## Problem Set # 3 Answers

1. "I don't see why a thirty-year U.S. Treasury Bond pays a higher interest rate than the risk-free rate. After all, if you hold it for thirty years you are guaranteed to get all your money." Criticize this statement.

• Suppose you want to sell it before the thirty years are up. Its price could be high, could be low. That's risk--and if it is systematic risk it will be priced. Similarly, even if you do hold it for thirty years differing rates of inflation may make the principal you receive of higher or lower real value. That's risk too.

2. Suppose you find a company, Abbott Aeronautics, with a beta of -1. Suppose, further, that the risk-free (nominal) rate is 3% per year and that the required rate of return on the market is 10% per year.

• What is the required rate of return on Abbott Aeronautics?
• =-4% per year (from its beta of -1)
• What do you expect the rate of return on Abbott Aeronautics to be in a year in which the rate of return on the market is 10% per year?
• -4% per year (there is no excess return on the market)
• What do you expect the rate of return on Abbott Aeronautics to be in a year in which the rate of return on the market is 3% per year? In which the rate of return on the market is 30% per year?
• +3% per year, and -24% per year
• Suppose Abbott Aeronautics pays a dividend equal to 3% of its stock price every year, and that it is selling this year for \$100 a share. Approximately how many years will it take before you expect it to be selling for less than \$10 a share?
• Expected return of -4% per year and dividend of 3% per year together mean that the expected price decline is 7% per year. Hence after ten years the stock will have halved in price. After thirty years the expected value of the stock will be down to \$12.50--and after thirty-three or thirty-four years the expected value of the stock will be less than \$10.

3. Suppose you invest 25% of your funds in Interesting Industries and 75% of your funds in Boring Buildings. The standard deviation of the annual return on the first is 15%; the standard deviation of the annual return on the second is 5%.

• What is the variance of your portfolio (in percentage points of the annual return) if the correlation between the two stocks is zero?
• Variance = 450/16 percent-squared, or .0028; std dev=5.3%
• What is the variance of your portfolio (in percentage points of the annual return) if the correlation between the two stocks is 0.5?
• Variance = .001406+.001406+.001406=.00422; std dev=6.5%
• What is the variance of your portfolio (in percentage points of the annual return) if the correlation between the two stocks is one?
• Variance = .00562; std dev=7.5%

4. Suppose that the standard deviation of the market return per year is 0.2 (20%), the standard deviation of the annual return on Zed Industries is 0.8 (40%) [oops! Let's solve for the case where it is 80%], and that the correlation between the excess return on the market and the excess return on Zed industries is 0.5.

• What is Zed Industries' beta?
• Covariance=.5 x .8 x .2 = .08; Variance of Market = .04; beta=2
• Suppose you find a large number of stocks with the same beta as Zed Industries, and suppose you form a diversified portfolio made up entirely of such stocks. What would the standard deviation of annual returns of such a portfolio be?
• Std dev of a diversified portfolio with a beta of two is twice the mkt std dev= 0.4 (40% per year)
• Suppose the risk-free rate is three percent per year and the market required rate of return is ten percent per year. What return do you expect on your investment in Zed Industries?
• 3+7+7=17% per year expected return

5. Suppose that you can invest in any of the following eight portfolios:

A
 B C D E F G H Expected Return 12% 8% 6% 3% 2% 5% 21% 0% Standard Deviation 25% 15% 12% 0% 3% 10% 50% 8%

Which portfolios are "efficient"? Which portfolios are "inefficient"? If security returns follow the CAPM, and portfolio A represents the market, which portfolios are fully diversified? What are the betas of the fully diversified portfolios?

• Portfolio D dominate E and H, so E & H are inefficient; 1/3 x A plus 2/3 x D dominates C, so C is inefficient; 5/9 x A + 4/9 x D dominates B, so B is inefficient; 7/9 x D + 2/9 x A dominates F, so F is inefficient. A, D, and G are efficient--and all are fully diversified, with G having a beta of 2.

6. Thermo-Electron Corporation has a beta of 1.4; Tyson Foods (a major campaign contributor to former Governor Clinton) has a beta of 1.0. Tyson Foods has a standard deviation of annual returns of about 26%; Thermo-Electron has a standard deviation of annual returns of about 24%. Which stock contributes more risk if added to a diversified portfolio? Which stock does the CAPM predict will have a higher expected return?

• Thermo-Electron is the riskier (and has the higher expected return)

7. Suppose that Bankers' Trust Corporation announces that it intends to acquire Bank of America, paying the Bank of America's shareholders a premium of \$25 a share over Bank of America's most recent stock price. In response to this announcement, Bank of America stock jumps by \$12.50 a share. Your assessment--and it is a good assessment: you are a Wall Street professional with better information than anyone else--is that there is a 60% chance that the acquisition will go through.

• From your perspective, is an investment in Bank of America a positive risk-adjusted net present value investment or a negative risk-adjusted net present value investment? Why?
• The control transaction has a zero-beta risk factor, so you don't have to reduce the value of the transaction to its certainty-equivalent; 60% of 25 is more than 12.50, so it is a positive NPV investment.
• Suppose that you have taken the plunge, sold \$10,000,000 worth of your fully-diversified beta-of-one portfolio, and invested that \$10,000,000 in Bank of America stock on hearing this news, and suppose further that Bank of America has a beta of 1.5. Is there anything else you need to do to keep the overall beta of your entire portfolio at one? If so, what?
• Sell an extra \$5,000,000 from the market and put it in Treasury bills.

8. The benefits of diversification from a portfolio of one to a portfolio of two stocks are greatest:

• when the returns to the two stocks are perfectly correlated.
• when the returns to the two stocks are somewhat correlated.
• when the returns to the two stocks are uncorrelated.
• when the returns to the two stocks are negatively correlated.
• When the returns to the two stocks are negatively correlated.

9. Suppose that your corporation's securities have a required rate of return of 8% per year, the risk-free rate is 3%, and the market's required rate of return is 10%. You have the opportunity to undertake an investment with the following set of expected cash flows (with all numbers in thousands):

 Year 0 1 2 3 4 5 6 7 Cash Flow -\$10,000 \$1,200 \$6,200 \$600 \$600 \$600 \$3,100 \$2,800 NPV -1336 10137 10457 4980 5125 5295 5493

Your financial advisors say that this investment carries an incremental beta of two. Will undertaking this project raise or lower your company's total stock market value? By how much?

• You should discount a beta of two project at 17% per year; NPV of -\$1,336

10. Suppose that the Nordhaus Lighting Corporation will pay a dividend of \$4 a share next year, and that thereafter its dividend will grow at 7% per year; suppose further that the risk-free rate is 3%, the market required rate of return is 10%, and that the Nordhaus Lighting Corporation has a beta of two. What does the CAPM predict for the current price of a share of the Nordhaus Lighting Corporation?

• Required rate of return of 17% per year; D/(r-g) gives us a price of \$40 a share