1. 2.4 Perpetuities and Annuities 2.5 Effective Annual Interest Rate
Outline 2: Time Value of Money & Introduction to Discount Rates & Rate of Return
2.1 Future Values
2.2 Present Values
2.3 Multiple Cash Flows
2.4 Perpetuities and Annuities
2.5 Effective Annual Interest Rate
2.6 Loan Amortization
Appendix on Time Value of Money 2

2. Future Values
Future Value - Amount to which an investment will grow after earning interest.
Compound Interest - Interest earned on interest.
Simple Interest - Interest earned only on the original investment. 3

3. Future Values Example - Simple Interest
Interest earned at a rate of 6% for five years on a principal balance of $100.
Interest Earned Per Year = 100 x = $ 6 5

4. Future Values Example - Simple Interest
Interest earned at a rate of 6% for five years on a principal balance of $100. 6

5. Future Values Example - Simple Interest
Interest earned at a rate of 6% for five years on a principal balance of $100.
Today Future Years
Interest Earned
Value 100 7

6. Future Values Example - Simple Interest
Interest earned at a rate of 6% for five years on a principal balance of $100.
Today Future Years
Interest Earned 6
Value 8

7. Future Values Example - Simple Interest
Interest earned at a rate of 6% for five years on a principal balance of $100.
Today Future Years
Interest Earned
Value 9

8. Future Values Example - Simple Interest
Interest earned at a rate of 6% for five years on a principal balance of $100.
Today Future Years
Interest Earned
Value 10

9. Future Values Example - Simple Interest
Interest earned at a rate of 6% for five years on a principal balance of $100.
Today Future Years
Interest Earned
Value 11

10. Future Values Example - Simple Interest
Interest earned at a rate of 6% for five years on a principal balance of $100.
Today Future Years
Interest Earned
Value
Value at the end of Year 5 = $130 12

11. Future Values Example - Compound Interest
Interest earned at a rate of 6% for five years on the previous year’s balance. 13

12. Future Values Example - Compound Interest
Interest earned at a rate of 6% for five years on the previous year’s balance.
Interest Earned Per Year =Prior Year Balance x .06 14

13. Future Values Example - Compound Interest
Interest earned at a rate of 6% for five years on the previous year’s balance.
Today Future Years
Interest Earned
Value 100 15

14. Future Values Example - Compound Interest
Interest earned at a rate of 6% for five years on the previous year’s balance.
Today Future Years
Interest Earned
Value 16

15. Future Values Example - Compound Interest
Interest earned at a rate of 6% for five years on the previous year’s balance.
Today Future Years
Interest Earned
Value 17

16. Future Values Example - Compound Interest
Interest earned at a rate of 6% for five years on the previous year’s balance.
Today Future Years
Interest Earned
Value 18

17. Future Values Example - Compound Interest
Interest earned at a rate of 6% for five years on the previous year’s balance.
Today Future Years
Interest Earned
Value 19

18. Future Values Example - Compound Interest
Interest earned at a rate of 6% for five years on the previous year’s balance.
Today Future Years
Interest Earned
Value
Value at the end of Year 5 = $133.82 20

19. Future Values
Future Value of $100 = FV 21

20. Future Values Future Value of any Present Value = FV
where t= number of time periods
r=is the discount rate 21

21. Future Values if t=4: FV = PV(1+r)(1+r) (1+r)(1+r) = PV(1+r)4 if t=10:
FV = PV(1+r)(1+r)(1+r)(1+r)(1+r)(1+r)(1+r) (1+r)(1+r)(1+r)
= PV(1+r)10 21

22. Future Values if t=n: FV = PV(1+r)(1+r) (1+r)(1+r)…(1+r) = PV(1+r)n
FV = PV(1+r) = PV(1+r)0 = PV 21

23. Future Values Example - FV
What is the future value of $100 if interest is compounded annually at a rate of 6% for five years? 22

24. Future Values Example - FV
What is the future value of $100 if interest is compounded annually at a rate of 6% for five years? 23

25. Future Values: FV with Compounding
Interest Rates 24

26. Future Value: Manhattan Island Sale
Peter Minuit bought Manhattan Island for $24 in Was this a good deal?
To answer, determine $24 is worth in the year 2006, compounded at 12.5% (long-term average annual return on S&P 500):
FYI - The value of Manhattan Island land is a very small fraction of this number.

