Discounted cash flow valuation

1. Discounted Cash Flow
Valuation

2. 2
BASIC PRINCIPAL
 Would you rather have $1,000 today or
$1,000 in 30 years?
Why?

3. 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

4. 4
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)

5. 5
PV Example
 What is the value today of $100 in one year, if
r = 15%?

6. 6
FV Example
 What is the value in one year of $100, invested
today at 15%?

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 =
 FV =
7

8. 8
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

9. 9
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 C0will be negative, and C1will be
positive
For Loans C0will be positive, and C1will be
negative

10. 10
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?

11. 11
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?
 At what price are we indifferent?

12. 12
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?

13. 13
Step 1: Draw out the cash flows
Today Year 1

14. 14
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?

15. 15
Step 3: Find NPV
 The NPV of the project is?

16. 16
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

17. 17
The Discount Rate
 Should take account of two things:
1. Time value of money
2. Riskiness of cash flow
 The appropriate discount rate is the
opportunity cost of capital
 This is the return that is offer on comparable
investments opportunities

18. 18
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?

19. 19
Calculations
 Need to recalculate the NPV

20. 20
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

21. 21
Expectation Calculation
 The expected value is the weighted average of
X’s possible values, where the probability of
any outcome is p
 E(X) = p1
X1
+ p2
X2
+ …. ps
Xs
E(X) – Expected Value of X
Xi
− Outcome of X in state i
pi
– Probability of state i
s – Number of possible states
 Note that = p1
+ p2
+….+ ps
= 1

22. 22
Risky Coffee Shop 2
 Now the Starbucks offer depends on the state
of the economy
Recession Normal Boom
Value 300,000 400,000 700,000
Probability 0.25 0.5 0.25

23. 23
Calculations
 Discount Rate = 12%
 Expected Future Cash Flow =
 NPV =
 Do we still build the coffee shop?

24. 24
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

25. 25
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!!

26. 26
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

27. 27
Four Related Questions
1. How much must you deposit today to have $1
million in 25 years? (r=12%)
2. If a $58,823.31 investment yields $1 million in 25
years, what is the rate of interest?
3. How many years will it take $58,823.31 to grow to
$1 million if r=12%?
4. What will $58,823.31 grow to after 25 years if
r=12%?

28. 28
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?
0 1 2 3 4 5
$10

29. 29
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%?
0 1 2 3 4 5
$20,000PV

30. 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

31. 31
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
=

32. 32
Compounding Periods
We have been assuming that compounding
and discounting occurs annually, this does not
need to be the case

33. 33
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
FVN = PV * ( 1+ r / M) M*N
PV = FVN / ( 1+ r / M) M*N
 M is number of compounding periods per year
 N is the number of years

34. 34
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?

35. 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

36. 36
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

37. 37
Compounding Example 2
 If you invest $50 for 3 years at 12%
compounded semi-annually, your investment
will grow to:

38. Compounding Example 2: Alt.
 If you invest $50 for 3 years at 12%
compounded semi-annually, your investment
will grow to:
 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
$70.93

39. 39
EAR Example
 Find the Effective Annual Rate (EAR) of an 18% loan
that is compounded weekly.

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 =
40

41. 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

42. 42
Present Value Of a Cash Flow Stream
 Discount each cash flow back to the present
using the appropriate discount rate and then
sum the present values.
PV
C
r
C
r
C
r
C
r
C
r
N
N
N
t
t
t
t
N
=
+
+
+
+
+
+ +
+
+=
∑
1
1
2
2
2
3
3
3
1
1 1 1 1
1
( ) ( ) ( )
...
( )
( )
=

43. 43
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?

44. 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

45. 45
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?

46. 46
Multiple Cash Flows (Given)
0 1 2 3 4
200 400 600 800
178.57
318.88
427.07
508.41
1,432.93
Don’t buy

47. 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

48. 48
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

49. 49
Perpetuity
 A stream of cash flows that lasts forever
 PV: = C/r
 What is PV if C=$100 and r=10%:
…0 1
C
2
C
3
C
+
+
+
+
+
+
= 32
)1()1()1( r
C
r
C
r
C
PV

50. 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

51. 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

52. 52
Growing Perpetuities
 Annual payments grow at a constant rate, g
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%?
…
0 1 2 3
C1 C2(1+g) C3(1+g)2

53. 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

54. 54
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?
0
…
1
$1.30
2
$1.30×(1.05)
= $1.37
3
$1.30 ×(1.05)2
= $1.43
PV = 1.30 / (0.10 – 0.05) = $26

55. 55
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?

56. 56
Annuity
A constant stream of cash flows with a fixed maturity
0 1
C
2
C
3
C
T
r
C
r
C
r
C
r
C
PV
)1()1()1()1( 32
+
+
+
+
+
+
+
= 






+
−= T
rr
C
PV
)1(
1
1
T
C


57. 57
Annuity Formula
T
r
r
C
r
C
PV
)1( +
−=
 Simply subtracting off the PV of the rest of the
perpetuity’s cash flows
0 1
C
2
C
3
C
T
C
T+1
C
T+2
C
T+3
C

58. 58
Annuity Example 1
 Compute the present value of a 3 year ordinary
annuity with payments of $100 at r=10%
 Answer:

59. 59
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

60. 60
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

61. 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?
What is the discount rate?

62. 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 do the payments look like?
0 2 4 6 8 10
PV $600$600 $600$600$600

63. 63
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?

64. 64
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 do the payments look like?
0 1 2 3 4 5

65. 65
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?

66. 66
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

67. 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

68. 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

69. 69
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?

70. 70
Growing Annuity
A growing stream of cash flows with a fixed maturity
0 1
C
T
T
r
gC
r
gC
r
C
PV
)1(
)1(
)1(
)1(
)1(
1
2
+
+×
++
+
+×
+
+
=
−















+
+
−
−
=
T
r
g
gr
C
PV
)1(
1
1

2
C×(1+g)
3
C ×(1+g)2
T
C×(1+g)T-1

71. 71
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%?
0 1
$20,000

2
$20,000×(1.03)
40
$20,000×(1.03)39

72. 72
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%?
PV = (8,500/(.12-.07)) * [ 1- {1.07/1.12}5
] = $34,706.26
0 1 2 3 4 5

73. 73
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:

74. 74
Valuation Formulas














+
+
−
−
=
T
r
g
gr
C
PV
)1(
1
1
1





+
−= T
rr
C
PV
)1(
1
1
gr
C
PV
−
=
1
r
C
PV =
n
n
r
FV
PV
)1( +
=
n
n rPVFV )1(* +=

75. 75
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

76. 76
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.

77. 77
Quick Quiz
1. How is the future value of a single cash flow
computed?
2. How is the present value of a series of cash flows
computed.
3. What is the Net Present Value of an investment?
4. What is an EAR, and how is it computed?
5. What is a perpetuity? An annuity?

78. 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