13. 1/10/2007
Chapter Tool Kit for Real Options
THE INVESTMENT TIMING OPTION: AN ILLUSTRATION (Section 13.2)
Murphy Systems is considering a project that will create a new type of hand-held device for connecting to the Internet.
The cost of the project is $50 million, but the future cash flows are uncertain. Murphy estimates a 25% probability that
the new Internet device will be very popular in which case the project will generate cash flows of $30 million each year
for three years. There is a 50% probability generating cash flows of $25 million each year for three years.
Unfortunately, there is a 25% chance that the Internet device will not be popular, which means that the project will
generate only $5 million per year in cash flows. The cost of capital for this project is 14%.
WACC=
Risk-free rate =
Initial cost of project=
14%
6%
$50
DCF Analysis
Expected annual cash flows (in millions):
Probability Cash Flow Prob. x CF
25%
$33
$8.25
50%
$25
$12.50
25%
$5
$1.25
Expected CF =
Time Line
Year
Expected CF
NPV =
Figure 13-1
0
($50)
$22.00
1
$22.00
2
$22.00
$1.08
DCF and Decision Tree Analysis for the Investment Timing Option
(Millions of Dollars)
Part 1. Scenario Analysis: Proceed with Project Today
Future Cash Flows
Now: Year 0
Year 1
Year 2
Year 3
0.25
-$50
3
$22.00
High
0.50
Average
NPV of this
Probability
Scenarioc Probability
x NPV
$33
$33
$33
$26.61
0.25
$6.65
$25
$25
$25
$8.04
0.50
$4.02
$5
$5
$5
-$38.39
0.25
1.00
-$9.60
Low
0.25
Expected value of NPVs =
$1.08
Standard Deviation =
$24.02
Coefficient of Variation =
22.32
a
b
Std Deviation
Part 2. Decision Tree Analysis: Implement in One Year Only if Optimal
Future Cash Flows
Now: Year 0
Year 1
Year 2
Year 3
Year 4
0.25
High
Wait
Average
Low
0.50
0.25
NPV of this
Probability
Scenariod Probability
x NPV
-$50
$33
$33
$33
$23.35
0.25
$5.84
-$50
$25
$25
$25
$7.05
0.50
$3.53
$0
$0
$0
$0
$0.00
0.25
1.00
$0.00
Expected value of NPVs =
$9.36
Standard Deviationa =
$8.57
Coefficient of Variationb =
Notes:
Std Deviation
0.92
a
The standard deviation is calculated as in Chapter 6.
b
The coefficient of variation is the standard deviation divided by the expected value.
The WACC is 14%.
d
The NPV in Part 2 is as of Year 0. Therefore, each of the project cash flows is discounted back one more
year than in Part 1.
c
Figure 13-2 Sensitivity Analysis for the Investment Timing Option Decision Tree
(Millions of Dollars)
Part 1. Decision Tree Analysis: Implement in One Year Only if Optimal (Discount Cost at the Risk-Free Rate and
Operating Cash Flows at the WACC)
Now: Year 0
Year 1
Year 2
Future Cash Flows
Year 3
Year 4
NPV of this
Probability
Scenarioc Probability
x NPV
-$50
$33
$33
$33
$20.04
0.25
$5.01
-$50
$25
$25
$25
$3.74
0.50
$1.87
$0
High
Average
$0
$0
$0
$0.00
0.25
1.00
$0.00
Low
Expected value of NPVs =
$6.88
Standard Deviationa =
$7.75
Coefficient of Variationb =
1.13
Part 2. Sensitivity Analysis of NPV to Changes in the Cost of Capital Used to Discount Cost and Cash Flows
Cost of Capital Used to Discount
the Year 2 through Year 4
Operating Cash Flows
$6.88
8.0%
9.0%
10.0%
11.0%
12.0%
13.0%
14.0%
15.0%
16.0%
17.0%
18.0%
Notes:
3.0%
$13.11
$11.78
$10.50
$9.27
$8.09
$6.95
$5.85
$4.79
$3.77
$2.78
$1.83
Cost of Capital Used to Discount the Year 1 Cost
4.0%
5.0%
6.0%
7.0%
$13.46
$13.80
$14.14
$14.47
$12.13
$12.47
$12.81
$13.14
$10.85
$11.20
$11.53
$11.86
$9.62
$9.97
$10.30
$10.64
$8.44
$8.78
$9.12
$9.45
$7.30
$7.64
$7.98
$8.31
$6.20
$6.54
$6.88
$7.21
$5.14
$5.48
$5.82
$6.15
$4.12
$4.46
$4.80
$5.13
$3.13
$3.47
$3.81
$4.14
$2.18
$2.52
$2.86
$3.19
8.0%
$14.79
$13.47
$12.19
$10.96
$9.78
$8.64
$7.54
$6.48
$5.45
$4.46
$3.51
9.0%
$15.11
$13.78
$12.51
$11.28
$10.09
$8.95
$7.85
$6.79
$5.77
$4.78
$3.83
The standard deviation is calculated as in Chapter 6.
