1. The three parameters required to specify a triangular distribution are the minimum, mean and maximum.
a. True
b. False
2. The binomial distribution is a discrete distribution that is applied to situations where n i
...
1. The three parameters required to specify a triangular distribution are the minimum, mean and maximum.
a. True
b. False
2. The binomial distribution is a discrete distribution that is applied to situations where n independent and
identical “trials” occur, with each trial resulting in a “success” or “failure,” and we want to generate the
random number of successes in the n trials.
a. True
b. False
3. The binomial distribution can be well approximated by the normal distribution when the number of trials n is
sufficiently small and the probability of success p is not too close to 0 or 1.
a. True
b. False
alse
4. When you try to find the most appropriate input probability distribution in a simulation model, you first have
to choose the most appropriate family, and then you have to select the most appropriate member of that
family
a. True
b. False
5. The flaw of averages is the reason deterministic models can be very misleading.
a. True
b. False
A company is about to develop and then market a new product. It wants to build a simulation model for the
entire process, and one key uncertain input is the development time, which is measured in an integer
number of months. For each of the scenarios in the questions below, choose an “appropriate” distribution,
together with its parameters, and explain your choice.
6. Company experts believe the development time will be from 5 to 9 months. They believe that 7 months is
twice as likely as either 6 months or 8 months and that either of these latter possibilities is three times as
likely as either 5 months or 9 months.
ANSWER: This is another general discrete distribution, where we have to choose the probabilities so that
they have the specified ratios, and add up to 1: P(5)=0.072, P(6)=0.215, P(7)=0.43,
P(8)=0.215, P(9)=0.072.
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7. The deterministic (non-simulation) approach, using best guesses for the uncertain inputs, is:
a. better to use in complicated real world applications.
b. a good estimate of what the answer will be using a simulation approach.
c. generally not the appropriate model.
d. the preferred approach when there is correlation between input variables.
8. Assume that x is a random number between 0 and 1, and that the number of units expected to be sold is
uniformly distributed between 300 and 500. Then, sales are given by the expression
a. 300 + x
b. 500 – x
c. 300 + 200 x
d. 500 – 200 x
e. 300 + 500 x
9. The “building blocks” of all spreadsheet simulation models are:
a. deterministic inputs
b. random numbers between 0 and 1
c. decision variables
d. probability distributions
10. Which of the following statements is correct regarding the graph of a discrete probability distribution?
a. It is a series of spikes.
b. The height of each spike is the probability of the corresponding value.
c. There is an empty space between adjacent spikes.
d. All of these options
11. Some important characteristics of probability distributions in general include the following distinctions:
a. Discrete versus continuous
b. Symmetric versus skewed
c. Bounded versus unbounded
d. Positive (or nonnegative) versus unrestricted
e. All of these options
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12. Which of the following statements are false regarding the graph of a continuous probability distribution?
a. It is characterized by a density function, a smooth curve.
b. It is a series of spikes
c. The height of the density curve above any point is not actually a probability—that is, it is not
necessarily between 0 and 1.
d. Heights above the density function indicate relative likelihoods but are not necessarily values
between 0 and 1.
13. Which of the following statements are true?
a. A probability distribution is symmetric (around some point) if the distribution to the left of the point is a
mirror image of the distribution to the right of the point.
b. We say a distribution is skewed to the right (or positively skewed) if the “longer tail” is the right tail.
c. We say a distribution is skewed to the left (or negatively skewed) if the “longer tail” is the left tail.
d. All of the above
14. A continuous probability distribution is characterized by a:
a. mean and a standard deviation.
b. mean and a variance.
c. density function.
d. histogram
15. We sometimes use discrete distributions in place of continuous distributions:
a. because they are more accurate.
b. because they are more simple.
c. when we don’t know the mean and variance of the distribution.
d. when we need to generate a histogram
16. A probability distribution is bounded if there are values A and B such that:
a. A and B represent the 95% confidence interval
b. A and B are the mean and standard deviation, respectively.
c. A and B are the mean and variance, respectively.
d. no value can be less than A or greater than B.
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17. One important special use of bounded distributions is when the only possible values are:
a. less than zero.
b. uniformly distributed around the mean.
c. skewed to the right.
d. nonnegative.
18. We can think of the uniform distribution as:
a. the “I have no idea” distribution.
b. a skewed distribution.
c. only modeling positive values.
d. nonnegative.
