Discounted Cash Flow
Valuation
BASIC PRINCIPAL
 Would
you rather have $1,000 today or
$1,000 in 30 years?
Why?
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%?
5
FV Example

What is the value in one year of $100, invested
today at 15%?
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
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
8
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
9
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?
10
Net Present Value
 Since
we cannot compare cash flow we
need to calculate the NPV of the
investment
If
 At
the discount rate is 5%, then NPV is?
what price are we indifferent?
11
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?
12
Step 1: Draw out the cash flows
Today
Year 1
13
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?
14
Step 3: Find NPV

The NPV of the project is?
15
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
16
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
17
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?
18
Calculations

Need to recalculate the NPV
19
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
20
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
21
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
22
Calculations
Discount Rate = 12%
 Expected Future Cash Flow =


NPV =

Do we still build the coffee shop?
23
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

24
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!!
25
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
26
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%?
27
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
0
1
2
3
4
5
28
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
29
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
30
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 =
31
Compounding Periods
We
have been assuming that compounding and
discounting occurs annually, this does not need
to be the case
32
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

33
Compounding Examples

What is the FV of $500 in 5 years, if the
discount rate is 12%, compounded monthly?

What is the PV of $500 received in 5 years, if
the discount rate is 12% compounded
monthly?
34
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?
35
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
36
Compounding Example 2

If you invest $50 for 3 years at 12%
compounded semi-annually, your investment
will grow to:
37
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


So, investing at
compounded annually
is the same as investing at 12% compounded
semi-annually
38
EAR Example

Find the Effective Annual Rate (EAR) of an 18% loan
that is compounded weekly.
39
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 =
40
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 =
41
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.
42
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?
43
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?
44
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?

45
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
46
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
47
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
48
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%:

49
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?
50
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?
51
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%?

52
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?
53
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
54
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?
55
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 


56
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
57
Annuity Example 1


Compute the present value of a 3 year ordinary
annuity with payments of $100 at r=10%
Answer:
58
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
59
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
60
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?
61
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
2
$600
4
$600
6
$600
8
$600
10
62
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?
63
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
64
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?
65
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:
1.
Get value of annuity at t= 5 (year end)
2.
Bring value in step 1 to t=0
66
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%?
67
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%?
68
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?
69
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
70
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
71
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
72
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:
73
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 )  


74
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

75
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.

76
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?
77
Why We Care
The Time Value of Money is the basis for all
of finance
 People will assume that you have this down
cold

78