CHAPTER 9 Time Value of Money Future value Present value Annuities Rates of return Amortization 9-‹#› 1 Time lines Show the timing of cash flows. Tick marks occur at the end of periods, so Time 0 is today; Time 1 is the end of the first period (year, month, etc.) or the beginning of the second period. CF0 CF1 CF3 CF2 0 1 2 3 I% 9-‹#› 2 Drawing time lines 100 100 100 0 1 2 3 I% 3 year $100 ordinary annuity 100 0 1 2 I% $100 lump sum due in 2 years 9-‹#› 3 Future Value of Money If you deposit $1,000 today at 10%, how much will you have after 15 years? Interest($) = Principal ∙ Interest Rate(%) Simple Interest The original principal stays the same. There is no interest on interest. The interest is only on the original principal. Compound Interest The principal changes through time. There is “interest on interest”. The interest is on the new principal. 9-‹#› 9-‹#› 9-‹#› Simple Interest Interest($) = Principal($) ∙ Interest Rate(%) = V0 ∙ I V1 = V0 + Interest = V0 + V0 ∙ I = V0(1 + I) V2 = V1 + Interest = V1 + V0 ∙ I = V0(1 + I) + V0 ∙ I = V0(1 + I + I) = V0(1 + 2I) V3 = V2 + Interest = V2 + V0 ∙ I = V0(1 + 2I) + V0 ∙ I = V0(1 + 2I + I) = V0(1 + 3I) . . Vn = V0(1 + nI) FVn = PV(1 + nI) 9-‹#› Compound Interest Interest ($) = Principal ($) ∙ Interest Rate (%) = V ∙ I V1 = V0 + Interest = V0 + V0 ∙ I = V0(1 + I) V2 = V1 + Interest = V1 + V1 ∙ I = V1(1 + I) V3 = V2 + Interest = V2 + V2 ∙ I = V2(1 + I) V2 = V1(1 + I) = V0(1 + I)(1 + I) = V0(1 + I)2 V3 = V2(1 + I) = V0(1 + I)2(1 + I) = V0(1 + I)3 Vn = V0 (1 + I)n FVn = PV(1 + I)n = PV∙FVIF V2 = V1 + Interest = V1 + (V0 + Interest) ∙ I 9-‹#› Example What is the future value of $20 invested for 2 years at 10%? Simple: FV = PV(1+nI) = 20(1+2I) = 20(1+0.2) = $24 Compound: FV = PV(1+I)n = 20(1+I)2 = 20(1+0.1)2 = $24.2 What is the future value of $20 invested for 100 years at 10%? Simple: FV = 20(1+ ) = Compound : FV = 20(1.1)100 = 275,612.25 9-‹#› The Power of Compounding The Value of Manhattan In 1626, the land was bought from American Indians at $24. In 2018, value = $24(1+I)392 9-‹#› Solving for FV: The formula method Solve the general FV equation: FVN = PV∙(1 + I)N = PV ∙ FVIF FV15 = PV∙(1 + I)15 = $1,000∙(1.10)15 = $4,177.25 = $1,000∙4.177 = $4,177 (Table A) 9-‹#› Present Value of Money If you want to have $4,177.25 after 15 years, how much do you have to deposit today at 10%? 9-‹#› PV = ? 4,177.25 Present Value of Money Finding the PV of a cash flow or series of cash flows is called discounting (the reverse of compounding). 0 1 2 … 15 10% 9-‹#› 13 Solving for PV: The formula method Solve the general FV equat.

1. CHAPTER 9
Time Value of Money
Future value
Present value
Annuities
Rates of return
Amortization
9-‹#›
1
Time lines
Show the timing of cash flows.
Tick marks occur at the end of periods, so Time 0 is today;
Time 1 is the end of the first period (year, month, etc.) or the
beginning of the second period.
CF0
CF1
CF3
CF2
0

2. 1
2
3
I%
9-‹#›
2
Drawing time lines
100
100
100
0
1
2
3
I%
3 year $100 ordinary annuity
100
0
1
2
I%

3. $100 lump sum due in 2 years
9-‹#›
3
Future Value of Money
If you deposit $1,000 today at 10%, how much will you have
after 15 years?
Interest($) = Principal ∙ Interest Rate(%)
Simple Interest
The original principal stays the same.
There is no interest on interest. The interest is only on the
original principal.
Compound Interest
The principal changes through time.
There is “interest on interest”. The interest is on the new
principal.
9-‹#›

4. 9-‹#›
9-‹#›
Simple Interest
Interest($) = Principal($) ∙ Interest Rate(%) = V0 ∙ I
V1 = V0 + Interest = V0 + V0 ∙ I = V0(1 + I)
V2 = V1 + Interest = V1 + V0 ∙ I = V0(1 + I) + V0 ∙ I
= V0(1 + I + I) = V0(1 + 2I)
V3 = V2 + Interest = V2 + V0 ∙ I = V0(1 + 2I) + V0 ∙ I
= V0(1 + 2I + I) = V0(1 + 3I)
.
.
Vn = V0(1 + nI)
FVn = PV(1 + nI)
9-‹#›
Compound Interest
Interest ($) = Principal ($) ∙ Interest Rate (%) = V ∙ I
V1 = V0 + Interest = V0 + V0 ∙ I = V0(1 + I)
V2 = V1 + Interest = V1 + V1 ∙ I = V1(1 + I)

5. V3 = V2 + Interest = V2 + V2 ∙ I = V2(1 + I)
V2 = V1(1 + I) = V0(1 + I)(1 + I) = V0(1 + I)2
V3 = V2(1 + I) = V0(1 + I)2(1 + I) = V0(1 + I)3
Vn = V0 (1 + I)n
FVn = PV(1 + I)n = PV∙FVIF
V2 = V1 + Interest = V1 + (V0 + Interest) ∙ I
9-‹#›
Example
What is the future value of $20 invested for 2 years at 10%?
Simple: FV = PV(1+nI)
= 20(1+2I) = 20(1+0.2) = $24
Compound: FV = PV(1+I)n
= 20(1+I)2 = 20(1+0.1)2 = $24.2
What is the future value of $20 invested for 100 years at 10%?
Simple: FV = 20(1+ ) =
Compound : FV = 20(1.1)100 = 275,612.25
9-‹#›
The Power of Compounding
The Value of Manhattan
In 1626, the land was bought from American Indians at $24.

6. In 2018, value = $24(1+I)392
9-‹#›
Solving for FV:
The formula method
Solve the general FV equation:
FVN = PV∙(1 + I)N = PV ∙ FVIF
FV15 = PV∙(1 + I)15 = $1,000∙(1.10)15 = $4,177.25
= $1,000∙4.177 = $4,177
(Table A)
9-‹#›
Present Value of Money
If you want to have $4,177.25 after 15 years, how much do you
have to deposit today at 10%?
9-‹#›

7. PV = ?
4,177.25
Present Value of Money
Finding the PV of a cash flow or series of cash flows is called
discounting (the reverse of compounding).
0
1
2 …
15
10%
9-‹#›
13
Solving for PV:
The formula method
Solve the general FV equation for PV:
PV = = ∙
= FVN ∙ PVIF
PV = = = $4,177.25 ∙
= $4,177.25∙0.239
(Table C)
= $998.36, but $4,177.25∙0.2394 = $1,000

8. 9-‹#›
14
Examples
If you want $10,000 after 10 years, how much do you have to
deposit today at 5%?
If you want $100,000 someday for a world tour, how long will it
take at 7% if you deposit $10,000 today?
If you want $50,000 in 10 years, at what rate do you have to
invest your money if you have $10,000 today?
9-‹#›
9-‹#›