19. The RAND() function in excel models which of the following probability distributions:
a. Normal(0,1)
b. Uniform(0,1)
c. Normal(-1,1)
d. Uniform(-1,1).
20. Which of the following statements is (are) false regarding the numbers generated by the RAND function in
Excel?
a. Approximately 10% of the numbers will be between 0.0 and 1.0
b. Approximately 20% of the numbers will be between 0.50 and 0.70
c. Approximately 40% of the numbers will be between 0.20 and 0.60
d. Approximately 60% of the numbers will be between 0.15 and 0.75
e. All of these options are false
21. Which of the following statements are false regarding the numbers generated by the RAND function in
Excel?
a. The numbers are random between 0 and 1
b. The numbers are probabilistically dependent
c. The numbers are probabilistically independent
d. The numbers are uniformly distributed between 0 and 1
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22. In order to generate random numbers in Excel from a discrete distribution with a finite number of possible
values and corresponding probabilities, we can use
a. only the RAND function
b. only the VLOOKUP function
c. only the VLOOKDOWN function
d. the RAND function along with a VLOOKUP function
23. If we want to model the monthly return on a stock, we might choose
a. symmetric distribution around 0
b. positively skewed distribution
c. negatively skewed distribution
d. All of these options
24. Which of the following statements is true regarding the Normal distribution?
a. It is always the appropriate distribution in simulation modeling
b. It does not permit negative values
c. There is a 95% chance that values will be within ± 2 standard deviations of the mean
d. All of these options
25. Which of the following statements is true regarding the Triangular distribution?
a. It is a discrete distribution with a minimum, maximum and most likely value
b. It is more flexible and intuitive than the normal distribution
c. It is a symmetric distribution
d. All of these options
26. When n is reasonably large and p isn’t too close to 0 or 1, the binomial distribution can be well
approximated by which of the following distributions?
a. Uniform distribution
b. Normal distribution
c. Triangular distribution
d. None of these options
27. Excel’s built-in functions, along with the RAND function, can be used to generate random numbers from
many different types of probability distributions.
a. True
b. False
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28. The Excel RAND() function generates random numbers from a Normal(0,1) distribution.
a. True
b. False
alse
29. The “random” numbers generated by the RAND function (or by any other package’s random number
generator) are not really random.
a. True
b. False
30. Sometimes it is convenient to treat a discrete probability distribution as continuous, and vice versa.
a. True
b. False
31. A probability distribution is continuous if its possible values are essentially some continuum.
a. True
b. False
32. We typically choose between a symmetric and skewed distribution on the basis of practical modeling
issues.
a. True
b. False
33. A probability distribution is bounded if there are values A and B such that only one possible value can be
less than A or greater than B.
a. True
b. False
alse
34. The uniform distribution is bounded by a minimum and a maximum, and all values between these two
extremes are equally likely.
a. True
b. False
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35. The normal distribution is often used in simulation models because it is the most common distribution in
statistics and it does not allow negative values.
a. True
b. False
alse
36. The triangular distribution is sometimes used in simulation models because it is more flexible and intuitive
than the normal distribution.
a. True
b. False
37. Which of the following statements are false?
a. A probability distribution is discrete if it has a finite number of possible values.
b. A probability distribution is continuous if its possible values are essentially some continuum.
c. An example of a discrete probability distribution is the amount of rain that falls during the month of
June in Michigan.
d. None of these options
38. If we want to model the time it takes to serve a customer at a bank, we will probably choose
a. symmetric distribution
b. positively skewed distribution
c. negatively skewed distribution
d. All of these options
39. It is simple to generate a uniformly distributed random number with a minimum and maximum other than 0
and 1. For example, the formula =150+100 RAND() generates a number uniformly distributed between
a. 100 and 150
b. 150 and 250
c. 50 and 100
d. 50 and 250
40. If we want to model a random stock price, we should do so with an unbounded symmetric probability
distribution.
a. True
b. False
alse
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41. A discrete distribution is useful for many situations, either when the uncertain quantity is not really
continuous (the number of televisions demanded, for example) or when you want a discrete approximation
to a continuous variable.
a. True
b. False
Generate a set of 40 random numbers in a column in an Excel spreadsheet by using RAND function. Fix
the set of random numbers by copying the column to another column and using the “Paste Special”
command with the “Values” option selected.