The coefficient of variation is the standard deviation divided by the expected value.
c
The Year 2 through Year 4 operating cash flows in are discounted at the WACC of 14%. The cost in
Year 1 is discounted at the risk-free rate of 6%.
a
b
Std Deviation
Real Option Analysis
the project, so the time to maturity of the option is one year. If the company exercises the option, it must pay a strike
price equal to the cost of implementing the project. If the company does implement the project, it gains the value of the
project. If you exercise a call option, you will own a stock that is worth whatever its price is. If the company implements
the project, it will gain a project, whose value is equal to the present value of its cash flows. Therefore, the present value
of a project's future cash flows is analogous to the current value of a stock. The rate of return on the project is equal to
its cost of capital. To find the value of a call option, we need the standard deviation of its rate of return; to find the value
of this real option, we need the standard deviation of the projects expected rate of return.
The first step is to find the value of the project's future cash flows, as of the time the option must be exercised. We also
need the standard deviation of the project's value as of the date it must be exercised. Finally, we need the present value of
the project's future cash flows.
Figure 13-3
Estimating the Input for "Stock Price" in the Option Analysis
of the Investment Timing Option (Millions of Dollars)
Now: Year 0
Year 1
Year 2
Future Cash Flows
Year 3
Year 4
PV of this
Scenarioc
Probability
Probability
x PV
$33
$33
$33
$67.21
0.25
$16.80
$25
$25
$25
$50.91
0.50
$25.46
$5
High
Average
$5
$5
$10.18
0.25
1.00
Std Deviation
$2.55
Low
Expected value of PVs =
$44.80
Standard Deviation =
$21.07
Coefficient of Variation =
0.47
a
b
Notes:
a
b
c
Figure 13-4
The standard deviation is calculated as in Chapter 6.
The coefficient of variation is the standard deviation divided by the expected value.
The WACC is 14%. All cash flows in this scenario are discounted back to Year 0.
Estimating the Input for "Stock Return Variance" in the Option Analysis
of the Investment Timing Option (Millions of Dollars)
Part 1. Find the Value and Risk of Future Cash Flows at the Time the Option Expires
Now: Year 0
Year 1
PV in Year 1
for this
Probability
x PVYear 1
Scenarioc Probability
Future Cash Flows
Year 2
Year 3
Year 4
$33
$33
$33
$76.61
0.25
$19.15
$25
$25
$25
$58.04
0.50
$29.02
$5
$5
$5
$11.61
0.25
1.00
$2.90
High
Average
Low
Expected value of PVYear 1 =
$51.08
Standard Deviation of PVYear 1a =
$24.02
Coefficient of Variation of PVYear 1b =
0.47
Std Deviation
Part 2. Direct Method: Use the Scenarios to Directly Estimate the Variance of the Project's Return
PriceYear 0d
Probability
Probability x ReturnYear 1
PVYear 1e
ReturnYear 1f
$76.61
71.0%
0.25
17.8%
$58.04
29.5%
0.50
14.8%
$11.61
-74.1%
0.25
1.00
-18.5%
Std Deviation
High
$44.80
Average
Low
Expected return =
14.0%
Standard deviation of return =
53.6%
Variance of return =
28.7%
a
g
Part 3. Indirect Method: Use the Scenarios to Indirectly Estimate the Variance of the Project's Return
Expected "price" at the time the option expiresh =
Standard deviation of expected "price" at the time the option expiresi =
Coefficient of variation (CV) =
Time (in years) until the option expires (t) =
Variance of the project's expected return = ln(CV2+1)/t =
Notes:
$51.08
$24.02
0.47
1
20.0%
The standard deviation is calculated as in Chapter 6.
