Question

Suppose the risk-free rate is 3%. And the market interest rate is 5%. There are two stocks A and B. Both pay annual dividend per share of $2 and $3, respectively. The correlations (denoted as corr(x,y)) between their returns and the market return are corr(r1,rm) = 0.2 and corr(r2,rm) = 0.7, respectively. The standard deviations (denoted as σ) of their returns and the market return are σ 1 = 0.4, σ 2 = 0.6, and σ m = 0.3, respectively. (Note: The x and y above are two arbitrary random variables representing time series of returns.)

a. What are the returns (r1 and r2) for these two stocks from CAPM? (Hint: when you are computing beta, be aware that Cov(x,y)= corr(x,y)*σx*σy , Cov denotes covariance here.)

b. What are the prices of these two stocks if the dividend is not growing?

c. What are the Sharp ratios for these stocks and the market portfolio?

d. If 30% of my portfolio is stock A, the rest of it is stock B. What is the variance of my portfolio if E (r1 * r2) = 0.2? (Hint: Cov(x,y) = E(x*y) – E(x)*E(y))

Answer 1

As per CAPM , required rate of return = risk free rate + ( market rate - risk free rate )* beta

beta = correlation between the market and stock * standard deviation of stock / standard deviation of market

beta of stock A = 0.2 * 0.4/0.3 = 0.2667

beta of stock B = 0.7 * 0.6 /0.3 = 1.4

Required rate of return for stock A = 3 + ( 5-3) *0.2667 = 3.53%

Reqired rate of return for stock B = 3 +( 5-3) * 1.4 = 5.8%

b)

Price of stock = dividend / required rate of return

price of stock A = 2/ 0.0353 = 56.60

price of stock B = 3/0.058 = 51.72

c)

sharpe ratio = (required return - risk free rate ) / standard deviation

sharpe ratio for stock A = ( 3.53 - 3 ) / 0. 4 = 1.325

sharpe ratio for stock B = ( 5.8 - 3) / 0.6 = 4.67

sharpe ratio for mrket = ( 5 -3 ) / 0.3 = 6.67

d)

variance of portfolio = ( weight of stock a^2* variance of stock a + weight of stock B ^2 *variance of stock b +2*weight of stock a * weight of stock b * covariance between a &b)

= 0.3^2* 0.4^2 + 0.7^2 *0.6^2 + 2 * 0.3*0.7*Cov