Chapter 13 (12ed: 12-11 )
Real Options
th
MINI CASE
Assume that you have just been hired as a financial analyst by Tropical Sweets Inc., a midsized California company that specializes in creating exotic candies from tropical fruits
such as mangoes, papayas, and dates. The firm's CEO, George Yamaguchi, recently
returned from an industry corporate executive conference in San Francisco, and one of the
sessions he attended was on real options. Since no one at Tropical Sweets is familiar with
the basics of real options, Yamaguchi has asked you to prepare a brief report that the
firm's executives could use to gain at least a cursory understanding of the topics.
To begin, you gathered some outside materials the subject and used these materials to
draft a list of pertinent questions that need to be answered. In fact, one possible approach
to the paper is to use a question-and-answer format. Now that the questions have been
drafted, you have to develop the answers.
a.
What are some types of real options?
Answer: 1. Investment timing options
2. Growth options
a.
Expansion of existing product line
b.
New products
c.
New geographic markets
3. Abandonment options
a.
Contraction
b.
Temporary suspension
c.
Complete abandonment
4. Flexibility options.
b.
What are five possible procedures for analyzing a real option?
Answer: 1.
2.
3.
4.
5.
DCF analysis of expected cash flows, ignoring option.
Qualitatively assess the value of the real option.
Decision tree analysis.
Use a model for a corresponding financial option, if possible.
Use financial engineering techniques if a corresponding financial option is not
available.
Mini Case: 12- 1
c.
Tropical Sweets is considering a project that will cost $70 million and will
generate expected cash flows of $30 per year for three years. The cost of capital
for this type of project is 10 percent and the risk-free rate is 6 percent. After
discussions with the marketing department, you learn that there is a 30 percent
chance of high demand, with future cash flows of $45 million per year. There is
a 40 percent chance of average demand, with cash flows of $30 million per year.
If demand is low (a 30 percent chance), cash flows will be only $15 million per
year. What is the expected NPV?
Answer: Initial Cost = $70 Million
Expected Cash Flows = $30 Million Per Year For Three Years
Cost Of Capital = 10%
PV Of Expected CFs = $74.61 Million
Expected NPV = $74.61 - $70
= $4.61 Million
Alternatively, one could calculate the NPV of each scenario:
Demand Probability
Annual Cash Flow
High
30%
$45
Average
40%
$30
Low
30%
$15
Find NPV of each scenario:
PV High:
N=3 I=10
PV=?
PMT=-45
PV= 111.91
NPV High = $111.91 - $70 = $41.91 Million.
PV Average: N=3 I=10
PV=?
PMT=-30
PV= 74.61
NPV Average = $74.61 - $70 = $4.71 Million.
PV Low: N=3
I=10
PV=?
PMT=-15
PV= 37.30
NPV Low = $37.30 - $70 = -$32.70 Million.
Find Expected NPV:
E(NPV)=.3($41.91)+.4($4.61)+.3(-$32.70)
E(PV)= $4.61.
Mini Case: 12 - 2
FV=0
FV=0
FV=0
d.
Answer:
Now suppose this project has an investment timing option, since it can be
delayed for a year. The cost will still be $70 million at the end of the year, and
the cash flows for the scenarios will still last three years. However, Tropical
Sweets will know the level of demand, and will implement the project only if it
adds value to the company. Perform a qualitative assessment of the investment
timing option’s value.
If we immediately proceed with the project, its expected NPV is $4.61 million.
However, the project is very risky. If demand is high, NPV will be $41.91
million. If demand is average, NPV will be $4.61 million. If demand is low, NPV
will be -$32.70 million. However, if we wait one year, we will find out additional
information regarding demand. If demand is low, we won’t implement project. If
we wait, the up-front cost and cash flows will stay the same, except they will be
shifted ahead by a year.
The value of any real option increases if the underlying project is very risky or if
there is a long time before you must exercise the option.
This project is risky and has one year before we must decide, so the option to wait
is probably valuable.
e.
Use decision tree analysis to calculate the NPV of the project with the investment
timing option.
Answer: The project will be implemented only if demand is average or high.
Here is the time line:
0
1
2
3
4
High
$0
-$70
$45
$45
$45
Average
$0
-$70
$30
$30
$30
Low
$0
$0
$0
$0
$0
To find the NPVC, discount the cost at the risk-free rate of 6 percent since it is known
for certain, and discount the other risky cash flows at the 10 percent cost of capital.