The coefficient of variation is the standard deviation divided by the expected value.
c
The WACC is 14%. The Year 2 through Year 4 cash flows are discounted back to Year 1.
d
The Year 0 price is the expected PV from Figure 13-3.
e
The Year 1 PVs are from Part 1.
f
The returns for each scenario are calculated as (PVYear 1 - PriceYear 0)/PriceYear 0.
g
The variance of return is the standard deviation squared.
h
The expected "price" at the time the option expires is taken from Part 1.
i
The standard deviation of expected "price" at the time the option expires is taken from Part 1.
a
b
Figure 13-5
Estimating the Value of the of the Investment Timing Option Using
a Standard Financial Option (Millions of Dollars)
Part 1. Find the Value of a Call Option Using the Black-Scholes Model
rRF =
t=
X=
P=
2 =
Real Option
Risk-free interest rate
Time until the option expires
Cost to implement the project
Current value of the project
Variance of the project's rate of return
=
=
=
=
=
6%
1
$50.00
$44.80 a
20.0% b
d1 =
d2 =
N(d1)=
N(d2)=
{ ln (P/X) + [rRF + 2 /2) ] t } / ( t1/2 )
d1 - (t 1 / 2)
=
=
=
=
0.112
-0.33
0.54
0.37
V=
P[ N (d1) ] - Xe-rRF t [ N (d2) ]
=
$7.04
Part 2. Sensitivity Analysis of Option Value to Changes in Variance
Variance
Option Value
$7.04
12.0%
$5.24
14.0%
$5.74
16.0%
$6.20
18.0%
$6.63
20.0%
$7.04
22.0%
$7.42
24.0%
$7.79
26.0%
$8.15
28.0%
$8.49
30.0%
$8.81
32.0%
$9.13
Notes:
a
b
The current value of the project is taken from Figure 13-3.
The variance of the project's rate of return is taken from Part 3 of Figure 13-4.
THE GROWTH OPTION: AN ILLUSTRATION (Section 13.3)
Kidco Corporation designs and produces products aimed at the pre-teen market. Most of these products have a very
short life cycle, given the rapidly changing tastes of pre-teens. Kidco is considering one such project, which will cost $30
million to implement and will last only two years. Kidco believes there is a 25 percent chance that the project will catch
the fancy of pre-teens. In this scenario, it will generate cash flows of $34 million in each of the next two years, after which
pre-teen tastes are likely to change. There is a 50 percent chance of average demand for the new product, which
produces expected cash flows of $19 million annually for two years. There is a 25% chance that the pre-teens wont like
the product, and it will generate cash flows of only $3 million per year. The cost of capital for this project is 14%.
