The Time Value of Money

Dr. Salman Khan

 Session(s) Outline
4.1 The Timeline
4.2 The Three Rules of Time Travel
4.3 Valuing a Stream of Cash Flows
4.4 Calculating the Net Present Value
4.5 Perpetuities, Annuities, and Other
Special Cases
 Chapter Outline (cont’d)
4.6 Solving Problems with a Spreadsheet
Program
4.7 Solving for Variables Other Than Present
Value or Future Value
 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:
4. A single sum.
5. An uneven stream of cash flows, starting either
now or sometime in the future.
6. An annuity, starting either now or sometime in
the future.
 Learning Objectives
7. Several cash flows occurring at regular intervals
that grow at a constant rate each period.
8. Calculate the present value of:
9. A single sum.
10.An uneven stream of cash flows, starting either
now or sometime in the future.
11.An infinite stream of identical cash flows.
12.An annuity, starting either now or sometime in
the future.
 Learning Objectives
13. 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.
14. 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.
15. Given cash flows and present or future value, compute the
internal rate of return for a series of cash flows.
 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.
 4.1 The Timeline (cont’d)
• Assume that you made a loan to a friend. You
will be repaid in two payments, one at the end
of each year over the next two years.
 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.
 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 – a month, a week, a day, etc.
Textbook Example 4.1
Textbook Example 4.1 (cont’d)
 4.2 Three Rules of Time Travel
• Financial decisions often require combining
cash flows or comparing values. Three rules
govern these processes.

Table 4.1 The Three Rules of Time Travel

 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?”
 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:
 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
 Using a Financial Calculator: The
Basics
• TI BA II Plus
– Future Value FV

PV
– Present Value
I/Y

– I/Y
• Interest Rate per Year
• Interest is entered as a percent, not a decimal
– For 10%, enter 10, NOT .10
 Using a Financial Calculator:
The Basics (cont'd)
• TI BA II Plus
– Number of Periods N

2ND FV
– 2nd → CLR TVM

• Clears out all TVM registers

• Should do between all problems
 Using a Financial Calculator:
Setting the keys
• TI BA II Plus
2ND I/Y
– 2ND → P/Y
• Check P/Y

– 2ND → P/Y → # → ENTER

• Sets Periods per Year to # 2ND I/Y # ENTER

– 2ND → FORMAT → # → ENTER

• Sets display to # decimal places
2ND . # ENTER
 Using a Financial Calculator
• TI BA II Plus
– Cash flows moving in opposite directions must
have opposite signs.
 Financial Calculator Solution
• Inputs:
2 N
–N=2
– I = 10 10 I/Y
– PV = 1,000
1,000 PV
• Output:
– FV = −1,210 CPT FV -1,210
Figure 4.1 The Composition of
Interest Over Time
Textbook Example 4.2
Textbook Example 4.2 (cont’d)
 Textbook Example 4.2 Financial
Calculator Solution for n=7 years
• Inputs:
–N=7 7 N
– I = 10
10 I/Y
– PV = 1,000
• Output: 1,000 PV
– FV = –1,948.72
CPT FV -1,948.72
 Alternative Example 4.2
• Problem
– 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.
 Alternative Example 4.2 (cont’d)
• Solution
– The time line looks like this:
0 1 2 3 4 5

$5,000 x 1.10 $5, 500 x 1.10 $6,050 x 1.10 $6,655 x 1.10 $7,321 x 1.10 $8,053
– In five years, the $5,000 will grow to:
$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.
 Alternative Example 4.2
Financial Calculator Solution
• Inputs:
–N=5 5 N
– I = 10
10 I/Y
– PV = 5,000
• Output: 5,000 PV
– FV = –8,052.55
CPT FV -8,052.55
 The 3rd Rule of Time Travel
• To move a cash flow backward in time, we
must discount it.

• Present Value of a Cash Flow

C
PV  C  (1  r )  n

(1  r ) n
Textbook Example 4.3
Textbook Example 4.3
 Textbook Example 4.3
Financial Calculator Solution
• Inputs:
– N = 10 10 N
–I=6
– FV = 15,000 6 I/Y

