Question

1. 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%.                                                                                                                                      (15 marks total)

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?                                                                                                                     (1.5 marks)

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?                                                                                                                     (1.5 marks)

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

(a) Expected returns and betas of portfolios:

(i) Where x = 0%

It implies that entire amount is invested in Risk free asset.

Weight of investment in risk free asset = 1 and weight in stock I = 0

Therefore, expected return = (6% * 1) + (25% * 0) = 6%

Beta of Stock I = 2 and beta of risk free asset = 0 (Because risk free assets beta are always zero as the risk is zero)

Beta of the portfolio = (0 * 1) + (2 * 0) = 0

(ii) where x = 25%

It implies that 75% is invested in Risk free asset.

Weight of investment in risk free asset = 0.75 and weight in stock I = 0.25

Therefore, expected return = (6% * 0.75) + (25% * 0.25) = 10.75%

Beta of Stock I = 2 and beta of risk free asset = 0 (Because risk free assets beta are always zero as the risk is zero)

Beta of the portfolio = (0 * 0.75) + (2 * 0.25) = 0.50

(iii) where x = 75%

It implies that 25% is invested in Risk free asset.

Weight of investment in risk free asset = 0.25 and weight in stock I = 0.75

Therefore, expected return = (6% * 0.25) + (25% * 0.75) = 20.25%

Beta of Stock I = 2 and beta of risk free asset = 0 (Because risk free assets beta are always zero as the risk is zero)

Beta of the portfolio = (0 * 0.25) + (2 * 0.75) = 1.50

(iv) where x = 100%

It implies that 0% is invested in Risk free asset.

Weight of investment in risk free asset = 0 and weight in stock I = 1

Therefore, expected return = (6% * 0) + (25% * 1) = 25%

Beta of Stock I = 2 and beta of risk free asset = 0 (Because risk free assets beta are always zero as the risk is zero)

Beta of the portfolio = (0 * 0) + (2 * 1) = 2

(v) where x = 125%

It implies that 25% is borrowed in Risk free asset.

Weight of investment in risk free asset = -0.25 and weight in stock I = 1.25

Therefore, expected return = (6% * -0.25) + (25% * 1.25) = 29.75%

Beta of Stock I = 2 and beta of risk free asset = 0 (Because risk free assets beta are always zero as the risk is zero)

Beta of the portfolio = (0 * 0) + (2 * 1.25) = 2.50

(vi) where x = 150%

It implies that -50% is borrowed in Risk free asset.

Weight of investment in risk free asset = -0.50 and weight in stock I = 1.50

Therefore, expected return = (6% * -0.50) + (25% * 1.50) = 34.5%

Beta of Stock I = 2 and beta of risk free asset = 0 (Because risk free assets beta are always zero as the risk is zero)

Beta of the portfolio = (0 * -0.50) + (2 * 1.50) = 3

(b) Reward to risk ratio of stock I = Risk / Return = 2/25 = 8%

The risk to reward is favorable in terms of investment.

(c) (i) Where x = 0%

It implies that entire amount is invested in Risk free asset.

Weight of investment in risk free asset = 1 and weight in stock J = 0

Therefore, expected return = (6% * 1) + (20% * 0) = 6%

Beta of Stock J = 1.7 and beta of risk free asset = 0 (Because risk free assets beta are always zero as the risk is zero)

Beta of the portfolio = (0 * 1) + (1.7 * 0) = 0

(ii) where x = 25%

It implies that 75% is invested in Risk free asset.

Weight of investment in risk free asset = 0.75 and weight in stock J = 0.25

Therefore, expected return = (6% * 0.75) + (20% * 0.25) = 9.5%

Beta of Stock J = 1.7 and beta of risk free asset = 0 (Because risk free assets beta are always zero as the risk is zero)

Beta of the portfolio = (0 * 0.75) + (1.7 * 0.25) = 0.425

(iii) where x = 75%

It implies that 25% is invested in Risk free asset.

Weight of investment in risk free asset = 0.25 and weight in stock J = 0.75

Therefore, expected return = (6% * 0.25) + (20% * 0.75) = 16.5%

Beta of Stock J = 1.7 and beta of risk free asset = 0 (Because risk free assets beta are always zero as the risk is zero)

Beta of the portfolio = (0 * 0.25) + (1.7 * 0.75) = 1.275

(iv) where x = 100%

It implies that 0% is invested in Risk free asset.

Weight of investment in risk free asset = 0 and weight in stock J = 1

Therefore, expected return = (6% * 0) + (20% * 1) = 20%

Beta of Stock J = 1.7 and beta of risk free asset = 0 (Because risk free assets beta are always zero as the risk is zero)

Beta of the portfolio = (0 * 0) + (1.7 * 1) = 1.7

(v) where x = 125%

It implies that 25% is borrowed in Risk free asset.

Weight of investment in risk free asset = -0.25 and weight in stock J = 1.25

Therefore, expected return = (6% * -0.25) + (20% * 1.25) = 23.5%

Beta of Stock J = 1.7 and beta of risk free asset = 0 (Because risk free assets beta are always zero as the risk is zero)

Beta of the portfolio = (0 * 0) + (1.7 * 1.25) = 2.125

(vi) where x = 150%

It implies that -50% is borrowed in Risk free asset.

Weight of investment in risk free asset = -0.50 and weight in stock J = 1.50

Therefore, expected return = (6% * -0.50) + (20% * 1.50) = 27%

Beta of Stock J = 1.7 and beta of risk free asset = 0 (Because risk free assets beta are always zero as the risk is zero)

Beta of the portfolio = (0 * -0.50) + (1.7 * 1.50) = 2.55

(d) Reward to risk ratio of stock J = Risk / Return = 1.7/20 = 8.5%

The risk to reward is favorable in terms of investment.