9. 9-‹#›
9-‹#›
9-‹#›
Calculation
You deposit $1000 today at 6%. After one year (t=1), you
withdraw $300, after two years (t=2), you deposit $500 more,
and no deposit or withdrawal after that, then how much will you
have in year 5 (t=5) ?
9-‹#›
Answer
Step by step solution
0 1 2 5
$1,000 -$300 $500 ?
1000 PV, 6 I/Y, 1 N, CPT FV =>1060
1060

10. ( )PV, 6 I/Y, ( )N, FV => ( )
( ) + ( ) = ( )
( ) PV, 6 I/Y, ( ) N, FV => ( )
9-‹#›
Annuity
A series of cash flows of the same amount with fixed intervals
for a specified number of periods.
0 1 2 3 4
$20 $20 $30 $20
$20 $30 $40 $50
$20 $20 $0 $20
$0 $0 $20 $20
$20 $30 $20 $30
9-‹#›
FV of Annuity
100
100
100

11. 0
1
2
3
I%
9-‹#›
9-‹#›
9-‹#›
9-‹#›
FV of Annuity
FVA = 100(1+I)2 + 100(1+I) + 100

12. = 100[(1+I)2 + (1+I) + 1]
= 100∙
For n periods,
FVA = PMT∙
= PMT∙FVAIF
9-‹#›
PV of Annuity
100
100
100
0
1
2
3
I%
9-‹#›
PV of Annuity

13. PVA = + +
= 100 [ + + ]
= 100
For n periods,
PVA = PMT
= PMT∙PVAIF
9-‹#›
9-‹#›
9-‹#›
9-‹#›
Examples
If you deposit $3,000 a year for 10 years at 7%, how much will

14. you have after 10 years?
If you want to receive $5o,000 per year for 20 years, how much
do you have to deposit today at 5%?
9-‹#›
More examples
You need $100,000 in year 15 to start your own business. If
your bank’s interest rate is 6%, how much do you have to
deposit each year to get $100,000?
You need $100,000 for a world tour. If you deposit $10,000
each year, how long will it take for you to accumulate $100,000
at 7%?
9-‹#›
9-‹#›
Investment Choice
You have $10,000 to invest. There are two choices for your

15. investment.
Choice A: Buying an annuity at $10,000 and receiving $1,000
for 20 years.
Choice B: Depositing $10,000 in a bank that pays 8% interest
rate.
9-‹#›
9-‹#›
Harder Problem
You need to accumulate $11,000. To do so, you plan to deposit
$1,350 per year in a bank that pays 6% interest. Your last
deposit will be less than $1,350 if less is needed to round out to
$11,000.
A. How many years will it take to reach your goal?
B. How large will the last deposit be for you to have exactly
$11,000 in your account?
9-‹#›

16. 9-‹#›
Answer
Two ways
A: The FV at year 6 will be $9,416.68, and the money will grow
in the account for a year to $9,981.68.
B: The FV at year 7 will be $11,331.68.
9-‹#›
9-‹#›
What is the difference between an ordinary annuity and an
annuity due?
Ordinary Annuity
PMT
PMT
PMT

17. 0
1
2
3
i%
PMT
PMT
0
1
2
3
i%
PMT
Annuity Due
9-‹#›
42
Ordinary Annuity and Annuity Due
FVADUE = FVA(1+I)

18. PVADUE = PVA(1+I)
9-‹#›
PV of Annuity Due
What is the PV of 3-year annuity due of $100 payments at 10%?
Now, $100 payments occur at the beginning of each period.
PVAdue= PVA (1+I) = $248.69(1.1) =$273.55.
Alternatively, set calculator to “BEGIN” mode and solve for the
FV of the annuity:
9-‹#›
FV of Annuity Due
What is the FV of 3-year annuity due of $100 payments at 10%?
Now, $100 payments occur at the beginning of each period.
FVAdue= FVA (1+I) = $331(1.1) = $364.10.
Alternatively, set calculator to “BEGIN” mode and solve for the
FV of the annuity:

19. 9-‹#›
9-‹#›
What is the (future) value of this annuity at t = 1? At t = 2?
100(1+I) + 100 +
9-‹#›
9-‹#›
9-‹#›

20. 9-‹#›
Question
If you can receive $50,000 per year forever, how much are you
willing to pay for that?
9-‹#›
Perpetuity
Perpetuity
PV =
If I = 10%, PV = = $0.5M
If I= 5%, PV = = $1M
9-‹#›
52
Growing Perpetuity
If the payments grow at a constant rate, g, it is a growing

21. perpetuity.
PV =
=
Example: If I = 10%, g = 5%,
PMT0 = $50,000,
PV = = $1.05M
9-‹#›
The Power of Compound Interest
A 20-year-old student wants to save $3 a day for her retirement.
Every day she places $3 in a drawer. At the end of the year, she
invests the accumulated savings ($1,095) in a brokerage account
with an expected annual return of 12%.
How much money will she have when she is 65 years old?
9-‹#›
Solving for FV:
If she begins saving today, how much will she have when she is
65?
If she sticks to her plan, she will have $( ) when she

22. is 65.
( ) N, 12 I/YR, -1095 PMT, FV =>
( )
9-‹#›
Solving for FV:
If you don’t start saving until you are 40 years old, how much
will you have at 65?
If a 40-year-old investor begins saving today, and sticks to the
plan, he or she will have $146,000.59 at age 65. This is $1.3
million less than if starting at age 20.
Lesson: It pays to start saving early.
25 N, 12 I/Y, 1095 PMT,
FV =› 146,000.59
9-‹#›
Solving for PMT:
How much must the 40-year old deposit annually to catch the
20-year old?
To find the required annual contribution, enter the number of
years until retirement and the final goal of $1,487,261.89, and
solve for PMT.
25 N, 12 I/Y, 1,487,261.89 FV,
PMT = › 11,154.42

23. 9-‹#›
9-‹#›
What is the PV of this uneven cash flow stream?
0
100
1
300
2
300
3
10%
-50
4

24. 90.91
247.93
225.39
-34.15
530.08 = PV
9-‹#›
59
Solving for PV:
Uneven cash flow stream
Input cash flows in the calculator’s “CF” register:
CF0 = 0
C01 = 100
F01 = 1
C02 = 300
F02 = 2
C03 = -50
Press NPV button, then enter I = 10, and hit CPT=> $530.087.
(Here NPV = PV.)
9-‹#›

25. NPV (Net Present Value)
NPV is calculated net of costs.
If your project’s PV of cash inflows is greater than the PV of
cash outflows, the project will enhance your company’s
profitability.
In chapter 9, generally there is no cost at time 0, so the NPVs
are positive, but in chapter 12, when we evaluate projects, the
NPVs can be negative.
Example: 0 1 2 3 (I = 10%)
-1000 300 300 500
9-‹#›
9-‹#›
9-‹#›

26. 9-‹#›
CHAPTER 9
Time Value of Money
Future value
Present value
Annuities
Rates of return
Amortization
9-‹#›
1
Time lines
Show the timing of cash flows.
Tick marks occur at the end of periods, so Time 0 is today;
Time 1 is the end of the first period (year, month, etc.) or the
beginning of the second period.
CF0
CF1
CF3
CF2

27. 0
1
2
3
I%
9-‹#›
2
Drawing time lines
100
100
100
0
1
2
3
I%
3 year $100 ordinary annuity
100
0
1
2
I%

28. $100 lump sum due in 2 years
9-‹#›
3
Future Value of Money
If you deposit $1,000 today at 10%, how much will you have
after 15 years?
Interest($) = Principal ∙ Interest Rate(%)
Simple Interest
The original principal stays the same.
There is no interest on interest. The interest is only on the
original principal.
Compound Interest
The principal changes through time.
There is “interest on interest”. The interest is on the new
principal.
9-‹#›