42. (A) What fraction of the random numbers are smaller than 0.5?
(B) What fraction of the time is a random number less than 0.5 followed by another random number less
than 0.5?
(C) What fraction of the random numbers are larger than 0.8?
(D) What do you expect the answers to (A), (B) and (C) to be before simulating? Do the answers you
provided to those questions match your expectations? Explain why or why not.
(E) Suppose your answers to (A), (B) and (C) are not close to the expected answers. What can you do to
obtain answers from the simulation that are closer to the expected answers?
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(D) The results will vary from simulation to simulation, but we expect P(<0.5) to be about 50%
for both Questions 65 and 66 (since the random numbers are independent), and we expect
P(>0.8) to be 20%. The numbers above are not too far off from those expectations.
(E) We can increase the sample size (e.g., to 1000) so that we average more values into the
answers.
It is surprising (but true) that if 23 randomly selected people are in the same room, there is about a 50%
chance that at least two people will have the same birthday. Suppose you want to estimate the probability
that if 30 people are in the same room, at least two of them will have the same birthday. You can proceed
as follows:
43. (A) Generate the “birthdays” of 30 different people, assuming that each person has a 1/365 chance of
having a given birthday (call the days of the year 1, 2, 3, ……..,365). You can use a formula involving the
DISCRETE and RAND functions to generate birthdays.
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(B) Once you have generated 30 people’s birthdays, you can tell whether at least two people have the
same birthday using Excel’s RANK function (i.e., in the case of a tie, two numbers are given the same
rank). Do you see any people with the same birthday in your sample?
(C) Obtain at least 20 samples of the 30 person group using the F9 key. What do you estimate the
probability of finding two people with the same birthday in a sample of 30 people to be?
44. (A) Company experts have no idea what the distribution of the development cost is. All they can state is
that “we are 90% sure it will be somewhere between $450,000 and $650,000.”
(B) Company experts can still make the same two statements as in (A), but now they can also state that
“we believe the distribution is symmetric and its most likely value is about $550,000.”
(C) Company experts can still make the same two statements as in (A), but now they can also state that
“we believe the distribution is skewed to the right, and its most likely value is about $500,000.”
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45. If you add several normally distributed random numbers, the result is normally distributed, where the mean
of the sum is the sum of the individual means, and the variance of the sum is the sum of the individual
variances. This result is difficult to prove mathematically, but it is easy to demonstrate with simulation. To
do so, run a simulation where you add three normally distributed random numbers, each with mean 100
and standard deviation 10. Your single output variable should be the sum of these three numbers. Verify
with @RISK that the distribution of this output is approximately normal with mean 300 and variance 300
(hence, standard deviation = 17.32).
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In August 2009, a car dealer is trying to determine how many 2010 cars to order. Each car ordered in
August 2009 costs $16,000. The demand for the dealer’s 2010 models has the probability distribution
shown in the table below. Each car sells for $21,000. If the demand for 2010 cars exceeds the number of
cars ordered in August 2009, the dealer must reorder at a cost of $18,000 per car. Excess cars can be
disposed of at $13,000 per car.
Cars demanded Probability
25 0.25
30 0.20
35 0.15
40 0.20
45 0.20
47. (A) Use simulation to determine how many cars the dealer should order in August, 2009 to maximize his
expected profit.
(B) For the optimal order quantity, find a 95% confidence interval for the expected profit.
(C) Why is it important to develop the confidence interval in (B)?
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48. Suppose that the demand for cars is normally distributed with mean of 120 and standard deviation of 20.
Use @Risk simulation add-in to determine the “best” order quantity; that is, the one that has the largest
expected profit. Using the statistics and/or graphs from @Risk, discuss whether this order quantity would
not be considered the “best” by the car dealer.
49. Obtain another set random numbers by pressing the F9 (recalculate) key. Do your results change
significantly? Do the changes match your expectations? Explain your answer.
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A company is about to develop and then market a new product. It wants to build a simulation model for the
entire process, and one key uncertain input is the development time, which is measured in an integer
number of months. For each of the scenarios in the questions below, choose an “appropriate” distribution,
together with its parameters, and explain your choice.
50. Company experts believe the development time will from 5 to 9 months. They believe the probabilities of
the extremes (5 and 9 months) are both 10%, and the probabilities will vary linearly from those endpoints to
a most likely value at 7 months.
