Chapter 4
The Time Value of Money
Chapter Outline
4.1 The Timeline
4.2 The Three Rules of Time Travel
4.3 The Power of Compounding
4.4 Valuing a Stream of Cash Flows
4.5 The Net Present Value of a Stream of
Cash Flows
4.6 Perpetuities, Annuities, and other
Special Cases
2
Chapter Outline (cont’d)
4.7 Solving Problems with a Spreadsheet
Program
4.8 Solving for Variables Other Than Present
Value or Future Value
3
Learning Objectives
1.
Draw a timeline illustrating a given set of cash
flows.
2.
List and describe the three rules of time travel.
3.
Calculate the future value of:
a.
A single sum.
b.
An uneven stream of cash flows, starting either now or
sometime in the future.
c.
An annuity, starting either now or sometime in the future.
d.
Several cash flows occurring at regular intervals that
grow at a constant rate each period.
4
Learning Objectives (cont'd)
Calculate the present value of:
4.
a.
A single sum.
b.
An uneven stream of cash flows, starting either now or
sometime in the future.
c.
An infinite stream of identical cash flows.
d.
An annuity, starting either now or sometime in
the future.
e.
An infinite stream of cash flows that grow at a constant
rate each period.
f.
Several cash flows occurring at regular intervals that
grow at a constant rate each period.
5
Learning Objectives (cont'd)
5.
Given four out of the following five inputs for an
annuity, compute the fifth: (a) present value, (b)
future value, (c) number of periods, (d) periodic
interest rate, (e) periodic payment.
6.
Given three out of the following four inputs for a
single sum, compute the fourth: (a) present value,
(b) future value, (c) number of periods, (d) periodic
interest rate.
7.
Given cash flows and present or future value,
compute the internal rate of return for a series of
cash flows.
6
4.1 The Timeline

A timeline is a linear representation of the
timing of potential cash flows.

Drawing a timeline of the cash flows will help
you visualize the financial problem.
7
4.1 The Timeline (cont’d)

Assume that you loan $10,000 to a friend.
You will be repaid in two payments, one at
the end of each year over the next two years.
8
4.1 The Timeline (cont’d)

Differentiate between two types of cash flows

Inflows are positive cash flows.

Outflows are negative cash flows, which are
indicated with a – (minus) sign.
9
4.1 The Timeline (cont’d)

Assume that you are lending $10,000 today and that
the loan will be repaid in two annual $6,000
payments.

The first cash flow at date 0 (today) is represented
as a negative sum because it is an outflow.

Timelines can represent cash flows that take place
at the end of any time period.
10
Example 4.1
11
Example 4.1 (cont’d)
12
4.2 Three Rules of Time Travel

Financial decisions often require combining
cash flows or comparing values. Three rules
govern these processes.
13
The 1st Rule of Time Travel

A dollar today and a dollar in one year are
not equivalent.

It is only possible to compare or combine
values at the same point in time.

Which would you prefer: A gift of $1,000 today or
$1,210 at a later date?

To answer this, you will have to compare the
alternatives to decide which is worth more. One
factor to consider: How long is “later?”
14
The 2nd Rule of Time Travel

To move a cash flow forward in time, you
must compound it.

Suppose you have a choice between receiving
$1,000 today or $1,210 in two years. You believe
you can earn 10% on the $1,000 today, but want
to know what the $1,000 will be worth in two years.
The time line looks like this:
15
The 2nd Rule of Time Travel (cont’d)

Future Value of a Cash Flow
FVn  C  (1  r )  (1  r ) 
 (1  r )  C  (1  r ) n
n times
16
Using a Financial Calculator: The Basics

TI BA II Plus


Future Value
Present Value
FV
PV
I/Y

I/Y

Interest Rate per Year

Interest is entered as a percent, not a decimal

For 10%, enter 10, NOT .10
17
Using a Financial Calculator:
The Basics (cont'd)

