Completed
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
rg
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
rg
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
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