Vertex42 The Excel Nexus
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.
Covariance with |
Mean Return | |||
Stock |
AOL |
Microsoft |
Intel | |
AOL |
.002 |
.001 |
0 |
15% |
Microsoft |
.001 |
.002 |
.001 |
12 |
Intel |
0 |
.001 |
.002 |
10 |
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:
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?
5.9. 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.
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.
Correlation |
Correlation | |||
Standard |
Mean |
with |
with | |
Stock Deviation |
Return |
Bell South |
Caterpillar | |
Apple |
.20 |
.15 |
.8 |
-.1 |
Bell South |
.30 |
.10 |
1.0 |
.2 |
Caterpillar |
.25 |
.12 |
.2 |
1.0 |
5.11. The Alumina Corporation has the following simplified balance sheet (based on market values)
Assets
Liabilities and Equity
Assets
Liabilities and Equity
Debt | |
$10 billion |
$6 billion |
$4 billion |
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.
If the CAPM holds, what is the mean return of the market portfolio?
Part II Valuing Financial Assets c. How does your answer to part a change if the debt is risky, has returns with a mean of 7 percent, has a standard deviation of 10 percent, a beta of .2, and has a correlation of .3 with the return of the common stock of Alumina?
5.12. The following are the returns for Exxon (which later merged with Mobil) and the corresponding returns of the S&P 500 market index for each month in 1994.
Month |
Exxon Return (%) |
S&P 500 Return (%) |
January |
5.35% |
3.35% |
February |
-1.36 |
-2.70 |
March |
-3.08 |
-4.35 |
April |
0.00 |
1.30 |
May |
-1.64 |
1.63 |
June |
-7.16 |
- 2.47 |
July |
4.85 |
3.31 |
August |
1.21 |
4.07 |
September |
-3.36 |
- 2.41 |
October |
9.35 |
2.29 |
November |
-2.78 |
-3.67 |
December |
0.62 |
1.46 |
Using a spreadsheet, compute Exxon's beta. Then apply the Bloomberg adjustment to derive the adjusted beta.
5.13. What value must ACYOU Corporation's expected return be in Example 5.4 to prevent us from forming a combination of Henry's portfolio, ACME, ACYOU, and the risk-free asset that is mean-variance superior to Henry's portfolio?
5.14. Assume that the tangency portfolio for stocks allocates 80 percent to the S&P 500 index and 20 percent to the Nasdaq composite index. This tangency portfolio has an expected return of 13 percent per year and a standard deviation of 8.8 percent per year. The beta for the S&P 500 index, computed with respect to this tangency portfolio, is .54. Compute the expected return of the S&P 500 index, assuming that this 80%/20% mix really is the tangency portfolio when the risk-free rate is 5 percent.
5.15. Exercise 5.14 assumed that the tangency portfolio allocated 80 percent to the S&P 500 index and 20 percent to the Nasdaq composite index. The beta for the S&P 500 index with this tangency portfolio is .54. Compute the beta of a portfolio that is 50 percent invested in the tangency portfolio and 50 percent invested in the S&P 500 index.
5.16. Using data only from 1991-1995, redo Example 5.9. Which differs more from the answer given in Example 5.9: the expected return estimated by averaging the quarterly returns or the expected return obtained by estimating beta and employing the risk-expected return equation? Why?
5.17. Estimate the Bloomberg-adjusted betas for the following companies.
Unadjusted Beta | |
Delta Air Lines |
0.84 |
Procter & Gamble |
1.40 |
Coca-Cola |
0.88 |
Gillette |
0.90 |
Citigroup |
1.32 |
Caterpillar |
1.00 |
ExxonMobil |
0.64 |
5.18. Compute the tangency and minimum variance portfolios assuming that there are only two stocks: Nike and McDonald's. The expected returns of Nike and McDonald's are .15 and .14, respectively. The variances of their returns are .04 and .08, respectively. The covariance between the two is .02. Assume the risk-free rate is 6%.
5.19. There exists a portfolio P, whose expected return is 11%. Stock I has a covariance with P of .004, and Stock II has a covariance with P of .005. If the expected returns on Stocks I and II are 9% and 12%, respectively, and the risk-free rate is 5%, then is it possible for portfolio P to be the tangency portfolio?
5.20. The expected return of the S&P 500, which you can assume is the tangency portfolio, is 16% and has a standard deviation of 25% per year. The expected return of Microsoft is unknown, but it has a standard deviation of 20% per year and a covariance with the S&P 500 of 0.10. If the risk-free rate is 6 percent per year, a. Compute Microsoft's beta.
b. What is Microsoft's expected return given the beta computed in part a?
c. If Intel has half the expected return of Microsoft, then what is Intel's beta?
d. What is the beta of the following portfolio?
.25 in Microsoft .10 in Intel .75 in the S&P 500 - .20 in GM (where jSGM = .80) .10 in the risk-free asset e. What is the expected return of the portfolio in part d?
Chapter 5 Mean-Variance Analysis and the Capital Asset Pricing Model 173
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