TI BA II Plus

Number of Periods

2nd → CLR TVM
N
2ND
FV

Clears out all TVM registers

Should do between all problems
18
Using a Financial Calculator:
Setting the keys

TI BA II Plus

2ND → P/Y


I/Y
Check P/Y
2ND → P/Y → # → ENTER


2ND
Sets Periods per Year to #
2ND
I/Y
#
.
#
ENTER
2ND → FORMAT → # → ENTER

Sets display to # decimal places
2ND
ENTER
19
Using a Financial Calculator

TI BA II Plus

Cash flows moving in opposite directions must
have opposite signs.
20
Financial Calculator Solution

Inputs:




N=2
I = 10
PV = 1,000
Output:

FV = −1,210
2
N
10
I/Y
1,000
PV
CPT
FV
-1,210
21
The 2nd Rule of Time Travel—
Alternative Example

To move a cash flow forward in time, you
must compound it.

Suppose you have a choice between receiving
$5,000 today or $10,000 in five years. You believe
you can earn 10% on the $5,000 today, but want
to know what the $5,000 will be worth in five years.
The time line looks like this:
22
The 2nd Rule of Time Travel—
Alternative Example (cont’d)
1
0
$5,000

$5, 500
x 1.10
$6,050
3
x 1.10
$6,655
4
x 1.10
$7,321
5
x 1.10
$8,053
In five years, the $5,000 will grow to:


x 1.10
2
$5,000 × (1.10)5 = $8,053
The future value of $5,000 at 10% for five years
is $8,053.
You would be better off forgoing the gift of
$5,000 today and taking the $10,000 in five
years.
23
Financial Calculator Solution

Inputs:




N=5
I = 10
PV = 5,000
Output:

5
N
10
I/Y
5,000
PV
CPT
FV
FV = –8,052.55
-8,052.55
24
The 3rd Rule of Time Travel

To move a cash flow backward in time, we
must discount it.

Present Value of a Cash Flow
PV  C  (1  r )
n
C

n
(1  r )
25
Example 4.2
26
Example 4.2 (cont’d)
27
Example 4.2 Financial Calculator Solution

Inputs:




N = 10
I=6
FV = 15,000
Output:

PV = –8,375.92
10
N
6
I/Y
15,000 FV
CPT
PV
-8,375.92
28
The 3rd Rule of Time Travel—
Alternative Example

Suppose you are offered an investment that
pays $10,000 in five years. If you expect to
earn a 10% return, what is the value of this
investment?
29
The 3rd Rule of Time Travel—
Alternative Example (cont’d)

The $10,000 is worth:

$10,000 ÷ (1.10)5 = $6,209
30
Alternative Example: Financial Calculator
Solution

Inputs:




N=5
I = 10
FV = 10,000
Output:

5
N
10
I/Y
10,000 FV
PV = –6,209.21
CPT
PV
-6,209.21
31
Combining Values Using the Rules of Time
Travel

Recall the 1st rule: It is only possible to
compare or combine values at the same point
in time. So far we’ve only looked at
comparing.

Suppose we plan to save $1000 today, and $1000
at the end of each of the next two years. If we can
earn a fixed 10% interest rate on our savings, how
much will we have three years from today?
32
Combining Values Using
the Rules of Time Travel (cont'd)

The time line would look like this:
33
Combining Values Using
the Rules of Time Travel (cont'd)
34
Combining Values Using
the Rules of Time Travel (cont'd)
35
Combining Values Using
the Rules of Time Travel (cont'd)
36
Example 4.3
37
Example 4.3 (cont'd)
38
Example 4.3 Financial Calculator Solution
CF
1,000
ENTER
↓
1,000
ENTER
↓
2
ENTER
NPV
10
ENTER
↓
CPT
2,735.54
39
Combining Values Using the Rules of Time
Travel—Alternative Example

Assume that an investment will pay you
$5,000 now and $10,000 in five years.

