The original Seven Essays, in a cool, new outfit ...
Email - [email protected]

All posts in General Questions

Please help with the following problem. Include calculations.

Suppose that the stock now sells at $80, and the price will go up by 5% or down by 5% at the end of first six month (t = ½). Then, the price will either go up by 10% or down by 10% at the end of year (t = 1). A call option on the stock has an exercise price of $75 and a time to expiration of one year. Also, assume 10% annual interest rate and no dividend payment for this year. Calculate the put price at t=0.

Use a two stage binominal model.

Dude Ranches Incorporated was founded on the idea that many families in the eastern and southern areas of the United States do not have a sufficient amount of vacation time to drive to the dude ranches in the Southwest and Rocky Mountain areas for their vacations. Various surveys indicated, however, that there was a considerable interest in this type of family vacation, which includes horseback riding, cattle drives, swimming, fishing, and the like. Dude Ranches Incorporated bought a large farm near several eastern cities and constructed a lake, a swimming pool, and other facilities.

However, to build a number of family cottages on the ranch would have required a considerable investment. Furthermore, they reasoned that most of this investment would be lost should the ranch-farm complex be a financial failure. Instead, they decided to enter into an agreement with the Mobile Homes Manufacturing Company to supply a very attractive authentic ranch-type mobile home. Mobile Homes agreed to deliver a mobile home on Saturday for $300 a week.

Mobile Homes must know early Saturday morning how many mobile homes Dude Ranches Incorporated wants for the forthcoming week. They have other customers to supply and can only deliver the homes on Saturday. This presents a problem. Dude Ranches will have some reservations by Saturday, but indications are that many families do not make them. Instead, they prefer to examine the facilities before making a decision. An analysis of the various costs involved indicated that $350 a week should be charged for a ranch home, including all privileges. The basic problem is how many mobile ranch homes to order from Mobile Homes each week.

Should Dude Ranches Incorporated order 10 (considered the minimum), 11, 12, 13, or 14 (considered the maximum)?

Any decision made solely on the information in the payoff table would ignore. However, the valuable experience that Dude Ranches Incorporated has acquired in the past four years (about 200 weeks) actually operating a dude ranch in the Southwest. Their records showed that they always had nine advance reservations. Also, they never had a demand for 15 or more cottages. The occupancy of 10, 11, 12, 13, or 14 ranch cottages, in part, represented families who drove in and inspected the facilities before renting. A frequency distribution showing the number of weeks in which 10, 11 -14 ranch cottages were rented during the 200-week period is found in the following table.
Number of Cottages Number of Rented Weeks
10 26
11 50
12 60
13 44
14 20

a. Construct a payoff table.
b. Determine the expected payoffs, and arrive at a decision.
c. Set up an opportunity loss table.
d. Compute the expected opportunity losses, and arrive at a decision.
e. Determine the value of perfect information.

Cooper Realty is a small real estate company located in Albany, New York, specializing primarily in residential listings. They recently became interested in
determining the liklihood of one of their listings being sold within a certain number of days. An analysis of company sales of 800 homes in previous years produced
the following data.

Days Listed Until Sold
Under 30 31-90 Over 90 Total
Under $150,000 50 40 10 100
Initial Asking Price $150,000 – $199,000 20 150 80 250
$200,000 – $250,000 20 280 100 400
Over $250,000 10 30 10 50
Total 100 500 200 800

a. If A is defined as the event that a home is listed for more than 90 days before being sold, estimate the probability of A.

b. If B is defined as the event that the initial asking price is under $150,000, estimate the probability of B.

c. What is the probability of A (upside down U meaning “intersecting” B?
P(A) + P(B) – P(A upside down u B )

d. Assuming that a contract was just signed to list a home with an initial asking price of less than $150,000, what is the probability that the home will take Cooper
Realty more than 90 days to sell?

e. Are events A and B independent?

In your opinion, the best outcome would be to have the communications contract awarded to your company but to have the construction contract awarded to a different company. This is not because you are fond of your competition, but instead because if you were to win both, the result would place some strain on your (relatively) small firm’s resources. Of course, if the construction contract were the only one you received, it would be good for the firm. You have placed subjective probabilities of 0.60 and 0.45 on the communications and construction contracts respectively, and believe that these events may be considered statistically independent for the purpose of these calculations.

—Find the probability that the result is your best outcome.

—Given that you are awarded at least one contract, find the conditional probability that (in particular) you are awarded the communications contract. This may not be as straightforward as it might appear.

It is difficult to make budgetary projections in the current economic climate. In particular, a decision regarding long-range planning will be made, in part, using the likelihood that the current recession will last less than 1.5 years. To estimate this probability, information from all previous recessions in the United States has been gathered. Of the 18 previous recessions, exactly 6 lasted less than 1.5 years. Assuming that the current recession is an observation from the same random experiment that produced the previous recessions (while admitting that this assumption is probably not strictly true) please produce a reasonable estimate of the probability that the current recession will last less than 1.5 years and justify the estimate.



1. A recent survey conducted by Towers Perrin and published in the Financial Times showed that among 460 organizations in 13 European countries, 93% have bonus plans, 55% have cafeteria-style benefits, and 70% employ home-based workers. If the types of benefits are independent, what is the probability that an organization selected at random will have at least one of the three types of benefits?
2. In a takeover bid for a certain company, management of the raiding firm believes that the takeover has a 0.65 probability of success if a member of the board of the raided firm resigns, and a 0.30 chance of success if she does not resign. Management of the raiding firm further believes that the chances for a resignation of the member in question are 0.70. What is the probability of a successful takeover?

1. The graph shows the results of a survey of drivers who were asked to name the most annoying habit of other drivers. You randomly select six people who participated in the survey and ask each one of them to name the most annoying habit of other drivers. Let x represent the number who named talking on cell phones as the most annoying habit. Compute parts (a) through (c)
a) construct a binomial distribution
Find the probability of each value of x to construct the binomial distribution.
b) Find the probability that exactly two people will name ” talking on cell phones”

2. 46% of adults say cashews are their favorite kind of nut. You randomly select 12 adults and ask each to name his or her favorite nut. Find the probability that the number who say cashews are their favorite nut is (a) exactly three, (b) at least four, and (c) at most two.
6. A company gave psychological tests to prospective employees. The random variable x represents the possible test scores. Use the histogram to find the probabilities in parts (a) and (b).