29. 9-‹#›
9-‹#›
Simple Interest
Interest($) = Principal($) ∙ Interest Rate(%) = V0 ∙ I
V1 = V0 + Interest = V0 + V0 ∙ I = V0(1 + I)
V2 = V1 + Interest = V1 + V0 ∙ I = V0(1 + I) + V0 ∙ I
= V0(1 + I + I) = V0(1 + 2I)
V3 = V2 + Interest = V2 + V0 ∙ I = V0(1 + 2I) + V0 ∙ I
= V0(1 + 2I + I) = V0(1 + 3I)
.
.
Vn = V0(1 + nI)
FVn = PV(1 + nI)
9-‹#›
Compound Interest
Interest ($) = Principal ($) ∙ Interest Rate (%) = V ∙ I
V1 = V0 + Interest = V0 + V0 ∙ I = V0(1 + I)

30. V2 = V1 + Interest = V1 + V1 ∙ I = V1(1 + I)
V3 = V2 + Interest = V2 + V2 ∙ I = V2(1 + I)
V2 = V1(1 + I) = V0(1 + I)(1 + I) = V0(1 + I)2
V3 = V2(1 + I) = V0(1 + I)2(1 + I) = V0(1 + I)3
Vn = V0 (1 + I)n
FVn = PV(1 + I)n = PV∙FVIF
V2 = V1 + Interest = V1 + (V0 + Interest) ∙ I
9-‹#›
Example
What is the future value of $20 invested for 2 years at 10%?
Simple: FV = PV(1+nI)
= 20(1+2I) = 20(1+0.2) = $24
Compound: FV = PV(1+I)n
= 20(1+I)2 = 20(1+0.1)2 = $24.2
What is the future value of $20 invested for 100 years at 10%?
Simple: FV = 20(1+ ) =
Compound : FV = 20(1.1)100 = 275,612.25
9-‹#›
The Power of Compounding
The Value of Manhattan

31. In 1626, the land was bought from American Indians at $24.
In 2018, value = $24(1+I)392
9-‹#›
Solving for FV:
The formula method
Solve the general FV equation:
FVN = PV∙(1 + I)N = PV ∙ FVIF
FV15 = PV∙(1 + I)15 = $1,000∙(1.10)15 = $4,177.25
= $1,000∙4.177 = $4,177
(Table A)
9-‹#›
Present Value of Money
If you want to have $4,177.25 after 15 years, how much do you
have to deposit today at 10%?
9-‹#›

32. PV = ?
4,177.25
Present Value of Money
Finding the PV of a cash flow or series of cash flows is called
discounting (the reverse of compounding).
0
1
2 …
15
10%
9-‹#›
13
Solving for PV:
The formula method
Solve the general FV equation for PV:
PV = = ∙
= FVN ∙ PVIF
PV = = = $4,177.25 ∙
= $4,177.25∙0.239
(Table C)
= $998.36, but $4,177.25∙0.2394 = $1,000

33. 9-‹#›
14
Examples
If you want $10,000 after 10 years, how much do you have to
deposit today at 5%?
If you want $100,000 someday for a world tour, how long will it
take at 7% if you deposit $10,000 today?
If you want $50,000 in 10 years, at what rate do you have to
invest your money if you have $10,000 today?
9-‹#›
9-‹#›

34. 9-‹#›
9-‹#›
9-‹#›
Calculation
You deposit $1000 today at 6%. After one year (t=1), you
withdraw $300, after two years (t=2), you deposit $500 more,
and no deposit or withdrawal after that, then how much will you
have in year 5 (t=5) ?
9-‹#›
Answer
Step by step solution
0 1 2 5
$1,000 -$300 $500 ?
1000 PV, 6 I/Y, 1 N, CPT FV =>1060

35. 1060
( )PV, 6 I/Y, ( )N, FV => ( )
( ) + ( ) = ( )
( ) PV, 6 I/Y, ( ) N, FV => ( )
9-‹#›
Annuity
A series of cash flows of the same amount with fixed intervals
for a specified number of periods.
0 1 2 3 4
$20 $20 $30 $20
$20 $30 $40 $50
$20 $20 $0 $20
$0 $0 $20 $20
$20 $30 $20 $30
9-‹#›
FV of Annuity
100
100

36. 100
0
1
2
3
I%
9-‹#›
9-‹#›
9-‹#›
9-‹#›
FV of Annuity

37. FVA = 100(1+I)2 + 100(1+I) + 100
= 100[(1+I)2 + (1+I) + 1]
= 100∙
For n periods,
FVA = PMT∙
= PMT∙FVAIF
9-‹#›
PV of Annuity
100
100
100
0
1
2
3
I%
9-‹#›

38. PV of Annuity
PVA = + +
= 100 [ + + ]
= 100
For n periods,
PVA = PMT
= PMT∙PVAIF
9-‹#›
9-‹#›
9-‹#›
9-‹#›
Examples

39. If you deposit $3,000 a year for 10 years at 7%, how much will
you have after 10 years?
If you want to receive $5o,000 per year for 20 years, how much
do you have to deposit today at 5%?
9-‹#›
More examples
You need $100,000 in year 15 to start your own business. If
your bank’s interest rate is 6%, how much do you have to
deposit each year to get $100,000?
You need $100,000 for a world tour. If you deposit $10,000
each year, how long will it take for you to accumulate $100,000
at 7%?
9-‹#›
9-‹#›
Investment Choice

40. You have $10,000 to invest. There are two choices for your
investment.
Choice A: Buying an annuity at $10,000 and receiving $1,000
for 20 years.
Choice B: Depositing $10,000 in a bank that pays 8% interest
rate.
9-‹#›
9-‹#›
Harder Problem
You need to accumulate $11,000. To do so, you plan to deposit
$1,350 per year in a bank that pays 6% interest. Your last
deposit will be less than $1,350 if less is needed to round out to
$11,000.
A. How many years will it take to reach your goal?
B. How large will the last deposit be for you to have exactly
$11,000 in your account?
9-‹#›

41. 9-‹#›
Answer
Two ways
A: The FV at year 6 will be $9,416.68, and the money will grow
in the account for a year to $9,981.68.
B: The FV at year 7 will be $11,331.68.
9-‹#›
9-‹#›
What is the difference between an ordinary annuity and an
annuity due?
Ordinary Annuity
PMT
PMT

42. PMT
0
1
2
3
i%
PMT
PMT
0
1
2
3
i%
PMT
Annuity Due
9-‹#›
42
Ordinary Annuity and Annuity Due
FVADUE = FVA(1+I)

43. PVADUE = PVA(1+I)
9-‹#›
PV of Annuity Due
What is the PV of 3-year annuity due of $100 payments at 10%?
Now, $100 payments occur at the beginning of each period.
PVAdue= PVA (1+I) = $248.69(1.1) =$273.55.
Alternatively, set calculator to “BEGIN” mode and solve for the
FV of the annuity:
9-‹#›
FV of Annuity Due
What is the FV of 3-year annuity due of $100 payments at 10%?
Now, $100 payments occur at the beginning of each period.
FVAdue= FVA (1+I) = $331(1.1) = $364.10.
Alternatively, set calculator to “BEGIN” mode and solve for the
FV of the annuity:

44. 9-‹#›
What is the (future) value of this annuity at t = 1? At t = 2?
100(1+I) + 100 +
9-‹#›
9-‹#›
9-‹#›
Question
If you can receive $50,000 per year forever, how much are you
willing to pay for that?

