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26-8 P = PV as of time zero of all expected future cash flows if the project is repeated starting
in year 2. Note it includes both the good cash flows and the bad cash flows since as of
now, we don’t know which outcome will result, and P excludes the $20,000 investment in
the franchise.
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The strike price, X, is the cost to extend the franchise at the end of year 2, and is $20,000.
The time to expiration is the time you decide whether or not to extend the franchise, and
is at the end of year 2.
Although the problem stated to assume the variance of the project’s rate of return was
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Answers and Solutions: 26 - 1
© 2017 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible
website, in whole or in part.
r = 10%
The expected value of these two returns is 52.54(0.40) – 31.78(0.60) = 1.95% [Note: this
The variance is
To calculate the variance of the project’s returns using the indirect method, first calculate
the standard deviation of the value at year 2. The value is either 43,388 (probability
Then calculate the coefficient of variation:
Notice that in this case the direct method and the indirect method give very similar results
for σ.
P = $18,646
From Excel function NORMSDIST, or approximated from the table in Appendix A:
Using the Black-Scholes Option Pricing Model, you calculate the option’s value as:
Answers and Solutions: 26 - 2
© 2017 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible
website, in whole or in part.
V = P[N(d1)] -
tr
RF
Xe
[N(d2)]
SOLUTION TO SPREADSHEET PROBLEMS
26-9 The detailed solution for the problem is available in the file, Ch 26 P9 Build a Model
Solution.xlsx, on the textbook’s Web site.
Answers and Solutions: 26 - 3
© 2017 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible
website, in whole or in part.
MINI CASE
Assume that you have just been hired as a financial analyst by Tropical Sweets Inc., a
mid-sized 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
3. Abandonment options
b. What are five possible procedures for analyzing a real option?
Answer: 1. DCF analysis of expected cash flows, ignoring option.
2. Qualitatively assess the value of the real option.
Answers and Solutions: 26 - 4
© 2017 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible
website, in whole or in part.
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
Alternatively, one could calculate the NPV of each scenario:
Demand Probability Annual Cash Flow
High 30% $45
Find Expected NPV:
Answers and Solutions: 26 - 5
© 2017 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible
website, in whole or in part.
d. 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.
Answer: 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
This project is risky and has one year before we must decide, so the option to wait
is probably valuable.
Answers and Solutions: 26 - 6
© 2017 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible
website, in whole or in part.
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:
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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.
Since this is much greater than the NPV of immediate implementation (which is
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 a strike price of $70 million and an expiration date of
one year.
Answers and Solutions: 26 - 7
© 2017 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible
website, in whole or in part.
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
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.
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The current expected present value, P, is:
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.
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
Answers and Solutions: 26 - 8
© 2017 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible
website, in whole or in part.
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:
Current Value Value At Expiration
Year 0 Year 1
The annual rate of return is:
The direct approach gives an estimate of 0.182 for the variance of the project’s return.
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
Answers and Solutions: 26 - 9
© 2017 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible
website, in whole or in part.
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.
Value At Expiration
Year 1
Coefficient Of Variation = CV = Expected Value / value
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:
Now, we proceed to use the OPM:
therefore,
Answers and Solutions: 26 - 10
© 2017 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible
website, in whole or in part.
