B .8, what must be the risk-free rate? 6. Assume that stock market returns have the market index as a common factor, and that all stocks in the economy have a beta of 1 on the market index. Firm-specific returns all have a standard deviation of 30%. III. Equilibrium In Capital Markets 11. Arbitrage Pricing Theory The McGraw−Hill Companies, 2001           336 PART III Equilibrium in Capital Markets     Suppose that an analyst studies 20 stocks, and finds that one-half have an alpha of 2%, and the other half an alpha of -2%. Suppose the analyst buys $1 million of an equally weighted portfolio of the positive alpha stocks, and shorts $1 million of an equally weighted portfolio of the negative alpha stocks. a. What is the expected profit (in dollars) and standard deviation of the analysts profit? b. How does your answer change if the analyst examines 50 stocks instead of 20 stocks? 100 stocks? 7. Assume that security returns are generated by the single-index model,   Ri i i RM ei   where Ri is the excess return for security i and RM is the markets excess return. The risk-free rate is 2%. Suppose also that there are three securities A, B, and C, character- ized by the following data:     Security   10% 25% 12% 10% 14% 20%       a. If M 20%, calculate the variance of returns of Securities A, B, and C. b. Now assume that there are an infinite number of assets with return characteristics identical to those of A, B, and C respectively. If one forms a well-diversified portfolio of type A securities, what will be the mean and variance of the portfolios excess