Question

Asset	Alpha	Beta with respect to market index	Idiosyncratic variance
1	0.02	0.9	0.33
2	0.005	1.6	0.19

In addition, there is the market portfolio, which has an expected return of 8.7% and a variance of 0.14, and the riskless asset, which returns 3.1%.

3A. Using the index model, what should the weights of Assets 1 and 2 be, assuming the beta of your actively managed portfolio is 1? 5 points

3B. Assuming the beta of your actively managed portfolio is 1, what proportion of the optimal risky portfolio should be composed of the market portfolio? 5 points

3C. What is the beta of your actively managed portfolio? 5 points

Answer 1

3A)

Asset 1

initial position = alpha/idiosyncratoc = 0.02/0.33 = 0.0606

Asset 2

initial position = 0.005/0.19 = 0.0263

Weight of asset 1 = 0.0606/(0.0606+0.0263) = 69.74%

Weight of asset 2 = 1-69.74% = 30.26%  

**since beta of active portfoio is assumed to be 1, the weightages remain same as above

3B)

Alpha active portfolio = 69.74%*0.02+30.26%*0.005 = 0.0155

Residual variance active portfolio = (69.74%^2)*0.33+(30.26%^2)*0.19 = 0.1779

% in optimal risky portfolio = (alpha active portfolio/Residual variance active portfolio)/(risk premium market/variance market)

% in optimal risky portfolio = (0.0155/0.1779)/((8.7%-3.1%)/0.14) = 21.78%

% in market portfolio = 100%-21.78% = 78.22%

3C)

Beta of optimal risky portfolio = 69.74%*0.9+30.26%*1.6 = 1.112

% of optimal risky portfolio = 21.78%

Beta of actively managed portfolio = 21.78%*1.112+78.22%*1 = 1.024