Discounted Cash Flow
Valuation
BASIC PRINCIPAL
 Would
you rather have $1,000 today or
$1,000 in 30 years?
Why?
Can invest the $1,000 today let it grow
This is a fundamental building block of
finance
2
Present and Future Value
Present Value: value of a future payment today
 Future Value: value that an investment will
grow to in the future
 We find these by discounting or compounding
at the discount rate

 Also
know as the hurdle rate or the opportunity
cost of capital or the interest rate
3
One Period Discounting

PV = Future Value / (1+ Discount Rate)
 V0
= C1 / (1+r)
Alternatively
 PV = Future Value * Discount Factor

 V0
= C1 * (1/ (1+r))
 Discount factor is 1/ (1+r)
4
PV Example

What is the value today of $100 in one year, if
r = 15%?
 PV
= 100 / 1.15 = 86.96
5
FV Example

What is the value in one year of $100, invested
today at 15%?
 FV
= 100 * (1.15)1 = $115
6
Discount Rate Example
Your stock costs $100 today, pays $5 in
dividends at the end of the period, and then
sells for $98. What is your rate of return?
 PV =
 FV =

7
Discount Rate Example
Your stock costs $100 today, pays $5 in
dividends at the end of the period, and then
sells for $98. What is your rate of return?
 PV = $100
 FV =

8
Discount Rate Example
Your stock costs $100 today, pays $5 in
dividends at the end of the period, and then
sells for $98. What is your rate of return?
 PV = $100
 FV = $103 = $98 + $5


($98 + $5)/$100 – 1 = 3%
9
NPV
 NPV
= PV of all expected cash flows
Represents
the value generated by the project
To compute we need: expected cash flows &
the discount rate
 Positive
NPV investments generate value
 Negative NPV investments destroy value
10
Net Present Value (NPV)

NPV = PV (Costs) + PV (Benefit)
 Costs:
are negative cash flows
 Benefits: are positive cash flows

One period example
 NPV
= C0 + C1 / (1+r)
 For Investments C0 will be negative, and C1 will
be positive
 For Loans C0 will be positive, and C1 will be
negative
11
Net Present Value Example
 Suppose
you can buy an investment that
promises to pay $10,000 in one year for
$9,500. Should you invest?
We
don’t know
We cannot simply compare cash flows
that occur at different times
12
Net Present Value
 Since
we cannot compare cash flow we
need to calculate the NPV of the
investment
If
the discount rate is 5%, then NPV is?
NPV = -9,500 + 10,000/1.05
NPV = -9,500 + 9,523.81
NPV = 23.81
 At
what price are we indifferent?
13
Net Present Value

Since we cannot compare cash flow we need to
calculate the NPV of the investment
 If
the discount rate is 5%, then NPV is?
 NPV = -9,500 + 10,000/1.05
 NPV = -9,500 + 9,523.81
 NPV = 23.81
 At
what price are we indifferent? $9,523.81
NPV
would be 0
14
Coffee Shop Example
If you build a coffee shop on campus, you can
sell it to Starbucks in one year for $300,000
 Costs of building a coffee shop is $275,000


Should you build the coffee shop?
15
Step 1: Draw out the cash flows
Today
-$275,000
Year 1
$300,000
16
Step 2: Find the Discount Rate
Assume that the Starbucks offer is guaranteed
 US T-Bills are risk-free and currently pay 7%
interest

 This

is known as rf
Thus, the appropriate discount rate is 7%
 Why?
17
Step 3: Find NPV

The NPV of the project is?
–
275,000 + (300,000/1.07)
 – 275,000 + 280,373.83
 NPV = $5,373.83
 Positive NPV → Build the coffee shop
18
If we are unsure about future?
 What
is the appropriate discount rate if
we are unsure about the Starbucks offer
 rd
= rf
 rd > rf
 rd < rf
19
If we are unsure about future?
 What
is the appropriate discount rate if
we are unsure about the Starbucks offer
 rd
= rf
 rd > rf
 rd < rf
20
The Discount Rate

Should take account of two things:
Time value of money
2. Riskiness of cash flow
1.

