1. Suppose that the market consists 2000 stocks, and the single index model holds. That is, the return of stock i is given by - T - Yf = a + B (T Tf) + & Cov(&,) = 0 i #j For simplicity, we assume the risk-free rate is 0, and the volatility of & is 20% for all stocks. Assume that stocks can be classified into several groups with different alpha and beta: Stock 1-5 6-105 106-1000 1001-1895 1896-1995 (895 stocks) (895 stocks) (100 stocks) (100 stocks) 20% 5% -5% -20% 1.2 0.9 0.9 1.2 (5 stocks) 50% 1.5 (1) Consider the portfolio with the following weights: Stock 1-5 106-1000 (895 stocks) 1/900 (5 stocks) Cov(&, &j) = 0, weight 1/1500 Stock 1-5 (5 stocks) weight 1/7.5 6-105 (100 stocks) 1/1200 1001-1895 (895 stocks) -1/900 6-105 (100 stocks) 0 1896-1995 (100 stocks) -1/1200 Calculate the expected return, the volatility and Sharpe ratio of this portfolio. (2) If we can not short stocks but we can short the index (for simplicity, assume that the index is tradable). In order to have a market neutral portfolio (that is, a portfolio without exposure to index risk), keep the long side the same as in (1), what should be the weight of the index? Calculate the expected return, the volatility and Sharpe ratio of this portfolio. (3) Consider the portfolio with the following weights: 106-1000 (895 stocks) 0 1001-1895 (895 stocks) 0 1996-2000 (5 stocks) -50% 1.5 1896-1995 (100 stocks) 0 1996-2000 (5 stocks) -1/1500 1996-2000 (5 stocks) 0 Again, suppose that we can not short stocks, what should be the weight of the index in order to have a market neutral portfolio? Calculate the expected return, the volatility and Sharpe ratio of this portfolio.