27. Present Values Present Value Value today of a future cash flow.
Discount Factor
Present value of a $1 future payment.
Discount Rate
Interest rate used to compute present values of future cash flows. 31

28. Present Values 32

29. Present Values
Since FV = PV (1+r) then solve for PV by dividing both sides by (1+r): 34

30. Present Values Example
You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years? 34

31. Present Values Example
You are twenty years old and want to have $1 million in cash when you are 80 years old (you can expect to live to one-hundred or more). If you expect to earn the long-term average 12.4% in the stock market how much do you need to invest now? 34

32. Present Values Discount Factor = DF = PV of $1
Discount Factors can be used to compute the present value of any cash flow.
r is the discount rate (of return) 36

33. Present Value
The PV formula has many applications. Given any variables in the equation, you can solve for the remaining variable. 38

34. Present Value: PV of Multiple Cash Flows
Example
Your auto dealer gives you the choice to pay $15,500 cash now, or make three payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money is 8%, which do you prefer? 42

35. Present Value: PV of Multiple Cash Flows
PVs can be added together to evaluate multiple cash flows. 43

36. Present Value: Perpetuities & Annuities
Perpetuity
A stream of level cash payments that never ends.
Annuity
Equally spaced level stream of cash flows for a limited period of time. 44

37. Present Value: Perpetuities & Annuities
PV of Perpetuity Formula
C = constant cash payment
r = interest rate or rate of return 45

38. Present Value: Perpetuities & Annuities
Example - Perpetuity
In order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today in the rate of interest is 10%? 47

39. Present Value: Perpetuities & Annuities
Example - continued
If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today? 49

40. Present Value: Perpetuities & Annuities
PV of Annuity Formula
C = cash payment
r = interest rate
t = Number of years (periods) cash payment is received 50

41. Present Value: Perpetuities & Annuities
If PV of Annuity Formula is:
Then formula for annuity payment is: 50

42. Present Value: Perpetuities & Annuities
Formula for annuity payment can be used to find loan payments. Just think of C as Payment, PV as loan amount, t as the number of months, and r must be the periodic loan r to coincide with the frequency of payments: 50

43. Present Value: Perpetuities & Annuities
PV Annuity Factor (PVAF) - The present value of $1 a year for each of t years. 51

44. Present Value: Perpetuities & Annuities
Example - Annuity
You are purchasing a car. You are scheduled to make 60 month installments of $500 for a $25,000 auto. Given an annual market rate of interest of 5% for a car loan, what is the price you are paying for the car (i.e. what is the PV)? 53

45. Present Value: Perpetuities & Annuities
Example - Annuity
You have just won the NJ lottery for $2 million over 25 years. How much is the “$2 million” NJ Lottery really worth at an opportunity cost rate of return of 12.4% - long-run annual stock market rate of return (ignoring income taxes)? 53

46. Present Value: Perpetuities & Annuities
Example - Annuity
Now what if you took the lump-sum based on a 5% discount rate by the State of New Jersey? 53

47. Perpetuities & Annuities
Example - Future Value of annual payments
You plan to save $4,000 every year for 20 years and then retire. Given a 10% rate of interest, what will be the FV of your retirement account? 56

48. Perpetuities & Annuities
Future Value of Ordinary Annuity: 56

49. Perpetuities & Annuities
Present Value of Ordinary Annuity: 56

50. Effective Interest Rates
Effective Annual Interest Rate - Interest rate that is annualized using compound interest.
r = annual or nominal rate of interest or return
m= number of compounding periods per year
rnom/m=also known as the periodic interest rate 26

51. Effective Interest Rates
example
Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)? 27

52. Effective Interest Rates
example
Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)? 28

53. Amortization
Amortization is the process by which a loan is paid off. During that process, the interest and contribution amounts change every month due to the mathematics of compounding.
Construct an amortization schedule
for a $1,000, 10% annual rate loan
with 3 equal payments.