ANSWER: This is a general discrete distribution. We just have to choose the probabilities of the values 5
to 9 so that they increase and then decrease linearly, and add up to 1: P(5)=0.1, P(6)=0.225,
P(7)=0.35, P(8)=0.225, P(9)=0.1.
51. If a model contains uncertain outputs, it can be very misleading to build a deterministic model by using the
means of the inputs to predict an output. This is called the:
a. Law of Large Numbers.
b. Flaw of Averages
c. Law of Inevitable Disappointment
d. Central Limit Theorem
52. @Risk introduces uncertainty explicitly into a spreadsheet model by allowing several inputs to have
probability distributions and then enabling the simulation of random values from these inputs.
a. True
b. False
53. Which of the following statements is true regarding a simulation model?
a. It explicitly models decision-making under uncertainty
b. It explicitly incorporates uncertainty in one or more input variables
c. It provides probability distributions for all outputs, rather than expected values
d. All of these options
54. Each different set of values obtained for the uncertain quantities in a simulation model can considered to
be:
a. the mean of the probability distribution.
b. a scenario.
c. a best guess.
d. all of these options.
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55. Simulation models are particularly useful for:
a. forecasting.
b. obtaining deterministic outputs.
c. evaluating constraints.
d. asking what-if questions.
56. One of the primary advantages of simulation models that they enable managers to answer what-if
questions about changes in systems without actually changing the systems themselves.
a. True
b. False
57. Spreadsheet simulation modeling is quite similar to the other modeling applications in that it begins with
input variables and then relates these with appropriate Excel formulas to produce output variables of
interest.
a. True
b. False
58. A primary difference between standard spreadsheet models and simulation models is that at least one of
the input variable cells in a simulation model contains random numbers.
a. True
b. False
59. Many companies have used simulation to determine which of several possible investment projects they
should choose. This is often referred to as
a. risk analysis
b. @RISK investment
c. unbounded risk
d. None of the above
60. Suppose you run a simulation model several times with different order quantities. What can we infer about
the quantity that maximizes the output, the company’s profit?
a. This quantity is the optimal order quantity.
b. This quantity might be the optimal order quantity, but we also need to consider the company’s
attitude toward risk.
c. This is not necessarily the optimal order quantity, because it may have produced the largest profit by
luck.
d. We can’t infer anything.
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61. Which of the following statements are true?
a. The @RISK contains a number of functions such as RISKNORMAL and RISKDISCRETE that make
it easy to generate observations from the most important probability distributions.
b. You can specify any cell or range of cells in your simulation model as output cells.
When you run the simulation, @RISK automatically keeps summary measures (averages, standard
deviation, percentiles, and others) from the values generated in these output cells across the
replications.
c. @RISK has a special function, RISKSIMTABLE, which allows you to run the same simulation
several times, using a different value of some key input variable each time.
d. All of these options.
62. RISKSIMTABLE is a function in @Risk for running several simulations simultaneously, one for each setting
of an input or decision variable.
a. True
b. False
63. When we maximize or minimize the value of a decision variable by running several simulations
simultaneously, we have found an optimal solution to the problem and attitude toward risk becomes
irrelevant.
a. True
b. False
alse
64. Different random numbers generated by the computer are probabilistically dependent. This implies that
when we generate a random number in a particular cell, it has some effect on the values of any other
random numbers generated in the spreadsheet.
a. True
b. False
alse
65. A correlation matrix must always have 1’s along its diagonal (because a variable is always perfectly
correlated with itself) and the correlations between variables elsewhere.
a. True
b. False
66. A correlation matrix must always be symmetric, so that the correlations above the diagonal are a mirror
image of those below it.
a. True
b. False
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67. Correlation between two random input variables might not change the mean of an output, but it can
definitely affect the variability and shape of an output disbribution.
a. True
b. False
68. It is usually not too difficult to predict the shape of the output distribution from the shape(s) of the input
distribution(s).
a. True
b. False
alse
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After Michigan State University reached the final four in the 2000 NCAA Basketball Tournament, a
sweatshirt supplier in Lansing is trying to decide how many sweatshirts to print for the upcoming
championships. The final four teams (Michigan State, Florida, Wisconsin, and North Carolina) have
emerged from the quarterfinal round, and there is a week left until the semifinals, which are then followed in
a couple of days by the finals. Each sweatshirt costs $12 to produce and sells for $24. However, in three
weeks, any leftover sweatshirts will be put on sale for half price, $12. The supplier assumes that the
demand (in thousands) for his sweatshirts during the next three weeks, when interest is at its highest,
follows the probability distribution shown in the table below. The residual demand, after the sweatshirts
have been put on sale, also has the probability distribution shown in the table below. The supplier realizes
that every sweatshirt sold, even at the sale price, yields a profit. However, he also realizes that any
sweatshirts produced but not sold must be thrown away, resulting in a $12 loss per sweatshirt.