High: NPV = -$70/1.06 + $45/1.102 + $45/1.103 +$45/1.104 = $35.70
Average: NPV = -$70/1.06 + $30/1.102 + $30/1.103 +$30/1.104 = $1.79
Low: NPV = $0.
Expected NPV = 0.3($35.70) + 0.4($1.79) + 0.3($0) = $11.42.
Since this is much greater than the NPV of immediate implementation (which is
$4.61 million) we should wait. In other words, implementing immediately gives an
expected NPV of $4.61 million, but implementing immediately means we give up the
option to wait, which is worth $11.42 million.
Mini Case: 12- 3
f.
Use a financial option pricing model to estimate the value of the investment
timing option.
Answer: The option to wait resembles a financial call option-- we get to “buy” the project for
$70 million in one year if value of project in one year is greater than $70 million.
This is like a call option with an exercise price of $70 million and an expiration date
of one year.
X = Exercise Price = Cost Of Implement Project = $70 Million.
RRF = Risk-Free Rate = 6%.
T = Time To Maturity = 1 year.
P = Current Price Of Stock = Current Value Of The Project’s Future Cash Flows.
σ 2 = Variance Of Stock Return = Variance Of Project’s Rate Of Return.
We explain how to calculate P and σ2 below.
Just as the price of a stock is the present value of all the stock’s future cash flows, the
“price” of the real option is the present value of all the project’s cash flows that occur
beyond the exercise date. Notice that the exercise cost of an option does not affect
the stock price. Similarly, the cost to implement the real option does not affect the
current value of the underlying asset (which is the PV of the project’s cash flows). It
will be helpful in later steps if we break the calculation into two parts. First, we find
the value of all cash flows beyond the exercise date discounted back to the exercise
date. Then we find the expected present value of those values.
Step 1: Find the value of all cash flows beyond the exercise date discounted back to
the exercise date. Here is the time line. The exercise date is year 1, so we discount
all future cash flows back to year 1.
0
1
2
3
4
High
$45
$45
$45
Average
$30
$30
$30
Low
$15
$15
$15
High: PV1 = $45/1.10 + $45/1.102 + $45/1.103 = $111.91
Average: PV1 = $30/1.10 + $30/1.102 + $30/1.103 = $74.61
Low: PV1 = $15/1.10 + $15/1.102 + $15/1.103 = $37.30
The current expected present value, P, is:
P = 0.3[$111.91/1.1] + 0.4[$74.61/1.1] + 0.3[$37.30/1.1] = $67.82.
For a stock option, σ2 is the variance of the stock return, not the variance of the stock
price. Therefore, for a real option we need the variance of the project’s rate of return.
There are three ways to estimate this variance. First, we can use subjective judgment.
Second, we can calculate the project’s return in each scenario and then calculate the
return’s variance. This is the direct approach. Third, we know the projects value at
each scenario at the expiration date, and we know the current value of the project.
Mini Case: 12 - 4
Thus, we can find a variance of project return that gives the range of project values
that can occur at expiration. This is the indirect approach.
Following is an explanation of each approach.
Subjective estimate:
The typical stock has σ2 of about 12%. Most projects will be somewhat riskier than
the firm, since the risk of the firm reflects the diversification that comes from having
many projects. Subjectively scale the variance of the company’s stock return up or
down to reflect the risk of the project. The company in our example has a stock with
a variance of 10%, so we might expect the project to have a variance in the range of
12% to 19%.
Direct approach:
From our previous analysis, we know the current value of the project and the value
for each scenario at the time the option expires (year 1). Here is the time line:
High
Average
Low
Current Value
Year 0
$67.82
$67.82
$67.82
Value At Expiration
Year 1
$111.91
$74.61
$37.30
The annual rate of return is:
High: Return = ($111.91/$67.82) – 1 = 65%.
High: Average = ($74.61/$67.82) – 1 = 10%.
High: Return = ($37.30/$67.82) – 1 = -45%.
Expected Return = 0.3(0.65) + 0.4(0.10) + 0.3(-0.45)
= 10%.
2 = 0.3(0.65-0.10)2 + 0.4(0.10-0.10)2 + 0.3(-0.45-0.10)2
= 0.182 = 18.2%.
The direct approach gives an estimate of 18.2% for the variance of the project’s
return.