54. Step 1: Find the required payments.
Amortization
Step 1: Find the required payments. 1 2 3
10%
-1,000
PMT
PMT
PMT
INPUTS N
I/YR PV
PMT FV
OUTPUT
402.11

55. Step 2: Find interest charge for Year 1.
Amortization
Step 2: Find interest charge for Year 1.
INTt = Beg balt (i)
INT1 = $1,000(0.10) = $100.
Step 3: Find repayment of principal in
Year 1.
Repmt = PMT – INT
= $ – $100
= $

56. Step 4: Find ending balance after year 1.
Amortization
Step 4: Find ending balance after year 1.
End bal = Beg bal – Repmt
= $1,000 – $ = $
Repeat these steps for Years 2 and 3
to complete the amortization table.

57. Amortization
BEG PRIN END
YR BAL PMT INT PMT BAL
1 $1,000 $402 $100 $302 $698
TOT 1, ,000
Interest declines and contribution to
principal grows. Tax implications from
lower interest paid.

58. 10% on loan outstanding, which is falling.$
402.11
Interest
302.11
Principal Payments 1 2 3
Level payments. Interest declines because outstanding balance declines. Lender earns
10% on loan outstanding, which is falling.

59. Amortization
Amortization tables are widely used--for home mortgages, auto loans, business loans, retirement plans, etc. They are very important!
Financial calculators (and spreadsheets) are great for setting up amortization tables.

60. Appendix on Time Value of Money
Future value
Present value
Rates of return

61. Future Value 1 2 3 i%
CF0
CF1
CF2
CF3
Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2. Time lines show timing of cash flows.

62. Time line for a $100 lump sum due at the end of Year 2.1
2 Year i%
100

63. Time line for an ordinary annuity of $100 for 3 years.1 2 3 i%
100
100
100

64. Time line for uneven CFs -$50 at t = 0 and $100, $75, and $50 at the end of Years 1 to 31 2 3 r%
-50
100 75 50

65. What’s the FV of an initial $100 after 3 years if r = 10%?1 2 3
10%
100
FV = ?
Finding FVs is compounding.

66. After 1 year:
FV1 = PV + INT1 = PV + PV(r)
= PV(1 + r)
= $100(1.10)
= $110.00
After 2 years:
FV2 = PV(1 + r)2
= $100(1.10)2
= $121.00

67. After 3 years:
FV3 = PV(1 + r)3
= 100(1.10)3
= $133.10
In general,
FVn = PV(1 + r)n

68. Four Ways to Find FVs
Solve the equation with a regular calculator.
Use tables.
Use a financial calculator.
Use a spreadsheet.

69. Financial Calculator Solution
Financial calculators solve this equation:
FVn = PV(1 + r)n
There are 4 variables. If 3 are known, the calculator will solve for the 4th.

70. Here’s the setup to find FV:
INPUTS
N r/YR PV PMT FV
133.10
OUTPUT
Clearing automatically sets everything to 0, but for safety enter PMT = 0.
Set: P/YR = 1, END

71. What is the PV of $100 due in 3 years if r=10%
What is the PV of $100 due in 3 years if r=10%? Finding PVs is discounting, and it’s the reverse of compounding. 1 2 3
10%
PV = ?
100

72. What interest rate would cause $100 to grow to $125.97 in 3 years?
Solve FVn = PV(1 + r )n for PV: . 3 1 æ ö ( ) PV =
$100 ç ÷ =
$100
PVIF è ø i, n
1.10 =
$100 (
0.7513 ) =
$75.13.

73. Financial Calculator Solution
N r/YR PV PMT FV
-75.13
INPUTS
OUTPUT
Either PV or FV must be negative. Here
PV = Put in $75.13 today, take
out $100 after 3 years.

74. If sales grow at 20% per year, how long before sales double?
Solve for n:
FVn = 1(1 + r)n;
2 = 1(1.20)n
Use calculator to solve, see next slide.

75. Graphical Illustration:
INPUTS
N r/YR PV PMT FV
3.8
OUTPUT
Graphical Illustration: FV 2
3.8 1
Year 1 2 3 4

76. What’s the difference between an ordinary annuity and an annuity due?1 2 3 r%
PMT
PMT
PMT
Annuity Due 1 2 3 r%
PMT
PMT
PMT

77. What’s the FV of a 3-year ordinary annuity of $100 at 10%?1 2 3
10%
100
100
100
110
121
FV = 331

78. Financial Calculator Solution
INPUTS
331.00 N
r/YR PV
PMT FV
OUTPUT
Have payments but no lump sum PV, so enter 0 for present value.