The time line would like this:
0
$5,000
1
2
3
4
5
$10,000
40
Combining Values Using the Rules of Time
Travel—Alternative Example (cont'd)

You can calculate the present value of the
combined cash flows by adding their values
today.
0
$5,000
$6,209
$11,209
1
2
3
4
÷ 1.105
The present value of both cash flows is
$11,209.
5
$10,000
41
Combining Values Using the Rules of Time
Travel—Alternative Example (cont'd)

You can calculate the future value of the
combined cash flows by adding their values
in Year 5.
1
0
$5,000

2
3
4
x 1.105
The future value of both cash flows is
$18,053.
5
$10,000
$8,053
$18,053
42
Combining Values Using the Rules of Time
Travel—Alternative Example (cont'd)
Present
Value
0
1
2
$11,209
0
$11,209
3
4
$18,053
÷ 1.105
1
2
3
x 1.105
5
4
Future
Value
5
$18,053
43
4.3 The Power of Compounding:
An Application

Compounding

Interest on Interest

As the number of time periods increases, the future
value increases, at an increasing rate since there is
more interest on interest.
44
Figure 4.1 The Power of Compounding
45
4.4 Valuing a Stream of Cash Flows

Based on the first rule of time travel we can
derive a general formula for valuing a stream
of cash flows: if we want to find the present
value of a stream of cash flows, we simply
add up the present values of each.
46
4.4 Valuing a Stream of Cash Flows (cont’d)

Present Value of a Cash Flow Stream
PV 
N
 PV (C )
n  0
n

N

n  0
Cn
(1  r ) n
47
Example 4.4
48
Example 4.4
(cont'd)
49
Example 4.4 Financial Calculator Solution
CF
0
ENTER
↓
5,000
ENTER
↓
8,000
ENTER
↓
3
ENTER
NPV
6
ENTER
↓
CPT
24,890.66
↓
50
Future Value of Cash Flow Stream

Future Value of a Cash Flow Stream with a Present Value of PV
FVn  PV  (1  r ) n
51
Future Value of Cash Flow Stream—
Alternative Example

What is the future value in three years of the
following cash flows if the compounding rate
is 5%?
0
1
2
$2,000
$2,000
$2,000
3
52
Future Value of Cash Flow Stream—
Alternative Example (cont'd)
1
0
2
3
$2,000
$2,315
x 1.05
x 1.05
x 1.05
$2,000
$2,205
x 1.05

x 1.05
$2,000
Or
x 1.05
0
1
2
$2,000
$2,000
$2,100
$4,100
$2,000
x 1.05
x 1.05
$2,100
$6,620
3
$4,305
$6,305
x 1.05
$6,620
53
4.5 Net Present Value of a Stream
of Cash Flows

Calculating the NPV of future cash flows
allows us to evaluate an investment decision.

Net Present Value compares the present
value of cash inflows (benefits) to the present
value of cash outflows (costs).
54
Example 4.5
55
Example
4.5 (cont'd)
56
Example 4.5 Financial Calculator Solution
CF
-1,000
ENTER
↓
500
ENTER
↓
3
ENTER
NPV
10
ENTER
↓
CPT
243.43
57
4.5 Net Present Value of a Stream
of Cash Flows—Alternative Example

Would you be willing to pay $5,000 for the
following stream of cash flows if the discount
rate is 7%?
0
1
2
3
$3,000
$2,000
$1,000
58
Compute the Present Value of the Benefits and
the Present Value of the Cost…

The present value of the benefits is:
3000 / (1.05) + 2000 / (1.05)2 + 1000 / (1.05)3 =
5366.91

The present value of the cost is $5,000,
because it occurs now.