Mini Case: 12- 5
The indirect approach:
Given a current stock price and an anticipated range of possible stock prices at some
point in the future, we can use our knowledge of the distribution of stock returns
(which is lognormal) to relate the variance of the stock’s rate of return to the range of
possible outcomes for stock price. To use this formula, we need the coefficient of
variation of stock price at the time the option expires. To calculate the coefficient of
variation, we need the expected stock price and the standard deviation of the stock
price (both of these are measured at the time the option expires). For the real option,
we need the expected value of the project’s cash flows at the date the real option
expires, and the standard deviation of the project’s value at the date the real option
expires.
We previously calculated the value of the project at the time the option expires, and
we can use this to calculate the expected value and the standard deviation.
High
Average
Low
Value At Expiration
Year 1
$111.91
$74.61
$37.30
Expected Value
value
=.3($111.91)+.4($74.61)+.3($37.3)
= $74.61.
= [.3($111.91-$74.61)2 + .4($74.61-$74.61)2
+ .3($37.30-$74.61)2]1/2
= $28.90.
Coefficient Of Variation = CV = Expected Value / value
CV = $74.61 / $28.90 = 0.39.
Here is a formula for the variance of a stock’s return, if you know the coefficient of
variation of the expected stock price at some point in the future. The CV should be
for the entire project, including all scenarios:
σ2 = LN[CV2 + 1]/T = LN[0.392 + 1]/1 = 14.2%.
Mini Case: 12 - 6
Now, we proceed to use the OPM:
V = $67.83[N(d1)] - $70e-(0.06)(1)[N(d2)].
d1 =
ln( $67.83/$70 )  [( 0.06  0.142/2 )](15)
0.5 0.5
(.142 ) (1)
= 0.2641.
d2 = d1 - (0.142)0.5(1)0.5 = 0.2641 - 0.3768
= -0.1127.
N(d1) = N(0.2641) = 0.6041.
N(d2) = N(-0.1127) = 0.4551.
therefore,
V = $67.83(0.6041) - $70e-0.06(0.4551)
= $10.98.
g.
Now suppose the cost of the project is $75 million and the project cannot be
delayed. But if Tropical Sweets implements the project, then Tropical Sweets
will have a growth option. It will have the opportunity to replicate the original
project at the end of its life. What is the total expected NPV of the two projects
if both are implemented?
Answer: Suppose the cost of the project is $75 million instead of $70 million, and there is no
option to wait.
NPV = PV of future cash flows - cost
= $74.61 - $75 = -$0.39 million.
The project now looks like a loser. Using NPV analysis:
NPV = NPV Of Original Project + NPV Of Replication Project
= -$0.39 + -$0.39/(1+0.10)3
= -$0.39 + -$0.30 = -$0.69.
Still looks like a loser, but you will only implement project 2 if demand is high. We
might have chosen to discount the cost of the replication project at the risk-free rate,
and this would have made the NPV even lower.
Mini Case: 12- 7
h.
Tropical Sweets will replicate the original project only if demand is high. Using
decision tree analysis, estimate the value of the project with the growth option.
Answer: The future cash flows of the optimal decisions are shown below. The cash flow in
year 3 for the high demand scenario is the cash flow from the original project and the
cost of the replication project.
High
Average
Low
0
-$75
-$75
-$75
1
$45
$30
$15
2
$45
$30
$15
3
$45 -$70
$30
$15
4
$45
$0
$0
5
$45
$0
$0
6
$45
$0
$0
To find the NPV, we discount the risky cash flows at the 10 percent cost of capital,
and the non-risky cost to replicate (i.e., the $75 million) at the risk-free rate.
NPV high = -$75 + $45/1.10 + $45/1.102 + $45/1.103 + $45/1.104
+ $45/1.105 + $45/1.106 - $75/1.063
= $58.02
NPV average = -$75 + $30/1.10 + $30/1.102 + $30/1.103 = -$0.39
NPV average = -$75 + $15/1.10 + $15/1.102 + $15/1.103 = -$37.70
Expected NPV = 0.3($58.02) + 0.4(-$0.39) + 0.3(-$37.70) = $5.94.
Thus, the option to replicate adds enough value that the project now has a positive
NPV.
i.
Use a financial option model to estimate the value of the growth option.
Answer: X = Exercise Price = Cost Of Implement Project = $75 million.