45. 9-‹#›
Perpetuity
Perpetuity
PV =
If I = 10%, PV = = $0.5M
If I= 5%, PV = = $1M
9-‹#›
50
Growing Perpetuity
If the payments grow at a constant rate, g, it is a growing
perpetuity.
PV =
=
Example: If I = 10%, g = 5%,
PMT0 = $50,000,
PV = = $1.05M
9-‹#›

46. The Power of Compound Interest
A 20-year-old student wants to save $3 a day for her retirement.
Every day she places $3 in a drawer. At the end of the year, she
invests the accumulated savings ($1,095) in a brokerage account
with an expected annual return of 12%.
How much money will she have when she is 65 years old?
9-‹#›
Solving for FV:
If she begins saving today, how much will she have when she is
65?
If she sticks to her plan, she will have $( ) when she
is 65.
( ) N, 12 I/YR, -1095 PMT, FV =>
( )
9-‹#›
Solving for FV:
If you don’t start saving until you are 40 years old, how much
will you have at 65?

47. If a 40-year-old investor begins saving today, and sticks to the
plan, he or she will have $146,000.59 at age 65. This is $1.3
million less than if starting at age 20.
Lesson: It pays to start saving early.
25 N, 12 I/Y, 1095 PMT,
FV =› 146,000.59
9-‹#›
Solving for PMT:
How much must the 40-year old deposit annually to catch the
20-year old?
To find the required annual contribution, enter the number of
years until retirement and the final goal of $1,487,261.89, and
solve for PMT.
25 N, 12 I/Y, 1,487,261.89 FV,
PMT = › 11,154.42
9-‹#›
What is the PV of this uneven cash flow stream?
0
100
1

48. 300
2
300
3
10%
-50
4
90.91
247.93
225.39
-34.15
530.08 = PV
9-‹#›
56
Solving for PV:
Uneven cash flow stream

49. Input cash flows in the calculator’s “CF” register:
CF0 = 0
C01 = 100
F01 = 1
C02 = 300
F02 = 2
C03 = -50
Press NPV button, then enter I = 10, and hit CPT=> $530.087.
(Here NPV = PV.)
9-‹#›
NPV (Net Present Value)
NPV is calculated net of costs.
If your project’s PV of cash inflows is greater than the PV of
cash outflows, the project will enhance your company’s
profitability.
In chapter 9, generally there is no cost at time 0, so the NPVs
are positive, but in chapter 12, when we evaluate projects, the
NPVs can be negative.
Example: 0 1 2 3 (I = 10%)
-1000 300 300 500
9-‹#›
CHAPTER 9

50. Time Value of Money
Future value
Present value
Annuities
Rates of return
Amortization
9-‹#›
1
Time lines
Show the timing of cash flows.
Tick marks occur at the end of periods, so Time 0 is today;
Time 1 is the end of the first period (year, month, etc.) or the
beginning of the second period.
CF0
CF1
CF3
CF2
0
1
2
3
I%

51. 9-‹#›
2
Drawing time lines
100
100
100
0
1
2
3
I%
3 year $100 ordinary annuity
100
0
1
2
I%
$100 lump sum due in 2 years

52. 9-‹#›
3
Future Value of Money
If you deposit $1,000 today at 10%, how much will you have
after 15 years?
Interest($) = Principal ∙ Interest Rate(%)
Simple Interest
The original principal stays the same.
There is no interest on interest. The interest is only on the
original principal.
Compound Interest
The principal changes through time.
There is “interest on interest”. The interest is on the new
principal.
9-‹#›

53. 9-‹#›
9-‹#›
Simple Interest
Interest($) = Principal($) ∙ Interest Rate(%) = V0 ∙ I
V1 = V0 + Interest = V0 + V0 ∙ I = V0(1 + I)
V2 = V1 + Interest = V1 + V0 ∙ I = V0(1 + I) + V0 ∙ I
= V0(1 + I + I) = V0(1 + 2I)
V3 = V2 + Interest = V2 + V0 ∙ I = V0(1 + 2I) + V0 ∙ I
= V0(1 + 2I + I) = V0(1 + 3I)
.
.
Vn = V0(1 + nI)
FVn = PV(1 + nI)
9-‹#›
Compound Interest
Interest ($) = Principal ($) ∙ Interest Rate (%) = V ∙ I
V1 = V0 + Interest = V0 + V0 ∙ I = V0(1 + I)
V2 = V1 + Interest = V1 + V1 ∙ I = V1(1 + I)
V3 = V2 + Interest = V2 + V2 ∙ I = V2(1 + I)
V2 = V1(1 + I) = V0(1 + I)(1 + I) = V0(1 + I)2
V3 = V2(1 + I) = V0(1 + I)2(1 + I) = V0(1 + I)3

54. Vn = V0 (1 + I)n
FVn = PV(1 + I)n = PV∙FVIF
V2 = V1 + Interest = V1 + (V0 + Interest) ∙ I
9-‹#›
Example
What is the future value of $20 invested for 2 years at 10%?
Simple: FV = PV(1+nI)
= 20(1+2I) = 20(1+0.2) = $24
Compound: FV = PV(1+I)n
= 20(1+I)2 = 20(1+0.1)2 = $24.2
What is the future value of $20 invested for 100 years at 10%?
Simple: FV = 20(1+ ) =
Compound : FV = 20(1.1)100 = 275,612.25
9-‹#›
The Power of Compounding
The Value of Manhattan
In 1626, the land was bought from American Indians at $24.
In 2018, value = $24(1+I)392

55. 9-‹#›
Solving for FV:
The formula method
Solve the general FV equation:
FVN = PV∙(1 + I)N = PV ∙ FVIF
FV15 = PV∙(1 + I)15 = $1,000∙(1.10)15 = $4,177.25
= $1,000∙4.177 = $4,177
(Table A)
9-‹#›
Present Value of Money
If you want to have $4,177.25 after 15 years, how much do you
have to deposit today at 10%?
9-‹#›
PV = ?
4,177.25
Present Value of Money
Finding the PV of a cash flow or series of cash flows is called

56. discounting (the reverse of compounding).
0
1
2 …
15
10%
9-‹#›
13
Solving for PV:
The formula method
Solve the general FV equation for PV:
PV = = ∙
= FVN ∙ PVIF
PV = = = $4,177.25 ∙
= $4,177.25∙0.239
(Table C)
= $998.36, but $4,177.25∙0.2394 = $1,000

57. 9-‹#›
14
Examples
If you want $10,000 after 10 years, how much do you have to
deposit today at 5%?
If you want $100,000 someday for a world tour, how long will it
take at 7% if you deposit $10,000 today?
If you want $50,000 in 10 years, at what rate do you have to
invest your money if you have $10,000 today?
9-‹#›
9-‹#›
9-‹#›

58. 9-‹#›
9-‹#›
Calculation
You deposit $1000 today at 6%. After one year (t=1), you
withdraw $300, after two years (t=2), you deposit $500 more,
and no deposit or withdrawal after that, then how much will you
have in year 5 (t=5) ?
9-‹#›
Answer
Step by step solution
0 1 2 5
$1,000 -$300 $500 ?
1000 PV, 6 I/Y, 1 N, CPT FV =>1060
1060
( )PV, 6 I/Y, ( )N, FV => ( )
( ) + ( ) = ( )
( ) PV, 6 I/Y, ( ) N, FV => ( )

59. 9-‹#›
Annuity
A series of cash flows of the same amount with fixed intervals
for a specified number of periods.
0 1 2 3 4
$20 $20 $30 $20
$20 $30 $40 $50
$20 $20 $0 $20
$0 $0 $20 $20
$20 $30 $20 $30
9-‹#›
FV of Annuity
100
100
100
0
1
2
3

60. I%
9-‹#›
9-‹#›
9-‹#›
9-‹#›
FV of Annuity
FVA = 100(1+I)2 + 100(1+I) + 100
= 100[(1+I)2 + (1+I) + 1]
= 100∙
For n periods,

61. FVA = PMT∙
= PMT∙FVAIF
9-‹#›
PV of Annuity
100
100
100
0
1
2
3
I%
9-‹#›
PV of Annuity
PVA = + +
= 100 [ + + ]
= 100
For n periods,

62. PVA = PMT
= PMT∙PVAIF
9-‹#›
Examples
If you deposit $3,000 a year for 10 years at 7%, how much will
you have after 10 years?
If you want to receive $5o,000 per year for 20 years, how much
do you have to deposit today at 5%?
9-‹#›
More examples
You need $100,000 in year 15 to start your own business. If
your bank’s interest rate is 6%, how much do you have to
deposit each year to get $100,000?
You need $100,000 for a world tour. If you deposit $10,000
each year, how long will it take for you to accumulate $100,000
at 7%?

