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MGT 201 - Quantitative Analysis
Homework 4
1. You are trying to develop a strategy for investing in two different stocks. The anticipated
annual return for a $1,000 investment in each stock under four different economic conditions
has the following probability distribution:
Returns
Probability Economic Condition Stock X Stock Y
0.1 Recession -50 -100
0.3 Slow growth 20 50
0.4 Moderate growth 100 130
0.2 Fast growth 150 200
Compute the
(a) expected return for stock X and for stock Y.
(b) standard deviation for stock X and for stock Y.
(c) covariance of stock X and stock Y.
(d) Would you invest in stock X or stock Y? Explain.
2. Suppose that in the above problem you wanted to create a portfolio that consists of stock
X and stock Y. Compute the portfolio expected return and portfolio risk for each of the
following percentages invested in stock X:
(a) 30%.
(b) 50%.
(c) 70%.
(d) On the basis of the results of (a) through (c), which portfolio would you recommend?
Explain.
3. The Gourmet Cafe serves the exotic Random Salmon at lunch and dinner. The number of
customers ordering the salmon at lunch and dinner daily are given by the following distributions:
Lunch demand (X) 3 5 9
Probability 0.3 0.5 0.2
Dinner demand (Y ) 6 10 13
Probability 0.2 0.4 0.4
Assume the daily lunch and dinner demands are independent of each other. The chef orders
the fish in advance at a cost of $7.50 per serving. Any fish left over at the end of the day is
discarded.
(a) Give the joint distribution of X and Y .
(b) What is the expected total daily demand for the salmon?
(c) Suppose the chef orders 16 servings. What is the breakeven selling price (i.e., the price
at which the expected revenue from sales of the fish equals the cost of the fish ordered)?
Assume that a customer who would have ordered the fish but finds it sold out simply
leaves rather than order something else. (Hint: Find the probability distribution of the
number of units sold and observe that revenue = selling price times number of units
sold.)
4. Suppose you borrow $1000 for one year at a variable interest rate tied to the yield on government
bonds. As a result, the total interest you will pay is a random variable X1, having
mean $60 and standard deviation $2. You invest the borrowed money. Your earnings on the
investment, X2, have mean $85 and standard deviation $8. Suppose the correlation between
your earnings and the interest you pay on the loan is 0.3.
(a) Your net earnings at the end of the year are given by Y = X2 − X1. Find the expected
value of your net earnings.
(b) Find the standard deviation of your net earnings.
5. Let X, Y , and Z be random variables. Which one(s) of the following statements is/are true?
(a) if Cov(X, Y ) > Cov(Y, Z) then ρXY > ρY Z.
(b) if ρXY > ρY Z then σX < σZ.
(c) if ρXY > 0 then Cov(X, Y ) > 0.
6. Gizmo and Co. is a major player in the nationwide market for Gizmos. They manufacture
14,000 Gizmos every year. Their number of defects per year is normally distributed with
mean 3,000 units and standard deviation 500.
(a) What is the probability that the number of defects during 2013 will exceed 2,000 units?
(b) What is the probability that the 2013 non-defective production of Gizmos will be between
10,500 and 12,000? (The 2013 production is the number of non-defective Gizmos
produced during 2013.)
(c) Operations at Gizmo and Co. claims that their company can meet their total demand
for Gizmos with probability 95%. How large can their yearly demand be in order for
that statement to be correct?
7. From a study of its catalog customer base, the J. Uris clothing company has determined that
catalog orders placed by men are normally distributed with a mean of $62 and a standard
deviation of $12, whereas those placed by women are normally distributed with a mean of
$84 and a standard deviation of $16. During March and April, J. Uris wants to send its
special swimsuit catalog to women placing new orders. Rather than try to determine whether
an individual order was placed by a man or a woman, the company will simply send the
swimsuit catalog to all customers ordering more than x dollars of merchandise, with x to be
determined.
(a) Find the value of x that will ensure that 80% of women placing orders will be sent the
swimsuit catalog. (Hint: consider a woman who has placed a new order; the probability
that she is sent the catalog should be 80%.)
(b) If your answer to part (a) is adopted, what proportion of men placing orders will also
receive the swimsuit catalog?
8. An important subcomponent of a medical device produced by a manufacturing line is normally
distributed with mean 70.2” and standard deviation is 1.26”.
(a) What is the probability that one randomly selected unit has a length exactly 72”?
(b) What is the probability that one randomly selected unit has a length less than 72”?
(c) What is the probability that one randomly selected unit has a length greater than 72”?
(d) What is the probability that, if three units are randomly selected, none of them have
lengths exceeding 72”?

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