• Output: 15,000 FV
– PV = –8,375.92
CPT PV -8,375.92
 Alternative Example 4.3
• Problem
– 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 today?
 Alternative Example 4.3 (cont’d)
• Solution
– The $10,000 is worth:
• $10,000 ÷ (1.10)5 = $6,209
 Alternative Example 4.3:
Financial Calculator Solution
• Inputs:
5 N
–N=5
– I = 10 10 I/Y
– FV = 10,000
10,000 FV
• Output:
– PV = –6,209.21
CPT PV -6,209.21
 Applying 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?
 Applying the Rules of Time Travel
(cont'd)
• The time line would look like this:
Applying the Rules of Time Travel
(cont'd)
Applying the Rules of Time Travel
(cont'd)
Applying the Rules of Time Travel
(cont'd)
 Applying the Rules of Time Travel
Table 4.1 The Three Rules of Time Travel
Textbook Example 4.4
Textbook Example 4.4 (cont’d)
 Textbook Example 4.4
Financial Calculator Solution
CF 1,000 ENTER

↓ 1,000 ENTER

↓ 2 ENTER

NPV 10 ENTER

↓ CPT 2,735.54
 Alternative Example 4.4
• Problem
– Assume that an investment will pay you $5,000
now and $10,000 in five years.
– The time1 line would
2
like this:
3 4 5
0

$5,000 $10,000
 Alternative Example 4.4 (cont'd)
• Solution
– You can calculate the present value of the combined cash flows by
adding their values today.
0 1 2 3 4 5

$5,000
$6,209 5
$10,000
÷ 1.10
$11,209
– The present value of both cash flows is $11,209.
 Alternative Example 4.4 (cont'd)
• Solution
– You can calculate the future value of the
combined cash flows by adding their values
0 in Year1 5. 2 3 4 5

$10,000
$5,000 x 1.105 $8,053
$18,053

– The future value of both cash flows is $18,053.

 Alternative Example 4.4 (cont'd)
Present
Value
0 1 2 3 4 5

$11,209 $18,053
÷ 1.105

Future
Value
0 1 2 3 4 5

$11,209 $18,053
x 1.105
4.3 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.
 4.3 Valuing a Stream
of Cash Flows (cont’d)

• Present Value of a Cash Flow Stream

N N
Cn
PV   PV (C )
n  0
n  
n  0 (1  r )n
Textbook Example 4.5
Textbook Example 4.5 (cont’d)
 Textbook Example 4.5
Financial Calculator Solution
CF 0 ENTER

↓ 5,000 ENTER ↓

↓ 8,000 ENTER

↓ 3 ENTER

NPV 6 ENTER

↓ CPT 24,890.66
 Future Value of Cash Flow Stream
• Future Value of a Cash Flow Stream with a Present Value of PV

FVn  PV  (1  r )n
 Alternative Example 4.5
• Problem
– What is the future value in three years of the
following cash flows if the compounding rate
is 5%? 0 1 2 3

$2,000 $2,000 $2,000

 Alternative Example 4.5 (cont'd)
• Solution 0 1 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 $2,100
x 1.05
$6,620
• Or
0 1 2 3

$2,000 $2,000 $2,000

x 1.05
$2,100
$4,100
x 1.05
$4,305
$6,305
$6,620
x 1.05
 4.4 Calculating the Net Present
Value
• 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).
Textbook Example 4.6
Textbook Example 4.6 (cont'd)
 Textbook Example 4.6
Financial Calculator Solution
CF -1,000 ENTER

↓ 500 ENTER

↓ 3 ENTER

NPV 10 ENTER

↓ CPT 243.43
 Alternative Example 4.6
• Problem
– 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

 Alternative Example 4.6 (cont’d)

• Solution
– 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
 Alternative Example 4.6
Financial Calculator Solution
CF -5,000 ENTER
• On a present value
basis, the benefits
↓ 3,000 ENTER ↓ exceed the costs by
↓ 2,000 ENTER ↓ $366.91.
↓ 1,000 ENTER ↓

NPV 7 ENTER

↓ CPT 366.91
 4.5 Perpetuities, Annuities,
and Other Special Cases
• When a constant cash flow will occur at
regular intervals forever it is called a
perpetuity.
 4.5 Perpetuities, Annuities,
and Other Special Cases (cont’d)
• 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
Textbook Example 4.7
Textbook Example 4.7 (cont’d)
 Alternative Example 4.7
• Problem
– You want to endow a chair for a female professor
of finance at your alma mater. You’d like to attract
a prestigious faculty member, so you’d like the
endowment to add $100,000 per year to the
faculty member’s resources (salary, travel,
databases, etc.) If you expect to earn a rate of
return of 4% annually on the endowment, how
much will you need to donate to fund the chair?
 Alternative Example 4.7 (cont’d)
• Solution
– The timeline of the cash flows looks like this:

– This is a perpetuity of $100,000 per year. The

funding you would need to give is the present
value of that perpetuity. From the formula:
C $100,000
PV    $2,500,000
r .04
– You would need to donate $2.5 million to endow
 Annuities
• When a constant cash flow will occur at
regular intervals for a finite number of N
periods, it is called an annuity.