WACC=
Risk-free rate=
Initial cost=
14%
6%
$30
DCF Analysis
Expected annual cash flows (in millions):
Probability Cash Flow Prob. x CF
25%
$34
$8.50
50%
$20
$10.00
25%
$2
$0.50
Expected CF =
Time Line
Year
Expected CF
NPV =
0
($30)
$1.29
$19.00
1
$19.00
2
$19.00
Figure 13-6
Scenario Analysis and Decision Tree Analysis for the Kidco Project
(Millions of Dollars)
Part 1. Scenario Analysis of Kidco's First Generation Project
Future Cash Flows
Year 1
Year 2
Now: Year 0
NPV of this
Probability
Scenarioc Probability
x NPV
$34
$34
$25.99
0.25
$6.50
$20
$20
$2.93
0.50
$1.47
$2
-$30
High
Average
$2
-$26.71
0.25
1.00
Std Deviation
-$6.68
Low
Expected Value of NPVs =
$1.29
Standard Deviation =
$18.70
Coefficient of Variation =
14.54
a
b
Part 2. Decision Tree Analysis of the Growth Option
Now: Year 0
Future Cash Flows
Year 2d
Year 3
Year 1
NPV of this
Probability
Scenarioe Probability
x NPV
Year 4
$34
$34
-$30
$34
$34
$42.37
0.25
$10.59
$20
$20
-$30
$20
$20
$1.57
0.50
$0.79
$2
$2
$0
$0
-$26.71
0.25
1.00
-$6.68
High
-$30
Average
Low
Expected Value of NPVs =
$4.70
Standard Deviation =
$24.62
Coefficient of Variation =
5.24
a
b
Notes:
a
b
c
The standard deviation is calculated as in Chapter 6.
The coefficient of variation is the standard deviation divided by the expected value.
The operating cash flows are discounted by the WACC of 14%.
The total cash flows in Year 2 are equal to the operating cash flows for the first generation
product minus the $30 million cost to implement the second generation product, if it is
optimal to do so.
d
The operating cash flows in Year 1 through Year 4, which do not include the $30 million
cost of implementing the second generation project in Year 2 for the high demand and
average demand scenarios, are discounted at the WACC of 14%. The implementation cost in
Year 2 for the high demand and average demand scenarios is discounted at the risk-free rate
of 6%.
e
Std Deviation
Sensitivity Analysis of the Kidco Decision Tree Analysis in Figure 13-6
(Millions of Dollars)
Cost of Capital Used to Discount the Year 1 Year 4 Operating Cash Flows (These do not
include the $30 million implemetation cost of the
second generation project in Year 2.)
Table 13-1
Cost of Capital Used to Discount the $30 Million Implementation
Cost in Year 2 of the Second Generation Project
4.0%
5.0%
6.0%
7.0%
8.0%
$11.36
$11.76
$12.14
$12.51
$12.88
$10.01
$10.41
$10.79
$11.16
$11.52
$8.71
$9.10
$9.49
$9.86
$10.22
$4.70
8.0%
9.0%
10.0%
11.0%
$7.04
$7.45
$7.84
$8.23
$8.60
$8.96
$9.31
12.0%
13.0%
14.0%
$5.83
$4.65
$3.52
$6.23
$5.06
$3.92
$6.63
$5.45
$4.32
$7.01
$5.84
$4.70
$7.38
$6.21
$5.07
$7.75
$6.57
$5.44
$8.10
$6.92
$5.79
15.0%
$2.42
$2.83
$3.22
$3.61
$3.98
$4.34
$4.69
16.0%
$1.36
$1.77
$2.16
$2.54
$2.92
$3.28
$3.63
17.0%
$0.33
$0.74
$1.13
$1.52
$1.89
$2.25
$2.60
18.0%
Figure 13-7
3.0%
$10.96
$9.61
$8.30
9.0%
$13.23
$11.88
$10.57
-$0.66
-$0.25
$0.14
$0.52
$0.90
$1.26
$1.61
Estimating the Input for "Stock Price" in the Growth Option Analysis
of the Investment Option Timing (Millions of Dollars)
Now: Year 0
Year 1
Year 2
Future Cash Flows
Year 3
Year 4
PV of this
Scenarioc
Probability
Probability
x PV
$34
$34
$43.08
0.25
$10.77
$20
$20
$25.34
0.50
$12.67
$2
High
Average
$2
$2.53
0.25
1.00
Std Deviation
$0.63
Low
Expected value of PVs =
$24.07
Standard Deviation =
$14.39
Coefficient of Variation =
0.60
a
b
Notes:
a
b
c
Figure 13-8
The standard deviation is calculated as in Chapter 6.
