Question

Stocks A and B have the following probability distributions of expected future returns: Probability A B 0.1 (11 % ) (23 %) 0.2 6 0 0.4 15. 23 0.2 23 27 0.1 39 48 a. Calculate the expected rate of return, rB, for Stock B (rA two decimal places. 14.60 % . ) Do not round intermediate calculations. Round your answer to 11 % b. Calculate the standard deviation of expected returns, aA, for Stock A (oe = 18.66 % . ) Do not round intermediate calculations. Round your answer to two decimal places. 12.75 Now calculate the coefficient of variation for Stock B. Do not round intermediate calculations. Round your answer to two decimal places. 1.70 Is it possible that most investors might regard Stock B as being less risky than Stock A? I. If Stock B is less highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense. II. If Stock B is less highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be more risky in a portfolio sense. III. If Stock B is more highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be less risky in a portfolio sense. ith the market than A, then it might have a lower beta than Stock A, and hence be less risky IV. If Stock B is more highly correlated in a portfolio sense. V. If Stock B is more highly correlated with the market than A, then it might have the same beta as Stock A, and hence be just as risky in a portfolio sense. пI c. Assume the risk-free rate is 2.5 %. What are the Sharpe ratios for Stocks A and B? Do not round intermediate calculations. Round

Answer 1

The Expected return of a stock can be calculated as

$\bar{R}_{A}= \sum_{i=1}^{n} p_{i} * R_{i}$

where p_i are the individual probabilities and R_i are the indiividual returns

Therefore $\bar{R}$ _B = 0.1* (-23%) + 0.2* 0% +0.4*23% +0.2*27% +0.1*48%

=-2.3%+0%+9.2%+5.4%+4.8%

=17.1%

The Expected Standard Deviation of a stock can be calculated as

$\sigma _{A} = \sqrt{\sum_{i=1}^{n}p_{i} *(R_{i} - \bar{R_{A}})^{2}}$
Therefore, $\sigma _{A}$ = sqrt { 0.1 * (-11 - 14.6)² +0.2 * (6 - 14.6)² +0.4 * (15 - 14.6)² +0.2 * (23 - 14.6)² +0.1 * (39 - 14.6)² }

= sqrt (65.54+14.79+0.064+14.11+59.53)

= 12.41%

Coefficient of Variation is given by

$CV = \sigma /\bar{R}$

Therefore for B , CV_B = 18.66%/17.1% = 1.09

for A , CV_A = 12.41%/14.6% = 0.85

If the stock B has a less higher correlation with market than A , then its Beta may be less than that of A because Beta of a stock is given by

Beta = Covariance between stock returns and market returns / market returns' variance

= correlation coefficient * st.dev. of stock * st. dev of Market/ market std. dev²

= correlation coefficient * st.dev of stock / st.dev of market

Even though stock B has higher st.dev. than stock A, a less correlation coefficient may make its Beta lesser than that of A and hence less risky in portfolio sense (option I)

c) Sharpe Ratio = (R_p - R_f) /  $\sigma$

Sharpe Ratio for A = (14.6-2.5)/12.41 = 0.975

Sharpe Ratio for B = (17.1-2.5)/18.66 = 0.782

Stock A has less risk than stock B (standard deviation) . Same is also reflected in coefficient of variation

The sharpe ratio also confirm that A is better in terms of overall return per unit risk

As A is less risky than B, and even if stock B has higher st.dev. than stock A, a less correlation coefficient may make its Beta lesser than that of A and hence less risky in portfolio sense

Therefore Option V