Demand distribution at regular price Demand distribution at reduced price
Demand Probability Demand Probability
7 0.05 2 0.20
8 0.10 3 0.30
9 0.25 4 0.20
10 0.30 5 0.15
11 0.20 6 0.10
12 0.10 7 0.05
69. Use simulation to analyze the supplier’s problem. Determine how many sweatshirts he should produce to
maximize the expected profit.
ata table for average profit as function of order quantity
Order quantity Average profit
$115,680
11000 $115,440
11500 $115,680
12000 $118,080
12500 $110,520
13000 $116,160
13500 $107,520
14000 $106,800
Note that an order quantity of 12000 appears to maximize the average profit. Its average profit
of $118,080 is than the average profits from other order quantities. However, again keep in
mind that this is a simulation, so it is conceivable that some other order quantity could be best.
70. Use @Risk simulation add-in to analyze the sweatshirt sales. Do this for the discrete distributions given in
the problem.
ANSWER:
Single simulation replication
Summary measures of profit from @RISK
Number of replications 100
Smallest profit $90,000
Largest profit $180,625
Average profit $154,750
Std .Deviation of profit $18,750
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71. Use @Risk simulation add-in to analyze the sweatshirt sales. Do this for normal distributions, where we
assume that the regular demand is normally distributed with mean 10,000 and standard deviation 1500,
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A company produces six-packs of soda cans. Each can is supposed to contain at least 12 ounces of
soda. If the total weight in a six-pack is under 72 ounces, the company is fined $100 and receives no sales
revenue for the six-pack. Each six-pack sells for $3.00. It costs the company $0.02 per ounce of soda put
in the cans. The company can control the mean fill rate of its soda-filling machines. The amount put in
each can by a machine is normally distributed with standard deviation 0.10 ounce.
72. (A) Assume that the weight of each can in a six-pack has a 0,8 correlation with the weight of the other cans
in the six-pack. What mean fill quantity (within 0.05 ounce) maximizes expected profit per sic-pack?
(B) If the weights of the cans in the six-pack are probabilistically independent, what mean fill quantity (within
0.05 ounce) will maximize expected profit per six-pack?
(C) How can you explain the difference in the answers for (A) and (B)?
73. (A) The three stock returns are highly correlated. The correlation between each pair is 0.9
(B) The three stock returns are practically independent. The correlation between each pair is 0.1
(C) The first two stocks are moderately correlated. The correlation between their returns is 0.4. The third
stock’s return is negatively correlated with the other two. The correlation between its return and each of the
first two is -0.8.
(D) Compare the portfolio distributions from @RISK for the three scenarios in (A), (B) and (C). What do
you conclude?
(E) You might think of a fourth scenario, where the correlation between each pair of returns is a large
negative number such as -0.80. But explain intuitively why this makes no sense. Try running a simulation
with these negative correlations to see what happens.
74. (A) Use a simulation model to help the institute decide how many violins they must reserve with the
instrument company. Consider five different possible reservation quantities: 400, 500, 600, 700, 800.
Which of these quantities yields the highest total revenue, net of instrument costs?
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(B) Which simulation yields the largest median total revenue?
(C) Which simulation has the most risk as measured by spread or dispersion in the data? Please state
clearly what statistic you used to answer this question.
(D) Are there any simulations in which there is at least a 1 in 20 (i.e., 5%) chance of getting a negative total
revenue? Briefly explain in one sentence.
(E) For each simulation what is the probability of exceeding $175,000 in total revenue (approximate these
numbers as closely as possible from the data given in the above table). Please put your answer in the
following table:
Simulation 1 Simulation 2 Simulation 3 Simulation 4 Simulation 5
(F) Considering your answers for (A) through (E), please state how many instruments you think should be
reserved in advance and explain why.
