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7. 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%. Suppose that an analyst studies 20 stocks, and finds that one-half have an alpha of 12%, and the other half an alpha of 22%. 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 analyst’s profit? b) How does your answer change if the analyst examines 50 stocks instead of 20 stocks? 100 stocks?

The Answer that is given is:

a) Shorting an equally-weighted portfolio of the ten negative-alpha stocks and investing the proceeds in an equally-weighted portfolio of the ten positive-alpha stocks eliminates the market exposure and creates a zero-investment portfolio. Denoting the systematic market factor as RM, the expected dollar return is (noting that the expectation of non-systematic risk, e, is zero):

The sensitivity of the payoff of this portfolio to the market factor is zero because the exposures of the positive alpha and negative alpha stocks cancel out. (Notice that the terms involving RM sum to zero.) Thus, the systematic component of total risk is also zero. The variance of the analyst’s profit is not zero, however, since this portfolio is not well diversified.

The Answer that is given is:

a) Shorting an equally-weighted portfolio of the ten negative-alpha stocks and investing the proceeds in an equally-weighted portfolio of the ten positive-alpha stocks eliminates the market exposure and creates a zero-investment portfolio. Denoting the systematic market factor as RM, the expected dollar return is (noting that the expectation of non-systematic risk, e, is zero):

**$1,000,000 × [0.02 + (1.0 × Rm)] − $1,000,000 × [(−0.02) + (1.0 × Rm)] = $1,000,000 × 0.04 = $40,000**The sensitivity of the payoff of this portfolio to the market factor is zero because the exposures of the positive alpha and negative alpha stocks cancel out. (Notice that the terms involving RM sum to zero.) Thus, the systematic component of total risk is also zero. The variance of the analyst’s profit is not zero, however, since this portfolio is not well diversified.

**Conceptually I can make sense but I am not sure of the return/profit calculation**

I am not able to understand the calculation of [0.02 + (1.0 × Rm)] and what happens to the returns calculated by the analyst? Is it just a sampling bias?I am not able to understand the calculation of [0.02 + (1.0 × Rm)] and what happens to the returns calculated by the analyst? Is it just a sampling bias?

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