Dr. V.R. Bencivenga April 11, 2012 Economics 329} Spring 2012 MIDTERM EXAM #2 Instructions: Answer the questions below in a blue book. Show your work to receive credit. The exam has nine problems worth 350 points, plus an extra credit question. This is an open book, open notes exam. You have two hours for the exam. Good luck! (80 points) 1. The joint distribution of rates of return on a bond mutual fund and a stock mutual fund (in percentage points) is below. For example, a percentage rate of return of 3 on the bond fund means that for each dollar you put in, you get a 3% return, or $.03 (in addition to your original dollar). Bond mutual fund 1 3 5 1 .05 .1 .15 Stock mutual fund 4 .1 .2 .1 7 .15 .1 .05 a. What is the probability the bond fund will have a rate of return exceeding 3 percentage points? What is the probability the stock fund will have a rate of return exceeding 4 percentage points? b. What is the probability the bond fund will have a rate of return exceeding 3 percentage points and the stock fund will have a rate of return exceeding 4 percentage points? What is the probability the bond fund will have a rate of return exceeding 3 percentage points or the stock fund will have a rate of return exceeding 4 percentage points? c. Are rates of return on the bond fund and stock fund independent? Explain why or why not. d. Calculate the expected rate of return (mean rate of return) on the bond fund, and the variance of the rate of return on the bond fund. Also calculate the expected rate of return on the stock fund, and the variance of the rate of return on the stock fund. e. Give the expected rate of return (mean rate of return) on the stock fund given that the rate of return on the bond fund is 5. Give the variance of the rate of return on the stock fund given that the rate of return on the bond fund is 5. f. Compute the covariance between the rate of return on the bond fund and the rate of return on the stock fund. g. Suppose you invest equally in the two funds. What is the expected rate of return on your portfolio? What is the probability that the rate of return on your portfolio will exceed the expected rate of return? (Your answer explains why investors diversify rather than investing exclusively in one asset! Note that you do not need the dollar amounts invested in the two funds.) h. You decide to invest $1000 in the bond fund, and $2000 in the stock fund. Give the expected dollar value of your portfolio one year from now, and the variance of the dollar value of your portfolio one year from now. (Note that this question differs from part g in that you are not investing in the two funds equally, and you are being asked about the dollar value of the portfolio one year from now, not the rate of return. The dollar value of the portfolio is the funds invested, plus all returns.)(20 points) 2. Do not evaluate your answer—just write out the expressions you would evaluate. A computer manufacturer keeps hard drives in inventory. Right now, it has 30 hard drives in inventory. Of these, 8 are from supplier A, 10 are from supplier B, and 12 from supplier C. The computer manufacturer receives an order for six computers, and so six hard drives are taken from inventory in order to build six computers. Unbeknownst to the computer manufacturer, 4 of the 8 hard drives from supplier A are defective. Also, unbeknownst to the computer manufacturer, hard drives from supplier C are of extremely high quality. (Hard drives from supplier B and non-defective hard drives from supplier A are of average quality.) a. If the six hard drives are selected randomly from the manufacturer’s inventory, what is the probability that at least one defective hard drive will be selected? b. What is the probability that all of the selected hard drives will be of extremely high quality? (30 points) 3. Firms spend large sums of money on applied research and product development. In the 1950’s, the “parallel-path strategy” was developed by the Rand Corporation to help firms achieve their research and development goals at minimum cost. This strategy involves n teams working independently to achieve the same goal, such as development of a new product. (All teams are working to develop the same new product.) a. Suppose each team has a probability of .07 of developing the new product. If the firm sets up 4 teams, what is the probability the new product will be developed? b. If the firm wants the probability the new product will be developed to be at least .3, what is the smallest number of teams it must set up? c. Each team costs $1 million. If the new product is developed, profits associated with the new product are $20 million. How many teams should the firm set up? (Be sure to show how you arrive at your answer.) (Hint: If the number of teams is too small, the probability of developing the product and capturing the $20 million is too low. If the number of teams is too big, the costs of the teams cut into the $20 profit too much. Use trial and error. Start with your answer to part b. Ask yourself whether the firm would be better off setting up either one fewer team compared to part b, or one additional team compared to part b.) (30 points) 4. A microfinance institution opens a branch in a remote village in Indonesia. The branch makes ten loans. Each borrower either repays or defaults, and borrowers’ repayment/default decisions are independent. The probability any one borrower will default on their loan is .08. a. What is the probability of zero defaults? What is the probability of one default? What is the probability of more than one default? b. Would the probability of zero defaults be larger or smaller if the microfinance institution made 15 loans rather than ten? (Answer by calculating the probability of zero defaults if 15 loans were given.) c. What is the smallest sample size for which the probability of zero defaults is below .2?(30 points) 5. A fishing company in New England operates a search plane to find schools of fish. These schools of fish are randomly located in the North Atlantic. On average, there is one school per 100,000 square miles of sea. On a given day, the search plane can fly 1000 miles, searching out five miles on either side of its path (i.e., the plane can search a path 1000 miles long and 10 miles wide). a.