Asset Management: Exercises on Arbitrage Pricing Theory
Instructor: Dr. Youchang Wu
13. Suppose that two factors have been deemed appropriate to explain returns on stocks. You have the following information on two stocks, X and Y, and on the two factors, 1 and 2:
stock X: b1 = 1.1, b2 = 0.5, residual variance = 0.02
stock Y: b1 = 0.2, b2 = 0.8, residual variance = 0.05
variance of factor 1 = 0.15, variance of factor 2 = 0.10
What is the variance of a portfolio consisting of € 1.000 invested in X and € 3.000 invested in Y
a) if the covariance between the factors is zero?
b) if the covariance between the factors is 0.05?
14. You have the following information about the factor sensitivities of three stocks:
stock A: b1 = 0, b2 = -4
stock B: b1 = 3, b2 = 2
stock C: b1 = 1.5, b2 = 0
You want to form two factor portfolios using these three stocks: one with a beta of 1 with respect to the first factor, and a beta of 0 with respect to the second factor; the other one with a beta of 0 with respect to the first factor, and a beta of 1 with respect to the second factor. What are the portfolio weights for these two factor portfolios?
15. Assume a two-factor APT model is appropriate, and there is an infinite number of assets in the economy. The cross-sectional relationship between expected return and factor betas indicates the price (risk premium) of factor 1 is 0.15, and the price of factor 2 is -0.20.
You have estimated the factor betas for stocks X and Y as follows:
stock X: b1 = 1.4, b2 = 0.4
stock Y: b1 = 0.9, b2 = 0.2
Also, the expected return on an asset having zero betas with respect to both factors is 0.05. According to the APT, what are the approximate equilibrium returns on each of the two stocks?
16. Assume a one-factor model with a nonlinear relationship of expected return to beta.
stock A: b1 = 0.5, expected return = 0.06
stock B: b1 = 1.0, expected return = 0.09
stock C: b1 = 1.5, expected return = 0.11
First, find a portfolio of A and C with a beta of zero. What is the expected return?
Next, find a portfolio of B and C with a beta of zero. What is the expected return?
What actions would you suggest in such a situation?
17. You have found a simple factor structure for the Austrian stock market. Factor F1 is the term spread and factor F2 is the credit spread. Now consider an alternative factor structure with factor F*1 = F1 + F2 and F*2 = F2.
1) If betas under the “old” factor structure are b1 = 1 and b2 = 1 for stock 1, and b1 = 0 and b2 = 2 for stock 2. What are the betas under the “new” factor structure?
2) If the risk premia for the “old” factors are l1=0.05 and l2=0.02, respectively. What are the risk premia l* for the “new” factors?
18. Consider a three factor model with the following factor covariance matrix:
The factor risk premia are l1=0.05, l2=0.07 and l3=0.08.
How would you choose the factor sensitivities of a well diversified portfolio if you want to maximize expected return subject to a volatility constraint of 20%?
(hint: use the EXCEL tool SOLVER to solve this exercise numerically.)