Assume a market index represents the common factor and all stocks in the economy

Question

Assume a market index represents the common factor and all stocks in the economy have a beta of 1. Firm-specific returns all have a standard deviation of 35%.

Suppose an analyst studies 20 stocks and finds that one-half have an alpha of 4.1%, and one-half have an alpha of –4.1%. The analyst then buys $1.5 million of an equally weighted portfolio of the positive-alpha stocks and sells short $1.5 million of an equally weighted portfolio of the negative-alpha stocks.

a. What is the expected return (in dollars), and what is the standard deviation of the analyst’s profit? (Enter your answers in dollars not in millions. Do not round intermediate calculations. Round your answers to the nearest dollar amount.)

b-1. How does your answer change if the analyst examines 50 stocks instead of 20? (Enter your answer in dollars not in millions. Do not round intermediate calculations. Round your answer to the nearest dollar amount.)

Standard deviation            $

b-2. How does your answer change if the analyst examines 100 stocks instead of 20? (Enter your answer in dollars not in millions.)

Standard deviation            $

Explanation / Answer

a. Shorting equally the 10 negative-alpha stocks and investing the proceeds equally in the 10 positive-alpha stocks eliminates the market exposure and creates a zero-investment portfolio. Denoting the market factor as M, the expected dollar return is =$1,500,000´[0.041 + 1.0´M]–$1,500,000´[–0.041 + 1.0´M]= 1,500,000x0.082 = $123,000

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 cancels out. Thus, the systematic component of total risk also is zero.

The variance of the analyst's profit is not zero, however, since this portfolio is not well diversified. For n = 20 stocks (i.e., long 10 stocks and short 10 stocks) the investor will have a $150,000 position (either long or short) in each stock.Net market exposure is zero, but firm-specific risk has not been fully diversified.

The variance of dollar returns from the positions in the 20 firms is 20´[(150,000´ 0.35)^2] = 55,125,000,000

The standard deviation of dollar returns is $234,787.

b-1. If n = 50 stocks (25 long and 25 short), $60,000 is placed in each position, and the variance of dollar returns is 50´[(60,000´ 0.35)^2] = 22,050,000,000.

The standard deviation of dollar returns is $148,492.

b-2. Similarly, if n = 100 stocks (50 long and 50 short), $30,000 is placed in each position, and the variance of dollar returns is 100´[(30,000´.35)^2] = 11,025,000,000

The standard deviation of dollar returns is $105,000.

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