The appropriate discount rate is the
opportunity cost of capital
 This
is the return that is offer on comparable
investments opportunities
21
Risky Coffee Shop

Assume that the risk of the coffee shop is
equivalent to an investment in the stock market
which is currently paying 12%

Should we still build the coffee shop?
22
Calculations

Need to recalculate the NPV
= – 275,000 + (300,000/1.12)
 NPV = – 275,000 + 267,857.14
 NPV = -7,142.86
 Negative NPV → Do NOT build the coffee shop
 NPV
23
Future Cash Flows

Since future cash flows are not certain, we
need to form an expectation (best guess)
 Need
to identify the factors that affect cash flows
(ex. Weather, Business Cycle, etc).
 Determine the various scenarios for this factor (ex.
rainy or sunny; boom or recession)
 Estimate cash flows under the various scenarios
(sensitivity analysis)
 Assign probabilities to each scenario
24
Expectation Calculation
The expected value is the weighted average of
X’s possible values, where the probability of
any outcome is p
 E(X) = p1X1 + p2X2 + …. psXs

 E(X)
 Xi
 pi
s

– Expected Value of X
 Outcome of X in state i
– Probability of state i
– Number of possible states
Note that = p1 + p2 +….+ ps = 1
25
Risky Coffee Shop 2

Now the Starbucks offer depends on the state
of the economy
Recession Normal
Value
300,000 400,000
Probability
0.25
0.5
Boom
700,000
0.25
26
Calculations
Discount Rate = 12%
 Expected Future Cash Flow =

 (0.25*300)

+ (0.50*400) + (0.25*700) = 450,000
NPV =
 -275,000
+ 450,000/1.12
 -275,000 + 401,786 = 126,790

Do we still build the coffee shop?
 Build
the coffee shop, Positive NPV
27
Valuing a Project Summary
Step 1: Forecast cash flows
 Step 2: Draw out the cash flows
 Step 3: Determine the opportunity cost of
capital
 Step 4: Discount future cash flows
 Step 5: Apply the NPV rule

28
Reminder
Important to set up problem correctly
 Keep track of


•
Magnitude and timing of the cash flows
•
TIMELINES
You cannot compare cash flows @ t=3 and @
t=2 if they are not in present value terms!!
29
General Formula
PV0 = FVN/(1 + r)N OR FVN = PVo*(1 + r)N

Given any three, you can solve for the fourth
 Present
value (PV)
 Future value (FV)
 Time period
 Discount rate
30
Four Related Questions
1.
2.
3.
4.
How much must you deposit today to have $1
million in 25 years? (r=12%)
If a $58,823.31 investment yields $1 million in 25
years, what is the rate of interest?
How many years will it take $58,823.31 to grow to
$1 million if r=12%?
What will $58,823.31 grow to after 25 years if
r=12%?
31
FV Example


Suppose a stock is currently worth $10, and is
expected to grow at 40% per year for the next five
years.
What is the stock worth in five years?
$10
14
0
 $53.78
1
19.6
27.44
38.42
2
3
4
$53.78
5
= $10×(1.40)5
32
PV Example

How much would an investor have to set aside
today in order to have $20,000 five years from
now if the current rate is 15%?
$20,000
PV
0
1
2
3
4
5
33
PV Example

How much would an investor have to set aside
today in order to have $20,000 five years from
now if the current rate is 15%?
$20,000
9,943.53
0

1
2
3
4
5
20,000/(1+0.15)5 = 9,943.53
34
Historical Example


From Fibonacci’s Liber Abaci, written in the year
1202: “A certain man gave 1 denari at interest so that
in 5 years he must receive double the denari, and in
another 5, he must have double 2 of the denari and
thus forever. How many denari from this 1denaro
must he have in 100 years?”
What is rate of return? Hint: what does the investor
earn every 5 years
35
Historical Example



