Question

Suppose we have two risky assets, Stock I and Stock J, and a risk-free asset. Stock I has an expected return of 25% and a beta of 1.5. Stock J has an expected return of 20% and a beta of 0.8. The risk-free asset’s return is 5%.                                   

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%, 50%, 100%, and 150%.

b.   Using the four portfolio betas calculated in part (a), reverse engineer (i.e., derive mathematically) the portfolio weights for a portfolio consisting of only Stock J and the risk-free asset.                                                                     

Hint: For example, if we wished to obtain a portfolio beta of 0.5, then the weights on Stock J and the risk-free asset must be 62.5% and 37.5%, respectively, and the expected return for this portfolio must be 14.375%.                                                                                              

c.Calculate the reward-to-risk ratios for Stock I and Stock J.                                                                             

d.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. Ensure that the x-axis and y-axis are clearly labelled. (Hint: This can be done easily with the charting function in Microsoft Excel.)                                             

e.Using the graph in part (d) above, together with your answers in part (c) above, elaborate on the efficiency of the market containing Stock I and Stock J.

Answer 1

Assets	Expected return	Beta
Stock I	25%	1.5
Stock J	20%	0.8
Risk Free asset	5%	0

a.

Calculation of Expected Return & Beta of the portfolio Containing stock I and Risk free assets.

Note- Portfolio return =

1. if investment in stock I is 0%

Stock	Weight	Expected Return	Beta	Wighted Return(Weight * Expected return)	Wighted Beta( Weight * Beta)
I	0.00	25%	1.5	0	0
Risk Free return	1.00	5%	0	5%	0
Total	1.00			5%	0

Hence Expected return of the portfolio = 5% and beta =0

2. if investment in stock I is 50%

Stock	Weight	Expected Return	Beta	Wighted Return(Weight * Expected return)	Wighted Beta( Weight * Beta)
I	0.50	25%	1.5	12.5%	0.75
Risk Free return	0.50	5%	0	2.5%	0
Total	1.00			15%	0.75

Hence Expected return of the portfolio = 15% and beta =0.75

3. if investment in stock I is 100%

Stock	Weight	Expected Return	Beta	Wighted Return(Weight * Expected return)	Wighted Beta( Weight * Beta)
I	1.00	25%	1.5	25%	1.5
Risk Free return	0.00	5%	0	0	0
Total	1.00			25%	1.5

Hence Expected return of the portfolio = 25% and beta =1.5

4. if investment in stock I is 150%

Stock	Weight	Expected Return	Beta	Wighted Return(Weight * Expected return)	Wighted Beta( Weight * Beta)
I	1.50	25%	1.5	37.5	2.25
Risk Free return	-0.50	5%	0	-2.5%	0
Total	1.00			35%	2.25

Note- investment in stock I is 150% means , borowung should be made from risk free assets in order to extra investment in Stock I Hence Expected return of the portfolio = 35% and beta =2.25

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b.

1. Portfolio beta is 0% . Hence 100% amount should be invested in Risk free assets as the Beta of the risk free assets is Zero & 0% amount should be invested in stock J. Portfolio return =5%

2. Portfolio Beta needed is 0.75.

Let the Weight of Stock J is Wj. Hence weight of risk free asset is (1-Wj)

0.75 = (0.8*Wj) + (0*(1-Wj))

=>Wj = 0.9375 or 93.75%

Weight of risk free assets = 1- Wj = 1-0.9375 =0.0625 or 6.25 %

Portfolio return = 19.0625%

3.

Portfolio Beta needed is 1.5.

Let the Weight of Stock J is Wj. Hence weight of risk free asset is (1-Wj)

1.5 = (0.8*Wj) + (0*(1-Wj))

=>Wj = 1.5/0.8 =1.875 or 187.5%

Weight of risk free assets = 1- Wj = 1-1.875 =(-0.875) or -87.5%

Amount should be borrowed in Risk free and should be invested in stock J.

Portfolio return = 33.125%

4.Portfolio Beta needed is 2.25.

Let the Weight of Stock J is Wj. Hence weight of risk free asset is (1-Wj)

2.25 = (0.8*Wj) + (0*(1-Wj))

=>Wj = 2.25/0.8 =2.8125 or 281.25%

portfolio return = 47.1875%

Weight of risk free assets = 1- Wj = 1-2.8125 =(-1.8125) or -181.25%

Amount should be borrowed in Risk free and should be invested in stock J.

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C.

Reward to risk ratio =

# Reward to risk Ratio of Stock I = 25% /1.5 = 16.67

#Reward to risk Ratio of Stock J = 20% /0.8 = 25

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d.

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e. As ghe graph shows the upward direction , hence the market containing stock I & J are efficient.