79. What’s the PV of this ordinary annuity?1 2 3
10%
100
100
100
90.91
82.64
75.13
= PV

80. Have payments but no lump sum FV, so enter 0 for future value.
INPUTS N
r/YR PV
PMT FV
OUTPUT
Have payments but no lump sum FV, so enter 0 for future value.

81. Find the FV and PV if the annuity were an annuity due.1 2 3
10%
100
100
100

82. Switch from “End” to “Begin.”
Then enter variables to find PVA3 = $
INPUTS N
r/YR PV
PMT FV
OUTPUT
Then enter PV = 0 and press FV to find
FV = $

83. What is the PV of this uneven cash flow stream?1 2 3 4
10%
100
300
300
-50
90.91
247.93
225.39
-34.15
= PV

84. Input in “CFLO” register:
Enter r = 10, then press NPV button to get NPV = (Here NPV = PV.)

85. Finding the interest rate or growth rate
INPUTS N
r/YR PV
PMT FV
OUTPUT 8%

86. LARGER! If compounding is more
Will the FV of a lump sum be larger or smaller if we compound more often, holding interest rate constant? Why?
LARGER! If compounding is more
frequent than once a year--for example, semiannually, quarterly,
or daily--interest is earned on interest
more often.

87. 1 2 3
10%
100
133.10
Annually: FV3 = 100(1.10)3 = 1 2 3 1 2 3 4 5 6 5%
100
134.01
Semiannually: FV6 = 100(1.05)6 =

88. Rates of Return:We will deal with 3 different rates:
rNom = nominal, or stated, or
quoted, rate per year.
rPer = periodic rate.
EAR = EFF% =
effective annual
rate

89. rNom is stated in contracts. Periods per year (m) must also be given.
Examples:
8%; Quarterly
8%, Daily interest (365 days)

90. Periodic rate = rPer = rNom/m, where m is number of compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding.
Examples:
8% quarterly: rPer = 8%/4 = 2%.
8% daily (365): rPer = 8%/365 = %.

91. Effective Annual Rate (EAR = EFF%):
The annual rate that causes PV to grow to the same FV as under multi-period compounding.
Example: EFF% for 10%, semiannual:
FV = (1 + rNom/m)m
= (1.05)2 =
EFF% = 10.25% because
(1.1025)1 =
Any PV would grow to same FV at 10.25% annually or 10% semiannually.

92. An investment with monthly payments is different from one with quarterly payments. Must put on EFF% basis to compare rates of return. Use EFF% only for comparisons.
Banks say “interest paid daily.” Same as compounded daily.

93. Find EFF% for a nominal rate of10%, compounded semi-annually
Or use a financial calculator.

94. EAR = EFF% of 10% EARAnnual = 10%. EARQ = (1 + 0.10/4)4 – 1 = 10.38%.
EARM = ( /12)12 – 1 = %.
EARD(360) = ( /360)360 – 1 = %.

95. Can the effective rate ever be equal to the nominal rate?
Yes, but only if annual compounding is used, i.e., if m = 1.
If m > 1, EFF% will always be greater than the nominal rate.

96. When is each rate used? iNom:
Written into contracts, quoted by banks and brokers. Not used in calculations or shown
on time lines.

97. rPer:
Used in calculations, shown on time lines.
If rNom has annual compounding,
then rPer = rNom/1 = rNom.

98. EAR = EFF%:
Used to compare returns on investments with different payments per year.
(Used for calculations if and only if
dealing with annuities where payments don’t match interest compounding periods.)