From Fibonacci’s Liber Abaci, written in the year
1202: “A certain man gave 1 denari at interest so that
in 5 years he must receive double the denari, and in
another 5, he must have double 2 of the denari and
thus forever. How many denari from this 1denaro
must he have in 100 years?”
What is rate of return? Hint: what does the investor
earn every 5 years 100%
1 * (1+1)20 = 1,048,576 denari.
36
Simple vs. Compound Interest
Simple Interest: Interest accumulates only on
the principal
 Compound Interest: Interest accumulated on the
principal as well as the interest already earned


What will $100 grow to after 5 periods at 35%?
•
Simple interest
 FV2 = (PV0 * (r) + PV0 *(r)) + PV0 = PV0 (1 + 2r) =
• Compounded interest
 FV2 = PV0 (1+r) (1+r)= PV0 (1+r)2 =
37
Simple vs. Compound Interest
Simple Interest: Interest accumulates only on
the principal
 Compound Interest: Interest accumulated on the
principal as well as the interest already earned


What will $100 grow to after 5 periods at 35%?
•
Simple interest
 FV2 = (PV0 * (r) + PV0 *(r)) + PV0 = PV0 (1 + 2r) = $275
• Compounded interest
 FV2 = PV0 (1+r) (1+r)= PV0 (1+r)2 =
38
Simple vs. Compound Interest
Simple Interest: Interest accumulates only on the
principal
 Compound Interest: Interest accumulated on the
principal as well as the interest already earned


What will $100 grow to after 5 periods at 35%?
•
Simple interest
 FV5 = (PV0*(r) + PV0*(r)+…) + PV0 = PV0 (1 + 5r) = $275
• Compounded interest
 FV5 = PV0 (1+r) (1+r) * …= PV0 (1+r)5 = $448.40
39
Compounding Periods
We
have been assuming that compounding and
discounting occurs annually, this does not need
to be the case
40
Non-Annual Compounding
Cash flows are usually compounded over
periods shorter than a year
 The relationship between PV & FV when
interest is not compounded annually

= PV * ( 1+ r / M) M*N
 PV = FVN / ( 1+ r / M) M*N
 FVN
M is number of compounding periods per year
 N is the number of years

41
Compounding Examples

What is the FV of $500 in 5 years, if the
discount rate is 12%, compounded monthly?
 FV

= 500 * ( 1+ 0.12 / 12) 12*5 = 908.35
What is the PV of $500 received in 5 years, if
the discount rate is 12% compounded
monthly?
 PV
= 500 / ( 1+ 0.12 / 12) 12*5 = 275.22
42
Another Example

An investment for $50,000 earns a rate of
return of 1% each month for a year. How much
money will you have at the end of the year?

$50,000 * 1.0112 = $56,341
43
Interest Rates

The 12% is the Stated Annual Interest Rate
(also known as the Annual Percentage Rate)
 This
is the rate that people generally talk about
 Ex. Car Loans, Mortgages, Credit Cards
However, this is not the rate people earn or pay
 The Effective Annual Rate is what people
actually earn or pay over the year

 The
more frequent the compounding the higher the
Effective Annual Rate
44
Compounding Example 2

If you invest $50 for 3 years at 12%
compounded semi-annually, your investment
will grow to:
$70.93
FV = 50 * (1+(0.12/2))2*3 = $70.93
45
Compounding Example 2: Alt.
If you invest $50 for 3 years at 12%
compounded semi-annually, your investment
will grow to: $70.93
 Calculate the EAR: EAR = (1 + R/m)m – 1

 EAR
 FV

= (1 + 0.12 / 2)2 – 1 = 12.36%
= 50 * (1+0.1236)3 = $70.93
So, investing at 12.36% compounded annually
is the same as investing at 12% compounded
semi-annually
46
EAR Example

Find the Effective Annual Rate (EAR) of an 18% loan
that is compounded weekly.
 EAR
= (1 + 0.18 / 52)52 – 1 = 19.68%
47
Credit Card


A bank quotes you a credit card with an interest rate
of 14%, compounded daily. If you charge $15,000 at
the beginning of the year, how much will you have to
repay at the end of the year?
EAR =
48
Credit Card