(G) Suppose the institute is able to negotiate with the instrument company to reduce the cost for a violin
from $500 to $350. Re-run the simulation model using the same reservation quantities (but with $350 for
the unused instrument cost). Has the reservation quantity that yields the highest average revenue
changed? If so, please explain why this has occurred.
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Estimate of Primary Recovery
Minimum Value: 20 bbl./acre-ft
Maximum Value: 90 bbl./acre-ft
The amount of reserves that can be produced is then the product of the area, thickness, and recovery
factor:
Number of barrels = Productive Area x Pay Thickness x Primary Recovery Factor
75. (A) Use @RISK distributions to generate the three random variables and derive a distribution for the
amount of reserves. What is the amount we can expect to recover from this field?
(B) The production output is a product of three very different types of input distributions. What does the
output distribution look like? What are the implications of the shape of this distribution?
(C) What is the standard deviation of the recoverable reserves? What are the 5th and 95th percentiles of
this distribution? What does this imply about the uncertainty in estimating the amount of recoverable
reserves?
(D) Suppose you think oil price is normally distributed with a mean of $65 per barrel and a standard
deviation of $10. How much revenue do you expect the field produce (ignore discounting)?
(E) Finally, your engineer is uncertain about costs to drill wells to develop the field, but she thinks the most
likely cost will be $1.7Bn, although it could be as much as $3Bn or as little as $1Bn. What is your expected
profit from the field?
(F) What is the chance that you will lose money? Is this a risky venture?
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76. When we run simulation, the @Risk automatically keeps statistics such as averages and standard
deviations, and can also create graphs such as histograms based on the values generated in the output
cells in the simulation model.
a. True
b. False
77. It is common in computer simulations to estimate the mean of some distribution by the average of the
simulated observations. The usual practice is then to accompany this estimate with a confidence interval,
which indicates the accuracy of the estimate.
a. True
b. False
78. A common guideline in constructing confidence intervals for the mean is to place upper and lower bounds
one standard error on either side of the average to obtain an approximate 95% confidence interval.
a. True
b. False
alse
79. Analysts often plan a simulation so that the confidence interval for the mean of some important output will
be sufficiently narrow. The reasoning is that narrow confidence intervals imply more precision about the
estimated mean of the output variable.
a. True
b. False
80. Data tables in spreadsheet simulations are useful for taking a “prototype” simulation and replicating its key
results a desired number of times.
a. True
b. False
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The College of Arts and Sciences at a Midwestern university currently has three parking lots, each
containing 160 spaces. Two hundred faculty members have been assigned to each lot. On a peak day, the
probability of a lot 1 parking sticker holder showing up is 73%, a lot 2 parking sticker holders showing up is
75%, and a lot 3 parking sticker holder showing up is 77%.
81. (A) What are the appropriate probability distributions to model the number of faculty members showing up
in each lot?
(B) Given the current situation, estimate the probability that on a peak day, at least one faculty member with
a sticker will be unable to find a parking space. Assume that the number who shows up at each lot is
independent of the number who shows up at the other two lots.
(C) Suppose that faculty members are allowed to park in any lot. Does this help solve the problem? Why
or why not?
(D) Suppose that the numbers of faculty who show up at the three lots are correlated, with each correlation
equal to 0.80. Does your answer to (C) change? Why or why not?
Name: Class: Date:
Chapter 15
82. (A) Use @Risk with 100 replications, provide a summary statistics of portfolio return; namely, minimum,
maximum, mean, and standard deviation.
(B) Use your answers to (A) to estimate the probability that Mrs. Smart’s portfolio’s annual return will
exceed 20%.
(C) Use your answers to (A) to estimate the probability that Mrs. Smart’s portfolio will lose money during
the course of a year.
(D) Suppose that the current price of each stock is as follows: stock 1: $16; stock 2: $18; stock 3: $20; and
stock 4: $22. Mrs. Smart has just bought an option involving these four stocks. If the price of stock 1, six
months from now are is $18 or more, the option enables Mrs. Smart to buy, if she desires, one share of
each stock for $20 six months from now. Otherwise the option is worthless. For example, if the stock
prices six months from now are: stock 1: $18; stock 2: $20; stock 3: $21; and stock 4: $24, then Mrs.
Smart would exercise her option to buy stocks 3 and 4 and receive (21- 20) + (24-20) = $5 in each cash
flow. How much is this option worth if the risk-free rate is 8%?
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