Question

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4.    Suppose we have one risky asset Stock I and a risk-free asset. Stock I has an expected return of 25% and a beta of 2. The risk-free asset’s return is 6%.   

a. Calculate the expected returns and betas on portfolios with x% invested in Stock I and the rest invested in the risk-free asset, where x% = 0%, 25%, 75%, 100%, 125%, and 150%.

b. What reward-to-risk ratio does Stock I offer? How do you interpret this ratio?

c.   Suppose we have a second risky asset, Stock J. Stock J has an expected return of 20% and a beta of 1.7. Calculate the expected returns and betas on portfolios with x% invested in Stock J and the rest invested in the risk-free asset, where x% = 0%, 25%, 75%, 100%, 125%, and 150%.   

d. What reward-to-risk ratio does Stock J offer? How do you interpret this ratio?

e. Plot the portfolio betas against the portfolio expected returns for Stock I on a graph, and link all the points together with a line. Then plot the portfolio betas against the portfolio expected returns for Stock J on the same graph, and link all these points together with another line. (This can be done easily with the charting function in Microsoft Excel.)   

f.   Use the graph in part (e) above, together with your answers to parts (b) and (d) above to explain why Stock J is an inferior investment to Stock I.                

g. Can a situation in which one stock is inferior to another stock persist in a well-organized, active market? Why or why not?                                        

Answer 1

Part a and b are answered as under

Stock I Expected Return	25%
Stock I Beta	2
Risk-free rate	6%
Portfolio comprises of:
Stock I Weight	0%	25%	75%	100%	125%	150%
Risk free Asset Weight	100%	75%	25%	0%	-25%	-50%
Portfolio Expected Return	6.00%	10.75%	20.25%	25.00%	29.75%	34.50%
Portfolio Beta	0	0.5	1.5	2	2.5	3
Reward to Risk	NA	21.50%	13.50%	12.50%	11.90%	11.50%

Below are the formulas used:

Reward to risk ratio tells the expected return for 1 unit of risk. We can see the portfolio's expected return keeps increasing with the increase in weight of risky asset. But, the additional risk for the additional return is much more and that's why the return to risk ratio is decreasing.

Part c and d are answered as under

Stock J Expected Return	20%
Stock J Beta	1.7
Risk free rate	6%
Portfolio comprises of:
Stock J Weight	0%	25%	75%	100%	125%	150%
Risk free Asset Weight	100%	75%	25%	0%	-25%	-50%
Portfolio Expected Return	6.00%	9.50%	16.50%	20.00%	23.50%	27.00%
Portfolio Beta	0	0.425	1.275	1.7	2.125	2.55
Reward to Risk	NA	22.35%	12.94%	11.76%	11.06%	10.59%

Below are the formulas used

We can see the portfolio's expected return keeps increasing with the increase in weight of risky asset. But again, the additional risk for the additional return is much more and that's why the return to risk ratio is decreasing.