The coefficient of variation is the standard deviation divided by the expected value.
The WACC is 14%. All cash flows in this scenario are discounted back to Year 0.
Estimating the Input for "Stock Return Variance" in the
Growth Option Analysis (Millions of Dollars)
Part 1. Find the Value and Risk of Future Cash Flows at the Time the Option Expires
Now: Year 0
Year 1
PV in Year 2
for this
Probability
x PVYear 2
Scenarioc Probability
Future Cash Flows
Year 2
Year 3
Year 4
$34
$34
$55.99
0.25
$14.00
$20
$20
$32.93
0.50
$16.47
$2
$2
$3.29
0.25
1.00
$0.82
High
Average
Low
Expected value of PVYear 2 =
$31.29
Standard Deviation of PVYear 2a =
$18.70
Coefficient of Variation of PV
b
Year 2
=
0.60
Std Deviation
Part 2. Direct Method: Use the Scenarios to Directly Estimate the Variance of the Project's Return
PriceYear 0d
PVYear 2e
Probability
Probability x ReturnYear 2
ReturnYear 2f
$55.99
52.5%
0.25
13.1%
5.4%
$32.93
17.0%
0.50
8.5%
0.6%
$3.29
-63.0%
0.25
1.00
-15.8%
11.9%
Expected return =
5.9%
17.9%
Standard deviation of return =
42.3%
Variance of return =
17.9%
High
$24.07
Average
Low
a
g
Part 3. Indirect Method: Use the Scenarios to Indirectly Estimate the Variance of the Project's Return
Expected "price" at the time the option expiresh =
Standard deviation of expected "price" at the time the option expiresi =
Coefficient of variation (CV) =
Time (in years) until the option expires (t) =
Variance of the project's expected return = ln(CV2+1)/t =
Notes:
$31.29
$18.70
0.60
2
15.3%
The standard deviation is calculated as in Chapter 6.
The coefficient of variation is the standard deviation divided by the expected value.
c
The WACC is 14%. The Year 3 through Year 4 cash flows are discounted back to Year 2.
d
The Year 0 price is the expected PV from Figure 13-7.
e
The Year 2 PVs are from Part 1.
f
The returns for each scenario are calculated as (PVYear 2/ PriceYear 0)0.5 - 1.
g
The variance of return is the standard deviation squared.
h
The expected "price" at the time the option expires is taken from Part 1.
i
The standard deviation of expected "price" at the time the option expires is taken from Part 1.
a
b
Figure 13-9 Estimating the Value of the of the Growth Option Using
a Standard Financial Option (Millions of Dollars)
Part 1. Find the Value of a Call Option Using the Black-Scholes Model
rRF =
t=
X=
P=
2 =
Real Option
Risk-free interest rate
Time until the option expires
Cost to implement the project
Current value of the project
Variance of the project's rate of return
=
=
=
=
=
6%
2
$30.00
$24.07 a
15.3% b
d1 =
d2 =
N(d1)=
N(d2)=
{ ln (P/X) + [rRF + 2 /2) ] t } / ( t1/2 )
d1 - (t 1 / 2)
=
=
=
=
0.096
-0.46
0.54
0.32
V=
P[ N (d1) ] - Xe-rRF t [ N (d2) ]
=
$4.34
Part 2. Sensitivity Analysis of Option Value to Changes in Variance
Variance
Option Value
$4.34
11.3%
$3.60
13.3%
$3.98
15.3%
$4.34
17.3%
$4.68
19.3%
$4.99
21.3%
$5.29
23.3%
$5.57
25.3%
$5.84
27.3%
$6.10
29.3%
$6.35
31.3%
$6.59
Notes:
a
b
The current value of the project is taken from Figure 13-7.
