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

Answer A) Variance of Market:

E(R_M) = 0.5 x 10% + 0.5 x 30%

= 20%

Variance = (0.5 (10 - 20)%² + 0.5 (30 - 20)%²)^1/2 = ( 50 + 50)^1/2 = 10%

Variance of Stock

E(R_S) = 0.5 x 14% + 0.5 x 26% = 20%

Variance = (0.5(14-20)%² +0.5(26-20)%²)^1/2) = (18 + 18)^1/2 = 6%

BETA = Number of securities (sum of returns of stock and market) - (sum of return of stock)(sum of return of market)/number of securities( sum of square of return of market - sum of return of market

sum of return of stock and market = 10 x 1 4+ 30 x 26 = 920%

sum of return of stock= 40%

sum of return of market=40%

sum of square of return of market= 1000

BETA = 2(140) - 40 x 40/ 2(1000) - 40 = 0.122

Answer B) Expected return on stock = E(R_f) = R_f +(R_M-R_f)beta =6% + (20%-6%)(0.122) =7.708%

Here the expected return using CAPM is lower than investor required return then in this situation the security is overvalued and should be sold.

Answer C) The passive portfolio with beta as the fund should invest 12.25 in the market index and 88.8% in the money market .Expected rate of return of portfolio = 0.122 x 7.708% + 0.88 x 6% = 0.94% + 5.28% = 6.22%