Assume that all 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. The firm-specific components of stock returns are uncorrelated and have a standard deviation of 30%. Suppose that an analyst studies 50 stocks, and finds that one-half have an alpha of 2.6%, and the other half have an alpha of -4%. Suppose the analyst buys £1 million of an equally weighted portfolio of the positive alpha stocks, and sells £1 million of an equally weighted portfolio of the negative alpha stocks.
What are the expected profit (in pounds) and the standard deviation of the analyst’s profit (in pounds)?
What would your answer be if the number of stocks were 100 instead of 50? Explain your results.