The variance of the project's rate of return is taken from Part 3 of Figure 13-8.
Total value = NPV of first generation project + Value of growth option
Total value =
1.29
+
4.34
Total value =
5.63
Data for
Std Deviation
163
24
389
577
=Variance of PV
Data for
Std Deviation
49
3
22
73
=Variance of PV
Data for
Std Deviation
43
5
12
60
=Variance of PV
Data for
Std Deviation
125
19
300
444
=Variance of PV
Data for
Std Deviation
163
24
389
577
=Variance of PV
Data for
Std Deviation
8.1%
1.2%
19.4%
28.7% =Variance
of PV
Data for
Std Deviation
153
1
196
350
=Variance of NPV
Data for
Std Deviation
355
5
247
606
=Variance of NPV
Data for
Std Deviation
90
1
116
207
=Variance of PV
Data for
Std Deviation
153
1
196
350
=Variance of PV
Data for
Std Deviation
=Variance of PV
Web 13A: THE ABANDONMENT OPTION
Intial Cost=
WACC=
Risk free rate =
After-tax salvage value =
$26
12%
6%
$14
DCF Analysis
Table 13A-1 Expected Operating Cash Flows for Project at Synapse Systems
(Millions of Dollars)
Demand
High
Average
Low
Probability
25%
50%
25%
2007
$18
$7
-$8
Operating Cash Flow
2008
2009
$23
$28
$8
$9
-$9
-$10
2010
$33
$10
-$12
Expected operating cash flow =
$6.00
$7.50
$9.00
$10.25
Year
Now: Year 0
Expected CF
($26)
Year 1
$6.00
Year 2
$7.50
Year 3
$9.00
Year 4
$10.25
Time Line
NPV =
Figure 13A-1
-1.74
Scenario Analysis and Decision Tree Analysis for the Synapse Project
(Millions of Dollars)
Part 1. Scenario Analysis of the Project (Ignoring the Option to Abandon)
Now: Year 0
Year 1
Future Cash Flows
Year 2
Year 3
NPV of this
Probability
Scenarioc Probability
x NPV
Year 4
$18
$23
$28
$33
$49.31
0.25
$12.33
$7
$8
$9
$10
-$0.61
0.50
-$0.31
-$8
-$9
-$10
-$12
-$55.06
0.25
1.00
-$13.77
High
-$26
Average
Low
Expected Value of NPVs =
-$1.74
Standard Deviationa =
$36.92
Coefficient of Variationb =
-21.17
Part 2. Decision Tree Analysis of the Abandonment Option
Future Cash Flows
NPV of this
Probability
Now: Year 0
Year 1d
Year 2e
Year 3e
Year 4e
Scenariof
Probability
x NPV
$18
$23
$28
$33
$49.31
0.25
$12.33
$7
$8
$9
$10
-$0.61
0.50
-$0.31
$6
$0
$0
$0
-$19.94
0.25
1.00
-$4.98
High
-$26
Average
Low
Expected Value of NPVs =
Standard Deviationa =
$25.65
Coefficient of Variationb =
Notes:
$7.04
3.64
The standard deviation is calculated as in Chapter 6.
The coefficient of variation is the standard deviation divided by the expected value.
c
The operating cash flows are discounted by the WACC of 12%.
d
The cash flow in Year 1 for the low demand scenario is equal to the -$8 million
operating cash flow plus the after-tax salvage value of $14 million, since Synapse will
e bandon the proejct in this scenario.
a The cash flows in Years 2 though Year 4 for the low demand scenario are zero,
because Synapse abandons the project immediately after the -$8 million operating
f oss in Year 1.
l The operating cash flows in Year 1 though Year 4, which do not include the $14
million after-tax salvage value, are discounted at the WACC of 12%. The $14
million salvage value in the low demand scenario in Year 1 is discounted at the riskfree rate of 6%.
a
b
Cost of Capital Used to Discount the
Year 1 through Year 4 Operating
Cash Flows (These do not include the
$14 million after-tax salvage value in
Year 1.)