63. 9-‹#›
Investment Choice
You have $10,000 to invest. There are two choices for your
investment.
Choice A: Buying an annuity at $10,000 and receiving $1,000
for 20 years.
Choice B: Depositing $10,000 in a bank that pays 8% interest
rate.
9-‹#›
Harder Problem
You need to accumulate $11,000. To do so, you plan to deposit
$1,350 per year in a bank that pays 6% interest. Your last
deposit will be less than $1,350 if less is needed to round out to
$11,000.
A. How many years will it take to reach your goal?
B. How large will the last deposit be for you to have exactly
$11,000 in your account?
9-‹#›
Answer
Two ways
A: The FV at year 6 will be $9,416.68, and the money will grow
in the account for a year to $9,981.68.

64. B: The FV at year 7 will be $11,331.68.
9-‹#›
What is the difference between an ordinary annuity and an
annuity due?
Ordinary Annuity
PMT
PMT
PMT
0
1
2
3
i%
PMT
PMT
0
1
2
3

65. i%
PMT
Annuity Due
9-‹#›
35
Ordinary Annuity and Annuity Due
FVADUE = FVA(1+I)
PVADUE = PVA(1+I)
9-‹#›
PV of Annuity Due
What is the PV of 3-year annuity due of $100 payments at 10%?
Now, $100 payments occur at the beginning of each period.
PVAdue= PVA (1+I) = $248.69(1.1) =$273.55.
Alternatively, set calculator to “BEGIN” mode and solve for the
FV of the annuity:

66. 9-‹#›
FV of Annuity Due
What is the FV of 3-year annuity due of $100 payments at 10%?
Now, $100 payments occur at the beginning of each period.
FVAdue= FVA (1+I) = $331(1.1) = $364.10.
Alternatively, set calculator to “BEGIN” mode and solve for the
FV of the annuity:
9-‹#›
What is the (future) value of this annuity at t = 1? At t = 2?
100(1+I) + 100 +
9-‹#›
Question
If you can receive $50,000 per year forever, how much are you
willing to pay for that?

67. 9-‹#›
Perpetuity
Perpetuity
PV =
If I = 10%, PV = = $0.5M
If I= 5%, PV = = $1M
9-‹#›
41
Growing Perpetuity
If the payments grow at a constant rate, g, it is a growing
perpetuity.
PV =
=
Example: If I = 10%, g = 5%,
PMT0 = $50,000,
PV = = $1.05M

68. 9-‹#›
The Power of Compound Interest
A 20-year-old student wants to save $3 a day for her retirement.
Every day she places $3 in a drawer. At the end of the year, she
invests the accumulated savings ($1,095) in a brokerage account
with an expected annual return of 12%.
How much money will she have when she is 65 years old?
9-‹#›
Solving for FV:
If she begins saving today, how much will she have when she is
65?
If she sticks to her plan, she will have $( ) when she
is 65.
( ) N, 12 I/YR, -1095 PMT, FV =>
( )
9-‹#›
Solving for FV:

69. If you don’t start saving until you are 40 years old, how much
will you have at 65?
If a 40-year-old investor begins saving today, and sticks to the
plan, he or she will have $146,000.59 at age 65. This is $1.3
million less than if starting at age 20.
Lesson: It pays to start saving early.
25 N, 12 I/Y, 1095 PMT,
FV =› 146,000.59
9-‹#›
Solving for PMT:
How much must the 40-year old deposit annually to catch the
20-year old?
To find the required annual contribution, enter the number of
years until retirement and the final goal of $1,487,261.89, and
solve for PMT.
25 N, 12 I/Y, 1,487,261.89 FV,
PMT = › 11,154.42
9-‹#›
What is the PV of this uneven cash flow stream?
0
100

70. 1
300
2
300
3
10%
-50
4
90.91
247.93
225.39
-34.15
530.08 = PV
9-‹#›
47

71. Solving for PV:
Uneven cash flow stream
Input cash flows in the calculator’s “CF” register:
CF0 = 0
C01 = 100
F01 = 1
C02 = 300
F02 = 2
C03 = -50
Press NPV button, then enter I = 10, and hit CPT=> $530.087.
(Here NPV = PV.)
9-‹#›
NPV (Net Present Value)
NPV is calculated net of costs.
If your project’s PV of cash inflows is greater than the PV of
cash outflows, the project will enhance your company’s
profitability.
In chapter 9, generally there is no cost at time 0, so the NPVs
are positive, but in chapter 12, when we evaluate projects, the
NPVs can be negative.
Example: 0 1 2 3 (I = 10%)
-1000 300 300 500
9-‹#›

72. 9’-‹#›
Classifications of interest rates
Nominal rate (INOM) – also called the quoted or stated rate.
An annual rate that ignores compounding effects.
INOM is stated in contracts. Periods must also be given, e.g.
8% Quarterly or 8% Daily interest.
Periodic rate (IPER) – amount of interest charged each period,
e.g. monthly or quarterly.
IPER = INOM / M, where M is the number of compounding
periods per year. M = 4 for quarterly and M = 12 for monthly
compounding.
9’-‹#›
2
Compounding More than Once per Year
Annual Compounding
0 8% 1
|______________________|
Semiannual Compounding
0 4% 1 4% 2
|__________|___________|
Quarterly Compounding

73. 0 2% 1 2% 2 2% 3 2% 4
|_____|_____|_____|_____|
9’-‹#›
Effective Annual Rate (EAR)
Effective (or equivalent) annual rate (EAR = EFF%): The
annual rate of interest actually being earned, accounting for
compounding.
EFF% for 8% semiannual investment
EFF% = ( 1 + )M - 1
= (1 + )2 – 1 = 8.16%
Should be indifferent between receiving 8.16% annual interest
and receiving 8% interest, compounded semiannually. EAR is
used to compare investment returns.
9’-‹#›
4
9’-‹#›

74. Calculator
Use ICONV key
NOM = INOM
EFF = EAR
C/Y = # of compounding per year
9’-‹#›
What is the FV of $100 after 3 years under 10% semiannual
compounding? Quarterly compounding?
9’-‹#›
7
calculator
10% semi-annual compounding
5 I/Y, 100 PV, 6 N, FV => 134.01
Quarterly compounding
2.5 I/Y, 100 PV, 12 N, FV => 134.49

75. 9’-‹#›
What is the future value of an annuity with $100 monthly
payments at 7% after 5 years?
FV = PMT
= 100
OR,
100 PMT
( ) I/Y
( ) N
FV =
9’-‹#›
GM Incentives
You need $12,000 loan to buy a car.
There are two financing options to choose:
A: 2.9% financing with a 36 month loan
B: A rebate of $1,000 is available and the remaining $11,000 is
to be financed at 10% for 36 months.
Which option would you choose?
9’-‹#›
Quarterly Compounding
A. If you deposit $1,000 in a bank that pays 8% quarterly
compounding, what is the rate of return if you withdraw after 10

76. months?
B. How much in dollars will you get if you withdraw after 10
months?
9’-‹#›
9’-‹#›
9’-‹#›
What’s the FV of a 3-year $100 annuity, if the quoted interest
rate is 10%, compounded semiannually?
Payments occur annually, but compounding occurs every 6
months.
Cannot use normal annuity valuation techniques.

