Question

Stocks A and B have the following probability distributions of expected future returns:

Probability	A	B
0.2	(7%)	(27%)
0.2	3	0
0.1	10	23
0.3	20	29
0.2	39	41

1. Calculate the expected rate of return, , for Stock B ( = 14.00%.) Do not round intermediate calculations. Round your answer to two decimal places.
     %

2. Calculate the standard deviation of expected returns, σ_A, for Stock A (σ_B = 24.43%.) Do not round intermediate calculations. Round your answer to two decimal places.
     %

   Now calculate the coefficient of variation for Stock B. Round your answer to two decimal places.

   Is it possible that most investors might regard Stock B as being less risky than Stock A?

   1. If Stock B is more 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.
   2. 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.
   3. 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.
   4. 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.
   5. 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.
   
   
   -Select-IIIIIIIVVItem 4

3. Assume the risk-free rate is 3.0%. What are the Sharpe ratios for Stocks A and B? Do not round intermediate calculations. Round your answers to two decimal places.

   Stock A:

   Stock B:

   Are these calculations consistent with the information obtained from the coefficient of variation calculations in Part b?

   1. In a stand-alone risk sense A is less risky than B. 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.
   2. In a stand-alone risk sense A is more risky than B. 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.
   3. In a stand-alone risk sense A is more risky than B. 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.
   4. In a stand-alone risk sense A is less risky than B. 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.
   5. In a stand-alone risk sense A is less risky than B. 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.

Answer 1

for Expected return and Expected Standard Deviation are given by

  

where pi represents the individual probabilities in different scenarios

Ri represents the corresponding returns in different scenarios and

represents the expected return calculated as above.

Therefore , Expected Return of Stock B = 0.2*(-27%)+ 0.2*0%+ 0.1*23%+0.3*29%+0.2*41%

= 13.8%

b) Expected Standard Deviation of stock A = sqrt [ 0.2* (-7-14)2 + 0.2* (3-14)2 + 0.1* (10-14)2 + 0.3* (20-14)2+ 0.2* (39-14)2]

= sqrt (88.2+24.2+1.6+10.8+125)

= sqrt (249.8) = 15.81%

Coefficient of variation of Stock B = Standard deviation of B/Expected return of B

= 24.43%/13.8% = 1.77

Coefficient of variation of Stock A = Standard deviation of A/Expected return of A

= 15.81%/14% = 1.13

If Stock B is less highly correlated with market than A, it may have a lower Beta , and then , even though it has a lower return and higher standard deviation than A, it may still be attractive in the portfolio sense, i.e for a well diversified investor (option III is correct)

c) Sharpe Ratio of a stock =(Return of the stock - Risk free rate)/ standard deviation of the stock

So, Sharpe Ratio of A    = (14%-3%)/15.81% =0.69598 or 0.70

Sharpe Ratio of B    = (13.8%-3%)/24.43% =0.442 or 0.44

Coefficient of variation indicates that B has more risk per unit return than A and Sharpe ratio also indicates that B is providing less excess return per unit risk than A. So, calculations are consistent

In a stand-alone risk sense A is less risky than B (has lower coefficient of variation and higher sharpe ratio). 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. (option V is the correct option)