• Present Value of an Annuity

C C C C N C
PV     ...   
( 1  r ) ( 1  r )2 ( 1  r )3 ( 1  r )N n1 ( 1  r )n
 Present Value of an Annuity
• To find a simpler formula, suppose you invest
$100 in a bank account paying 5% interest. As
with the perpetuity, suppose you withdraw
the interest each year. Instead of leaving the
$100 in forever, you close the account and
withdraw the principal in 20 years.
 Present Value of an Annuity
(cont’d)
• You have created a 20-year annuity of $5 per
year, plus you will receive your $100 back in
20 years. So:
$100  PV(20  year annuity of $5 per year)  PV($100 in 20 years)

• Re-arranging terms:
PV(20  year annuity of $5 per year)  $100  PV($100 in 20 years)
100
 100  20
 $62.31
(1.05)
 Present Value of an Annuity
• For the general formula, substitute P for the
principal value and:
PV(annuity of Cfor N periods)
 P  PV(Pin period N)
P  1 
P  P 1  N 
(1  r) N
 (1  r) 
Textbook Example 4.8
Textbook Example 4.8
 Textbook Example 4.8
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
Textbook Example 4.8 Financial
Calculator Solution (cont'd)
– Then:
30 N

8 I/Y

1,000,000 PMT

CPT PV -12,158,406

• $15 million > $12.16 million, so take the lump sum.

 Future Value of an Annuity
• Future Value of an Annuity
FV (annuity)  PV  (1  r ) N
C  1 
 1    (1  r ) N

r  (1  r ) N

 C 
1
r
 (1  r ) N
 1
Textbook Example 4.9
Textbook Example 4.9 (cont’d)
 Textbook Example 4.9
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

Textbook Example 4.9 Financial
Calculator Solution (cont'd)
– Then
30 N

10 I/Y

10,000 PMT

CPT FV -1,644,940
 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
Textbook Example 4.10
Textbook Example 4.10 (cont’d)
 Alternative Example 4.10
• Problem
– In Alternative Example 4.7, you planned to donate
money to endow a chair at your alma mater to
supplement the salary of a qualified individual by
$100,000 per year. Given an interest rate of 4%
per year, the required donation was $2.5 million.
The University has asked you to increase the
donation to account for the effect of inflation,
which is expected to be 2% per year. How much
will you need to donate to satisfy that request?
 Alternative Example 4.10 (cont’d)
• Solution
The timeline of the cash flows looks like this:

The cost of the endowment will start at $100,000, and increase by 2% each
year. This is a growing perpetuity. From the formula:

C $100,000
PV    $5,000,000
r .04  .02
You would need to donate $5.0 million to endow the chair.
 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
1   1  g  
N

PV  C  1    
(r  g )   (1  r )  
Textbook Example 4.11
Textbook Example 4.11
 4.6 Solving Problems with a
Spreadsheet Program
• Spreadsheets simplify the calculations of TVM
problems
– NPER
– RATE
– PV
– PMT
– FV
• These functions all solve the problem:
1  1  FV
NPV  PV  PMT   1  NPER 
  0
RATE  (1  RATE )  (1  RATE ) NPER
Textbook Example 4.12
Textbook Example 4.12 (cont’d)
Textbook Example 4.13
Textbook Example 4.13 (cont’d)
 4.7 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.
Textbook Example 4.14
Textbook Example 4.14 (cont’d)
4.7 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.
Textbook Example 4.15
Textbook Example 4.15 (cont’d)
Textbook Example 4.16
Textbook Example 4.16 (cont’d)
4.7 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.
Textbook Example 4.17
Textbook Example 4.17
 Discussion of Data Case Key Topic
• Would your answer change if the certification
job would require her to stay in Seattle but an
MBA would allow her to move to New York
City, where Natasha has always wanted to
live?
– Hint: Consider the cost of living differences
between Seattle and New York. Would her salary
be different in Seattle versus New York?
– http://cgi.money.cnn.com/tools/costofliving/costo
fliving.html
 Chapter Quiz
1. Can you compare or combine cash flows at different times?
2. How do you calculate the present value of a cash flow
stream?
3. What benefit does a firm receive when it accepts a project
with a positive NPV?
4. How do you calculate the present value of a
a. Perpetuity?
b. Annuity?
c. Growing perpetuity?
d. Growing annuity?