APPS] [Finance]
[ENTER] 7
NPV(.08,0,L1)
[ENTER]

70. Example: Deferred annuity Spreadsheet solutionsB
Year
Cash flow 1 $0 2 3 4
$6000 5 6 7 8
=PV(0.08,3,0,PV(0.08,5,6000,0))
=PV(0.08,4,0,PV(0.08,5,6000,0,1))
=NPV(0.08,A1:A9)

71. Perpetuities
A perpetuity is an even cash flows that occurs at regular intervals of time, forever. The valuation of a perpetuity is simple: PV= 𝐶𝐹 1+𝑖 1 + 𝐶𝐹 1+𝑖 2 + 𝐶𝐹 1+𝑖 3 +… 𝐶𝐹 1+𝑖 ∞ PV= 𝐶𝐹 𝑖

72. Problem Set 3

73. Problem 3.1
Which do you prefer if the appropriate discount rate is 6% per year:
An annuity of $4,000 for four annual payments starting today.
An annuity of $4,100 for four annual payments, starting one year from today.
An annuity of $4,200 for four annual payments, starting two years from today.

74. 5.3 Nominal and effective rates%
𝐴𝑃𝑅=𝑖 ×𝑚
𝑬𝑨𝑹= 𝟏+ 𝑨𝑷𝑹 𝒎 𝒎
𝑬𝑨𝑹= 𝒆 𝑨𝑷𝑹 −𝟏 i
5.3 Nominal and effective rates

75. APR & EAR
The annual percentage rate (APR) is the nominal or stated annual rate.
The APR ignores compounding within a year.
The APR understates the true, effective rate.
The effective annual rate (EAR) incorporates the effect of compounding within a year.
Note to instructors: the post-subprime mortgage rules confuse the APR with the EAR, so the term “APR” is used as both an effective rate and a nominal rate, depending on the part of the paperwork.
Booth Cleary Drake

76. APR EAR
EAR = 1+ 𝐴𝑃𝑅 𝑚 𝑚 −1 Suppose interest is stated as 10% per years, compounded quarterly. EAR = −1= −1 EAR = %

77. EAR with continuous compounding
EAR = 𝑒 𝐴𝑃𝑅 −1 Suppose interest is stated as 10% per years, compounded continuously. EAR = e 0.1 −1= −1 EAR = %

78. Frequency of compounding
If interest is compounded more frequently than annually, then this is considered in compounding and discounting.
There are two approaches
Adjust the i and n; or
Calculate the EAR and use this

79. Example: EAR & compounding
Suppose you invest $2,000 in an investment that pays 5% per year, compounded quarterly. How much will you have at the end of 4 years?

80. Example: EAR & compounding
Method 1:
FV = $2,000 ( )16 = $2,439.78
Method 2:
EAR = ( )4 – 1 = %
FV = $2,000 ( )4 = $2,439.78

81. Try it: APR & EAR
Suppose a loan has a stated rate of 9%, with interest compounded monthly. What is the effective annual rate of interest on this loan?

82. Try it: Answer
EAR = −1
EAR = −1
EAR = %

83. Problem Set 4

84. Problem 4.1
What is the effective interest rate that corresponds to a 6% APR when interest is compounded monthly?

85. Problem 4.2
What is the effective interest rate that corresponds to a 6% APR when interest is compounded continuously?

86. 5.4 Applications

87. Saving for retirement
Suppose you estimate that you will need $60,000 per year in retirement. You plan to make your first retirement withdrawal in 40 years, and figure that you will need 30 years of cash flow in retirement. You plan to deposit funds for your retirement starting next year, depositing until the year before retirement. You estimate that you will earn 3% on your funds. How much do you need to deposit each year to satisfy your plans?

88. Deferred annuity time line1 2 3 4 5 6 7 8 … 39 40 41 42 43 79 | D W
D = Deposit (39 in total)
W = Withdrawal (30 in total)

89. Deferred annuity time line1 2 3 4 5 6 7 8 … 39 40 41 42 43 79 | W PV
← Ordinary annuity ↓
Ordinary annuity → FV D

90. Two steps Step 1: Present value of ordinary annuity
N = 30; i = 3%; PMT = $60,000
PV39 = $1,176,026.48
Step 2: Solve for payment in an ordinary annuity
N = 39; i = 3%; FV = $1,176,026.48
PMT = $16,280.74

91. What does this mean?
If there are 39 annual deposits of $16, each and the account earns 3%, there will be enough to allow for 30 withdrawals of $60,000 each, starting 40 years from today.

92. Balance in retirement account

93.   + = €  $
Practice problems

94. Problem 1
What is the future value of $2,000 invested for five years at 7% per year, with interest compounded annually?
Booth Cleary Drake

95. Problem 2
What is the value today of €10,000 promised in four years if the discount rate is 4%?
Booth Cleary Drake

96. Problem 3
What is the present value of a series of five end-of-year cash flows of $1,000 each if the discount rate is 4%?
Booth Cleary Drake

97. Problem 4
Suppose you plan to save $3,000 each year for ten years. If you earn 5% annual interest on your savings, how much more will you have at the end of ten years if you make your payments at the beginning of the year instead of the end of the year?
Booth Cleary Drake

98. Problem 5
Sue plans to deposit $5,000 in a savings account each year for thirty years, starting ten years from today. Yan plans to deposit $3,500 in a savings account each year for forty years, starting at the end of this year. If both Sue and Yan earn 3% on their savings, who will have the most saved at the end of forty years?
Booth Cleary Drake

99. Problem 6
Suppose you have two investment opportunities: Opportunity 1: APR of 12%, compounded monthly Opportunity 2: APR of 11.9%, compounded continuously Which opportunity provides the better return?
Booth Cleary Drake

100. Problem 7
If you can earn 5% per year, what would you have to deposit in an account today so that you have enough saved to allow withdrawals of $40,000 each year for twenty years, beginning thirty years from today?
Booth Cleary Drake

101. Problem 8
Suppose you deposit ¥50000 in an account that pays 4% interest, compounded continuously. How much will you have in the account at the end of ten years if you make no withdrawals?
Booth Cleary Drake

102. The end