# Homework 4

Turn in all of the following homework problems on Thurs Oct 28 (for Foster's section, Monday Nov 1st).

1. As personnel manager for Computeronics Inc. you may receive many applications from people who want jobs as systems programmers. You first have to decide whether or not an applicant should go through a thorough screening test which costs the company \$100. There are two reasons why an applicant should not be given the screening test: you are confident that he/she is qualified, in which case you will hire the applicant, or you are confident that he/she is unqualified, in which case you will not hire the applicant. If you hire an applicant, and he/she is qualified it is worth \$800 to the company. On the other hand, if an applicant is unqualified and you hire the applicant the company estimates their loss to be \$300 for low quality work. Finally, they gain/lose nothing if they do not hire the applicant.

From past experience, the test accuracy is summarized as follows:

• if an applicant is qualified, the probability that an applicant will pass the test is .9
• if an applicant is unqualified, the probability that an applicant will fail the test is .9

An applicant sends his resume to you and you feel that there is a .5 chance that he is qualified.

1. Set up a decision tree starting with the decision as to whether or not the applicant should be given the screen test.

2. Should you give the applicant the screen test? What is the best strategy?

3. What is the expected value of prefect information prior to the screen test?

4. How much is the screen test worth?

5. What other factors should you consider that are not included in the above analysis?

2. Do the following problems:

1. Text exercise 4.50 (page 153)
2. Text exercise 4.51 (page 154)
3. Text exercise 4.52 (page 154)
4. Text exercise 4.56 (page 161)
5. Text exercise 4.57 (page 161)
6. Text exercise 4.58 (page 161)
7. Text exercise 4.59 (page 161)

3. Do the following problems:

1. Text exercise 4.104 (page 176)
2. Text exercise 4.105 (page 177)
3. What are the mean and standard deviation for the total number of repairs made for number of motor/transmission repairs and number of body repairs?

4. Here are summary statistics on monthly returns for five US common stocks, based on 60 observations from January, 1991, through December, 1995:

Company Bristol Myers Ford IBM Merck USAir
mean 0.017 0.019 0.0025 0.0088 0.0097
Std. Dev. .062 0.077 0.081 0.050 0.16

Here is the correlations matrix:

Company Bristol Myers Ford IBM Merck USAir
Bristol Myers 1.0 0.12 0.02 0.09 0.20
Ford 0.12 1.0 0.27 0.04 -0.01
IBM 0.02 0.27 1.0 0.66 0.22
Merck 0.09 0.04 0.66 1.0 0.21
USAir 0.20 -0.01 0.22 0.21 1.0

Assume that these summary figures will still apply to future investments. Also assume that you may not make negative investments (such as short sales) in any of the stocks.

1. What portfolio (expressed as fractions of a dollar invested in each of the five stocks) would yield the highest expected return? (Hint: This is easy.)

2. What stock is the riskiest? Is it true that a portfolio with half the money invested in this stock and half invested in one of the other stocks will be riskier than a portfolio with all the money invested in the other stock?

3. What pair of stocks have the highest covariance (not necessarily the highest correlation)? What pair of stocks have the lowest covariance?

4. Consider two portfolios, one with equal amounts invested in the two highest-covariance stocks, the other with equal amounts invested in the two lowest-covariance stocks. Find the expected returns and standard deviations (per dollar invested) in each portfolio.