77. 9’-‹#›
Method 1:
Compound each cash flow
FV3 = $100(1.05)4 + $100(1.05)2 + $100
FV3 = $331.80
9’-‹#›
Method 2:
Financial calculator
Find the EAR and treat as an annuity.
EAR = ( 1 + )2 – 1 = 10.25%.
10.25 I/Y, 3 N, -100 PMT, ---
9’-‹#›
Calculator 2

78. 1 P/Y, 2 C/Y, 100 PMT, 3 N, 10 I/Y,
FV => 331.8006 => 331.80
9’-‹#›
Find the PV of this 3-year ordinary annuity.
Could solve by discounting each cash flow, or …
Use the EAR and treat as an annuity to solve for PV.
10.25 I/Y, 3 N, 100 PMT, --
9’-‹#›
Calculator 2
1 P/Y, 2 C/Y, 100 PMT, 3 N, 10 I/Y,
PV => 247.5947 => 247.59
9’-‹#›
P/Y, C/Y
What is the future value of a three-year annuity with quarterly
payments of $50 each at 7%, monthly compounding?

79. 9’-‹#›
Loan amortization
Amortization tables are widely used for home mortgages, auto
loans, business loans, retirement plans, etc.
Financial calculators and spreadsheets are great for setting up
amortization tables.
EXAMPLE: Construct an one-year amortization table for a
$100,000, 8%, semiannual payment, 30-year loan.
9’-‹#›
21
Step 1:
Find the required annual payment
All input information is already given.
60 N, 4 I/Y, 100,000 PV,
PMT = 4,420.18
9’-‹#›

80. 22
Step 2:
Find the interest paid in Period 1
The borrower will owe interest upon the initial balance at the
end of the first period. Interest to be paid in the first period can
be found by multiplying the beginning balance by the periodic
interest rate.
INTt = Beg balt (I)
INT1 = $100,000 (0.04) = $4,000
9’-‹#›
23
Step 3:
Find the principal repaid in Period 1
If a payment of $4,420.18 was made at the end of the first
period and $4,000 was paid toward interest, the remaining value
must represent the amount of principal repaid.
PRIN REPAYMENT = PMT – INT
= $4,420.18 - $4,000 = $420.18
9’-‹#›

81. 24
Step 4:
Find the ending balance after Period 1
To find the balance at the end of the period, subtract the amount
paid toward principal from the beginning balance.
END BAL= BEG BAL – PRIN REP.
= $100,000 - $420.18
= $99,579.82
9’-‹#›
25
Constructing an amortization table:
Repeat steps 1 – 4 until end of loanPBEG BALPMTINTPRIN
REPAYEND
BAL1$100,000$4,420.18$4,000$420.18$99,579.822 99,579.82
4,420.18
3,983.19 436.99 99,142.83
Interest paid declines with each payment as the balance
declines. What are the tax implications of this?
9’-‹#›

82. 26
Illustrating an amortized payment:
Where does the money go?
Constant payments.
Declining interest payments.
Declining balance.
$
0
1
2
3
4,420.18
Interest
420.18
Principal Repayments
9’-‹#›
27
Continuous Compounding
= ℮

83. 9’-‹#›
If compounding takes place continuously,
FVt = PV∙℮It
Alternatively,
PV = = FVt∙℮-It
Example 1: Suppose you invest $200 at 12% continuously
compounded for two years. How much are you going to receive
at the end of two years?
9’-‹#›
Continued …
Answer: It = 0.12×2 = 0.24, e0.24 = 1.2712. So, FVt = FV2 =
$200×1.2712 = $254.25
Example 2: What is the PV of $300 in one year’s time if I =
5%, and continuously compounded?
Answer: It = 0.05×1 = 0.05,
e-It = e-0.05= 0.9512,
so, PV = $300×0.9512 = $285.37
9’-‹#›
$134.49

84. (1.025)
$100
FV
$134.01
(1.05)
$100
FV
)
2
0.10
1
(
$100
FV
)
M
I
1

85. (
PV
FV
12
3Q
6
3S
3
2
3S
N
M
NOM
n
=
=
=
=
+
=
+
=
´
´
CHAPTER 9
Time Value of Money
Future value
Present value
Annuities
Rates of return

86. Amortization
9-‹#›
1
Time lines
Show the timing of cash flows.
Tick marks occur at the end of periods, so Time 0 is today;
Time 1 is the end of the first period (year, month, etc.) or the
beginning of the second period.
CF0
CF1
CF3
CF2
0
1
2
3
I%

87. 9-‹#›
2
Drawing time lines
100
100
100
0
1
2
3
I%
3 year $100 ordinary annuity
100
0
1
2
I%
$100 lump sum due in 2 years
9-‹#›

88. 3
Future Value of Money
If you deposit $1,000 today at 10%, how much will you have
after 15 years?
Interest($) = Principal ∙ Interest Rate(%)
Simple Interest
The original principal stays the same.
There is no interest on interest. The interest is only on the
original principal.
Compound Interest
The principal changes through time.
There is “interest on interest”. The interest is on the new
principal.
9-‹#›
9-‹#›

89. 9-‹#›
Simple Interest
Interest($) = Principal($) ∙ Interest Rate(%) = V0 ∙ I
V1 = V0 + Interest = V0 + V0 ∙ I = V0(1 + I)
V2 = V1 + Interest = V1 + V0 ∙ I = V0(1 + I) + V0 ∙ I
= V0(1 + I + I) = V0(1 + 2I)
V3 = V2 + Interest = V2 + V0 ∙ I = V0(1 + 2I) + V0 ∙ I
= V0(1 + 2I + I) = V0(1 + 3I)
.
.
Vn = V0(1 + nI)
FVn = PV(1 + nI)
9-‹#›
Compound Interest
Interest ($) = Principal ($) ∙ Interest Rate (%) = V ∙ I
V1 = V0 + Interest = V0 + V0 ∙ I = V0(1 + I)
V2 = V1 + Interest = V1 + V1 ∙ I = V1(1 + I)
V3 = V2 + Interest = V2 + V2 ∙ I = V2(1 + I)
V2 = V1(1 + I) = V0(1 + I)(1 + I) = V0(1 + I)2
V3 = V2(1 + I) = V0(1 + I)2(1 + I) = V0(1 + I)3
Vn = V0 (1 + I)n
FVn = PV(1 + I)n = PV∙FVIF
V2 = V1 + Interest = V1 + (V0 + Interest) ∙ I

90. 9-‹#›
Example
What is the future value of $20 invested for 2 years at 10%?
Simple: FV = PV(1+nI)
= 20(1+2I) = 20(1+0.2) = $24
Compound: FV = PV(1+I)n
= 20(1+I)2 = 20(1+0.1)2 = $24.2
What is the future value of $20 invested for 100 years at 10%?
Simple: FV = 20(1+ ) =
Compound : FV = 20(1.1)100 = 275,612.25
9-‹#›
The Power of Compounding
The Value of Manhattan
In 1626, the land was bought from American Indians at $24.
In 2018, value = $24(1+I)392
9-‹#›
Solving for FV:

91. The formula method
Solve the general FV equation:
FVN = PV∙(1 + I)N = PV ∙ FVIF
FV15 = PV∙(1 + I)15 = $1,000∙(1.10)15 = $4,177.25
= $1,000∙4.177 = $4,177
(Table A)
9-‹#›
Present Value of Money
If you want to have $4,177.25 after 15 years, how much do you
have to deposit today at 10%?
9-‹#›
PV = ?
4,177.25
Present Value of Money
Finding the PV of a cash flow or series of cash flows is called
discounting (the reverse of compounding).

