Answers and Solutions: 26 – 11
Then calculate the coefficient of variation:
CV = 17,004/22,562 = 0.7537.
σ2 = ln(CV2+1)/t = ln(0.75372+1)/2 = 0.2249
X = $20,000
t = 2.
rRF = 0.06.
σ2 = 0.2025
N(d1) = 0.6542
N(d2) = 0.4053
Using the Black-Scholes Option Pricing Model, you calculate the option’s value as:
V = P[N(d1)] –
trRF
Xe
[N(d2)]
= $18.646(0.6542) – $20e(-0.06)(2)(0.4053)
= $12.198 – $7.189
= $5.009 thousand.
Answers and Solutions: 26 – 12
Mini Case: 26 – 13
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
a. Expansion of existing product line
b. New products
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.
Mini Case: 26 – 14
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%
High 30% $45
Average 40% $30
Low 30% $15
Find NPV of each scenario:
PV Low: N=3 I=10 PV=? PMT=-15 FV=0
PV= 37.30
NPV Low = $37.30 – $70 = -$32.70 Million.
Find Expected NPV:
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.
shifted ahead by a year.
Mini Case: 26 – 16
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
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
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
T = Time To Maturity = 1 year.
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
“price” of the real option is the present value of all the project’s cash flows that occur
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
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.
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.
Mini Case: 26 – 18
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website, in whole or in part.
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
Current Value Value At Expiration
Year 0 Year 1
High $67.82 $111.91
Average $67.82 $74.61
Low $67.82 $37.30
Expected Return = 0.3(0.65) + 0.4(0.10) + 0.3(-0.45)
= 10%.
The direct approach gives an estimate of 0.182 for the variance of the project’s return.