99. FV of $100 after 3 years under 10% semiannual compounding? Quarterly?mn i æ ö FV = PV ç +
Nom ÷ n è ø m
2x3
0.10 æ ö FV =
$100 ç 1 + ÷ 3S è ø 2
= $100(1.05)6 = $
FV3Q = $100(1.025)12 = $

100. What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semiannually? 1 2 3 4 5
6 6-mos.
periods 5%
100
100
100

101. Payments occur annually, but compounding occurs each 6 months.
So we can’t use normal annuity valuation techniques.

102. 1st Method: Compound Each CF1 2 3 4 5 6 5%
100
100
100.00
110.25
121.55
331.80
FVA3 = 100(1.05) (1.05) =

103. Could you find FV with a financial calculator?
2nd Method: Treat as an Annuity
Could you find FV with a financial calculator?
Yes, by following these steps:
a. Find the EAR for the quoted rate:
EAR = ( ) – 1 = 10.25%.
0.10 2 2

104. Or, to find EAR with a calculator:
NOM% = 10.
P/YR = 2.
EFF% =

105. b. The cash flow stream is an annual annuity. Find rNom (annual) whose
EFF% = 10.25%. In calculator,
EFF% = 10.25
P/YR = 1
NOM% = 10.25 c.
INPUTS N
r/YR PV
PMT FV
OUTPUT
331.80

106. What’s the PV of this stream?1 2 3 5%
100
100
100
90.70
82.27
74.62
247.59

107. On January 1 you deposit $100 in an account that pays a nominal interest rate of 10%, with daily compounding (365 days).
How much will you have on October 1, or after 9 months (273 days)? (Days given.)

108. ( ) ( ) iPer = 10.0% / 365 = 0.027397% per day. ... FV = $1001 2
273 %
...
-100
FV = ? FV =
$100 ( )
273
273 =
$100 ( ) = $
Note: % in calculator, decimal in equation.

109. Leave data in calculator.
rPer = rNom/m
= 10.0/365
= % per day.
INPUTS
107.77 N
r/YR PV
PMT FV
OUTPUT
Enter i in one step.
Leave data in calculator.

110. Now suppose you leave your money in the bank for 21 months, which is 1
Now suppose you leave your money in the bank for 21 months, which is 1.75 years or = 638 days.
How much will be in your account at maturity?
Answer: Override N = 273 with N = 638.
FV = $

111. rPer = 0.027397% per day. ... ... FV = $100(1 + .10/365)638
365
638 days
...
...
-100
FV =
FV = $100( /365)638
= $100( )638
= $100(1.1910)
= $

112. You are offered a note that pays $1,000 in 15 months (or 456 days) for $850. You have $850 in a bank that pays a 7.0% nominal rate, with 365 daily compounding, which is a daily rate of % and an EAR of 7.25%. You plan to leave the money in the bank if you don’t buy the note. The note is riskless.
Should you buy it?

113. 1. Greatest future wealth: FV 2. Greatest wealth today: PV
rPer = % per day.
365
456 days
...
...
-850
1,000
3 Ways to Solve:
1. Greatest future wealth: FV
2. Greatest wealth today: PV
3. Highest rate of return: Highest EFF%

114. 1. Greatest Future Wealth
Find FV of $850 left in bank for
15 months and compare with
note’s FV = $1,000.
FVBank = $850( )456
= $ in bank.
Buy the note: $1,000 > $

115. Calculator Solution to FV: rPer = rNom/m = 7.0/365
= 7.0/365
= % per day.
INPUTS
927.67 N
r/YR PV
PMT FV
OUTPUT
Enter rPer in one step.

116. 2. Greatest Present Wealth
Find PV of note, and compare
with its $850 cost:
PV = $1,000/( )456
= $

117. 7/365 =
INPUTS N
r/YR PV
PMT FV
OUTPUT
PV of note is greater than its $850 cost, so buy the note. Raises your wealth.

118. 3. Rate of Return
Find the EFF% on note and compare with 7.25% bank pays, which is your opportunity cost of capital:
FVn = PV(1 + r)n
$1,000 = $850(1 + r)456
Now we must solve for r.

119. Convert % to decimal: Decimal = 0.035646/100 = 0.00035646.
INPUTS %
per day N
r/YR PV
PMT FV
OUTPUT
Convert % to decimal:
Decimal = /100 =
EAR = EFF% = ( )365 – 1
= 13.89%.

120. Using interest conversion:
P/YR = 365.
NOM% = (365) =
EFF% =
Since 13.89% > 7.25% opportunity cost,
buy the note.