Question

Assume that stock market returns have the market index as a common factor, and that all stocks in the economy have a beta of 1.7 on the market index. Firm-specific returns all have a standard deviation of 25%.

Suppose that an analyst studies 20 stocks and finds that one-half of them have an alpha of +2.5%, and the other half have an alpha of −2.5%. Suppose the analyst invests $1.0 million in 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 analyst’s profit?

How does your answer change if the analyst examines 50 stocks instead of 20 stocks? 100 stocks?

Answer 1

If alpha of the portfolio is +2.5%, then return in case of Long position,

R(p) = alpha(p) + beta(market risk premium)

= 2.5% + 1.7(market risk premium)

If alpha of the portfolio is -2.5%, then return in case of Short position,

R(p) = alpha(p) + beta(market risk premium)

= -2.5% + 1.7(market risk premium)

Aggregate return = [2.5% + 1.7(market risk premium)] - [ -2.5% + 1.7(market risk premium)]

= 2.5% + 1.7(market risk premium) + 2.5% - 1.7(market risk premium)]

= 5%

Hence, R(p) = 0.05*1,000,000

= $50,000

The answer does not change if instead of 20 stocks, 50 or 100 stocks are examined because the beta and standard deviation, or in other words, market sensitivity is the same for all stocks as given in question.