1. Suppose a three-factor model is appropriate to describe the returns of a stock. Information about those three factors is presented in the following chart:

Factor β Expected Value Actual Value GDP .0000479 $13,275 $13,601
Inflation -1.30 3.9% 3.2%
Interest rates -.67 5.2% 4.7%

a.What is the systematic risk of the stock return?

b.Suppose unexpected bad news about the firm was announced that causes the stock price to drop by 2.6 percent. If the expected return on the stock is 10.8 percent, what is the total return on this stock?

Multifactor Models

3. Suppose stock returns can be explained by a two-factor model. The firm specific risks for all stocks are independent. The following table shows the information for two diversified portfolios:

β_{1 } β_{2 } E(R)

Portfolio A .85 1.15 16%
Portfolio B 1.45 -.25 12

a.If the risk-free rate is 4 percent, what are the risk premiums for each factor in this model?

Portfolio Risk

3. You are forming an equally weighted portfolio of stocks. Many stocks have the same beta of .84 for factor 1 and the same beta of 1.69 for factor 2. All stocks also have the same expected return of 11 percent. Assume a two-factor model describes the return on each of these stocks.

a.Write the equation of the returns on your portfolio if you place only five stocks in it.

b.Write the equation of the returns on your portfolio if you place in it a very large number of stocks that all have the same expected returns and the same betas.

APT

4. Assume that the returns on individual securities are generated by thefollowing two-factor model:

F_{1t }and F_{2t} are market factors with zero expectation and zero covariance.

In addition, assume that there is a capital market for four securities, and the capital market for these four assets is perfect in the sense that there are no transaction costs and short sales (i.e., negative positions) are permitted. The characteristics of the four securities follow:

Security β_{1 } β_{2 } E(R)

1 1.0 1.5 20%

2 .5 2.0 20

3 1.0 .5 10

4 1.5 .75 10

a. Construct a portfolio containing (long or short) securities 1 and 2, with a return that does not depend on the market factor, F_{1t}, in any way. (Hint: Such a portfolio will have β_{1} = 0.)

b. Compute the expected return and β_{2} coefficient for this portfolio.

c. Following the procedure in (a), construct a portfolio containing securities 3 and 4 with a return that does not depend on the market factor, F_{1t}.

d. Compute the expected return and β_{2 }coefficient for this portfolio.

e. There is a risk-free asset with an expected return equal to 5 percent, β_{1} = 0, and β_{2} = 0. Describe a possible arbitrage opportunity in such detail that an investor could implement it.

f. What effect would the existence of these kinds of arbitrage opportunities have on the capital markets for these securities in the short run and long run? Graph your analysis.

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