92. 0
1
2 …
15
10%
9-‹#›
13
Solving for PV:
The formula method
Solve the general FV equation for PV:
PV = = ∙
= FVN ∙ PVIF
PV = = = $4,177.25 ∙
= $4,177.25∙0.239
(Table C)
= $998.36, but $4,177.25∙0.2394 = $1,000
9-‹#›
14

93. Examples
If you want $10,000 after 10 years, how much do you have to
deposit today at 5%?
If you want $100,000 someday for a world tour, how long will it
take at 7% if you deposit $10,000 today?
If you want $50,000 in 10 years, at what rate do you have to
invest your money if you have $10,000 today?
9-‹#›
9-‹#›
Calculation
You deposit $1000 today at 6%. After one year (t=1), you
withdraw $300, after two years (t=2), you deposit $500 more,
and no deposit or withdrawal after that, then how much will you
have in year 5 (t=5) ?
9-‹#›

94. Answer
Step by step solution
0 1 2 5
$1,000 -$300 $500 ?
1000 PV, 6 I/Y, 1 N, CPT FV =>1060
1060
( )PV, 6 I/Y, ( )N, FV => ( )
( ) + ( ) = ( )
( ) PV, 6 I/Y, ( ) N, FV => ( )
9-‹#›
Annuity
A series of cash flows of the same amount with fixed intervals
for a specified number of periods.
0 1 2 3 4
$20 $20 $30 $20
$20 $30 $40 $50
$20 $20 $0 $20
$0 $0 $20 $20
$20 $30 $20 $30
9-‹#›
FV of Annuity
100
100

95. 100
0
1
2
3
I%
9-‹#›
FV of Annuity
FVA = 100(1+I)2 + 100(1+I) + 100
= 100[(1+I)2 + (1+I) + 1]
= 100∙
For n periods,
FVA = PMT∙
= PMT∙FVAIF
9-‹#›
PV of Annuity
100

96. 100
100
0
1
2
3
I%
9-‹#›
PV of Annuity
PVA = + +
= 100 [ + + ]
= 100
For n periods,
PVA = PMT
= PMT∙PVAIF
9-‹#›
Examples
If you deposit $3,000 a year for 10 years at 7%, how much will

97. you have after 10 years?
If you want to receive $5o,000 per year for 20 years, how much
do you have to deposit today at 5%?
9-‹#›
More examples
You need $100,000 in year 15 to start your own business. If
your bank’s interest rate is 6%, how much do you have to
deposit each year to get $100,000?
You need $100,000 for a world tour. If you deposit $10,000
each year, how long will it take for you to accumulate $100,000
at 7%?
9-‹#›
Investment Choice
You have $10,000 to invest. There are two choices for your
investment.
Choice A: Buying an annuity at $10,000 and receiving $1,000
for 20 years.
Choice B: Depositing $10,000 in a bank that pays 8% interest
rate.

98. 9-‹#›
Harder Problem
You need to accumulate $11,000. To do so, you plan to deposit
$1,350 per year in a bank that pays 6% interest. Your last
deposit will be less than $1,350 if less is needed to round out to
$11,000.
A. How many years will it take to reach your goal?
B. How large will the last deposit be for you to have exactly
$11,000 in your account?
9-‹#›
Answer
Two ways
A: The FV at year 6 will be $9,416.68, and the money will grow
in the account for a year to $9,981.68.
B: The FV at year 7 will be $11,331.68.
9-‹#›
What is the difference between an ordinary annuity and an
annuity due?
Ordinary Annuity

99. PMT
PMT
PMT
0
1
2
3
i%
PMT
PMT
0
1
2
3
i%
PMT
Annuity Due

100. 9-‹#›
29
Ordinary Annuity and Annuity Due
FVADUE = FVA(1+I)
PVADUE = PVA(1+I)
9-‹#›
PV of Annuity Due
What is the PV of 3-year annuity due of $100 payments at 10%?
Now, $100 payments occur at the beginning of each period.
PVAdue= PVA (1+I) = $248.69(1.1) =$273.55.
Alternatively, set calculator to “BEGIN” mode and solve for the
FV of the annuity:
9-‹#›
FV of Annuity Due
What is the FV of 3-year annuity due of $100 payments at 10%?
Now, $100 payments occur at the beginning of each period.
FVAdue= FVA (1+I) = $331(1.1) = $364.10.

101. Alternatively, set calculator to “BEGIN” mode and solve for the
FV of the annuity:
9-‹#›
What is the (future) value of this annuity at t = 1? At t = 2?
100(1+I) + 100 +
9-‹#›
Question
If you can receive $50,000 per year forever, how much are you
willing to pay for that?
9-‹#›
Perpetuity
Perpetuity
PV =

102. If I = 10%, PV = = $0.5M
If I= 5%, PV = = $1M
9-‹#›
35
Growing Perpetuity
If the payments grow at a constant rate, g, it is a growing
perpetuity.
PV =
=
Example: If I = 10%, g = 5%,
PMT0 = $50,000,
PV = = $1.05M
9-‹#›
The Power of Compound Interest
A 20-year-old student wants to save $3 a day for her retirement.
Every day she places $3 in a drawer. At the end of the year, she
invests the accumulated savings ($1,095) in a brokerage account
with an expected annual return of 12%.
How much money will she have when she is 65 years old?

103. 9-‹#›
Solving for FV:
If she begins saving today, how much will she have when she is
65?
If she sticks to her plan, she will have $( ) when she
is 65.
( ) N, 12 I/YR, -1095 PMT, FV =>
( )
9-‹#›
Solving for FV:
If you don’t start saving until you are 40 years old, how much
will you have at 65?
If a 40-year-old investor begins saving today, and sticks to the
plan, he or she will have $146,000.59 at age 65. This is $1.3
million less than if starting at age 20.
Lesson: It pays to start saving early.
25 N, 12 I/Y, 1095 PMT,
FV =› 146,000.59

104. 9-‹#›
Solving for PMT:
How much must the 40-year old deposit annually to catch the
20-year old?
To find the required annual contribution, enter the number of
years until retirement and the final goal of $1,487,261.89, and
solve for PMT.
25 N, 12 I/Y, 1,487,261.89 FV,
PMT = › 11,154.42
9-‹#›
What is the PV of this uneven cash flow stream?
0
100
1
300
2
300
3
10%
-50

105. 4
90.91
247.93
225.39
-34.15
530.08 = PV
9-‹#›
41
Solving for PV:
Uneven cash flow stream
Input cash flows in the calculator’s “CF” register:
CF0 = 0
C01 = 100
F01 = 1
C02 = 300
F02 = 2
C03 = -50
Press NPV button, then enter I = 10, and hit CPT=> $530.087.
(Here NPV = PV.)

106. 9-‹#›
NPV (Net Present Value)
NPV is calculated net of costs.
If your project’s PV of cash inflows is greater than the PV of
cash outflows, the project will enhance your company’s
profitability.
In chapter 9, generally there is no cost at time 0, so the NPVs
are positive, but in chapter 12, when we evaluate projects, the
NPVs can be negative.
Example: 0 1 2 3 (I = 10%)
-1000 300 300 500
9-‹#›
9’-‹#›
Classifications of interest rates
Nominal rate (INOM) – also called the quoted or stated rate.