A bank quotes you a credit card with an interest rate
of 14%, compounded daily. If you charge $15,000 at
the beginning of the year, how much will you have to
repay at the end of the year?
EAR = is (1+0.14/365)365 – 1 = 15%
$15,000 * 1.15 = $17,250
49
Present Value Of a Cash Flow Stream
C1
C2
C3
CN
PV 


...
2
3
N
(1  r1 ) (1  r2 ) (1  r3 )
(1  rN )
N
Ct
=
t
(
1

r
)
t 1
t

Discount each cash flow back to the present
using the appropriate discount rate and then
sum the present values.
50
Insight Example
r = 10%
Year
Project A
Project B
1
100
300
2
400
400
3
300
100
PV
Which project is more valuable? Why?
51
Insight Example
r = 10%
Year
Project A
1
100
90.91
300
272.73
2
400
330.58
400
330.58
3
300
225.39
100
75.13
PV
Project B
646.88
678.44
Which project is more valuable? Why?
B, gets the cash faster
52
Various Cash Flows




A project has cash flows of $15,000, $10,000, and
$5,000 in 1, 2, and 3 years, respectively. If the
interest rate is 15%, would you buy the project if it
costs $25,000?
PV = 15,000/1.15+$10,000/1.152 +$5,000/1.153
PV = $23,892.50
NPV = –$25,000+$23,892.50 –$1,107.50
53
Example (Given)
Consider an investment that pays $200 one
year from now, with cash flows increasing by
$200 per year through year 4. If the interest
rate is 12%, what is the present value of this
stream of cash flows?
 If the issuer offers this investment for $1,500,
should you purchase it?

54
Multiple Cash Flows (Given)
0
1
200
2
3
4
400
600
800
178.57
318.88
427.07
508.41
1,432.93
Don’t buy
55
Various Cash Flow (Given)





A project has the following cash flows in periods 1
through 4: –$200, +$200, –$200, +$200. If the prevailing
interest rate is 3%, would you accept this project if you
were offered an up-front payment of $10 to do so?
PV = –$200/1.03 + $200/1.032 – $200/1.033 + $200/1.034
PV = –$10.99.
NPV = $10 – $10.99 = –$0.99.
You would not take this project
56
Common Cash Flows Streams



Perpetuity, Growing Perpetuity
 A stream of cash flows that lasts forever
Annuity, Growing Annuity
 A stream of cash flows that lasts for a fixed
number of periods
NOTE: All of the following formulas assume the
first payment is next year, and payments occur
annually
57
Perpetuity

A stream of cash flows that lasts forever
0
C
C
C
1
2
3
…
C
C
C
PV 



2
3
(1  r ) (1  r )
(1  r )
PV: = C/r
 What is PV if C=$100 and r=10%:
100/0.1 = $1,000

58
Perpetuity Example
What is the PV of a perpetuity paying $30
each month, if the annual interest rate is a
constant effective 12.68% per year?
 Monthly rate: 1.1268(1/2)– 1 = 1%
 PV = $30/0.01 = $3,000.

59
Perpetuity Example 2

What is the prevailing interest rate if a
perpetual bond were to pay $100,000 per year
beginning next year and costs $1,000,000
today?

r = C/PV = $100,000/$1,000,000 = 10%
60
Growing Perpetuities

Annual payments grow at a constant rate, g
0
C1
C2(1+g)
C3(1+g)2
1
2
3
…
PV= C1/(1+r) + C1(1+g)/(1+r)2 + C1(1+g)2(1+r)3 +…
PV = C1/(r-g)
 What is PV if C1 =$100, r=10%, and g=2%?
PV = 100 / (0.10 – 0.02) =1,250

61
Growing Perpetuity Example

What is the interest rate on a perpetual bond that pays
$100,000 per year with payments that grow with the
inflation rate (2%) per year, assuming the bond costs
$1,000,000 today?

r = C/PV+g = $100,000/$1,000,000+0.02 = 12%
62
Growing Perpetuity: Example (Given)


