Discounted Cash Flow Valuation

1
Discounted Cash Flow Valuation
2
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

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

4
One Period Discounting

• PV Future Value / (1 Discount Rate)
• V0 C1 / (1r)
• Alternatively
• PV Future Value Discount Factor
• V0 C1 (1/ (1r))
• Discount factor is 1/ (1r)

5
PV Example

• What is the value today of 100 in one year, if r
  15?
• PV 100 / 1.15 86.96

6
FV Example

• What is the value in one year of 100, invested
  today at 15?
• FV 100 (1.15)1 115

7
Discount Rate Example

• Your stock costs 100 today, pays 5 in dividends
  at the end of the period, and thensells for 98.
  What is your rate of return?
• PV
• FV

8
Discount Rate Example

• Your stock costs 100 today, pays 5 in dividends
  at the end of the period, and thensells for 98.
  What is your rate of return?
• PV 100
• FV

9
Discount Rate Example

• Your stock costs 100 today, pays 5 in dividends
  at the end of the period, and thensells for 98.
  What is your rate of return?
• PV 100
• FV 103 98 5
• (98 5)/100 1 3

10
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

11
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 / (1r)
• For Investments C0 will be negative, and C1 will
  be positive
• For Loans C0 will be positive, and C1 will be
  negative

12
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 dont know
• We cannot simply compare cash flows that occur at
  different times

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?

14
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

15
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?

16
Step 1 Draw out the cash flows
-275,000
300,000
17
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?

18
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

19
If we are unsure about future?

• What is the appropriate discount rate if we are
  unsure about the Starbucks offer
• rd rf
• rd gt rf
• rd lt rf

20
If we are unsure about future?

• What is the appropriate discount rate if we are
  unsure about the Starbucks offer
• rd rf
• rd gt rf
• rd lt rf

21
The Discount Rate

• Should take account of two things
• Time value of money
• 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

22
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?

23
Calculations

• Need to recalculate the NPV
• 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

24
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

25
Expectation Calculation

• The expected value is the weighted average of Xs
  possible values, where the probability of any
  outcome is p
• E(X) p1X1 p2X2 . psXs
• 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

26
Risky Coffee Shop 2

• Now the Starbucks offer depends on the state of
  the economy

27
Calculations

• Discount Rate 12
• Expected Future Cash Flow
• (0.25300) (0.50400) (0.25700) 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

28
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

29
Reminder

• Important to set up problem correctly
• Keep track of
• Magnitude and timing of the cash flows
• TIMELINES
• You cannot compare cash flows _at_ t3 and _at_ t2 if
  they are not in present value terms!!

30
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

31
Four Related Questions

• How much must you deposit today to have 1
  million in 25 years? (r12)
• 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 r12?
• What will 58,823.31 grow to after 25 years if
  r12?

32
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?
• 53.78 10(1.40)5

53.78
10
27.44
19.6
14
38.42
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
PV
34
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/(10.15)5 9,943.53

35
Historical Example

• From Fibonaccis 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

36
Historical Example

• From Fibonaccis 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 (11)20 1,048,576 denari.

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)
  
• Compounded interest
• FV2 PV0 (1r) (1r) PV0 (1r)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
• FV2 (PV0 (r) PV0 (r)) PV0 PV0 (1 2r)
  275
• Compounded interest
• FV2 PV0 (1r) (1r) PV0 (1r)2

39
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 (1r) (1r) PV0 (1r)5 448.40

40
Compounding Periods

• We have been assuming that compounding and
  discounting occurs annually, this does not need
  to be the case

41
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) MN
• PV FVN / ( 1 r / M) MN
• M is number of compounding periods per year
• N is the number of years

42
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) 125 908.35
• What is the PV of 500 received in 5 years, if
  the discount rate is 12 compounded monthly?
• PV 500 / ( 1 0.12 / 12) 125 275.22

43
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

44
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

45
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))23 70.93

46
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
• EAR (1 0.12 / 2)2 1 12.36
• FV 50 (10.1236)3 70.93
• So, investing at compounded annually
  is the same as investing at 12 compounded
  semi-annually

70.93
12.36
47
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

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

49
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 (10.14/365)365 1 15
• 15,000 1.15 17,250

50
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.

