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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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