The expected dividend next year is $1.30, and
dividends are expected to grow at 5% forever.
If the discount rate is 10%, what is the value of this
promised dividend stream?
2
$1.30
×(1.05)
$1.30×(1.05)
$1.30
= $1.43
= $1.37
…
0
1
2
3
PV = 1.30 / (0.10 – 0.05) = $26
63
Example
An investment in a growing perpetuity costs
$5,000 and is expected to pay $200 next year.
If the interest is 10%, what is the growth rate
of the annual payment?
5,000 = 200/ (0.10 – g)
5,000 * (0.10 – g) = 200
0.10 – g = 200 / 5,000
0.10 – (200 / 5,000) = g = 0.06 = 6%
64
Annuity
A constant stream of cash flows with a fixed maturity
C
C
C
C

0
1
2
3
T
C
C
C
C
PV 



2
3
T
(1  r ) (1  r ) (1  r )
(1  r )
C
PV 
r

1 
1  (1  r ) T 


65
Annuity Formula
C
C
PV   r T
r (1  r )
0

C
C
C
C
C
C
C
1
2
3
T
T+1
T+2
T+3
Simply subtracting off the PV of the rest of the
perpetuity’s cash flows
66
Annuity Example 1


Compute the present value of a 3 year ordinary
annuity with payments of $100 at r=10%
Answer:
Or
100 
1 
1  = $248.69
PVA3 =
3
0.1  (1.1 ) 
1
1
1
+ 100 2 + 100 3 = $248.69
PVA 3 = 100
1.1
1.1
1.1
67
Alternative: Use a Financial Calculator

Texas Instruments BA-II Plus, basic
N
= number of periods
 I/Y = periodic interest rate


P/Y must equal 1 for the I/Y to be the periodic rate
Interest is entered as a percent, not a decimal
 PV
= present value
 PMT = payments received periodically
 FV = future value
 Remember to clear the registers (CLR TVM) after each
problem
 Other calculators are similar in format
68
Annuity Example 2

You agree to lease a car for 4 years at $300 per month.
You are not required to pay any money up front or at the
end of your agreement. If your opportunity cost of
capital is 0.5% per month, what is the cost of the lease?
Work through on your financial calculators
N
= 4 * 12 = 48
I/Y
= 0.5
PV
= ????
PMT = 300
FV
=0
Solve = 12,774.10
69
Annuity Example 3

What is the value today of a 10-year annuity
that pays $600 every other year? Assume that
the stated annual discount rate is 10%.
 What
do the payments look like?
 What
is the discount rate?
70
Annuity Example 3

What is the value today of a 10-year annuity
that pays $600 every other year? Assume that
the stated annual discount rate is 10%.
 What
do the payments look like?
PV
$600
0
 We
2
$600
4
$600
6
$600
8
$600
10
receive 5 payments of $600
71
Annuity Example 3

What is the value today of a 10-year annuity
that pays $600 every other year? Assume that
the stated annual discount rate is 10%.
 What
is the discount rate?
 The discount rate is 10% each year, so over 2
years the discount rate is going to be
72
Annuity Example 3

What is the value today of a 10-year annuity that pays
$600 every other year? Assume that the stated annual
discount rate is 10%.
 What is the discount rate?
 The discount rate is 10% each year, so the two
year stated rate SBAR is 20%, and the effective
rate is
 EBAR = (1 + SBAR/m)m -1
 1.12 – 1 = 0.21 = 21%
73
Annuity Example 3

What is the value today of a 10-year annuity
that pays $600 every other year? Assume that
the stated annual discount rate is 10%.
N
=5
we receive 5 payment over 10 years
I/Y
= 21
PV
= ????
PMT
= 600
FV
=0
Solve = 1,755.59
74
Annuity Example 4

What is the present value of a four payment
annuity of $100 per year that makes its first
payment two years from today if the discount
rate is 9%?
 What
0
do the payments look like?
1
2
3
4
5
75
Annuity Example 4

What is the present value of a four-payment
annuity of $100 per year that makes its first
payment two years from today if the discount
rate is 9%?
1
100
100
2
3
100
4
100
5
76
Annuity Example 4