RRF = Risk-Free Rate = 6%.
T = Time To Maturity = 3 years.
P = Current Price Of Stock = Current Value Of The Project’s Future Cash Flows.
σ2 = Variance Of Stock Return = Variance Of Project’s Rate Of Return.
We explain how to calculate P and σ2 below.
Step 1: Find the value of all cash flows beyond the exercise date discounted back to
the exercise date. Here is the time line. The exercise date is year 1, so we discount
all future cash flows back to year 3.
0
High
Average
Low
Mini Case: 12 - 8
1
2
3
4
$45
$30
$15
5
$45
$30
$15
6
$45
$30
$15
High: PV3 = $45/1.10 + $45/1.102 + $45/1.103 = $111.91
Average: PV3 = $30/1.10 + $30/1.102 + $30/1.103 = $74.61
Low: PV3 = $15/1.10 + $15/1.102 + $15/1.103 = $37.30
The current expected present value, P, is:
P = 0.3[$111.91/1.13] + 0.4[$74.61/1.13] + 0.3[$37.30/1.13] = $56.05.
Direct approach for estimating σ2:
From our previous analysis, we know the current value of the project and the value
for each scenario at the time the option expires (year 3). Here is the time line:
High
Average
Low
Current Value
Year 0
$56.02
$56.02
$56.02
Value At Expiration
Year 3
$111.91
$74.61
$37.30
The annual rate of return is:
High: Return = ($111.91/$56.02)(1/3) – 1 = 25.9%.
High: Average = ($74.61/$56.02)(1/3) – 1 = 10%.
High: Return = ($37.30/$56.02)(1/3) – 1 = -12.7%.
Expected Return = 0.3(0.259) + 0.4(0.10) + 0.3(-0.127)
= 8.0%.
2 = 0.3(0..259-0.08)2 + 0.4(0.10-0.08)2 + 0.3(-0.127-0.08)2
= 0.182 = 2.3%.
This is lower than the variance found for the previous option because the dispersion
of cash flows for the replication project is the same as for the original, even though
the replication occurs much later. Therefore, the rate of return for the replication is
less volatile. We do sensitivity analysis later.
The indirect approach:
First, find the coefficient of variation for the value of the project at the time the option
expires (year 3).
Mini Case: 12- 9
We previously calculated the value of the project at the time the option expires, and
we can use this to calculate the expected value and the standard deviation.
High
Average
Low
Value At Expiration
Year 3
$111.91
$74.61
$37.30
Expected Value
value
=.3($111.91)+.4($74.61)+.3($37.3)
= $74.61.
= [.3($111.91-$74.61)2 + .4($74.61-$74.61)2
+ .3($37.30-$74.61)2]1/2
= $28.90.
Coefficient Of Variation = CV = Expected Value / value
CV = $74.61 / $28.90 = 0.39.
To find the variance of the project’s rate or return, we use the formula below:
σ2 = LN[CV2 + 1]/T = LN[0.392 + 1]/3 = 4.7%.
Now, we proceed to use the OPM:
V = $56.06[N(d1)] - $75e-(0.06)(3)[N(d2)].
d1 =
ln( $56.06/$75 )  [( 0.06  0.047/2 )]( 3)
0.5 0.5
(0.047 ) (3)
= -0.1085.
d2 = d1 - (0.047)0.5(3)0.5 = -.1085 - 0.3755
= -0.4840.
N(d1) = N(-0.1080) = 0.4568.
N(d2) = N(-0.4835) = 0.3142.
Therefore,
V = $56.06(0.4568) - $75e-(0.06)(3)(0.3142)
= $5.92.
Total Value = NPV Of Project 1 + Value Of Growth Option
=-$0.39 + $5.92
= $5.5 million
Mini Case: 12 - 10
j.
What happens to the value of the growth option if the variance of the project’s
return is 14.2 percent? What if it is 50 percent? How might this explain the
high valuations of many dot.com companies?
Answer: If risk, defined by σ2, goes up, then value of growth option goes up (see the file ch 12
mini case.xls for calculations):
σ2 = 4.7%, option value = $5.92
σ2 = 14.2%, option value = $12.10
σ2 = 50%, option value = $24.09
If the future profitability of dot.com companies is very volatile (i.e., there is the
potential for very high profits), then a company with a real option on those profits
might have a very high value for its growth option.
Mini Case: 12- 11