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 1.8 on the market index. Firm-specific returns all have a standard deviation of 40%.

Suppose that an analyst studies 20 stocks and finds that one-half of them have an alpha of +2.8%, and the other half have an alpha of −2.8%. 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? (Do not round intermediate calculations. Round your answers to the nearest whole dollar amount.)

b.How does your answer change if the analyst examines 60 stocks instead of 20 stocks? 120 stocks? (Do not round intermediate calculations. Round your answers to the nearest whole dollar amount.)

Answer 1

Answer (a) A zero investment portfolio is created because half of the stocks have a negative alpha and half of the stocks have positive alpha.Hence the exposure to the market is nullified or removed.

The expected market return is given by the formula

Return = Investment[ ₁+(w* R_m)] - Investment[ ₂ + (w * R_m)]

Where Investment means the amount invested in each of the stocks = 1000000

Alpha of first half of the stocks is ₁ = +2.8%

Alpha of second half of the stocks is ₂ = -2.8%

w represents the weight of each stocks = 1

Rm is the market return.

Return = 1000000[ .028 + (1*Rm)] - 1000000[ .028 +(1*Rm)]

= 1000000[.028+ Rm + .028 - Rm]

= 1000000*.056 = $ 56000

Hence the expected profit is $ 56000

Here we have to note that the systematic component of the total risk is zero. Also the SD of the analysts portfolio is not zero since the portfolio is not very well diversified. However the market beta component is nullified due to equal investment in the long and short positions.

The standard deviation of the analysts profit will be given by

SD = {n*(investment* SD of the firm)^2}^(1/2)

SD= { 20(100000*.40)^2}^(1/2) = 1788854

If 60 stocks are examined then

Amount invested in each (30 long and 30 short) = 1000000/30 = $33333

SD = {n*(investment* SD of the firm)^2}^(1/2) = {30*(33333*.40)^2}^(1/2) = 73028.94 = $73029