What is the present value of a four-payment annuity of $100
per year that makes its first payment two years from today if
the discount rate is 9%?
100
100
100
100
323.97
1
2
3
4
5
N
=4
I/Y
=9
PV
= ????
PMT = 100
FV
=0
PV = 323.97
But the $323.97 is a year 1 cash flow and we want to know the
77
year 0 value
Annuity Example 4

What is the present value of a four-year annuity of
$100 per year that makes its first payment two years
from today if the discount rate is 9%?
297.22

323.97
100
100
1
2
3
100
4
100
5
To get PV today we need to discount the $323.97
back one more year
 323.97
/ 1.09 = 297.22
78
Annuity Example 5
What is the value today of a 10-pymt annuity
that pays $300 a year if the annuity’s first
cash flow is at the end of year 6. The interest
rate is 15% for years 1-5 and 10% thereafter?

0
1
2
3
4
5
$300
$300
$300
$300
$300
$300
$300
$300
$300
$300
6
7
8
9
10
11
12
13
14
15
79
Annuity Example 5


What is the value today of a 10-pymt annuity that
pays $300 a year (at year-end) if the annuity’s first
cash flow is at the end of year 6. The interest rate is
15% for years 1-5 and 10% thereafter?
Steps:
Get value of annuity at t= 5 (year end)
N
= 10
I/Y
= 10
PV
= ???? = 1,843.37
PMT
= 300
FV
=0
2. Bring value in step 1 to t=0
1,843.37 / 1.155 = 916.48
1.
80
Annuity Example 6

You win the $20 million Powerball. The lottery
commission offers you $20 million dollars today or
a nine payment annuity of $2,750,000, with the first
payment being today. Which is more valuable is
your discount rate is 5.5%?
N
=9
I/Y
= 5.5
PV
= ????
PMT
= 2,750,000
FV
=0
PV = $19,118,536.94
When is the $19,118,536.94?
Year -1, so to bring it into today we?
81
Annuity Example 6

You win the $20 million Powerball. The lottery
commission offers you $20 million dollars today or
a nine payment annuity of $2,750,000, with the first
payment being today. Which is more valuable if
your discount rate is 5.5%?
When is the $19,118,536.94?
Year -1, so to bring it into today we?
19118536.94 * 1.055 = 20,170,056.47
Take the annuity
82
Alt: Annuity Example 6

You win the $20 million Powerball. The lottery
commission offers you $20 million dollars today or
a nine payment annuity of $2,750,000, with the first
payment being today. Which is more valuable if
your discount rate is 5.5%?
N
=8
I/Y
= 5.5
PV
= ????
PMT
= 2,750,000
FV
=0
PV = $17420056.47
Then add today’s payment $2,750,000
20,170,056.47
83
Delayed first payment: Perpetuity

What is the present value of a growing
perpetuity, that pays $100 per year, growing at
6%, when the discount rate is 10%, if the first
payment is in 12 years?
84
Delayed first payment: Perpetuity
What is the present value of a growing
perpetuity, that pays $100 per year, growing at
6%, when the discount rate is 10%, if the first
payment is in 12 years?
 Steps:

1. Get
value of perpetuity at t= 11 (year end)
Why year 11?
85
Delayed first payment: Perpetuity


What is the present value of a growing
perpetuity, that pays $100 per year, growing
at 6%, when the discount rate is 10%, if the
first payment is in 12 years?
Steps:
Get value of perpetuity at t= 11 (year end)
100/(0.10-0.06) = 2,500
1.
86
Delayed first payment: Perpetuity
What is the present value of a growing
perpetuity, that pays $100 per year, growing at
6%, when the discount rate is 10%, if the first
payment is in 12 years?
 Steps:

1. Get
value of perpetuity at t= 11 (year end)
100/(0.10-0.06) = 2,500
2. Bring value in step 1 to t=0
87
Delayed first payment: Perpetuity