107. An annual rate that ignores compounding effects.
INOM is stated in contracts. Periods must also be given, e.g.
8% Quarterly or 8% Daily interest.
Periodic rate (IPER) – amount of interest charged each period,
e.g. monthly or quarterly.
IPER = INOM / M, where M is the number of compounding
periods per year. M = 4 for quarterly and M = 12 for monthly
compounding.
9’-‹#›
2
Compounding More than Once per Year
Annual Compounding
0 8% 1
|______________________|
Semiannual Compounding
0 4% 1 4% 2
|__________|___________|
Quarterly Compounding
0 2% 1 2% 2 2% 3 2% 4
|_____|_____|_____|_____|
9’-‹#›
Effective Annual Rate (EAR)

108. Effective (or equivalent) annual rate (EAR = EFF%): The
annual rate of interest actually being earned, accounting for
compounding.
EFF% for 8% semiannual investment
EFF% = ( 1 + )M - 1
= (1 + )2 – 1 = 8.16%
Should be indifferent between receiving 8.16% annual interest
and receiving 8% interest, compounded semiannually. EAR is
used to compare investment returns.
9’-‹#›
4
9’-‹#›
Calculator
Use ICONV key
NOM = INOM
EFF = EAR
C/Y = # of compounding per year

109. 9’-‹#›
What is the FV of $100 after 3 years under 10% semiannual
compounding? Quarterly compounding?
9’-‹#›
7
calculator
10% semi-annual compounding
5 I/Y, 100 PV, 6 N, FV => 134.01
Quarterly compounding
2.5 I/Y, 100 PV, 12 N, FV => 134.49
9’-‹#›
What is the future value of an annuity with $100 monthly
payments at 7% after 5 years?
FV = PMT
= 100
OR,
100 PMT
( ) I/Y

110. ( ) N
FV =
9’-‹#›
GM Incentives
You need $12,000 loan to buy a car.
There are two financing options to choose:
A: 2.9% financing with a 36 month loan
B: A rebate of $1,000 is available and the remaining $11,000 is
to be financed at 10% for 36 months.
Which option would you choose?
9’-‹#›
Quarterly Compounding
A. If you deposit $1,000 in a bank that pays 8% quarterly
compounding, what is the rate of return if you withdraw after 10
months?
B. How much in dollars will you get if you withdraw after 10
months?
9’-‹#›

111. What’s the FV of a 3-year $100 annuity, if the quoted interest
rate is 10%, compounded semiannually?
Payments occur annually, but compounding occurs every 6
months.
Cannot use normal annuity valuation techniques.
9’-‹#›
Method 1:
Compound each cash flow
FV3 = $100(1.05)4 + $100(1.05)2 + $100
FV3 = $331.80
9’-‹#›
Method 2:
Financial calculator

112. Find the EAR and treat as an annuity.
EAR = ( 1 + )2 – 1 = 10.25%.
10.25 I/Y, 3 N, -100 PMT, ---
9’-‹#›
Calculator 2
1 P/Y, 2 C/Y, 100 PMT, 3 N, 10 I/Y,
FV => 331.8006 => 331.80
9’-‹#›
Find the PV of this 3-year ordinary annuity.
Could solve by discounting each cash flow, or …
Use the EAR and treat as an annuity to solve for PV.
10.25 I/Y, 3 N, 100 PMT, --
9’-‹#›
Calculator 2
1 P/Y, 2 C/Y, 100 PMT, 3 N, 10 I/Y,
PV => 247.5947 => 247.59

113. 9’-‹#›
P/Y, C/Y
What is the future value of a three-year annuity with quarterly
payments of $50 each at 7%, monthly compounding?
9’-‹#›
Loan amortization
Amortization tables are widely used for home mortgages, auto
loans, business loans, retirement plans, etc.
Financial calculators and spreadsheets are great for setting up
amortization tables.
EXAMPLE: Construct an one-year amortization table for a
$100,000, 8%, semiannual payment, 30-year loan.
9’-‹#›
19

114. Step 1:
Find the required annual payment
All input information is already given.
60 N, 4 I/Y, 100,000 PV,
PMT = 4,420.18
9’-‹#›
20
Step 2:
Find the interest paid in Period 1
The borrower will owe interest upon the initial balance at the
end of the first period. Interest to be paid in the first period can
be found by multiplying the beginning balance by the periodic
interest rate.
INTt = Beg balt (I)
INT1 = $100,000 (0.04) = $4,000
9’-‹#›
21
Step 3:
Find the principal repaid in Period 1

115. If a payment of $4,420.18 was made at the end of the first
period and $4,000 was paid toward interest, the remaining value
must represent the amount of principal repaid.
PRIN REPAYMENT = PMT – INT
= $4,420.18 - $4,000 = $420.18
9’-‹#›
22
Step 4:
Find the ending balance after Period 1
To find the balance at the end of the period, subtract the amount
paid toward principal from the beginning balance.
END BAL= BEG BAL – PRIN REP.
= $100,000 - $420.18
= $99,579.82
9’-‹#›
23
Constructing an amortization table:
Repeat steps 1 – 4 until end of loanPBEG BALPMTINTPRIN

116. REPAYEND
BAL1$100,000$4,420.18$4,000$420.18$99,579.822 99,579.82
4,420.18
3,983.19 436.99 99,142.83
Interest paid declines with each payment as the balance
declines. What are the tax implications of this?
9’-‹#›
24
Illustrating an amortized payment:
Where does the money go?
Constant payments.
Declining interest payments.
Declining balance.
$
0
1
2
3
4,420.18
Interest
420.18
Principal Repayments

117. 9’-‹#›
25
Continuous Compounding
= ℮
9’-‹#›
If compounding takes place continuously,
FVt = PV∙℮It
Alternatively,
PV = = FVt∙℮-It
Example 1: Suppose you invest $200 at 12% continuously
compounded for two years. How much are you going to receive
at the end of two years?
9’-‹#›
Continued …
Answer: It = 0.12×2 = 0.24, e0.24 = 1.2712. So, FVt = FV2 =
$200×1.2712 = $254.25

118. Example 2: What is the PV of $300 in one year’s time if I =
5%, and continuously compounded?
Answer: It = 0.05×1 = 0.05,
e-It = e-0.05= 0.9512,
so, PV = $300×0.9512 = $285.37
9’-‹#›
$134.49
(1.025)
$100
FV
$134.01
(1.05)
$100
FV
)
2
0.10
1

119. (
$100
FV
)
M
I
1
(
PV
FV
12
3Q
6
3S
3
2
3S
N
M
NOM
n
=
=
=
=
+
=
+

120. =
´
´
Term Paper on the TVM
The first part of your term paper should include a summary of
all the concepts of Time value money covered in class, for
example, Future value of single sum, present value of single
sum, annuity, etc., in your own words. You must include a
simple numerical example to explain each concept. Do not
simply copy the textbook, and you should follow the rule of
quotation if you want to quote a sentence from the book.
Include all the concepts covered in class.
The second part of the paper is about the application of Time
value money to your own financial problems. Try to use the
concepts you learn from the class to solve practical financial
issues in your life, for example, your retirement planning, your
mortgage payments or car loan payments, credit card interest
rates, to name a few. Since you use a numerical example for
each concept in the first part, it is encouraged to analyze one
big problem rather than several small problems in the second
part. It has to be your own and specific problem. If you cannot
think of your own problem, you may make up a case that is
interesting.
Do not mix the first part and the second part, that is, the second
part should have a separate heading. There is a penalty of 20%
for mixing the two parts. Two sample papers are on reserve in
the library for your perusal.
Include word count on the front page. The word count should be
at least 2,500. Simply type the word count shown on your MS
Word or other word processors.
Grading:
First Part 50%
Second Part 40%
Writing Quality 10%

121. The emphasis on the first part is thoroughness, that is, you have
to explain every concept covered from time lines to loan
amortization.
The second part is comprised of creativity/originality (10%),
correctness (10%), sophistication (10%), and overall content
(10%).