Question

4.      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%.              

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

2. 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.)                                 
3. 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

4. A.

The portfolio Beta's & Returns of stock 'I' & 'Risk free' asset can be prepared using the below table

			Portfolio Scenario with 'I' & Risk Free Assets
Stock	Expected Return	Beta	Portfolio Weight	Portfolio Weight	Portfolio Weight	Portfolio Weight
I	25%	1.5	0%	50%	100%	150%
J	20%	0.8
Risk Free Asset	5%	0	100%	50%	0%	-50%
Portfolio Beta( = Wa x Ba + Wb X Bb)			0	0.75	1.50	2.25
Portfolio Return (= Wa X Ra + Wb XRb)			5.00%	15.00%	25.00%	35.00%

The Portfolio Beta Bp= Wa x Ba + Wb x Bb

where Bp = Portfolio Beta

Wa = Weight of stock in A in portfolio

Wb = Weight of stock B in portfolio

Ba = Beta of stock A

Bb = Beta of stock B

The beta of risk free asset is zero = 0

Thus the portfolio with 50% in Stock I & 50% in Risk Free asset, will have a Portfolio Beta

= 50% X 1.5 + 50% X 0 = 0.75 + 0 = 0.75

Similarly other Portfolio betas can be calculated.

The Portfolio Return Rp= Wa x Ra + Wb x Rb

where Rp = Portfolio Return

Wa = Weight of stock in A in portfolio

Wb = Weight of stock B in portfolio

Ra = Return of stock A

Rb = Return of stock B

A portfolio with 50% in I & 50% in Risk free asset will have the return

= 50% X 25% + 50% x 5% = 0.125 + 0.025 = 0.15 = 15%

Similarly others can be tabulated as shown in the table above

B. We need to reconstruct portfolios of J and Risk free asset with Beta equivalent to 0.0, 0.75, 1.50, 2.25

The constraint is the weights should add upto 1

If Stock J has x weight, then risk free asset has 1-x weight

Beta of J = 0.80

Beta of Risk Free Asset = 0 (Since a risk free asset)

Portfolio Beta Bp= Wa x Ba + Wb x Bb

i). If Portfolio Beta = 0

=> 0 = x * 0.8 + (1-x) * 0 = 0.8 x +0

Thus x = 0% & 1-x = 100%

ii). If Portfolio Beta = 0.75

=> 0.75 = x * 0.8 + (1-x) * 0 = 0.8 x +0

Thus x = 0.75/ 0.80 = 0.9375 = 93.75% & 1-x = 6.25%

iii) If Portfolio Beta = 1.50

=> 1.50 = x * 0.8 + (1-x) * 0 = 0.8 x +0

Thus x = 1.50/ 0.80 = 1.875 = 187.50% & 1-x = -87.50% (Go short)

iv) If Portfolio Beta = 2.25

=> 2.25 = x * 0.8 + (1-x) * 0 = 0.8 x +0

Thus x = 2.25/ 0.80 = 2.8125 = 281.25% & 1-x = -181.25% (Go short)

C. The reward risk ratio can be calculated using Treynor's ratio (Uses Beta)

Tryenor Ratio = (Stock Return - Risk Free Return) / Stock Beta

Treynor Ratio for I = (25% -5%) /1.5 = 0.13

Treynor Ratio for J = (20% - 5%) / 0.8 = 0.19

D. The chart of Returns vs Portfolio Weights