What is the present value of a growing perpetuity,
that pays $100 per year, growing at 6%, when the
discount rate is 10%, if the first payment is in 12
years?
Steps:
Get value of perpetuity at t= 11 (year end)
100/(0.10-0.06) = 2,500
2. Bring value in step 1 to t=0
2,500 / 1.111 = 876.23
1.
88
Growing Annuity
A growing stream of cash flows with a fixed maturity
C
C×(1+g)
C ×(1+g)2
C×(1+g)T-1

0
1
2
3
T
C
C  (1  g )
C  (1  g )
PV 


2
T
(1  r )
(1  r )
(1  r )
T

 1 g  
C
 
PV 
1  
r  g   (1  r )  


T 1
89
Growing Annuity: Example
A defined-benefit retirement plan offers to pay $20,000 per
year for 40 years and increase the annual payment by 3% each
year. What is the present value at retirement if the discount rate
is 10%?
$20,000
$20,000×(1.03) $20,000×(1.03)39

0
1
2
40
90
Growing Annuity: Example
A defined-benefit retirement plan offers to pay $20,000 per
year for 40 years and increase the annual payment by 3% each
year. What is the present value at retirement if the discount rate
is 10%?
$20,000
$20,000×(1.03) $20,000×(1.03)39

0
PV
1
2
40
= (20,000/(.1-.03)) * [ 1- {1.03/1.1}40] = 265,121.57
91
Growing Annuity: Example (Given)
You are evaluating an income generating property. Net rent is
received at the end of each year. The first year's rent is
expected to be $8,500, and rent is expected to increase 7%
each year. What is the present value of the estimated income
stream over the first 5 years if the discount rate is 12%?
0
1
2
3
4
5
PV = (8,500/(.12-.07)) * [ 1- {1.07/1.12}5] = $34,706.26
92
Growing Perpetuity Example

What is the value today a perpetuity that makes
payments every other year, If the first payment is $100,
the discount rate is 12%, and the growth rate is 7%?
 r:
 g:
 Price:
93
Growing Perpetuity Example

What is the value today a perpetuity that makes
payments every other year, If the first payment is $100,
the discount rate is 12%, and the growth rate is 7%?
 r: is 12%/year so the 2-year is 25.44%

EBAR = (1 + 0.24/2)2 -1
 g:
 Price:
94
Growing Perpetuity Example

What is the value today a perpetuity that makes
payments every other year, If the first payment is $100,
the discount rate is 12%, and the growth rate is 7%?
 r: is 12%/year so the 2-year is 25.44%

 g:

EBAR = (1 + 0.24/2)2 -1
is 7%/year so the 2-year is 14.49%
EBAGR = (1 + 0.14/2)2 -1
 What

is half of infinity?
Infinity
 Price:

100/(0.2544-0.1449) = $913.24
95
Valuation Formulas
FVn
PV 
(1  r ) n
C
PV 
r
C
PV 
r

1 
1  (1  r ) T 


F V n  P V * (1  r ) n
C1
PV 
rg
T

 1 g  
C1
 
PV 
1  
r  g   (1  r )  


96
Valuation Formulas
Lump Sum
FVn
PV 
(1  r ) n
Lump Sum
F V n  P V * (1  r ) n
Growing Perpetuity
C1
PV 
rg
Perpetuity
C
PV 
r
Annuity
C
PV 
r

1 
1  (1  r ) T 


Growing Annuity
T

 1 g  
C1
 
PV 
1  
r  g   (1  r )  


97
Remember
That when you use one of these formula’s or
the calculator the assumptions are that:
 PV is right now
 The first payment is next year

98
What Is a Firm Worth?
Conceptually, a firm should be worth the
present value of the firm’s cash flows.
 The tricky part is determining the size, timing,
and risk of those cash flows.

99
Quick Quiz
1.
2.
3.
4.
5.
How is the future value of a single cash flow
computed?
How is the present value of a series of cash flows
computed.
What is the Net Present Value of an investment?
What is an EAR, and how is it computed?
What is a perpetuity? An annuity?
100
Why We Care
The Time Value of Money is the basis for all
of finance
 People will assume that you have this down
cold

101