The NPV = PV(benefits) – PV(cost)
= 5366.91 – 5000 = 366.91
59
Alternative Example
Financial Calculator Solution

CF
-5,000
ENTER
↓
3,000
ENTER
↓
↓
2,000
ENTER
↓
↓
1,000
ENTER
↓
NPV
7
ENTER
↓
CPT
366.91
On a present value
basis, the benefits
exceed the costs by
$366.91.
60
4.6 Perpetuities, Annuities, and Other Special
Cases

When a constant cash flow will occur at
regular intervals forever it is called a
perpetuity.

The value of a perpetuity is simply the cash
flow divided by the interest rate.

Present Value of a Perpetuity
C
PV (C in perpetuity) 
r
61
Example 4.6
62
Example 4.6 (cont'd)
63
Annuities

When a constant cash flow will occur at
regular intervals for N periods it is called an
annuity.

Present Value of an Annuity
PV (annuity of C for N periods with interest rate r )  C 
1
r


1
1


N 
(1

r
)


64
Example 4.7
65
Example 4.7
(cont'd)
66
Example 4.7 (cont'd)
Future Value of an Annuity
FV (annuity)  PV  (1  r ) N

C 
1

1 
r 
(1  r ) N
1
N
 C 
(1  r )

r

N
  (1  r )

 1
67
Example 4.7 Financial Calculator Solution

Since the payments begin today, this is an
Annuity Due.

First, put the calculator on “Begin” mode:
2ND
PMT
2ND
ENTER
2ND
CPT
68
Example 4.7
Financial Calculator Solution (cont'd)

Then:
30
N
8
I/Y
1,000,000
CPT

PMT
PV
-12,158,406
$15 million > $12.16 million, so take the lump sum.
69
Example 4.8
70
Example 4.8 (cont'd)
71
Example 4.8 Financial Calculator Solution

Since the payments begin in one year, this is
an Ordinary Annuity.

Be sure to put the calculator back on “End” mode:
2ND
PMT
2ND
ENTER
2ND
CPT
72
Example 4.8
Financial Calculator Solution (cont'd)

Then
30
N
10
I/Y
10,000
CPT
PMT
FV
-1,644,940
73
Growing Perpetuities

Assume you expect the amount of your
perpetual payment to increase at a constant
rate, g.

Present Value of a Growing Perpetuity
C
PV (growing perpetuity) 
r  g
74
Example 4.9
75
Example 4.9 (cont'd)
76
Growing Annuities

The present value of a growing annuity with
the initial cash flow c, growth rate g, and
interest rate r is defined as:

Present Value of a Growing Annuity
N

 1  g  
1
1  

PV  C 

(r  g ) 
 (1  r )  
77
Example 4.10
78
Example 4.10 (cont'd)
79
4.7 Solving Problems with a Spreadsheet
Program

Spreadsheets simplify the calculations of
TVM problems





NPER
RATE
PV
PMT
FV
NPV  PV  PMT 


1
1
FV
1


 0

NPER 
NPER
RATE 
(1  RATE )
(1  RATE )

80
Example 4.11
81
Example 4.11 (cont'd)
82
Example 4.12
83
Example 4.12 (cont'd)
84
4.8 Solving for Variables Other Than Present
Values or Future Values

Sometimes we know the present value or
future value, but do not know one of the
variables we have previously been given as
an input. For example, when you take out a
loan you may know the amount you would
like to borrow, but may not know the loan
payments that will be required to repay it.
85
Example 4.13
86
Example 4.13 (cont'd)
87
4.8 Solving for Variables Other Than Present
Values or Future Values (cont’d)

In some situations, you know the present
value and cash flows of an investment
opportunity but you do not know the internal
rate of return (IRR), the interest rate that
sets the net present value of the cash flows
equal to zero.
88
Example 4.14
89
Example 4.14 (cont'd)
90
Example 4.15
91
Example 4.15 (cont'd)
92
4.8 Solving for Variables Other Than Present
Values or Future Values (cont’d)

In addition to solving for cash flows or the
interest rate, we can solve for the amount of
time it will take a sum of money to grow to a
known value.
93
Example 4.16
94
Example 4.16 (cont'd)
95