Table 13A-2
Sensitivity Analysis of the Synapse Decision Tree Analysis in Figure 13A-1
(Millions of Dollars)
$7.04
8.0%
9.0%
10.0%
11.0%
12.0%
13.0%
14.0%
15.0%
16.0%
17.0%
18.0%
Cost of Capital Used to Discount the $14 Million After-tax Salvage Value in the
Low Demand Scenario of Year 1.
3.0%
4.0%
5.0%
6.0%
7.0%
8.0%
9.0%
$10.18
$10.15
$10.12
$10.08
$10.05
$10.02
$9.99
$9.38
$9.34
$9.31
$9.28
$9.25
$9.22
$9.19
$8.60
$8.57
$8.54
$8.50
$8.47
$8.44
$8.41
$7.85
$7.82
$7.79
$7.76
$7.73
$7.70
$7.67
$7.13
$7.10
$7.07
$7.04
$7.01
$6.98
$6.95
$6.44
$6.41
$6.38
$6.34
$6.31
$6.28
$6.25
$5.77
$5.74
$5.71
$5.67
$5.64
$5.61
$5.58
$5.13
$5.09
$5.06
$5.03
$5.00
$4.97
$4.94
$4.50
$4.47
$4.44
$4.41
$4.37
$4.34
$4.31
$3.90
$3.87
$3.84
$3.80
$3.77
$3.74
$3.71
$3.32
$3.29
$3.25
$3.22
$3.19
$3.16
$3.13
Real Option Analysis Where Firm Has the Option to Abandon
Break the project into two projects plus an option to abandon the second project. Project A starts at time zero and
lasts one year. It has the initial cost and first-year operating cash flows of the original project. Project B begins at
Year 2 and last through Year 4. It has no initial cost, but has the Year 2 through 4 operating cash flows of the
original project. There is also a real option that allows you to abandon Project B if the value of B at time 1 is less
than the abandonment amount.
Figure 13A-2
Breaking Synapse's Project into Two Separate Projects
of the Investment Timing Option (Millions of Dollars)
Part 1. DCF Analysis of Project A that Lasts One Year
Future Cash Flows
Now: Year 0
Year 1
NPV of this
Probability
Scenarioc Probability
x NPV
$18
-$9.93
0.25
-$2.48
$7
-$19.75
0.50
-$9.88
-$8
-$33.14
0.25
1.00
-$8.29
High
-$26
Average
Low
Expected value of PVs =
-$20.64
Standard Deviationa =
$8.26
Coefficient of Variationb =
(0.40)
Part 2. DCF Analysis of Project B that Starts at Year 1 If Project A is Already in Place
Future Cash Flows
NPV of this
Probability
Now: Year 0
Year 1
Year 2
Year 3
Year 4
Scenarioc Probability
x NPV
$23
$28
$33
$59.24
0.25
$14.81
$8
$9
$10
$19.14
0.50
$9.57
-$9
-$10
-$12
-$21.92
0.25
1.00
-$5.48
High
Average
Low
Expected value of PVs =
Standard Deviationa =
a
b
c
Figure 13A-3
$28.69
Coefficient of Variationb =
Notes:
$18.90
1.52
The standard deviation is calculated as in Chapter 6.
The coefficient of variation is the standard deviation divided by the expected value.
The WACC is 12%. All cash flows in this scenario are discounted back to Year 0.