51
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?
52
Insight Example
r 10
Year Project A Project B
1 100 90.91 300 272.73
2 400 330.58 400 330.58
3 300 225.39 100 75.13

PV 646.88 678.44
Which project is more valuable? Why? B, gets the
cash faster
53
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 15,000/1.1510,000/1.152 5,000/1.153
• PV 23,892.50
• NPV 25,00023,892.50 1,107.50

54
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?

55
Multiple Cash Flows (Given)
0
1
2
3
4
200
400
600
800
178.57
318.88
427.07
508.41
1,432.93
Dont buy
56
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

57
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

58
Perpetuity

• A stream of cash flows that lasts forever
• PV C/r
• What is PV if C100 and r10
• 100/0.1 1,000

59
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.

60
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

61
Growing Perpetuities

• Annual payments grow at a constant rate, g
• PV C1/(1r) C1(1g)/(1r)2 C1(1g)2(1r)3
• PV C1/(r-g)
• What is PV if C1 100, r10, and g2?
• PV 100 / (0.10 0.02) 1,250

62
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/PVg 100,000/1,000,0000.02 12

63
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?

1.30 (1.05)2 1.43
1.30(1.05) 1.37

2
3

• PV 1.30 / (0.10 0.05) 26

64
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

65
Annuity

• A constant stream of cash flows with a fixed
  maturity

66
Annuity Formula

• Simply subtracting off the PV of the rest of the
  perpetuitys cash flows

67
Annuity Example 1

• Compute the present value of a 3 year ordinary
  annuity with payments of 100 at r10
• Answer

Or
68
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

69
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

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?
• What is the discount rate?

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 do the payments look like?
• We receive 5 payments of 600

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 over 2
  years the discount rate is going to be

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.
• 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

74
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

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?
• What do the payments look like?

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
1 2 3 4
5
77
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?
• 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 year 0 value

100
100
100
100
323.97
1 2 3 4
5
78
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?
• To get PV today we need to discount the 323.97
  back one more year
• 323.97 / 1.09 297.22

100
100
100
100
323.97
297.22
1 2 3 4
5
79
Annuity Example 5

• What is the value today of a 10-pymt annuity that
  pays 300 a year if the annuitys first cash flow
  is at the end of year 6. The interest rate is 15
  for years 1-5 and 10 thereafter?

80
Annuity Example 5

• What is the value today of a 10-pymt annuity that
  pays 300 a year (at year-end) if the annuitys
  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 ????
• PMT 300
• FV 0
• Bring value in step 1 to t0
• 1,843.37 / 1.155 916.48

1,843.37
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
  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?

82
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

83
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 todays payment 2,750,000
• 20,170,056.47

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?

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)
• Why year 11?

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
• Get value of perpetuity at t 11 (year end)
• 100/(0.10-0.06) 2,500

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
• Bring value in step 1 to t0

88
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
• Bring value in step 1 to t0
• 2,500 / 1.111 876.23

89
Growing Annuity

• A growing stream of cash flows with a fixed
  maturity

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?

91
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?
• PV (20,000/(.1-.03)) 1- 1.03/1.140
  265,121.57

92
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-
1.07/1.125 34,706.26
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
• 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
• EBAR (1 0.24/2)2 -1
• g
• Price

95
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 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

96
Valuation Formulas
97
Valuation Formulas
Lump Sum
Lump Sum
Growing Perpetuity
Perpetuity
Growing Annuity
Annuity
98
Remember

• That when you use one of these formulas or the
  calculator the assumptions are that
• PV is right now
• The first payment is next year

99
What Is a Firm Worth?

• Conceptually, a firm should be worth the present
  value of the firms cash flows.
• The tricky part is determining the size, timing,
  and risk of those cash flows.

100
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?

101
Why We Care

• The Time Value of Money is the basis for all of
  finance
• People will assume that you have this down cold