Question

You have prepared the following scenario analysis for the returns of the market index portfolio, M, and a stock. Assume that each scenario is equally likely. 1. Rate of return Scenario Bust Boom Market 10% 30% Stock 14% 26% a. Find the variance of the market and the stock, and beta of the stock. b. What is the expected rate of return on the stock and the market index? If the T-bill rate is 6 percent, what does the CAPM say about the fair expected rate of return on the stock (i.e., the rate of return commensurate with the risk of the stock)? Is the stock overpriced or underpriced? What passive portfolio comprised of the market index and T-bills would have the same systematic risk as the stock? What would be the expected rate of return on that portfolio? C.

Answer 1

(a) As each scenario is equally likely, the probability of each scenario is 50%.

Expected Return on Market = Rm = 0.5 x 10 + 0.5 x 30 = 20% and Expected Return on Stock = Rs = 0.5 x 14 + 0.5 x 26 = 20 %

Variance of Market = Vm = 0.5 x (10 - 20)^(2) + 0.5 x (30 - 20)^(2) = 100 and Variance of Stock = Vs = 0.5 x (14 - 20)^(2) + 0.5 x (26 - 20)^(2) = 36

Covariance of Market and Stock = 0.5 x (10 - 20) x (14 - 20) + 0.5 x (30 - 20) x (26 - 20) = 60

Stock's Beta = Covariance / Vm = 30 / 100 = 0.3

(b) Expected Return on Stock = Expected Return on Market = 20 % (as calculated in part (a))

Risk-Free Rate = Rf = 6 %, Expected Return on market = Rm = 20 % and beta = B = 0.3

Then, expected return on stock as per CAPM = E(r) = Rf + B x (Rm - Rf) = 6 + 0.3 x (20 - 6) = 10.2 %

As the stock's actual expected return (of 20 %) is greater than the CAPM predicted return of 10.2 %, the stock can be tagged as been underpriced (as it gives a return greater than it otherwise should).

(c) A portfolio of the market index and risk-free asset (T-Bills) will have the same systematic risk as the stock if th portfolio's beta is equal to the stock's beta of 0.3. Let the weight of the market index be Y and the weight of the risk-free asset be (1-Y) in the portfolio.

Beta of Market Index = 1 and Beta of T-Bills = 0

Therefore, Portfolio Beta = 0.3 = Y x 1 + (1-Y) x 0

Y = 0.3

Therfore, a passive portfolio of 30% market index and 70% T-Bills will have systematic risk equal to the stock.

Expected Return on this portfolio = 0.3 x 20 + 0.7 x 6 = 10.2 %