Estimating the Input for "Stock Return Variance" in the
Abandonment Option Analysis (Millions of Dollars)
Part 1. Find the Value and Risk of Future Cash Flows at the Time the Option Expires
Now: Year 0
Year 1
Future Cash Flows
Year 2
Year 3
Year 4
PV in Year 1
for this
Probability
c
Scenario Probability x PVYear 1
$23
$33
$66.35
0.25
$16.59
$8
$9
$10
$21.44
0.50
$10.72
-$9
High
Average
$28
-$10
-$12
-$24.55
0.25
1.00
-$6.14
Low
Expected value of PVYear 1 =
$21.17
Standard Deviation of PVYear 1a =
$32.14
Coefficient of Variation of PVYear 1b =
1.518
Part 2. Direct Method: Use the Scenarios to Directly Estimate the Variance of the Project's Return
PriceYear 0d
PVYear 1e
ReturnYear 1f
Probability
Probabilityx ReturnYear 1
$66.35
High
Average
0.25
62.8%
$21.44
13.4%
0.50
6.7%
-$24.55
$18.90
251.1%
-229.9%
0.25
1.00
-57.5%
Low
Expected return =
12.0%
Standard deviation of returna =
170.0%
Variance of returng =
289.2%
Part 3. Indirect Method: Use the Scenarios to Indirectly Estimate the Variance of the Project's Return
Expected "price" at the time the option expiresh =
Standard deviation of expected "price" at the time the option expiresi =
Coefficient of variation (CV) =
Time (in years) until the option expires (t) =
Variance of the project's expected return = ln(CV2+1)/t =
Notes:
$21.17
$32.14
1.52
1
119.5%
The standard deviation is calculated as in Chapter 6.
The coefficient of variation is the standard deviation divided by the expected value.
c
The WACC is 12%. The Year 2 through Year 4 cash flows are discounted back to Year 1.
d
The Year 0 price is the expected NPV from Part 2 of Figure 13-12.
e The Year 1 PVs are from Part 1.
f
The returns for each scenario are calculated as (PVYear 1/ PriceYear 0) - 1.
g
The variance of return is the standard deviation squared.
h
The expected "price" at the time the option expires is taken from Part 1.
i
The standard deviation of expected "price" at the time the option expires is taken from Part 1.
a
b
Figure 13A-4
Estimating the Value of the of the Abandonment Option Using
a Standard Financial Option (Millions of Dollars)
Part 1. Find the Value of a Put Option Using the Black-Scholes Model
rRF =
t=
X=
P=
2 =
Real Option
Risk-free interest rate
Time until the option expires
Salvage value if abandon
Current value of the Project B
Variance of Project B's rate of return
d1 =
d2 =
N(d1)=
N(d2)=
V of Call =
V of Put =
V of Put =
V of Put =
=
=
=
=
=
6%
1
$14.00
$18.90 a
175.0% b
{ ln (P/X) + [rRF + 2 /2) ] t } / ( t1/2
d1 - (t 1 / 2)
0.82
0.35
=
=
0.934
-0.39
P[ N (d1) ] - Xe-rRF t [ N (d2) ]
=
$10.99
Call
$10.99
$5.28
-
P
$18.90
+
+
X e-rRF t
$13.18
Part 2. Sensitivity Analysis of Put Option Value to Changes in Variance
Variance
Option Value
$5.28
55.0%
$2.29
75.0%
$2.94
95.0%
$3.51
115.0%
$4.02
135.0%
$4.48
155.0%
$4.89
175.0%
$5.28
195.0%
$5.63
215.0%
$5.96
235.0%
$6.27
255.0%
$6.56
Notes:
a
b
The current value of the project is taken from Figure 13E-7.
The variance of the project's rate of return is taken from Part 3 of Figure 13E-7.
Total NPV=
Total NPV=
Total NPV=
NPV of A
-$20.64
$3.53
+
+
NPV of B
$18.90
+
+
Value of abandonment option
$5.28
Probability
Data for
Std Deviation
652
1
711
1,363
Probability
Data for
=Variance of NPV
Std Deviation
447
29
182
658
=Variance of NPV
Probability
Data for
Std Deviation
29
0
39
68
Probability
=Variance of PV
Data for
Std Deviation
407
0
417
823
Probability
=Variance of PV
Data for
Std Deviation
510
0
522
1,033
=Variance of PV
Data for
Std Deviation
142.9%
0.0%
146.3%
289.2% =Variance
of PV