Exercises 5.1. Here are some general questions and instructions to test your understanding of the mean...

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Exercises 5.1. Here are some general questions and instructions to test your understanding of the mean standard deviation diagram. a. Draw a mean-standard deviation diagram to illustrate combinations of a risky asset and the risk-free asset. b. Extend this concept to a diagram of the risk-free asset and all possible risky portfolios. c. Why does one line, the capital market line, dominate all other possible portfolio combinations? d. Label the capital market line and tangency portfolio. e. What condition must hold at the tangency portfolio? Exercises 5.2-5.9 make use of the following information about the mean returns and covariances for three stocks. The numbers used are hypothetical. b. Then compute the beta of an equally weighted portfolio of the three stocks. 5.8. Using the fact that the hyperbolic boundary of the feasible set of the three stocks is generated by any two portfolios: 5.9. a. Find the boundary portfolio that is uncorrelated with the tangency portfolio in exercise 5.2. b. What is the covariance with the tangency portfolio of all inefficient portfolios that have the same mean return as the portfolio found in part a? What is the covariance of the return of the tangency portfolio from exercise 5.2 with the return of all portfolios that have the same expected return as AOL? 5.10. Using a spreadsheet, compute the minimum variance and tangency portfolios for the universe of three stocks described below. Assume the risk-free return is 5 percent. Hypothetical data necessary for this calculation are provided in the table below. See exercise 5.6 for detailed instructions. Covariance with Stock AOL Microsoft Intel Mean Return AOL .002 .001 0 15% Microsoft .001 .002 .001 12 Stock Standard Mean Deviation Return Correlation Correlation with with Bell South Caterpillar Intel 0 .001 .002 10 Apple .20 .15 .8 -.1 Bell South Caterpillar .30 .10 1.0 .2 .25 .12 .2 1.0 5.2. Compute the tangency portfolio weights assuming a risk-free asset yields 5 percent. 5.3. How does your answer to exercise 5.2 change if the risk-free rate is 3 percent? 7 percent? 5.4. Draw a mean-standard deviation diagram and plot AOL, Microsoft, and Intel on this diagram as well as the three tangency portfolios found in exercises 5.2 and 5.3. 5.5. Show that an equally weighted portfolio of AOL, Microsoft, and Intel can be improved upon with marginal variance-marginal mean analysis. 5.6. Repeat exercises 5.2 and 5.3, but use a spreadsheet to solve for the tangency portfolio weights of AOL, Microsoft, and Intel in the three cases. The solution of the system of equations requires you to invert the matrix of covariances above, then post multiply the inverted covariance matrix by the column of risk premiums. The solution should be a column of cells, which needs to be rescaled so that the weights sum to 1. Hint: See footnote 11. 5.7. a. Compute the betas of AOL, Microsoft, and Intel with respect to the tangency portfolio found in exercise 5.2. 5.11. The Alumina Corporation has the following simplified balance sheet (based on market values) Assets $10 billion Liabilities and Equity Debt $6 billion Common Stock $4 billion a. The debt of Alumina, being risk-free, earns the risk-free return of 6 percent per year. The equity of Alumina has a mean return of 12 percent per year, a standard deviation of 30 percent per year, and a beta of .9. Compute the mean return, beta, and standard deviation of the assets of Alumina. Hint: View the assets as a portfolio of the debt and equity. b. If the CAPM holds, what is the mean return of the market portfolio? m Exercises 5.1. Here are some general questions and instructions to test your understanding of the mean standard deviation diagram. a. Draw a mean-standard deviation diagram to illustrate combinations of a risky asset and the risk-free asset. b. Extend this concept to a diagram of the risk-free asset and all possible risky portfolios. c. Why does one line, the capital market line, dominate all other possible portfolio combinations? d. Label the capital market line and tangency portfolio. e. What condition must hold at the tangency portfolio? Exercises 5.2-5.9 make use of the following information about the mean returns and covariances for three stocks. The numbers used are hypothetical. b. Then compute the beta of an equally weighted portfolio of the three stocks. 5.8. Using the fact that the hyperbolic boundary of the feasible set of the three stocks is generated by any two portfolios: 5.9. a. Find the boundary portfolio that is uncorrelated with the tangency portfolio in exercise 5.2. b. What is the covariance with the tangency portfolio of all inefficient portfolios that have the same mean return as the portfolio found in part a? What is the covariance of the return of the tangency portfolio from exercise 5.2 with the return of all portfolios that have the same expected return as AOL? 5.10. Using a spreadsheet, compute the minimum variance and tangency portfolios for the universe of three stocks described below. Assume the risk-free return is 5 percent. Hypothetical data necessary for this calculation are provided in the table below. See exercise 5.6 for detailed instructions. Covariance with Stock AOL Microsoft Intel Mean Return AOL .002 .001 0 15% Microsoft .001 .002 .001 12 Stock Standard Mean Deviation Return Correlation Correlation with with Bell South Caterpillar Intel 0 .001 .002 10 Apple .20 .15 .8 -.1 Bell South Caterpillar .30 .10 1.0 .2 .25 .12 .2 1.0 5.2. Compute the tangency portfolio weights assuming a risk-free asset yields 5 percent. 5.3. How does your answer to exercise 5.2 change if the risk-free rate is 3 percent? 7 percent? 5.4. Draw a mean-standard deviation diagram and plot AOL, Microsoft, and Intel on this diagram as well as the three tangency portfolios found in exercises 5.2 and 5.3. 5.5. Show that an equally weighted portfolio of AOL, Microsoft, and Intel can be improved upon with marginal variance-marginal mean analysis. 5.6. Repeat exercises 5.2 and 5.3, but use a spreadsheet to solve for the tangency portfolio weights of AOL, Microsoft, and Intel in the three cases. The solution of the system of equations requires you to invert the matrix of covariances above, then post multiply the inverted covariance matrix by the column of risk premiums. The solution should be a column of cells, which needs to be rescaled so that the weights sum to 1. Hint: See footnote 11. 5.7. a. Compute the betas of AOL, Microsoft, and Intel with respect to the tangency portfolio found in exercise 5.2. 5.11. The Alumina Corporation has the following simplified balance sheet (based on market values) Assets $10 billion Liabilities and Equity Debt $6 billion Common Stock $4 billion a. The debt of Alumina, being risk-free, earns the risk-free return of 6 percent per year. The equity of Alumina has a mean return of 12 percent per year, a standard deviation of 30 percent per year, and a beta of .9. Compute the mean return, beta, and standard deviation of the assets of Alumina. Hint: View the assets as a portfolio of the debt and equity. b. If the CAPM holds, what is the mean return of the market portfolio? m