(A Factor Model Question) Based on a single factor, model, you have performed the following regressions on the returns to stocks X and Y against the returns on a market index (MI) which you believe is a good proxy for all relevant risk factors:

The results you are given are as follows

Stock
X	0.02	0.36	0.0625
Y	0.005	0.85	0.16

In addition, you estimate the variance of the market index as, .09

What will be the unique risk ( ) of a portfolio composed of equal amounts of stock X and stock Y?

Instruction: 1. You need to write your solutions in details, including a. Formulas (in symbols, not Excel cell addresses) b. The same formulas but with the data from a problem plugged into these formulas c. A short verbal comments at the end of each step of each problem, explaining what you have accomplished with this step d. A short verbal summary at the end of each problem 2. Your solution should clearly explain your logic and show how you came up with quantitative results. THIS IS THE MOST IMPORTANT CRETERION for grading. 3. Although you will use Excel for calculations, please, don't copy your Excel spreadsheets into your solution directly. Once again, it is your logic that counts most, not your final answers. GOOD LUCK! Question 1 Consider the following data on four securities: Expected returns (annualized): E[r] 1 4.29% 2 8.52% 3 5.28% 4 6.51% 1 Variance - Covariance Matrix (also annualized, numbers are given in decimals) is below. 1 0.0001 0.0006 0.0005 0.0003 1 2 3 4 2 3 4 0.0009 0.0005 0.0003 0.0006 0.0001 0.0005 1. Use Solver to determine the weights of the above four securities in the Minimum Variance Efficient (MVE) portfolio and report these weights. To receive a full credit for this part of the question, please make sure you explain the optimization problem which you solve with the help of Solver. 2. What are the mean and standard deviation of the MVE portfolio constructed from these four assets? To receive a full credit for this question: a. Please, show the formulas that you used to find the mean and the standard deviation of the MVE portfolio of four assets (using symbols). b. Show the same formulas with the numbers plugged in. 3. Suppose that the short sales are not allowed. Please show how your optimization problem has changed in comparison to the part 1 of this question to accommodate these constraints. 4. What are the mean and the standard deviation of the MVE portfolio under short sale constraints? 5. Please comment on the effect that short sale constraints have on your investment, assuming that you want to invest into the MVE portfolio. Question 2 You are given the following information on two securities, the market portfolio, and the riskfree rate: Expected Return Correlation with Market Portfolio Standard Deviation Security 1 15.5% 0.9 20% Security 2 9.2% 0.8 9% Market Portfolio 12% 1 12% Risk-free Rate 5% 0 0% 2 For parts a, b, and c use the above table. a. Calculate the intercept and the slope of SML. b. Draw the SML clearly naming both axes and marking the intercept and the slope of the SML on your graph. c. What are the betas of the two securities? d. Plot the two securities, the risk free asset and the Market portfolio on the SML. Question 3 (A Factor Model Question) Based on a single factor, model, you have performed the following regressions on the returns to stocks X and Y against the returns on a market index (MI) which you believe is a good proxy for all relevant risk factors: r j a j j rMI j The results you are given are as follows Stock aj j 2 (r j ) X 0.02 0.36 0.0625 Y 0.005 0.85 0.16 In addition, you estimate the variance of the market index as, 2 =0.09 (rMI ) What will be the unique risk ( 2 ) of a portfolio composed of equal amounts of stock ( p ) X and stock Y? Question 4 You wish to invest into two mutual funds A and B. The analyst provided you with the following information on these two funds: Security Expected Return Fund A 5% Fund B 7% 3 Standard Deviation 4% 8% The correlation coefficient between the returns on two funds is 0.5. 1. What proportion of your funds should you invest in each fund in order to construct an optimal portfolio, which will earn 9% expected return? 2. What is the standard deviation of your portfolio? 3. Suppose that the correlation coefficient between returns on funds A and B increased to 0.8. What is the standard deviation of the optimal portfolio that pays 9% expected return now? 4. How does correlation coefficient effect the risk of the portfolio? Why? 4