An investor uses the Treynor-Black single-index model to set up his portfolio. The passive part of his portfolio is an index fund with an expected return of 12% and a standard deviation of 20%. The risk-free rate is 2%. The active part of his portfolio consists of the stock of two companies, A and B. The investor has estimated the α of stock A to be 5%, its β to be 1.6 and the firm specific error term to have a standard deviation of 30%. The investor has likewise estimated the α of stock B to be -3%, its β to be 0.8 and the firm specific error term to have a standard deviation of 40%.
What will be the portfolio weights of the passive index fund, A and B and what will be the expected return and standard deviation of the portfolio? What is the Sharpe ratio of the portfolio and how does that compare to the Sharpe ratio of the index fund?