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 1.6 on the market index. Firm-specific returns all have a standard deviation of 20%. Suppose that an analyst studies 20 stocks and finds that one-half of them have an alpha of +2.4%, and the other half have an alpha of −2.4%. 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 40 stocks instead of 20 stocks? 100 stocks? (Do not round intermediate calculations. Round your answers to the nearest whole dollar amount.)

Answer 1

Answer :

a - i) $48,000

a - ii) $89,443

b - i) $56,569

b - ii) $40,000

Explanation :

Part I

Calculate the expected profit and standard deviation of the analyst's profit as follows :

= $1,000,000 [ 0.024 + (1.6 x Rm) ] - $1,000,000 [ 0.024 + (1.6 x Rm) ]

= $1,000,000 x 0.048

= $ 48,000

The investor will have a $100,000 position i.e. ($1,000,000 / 10 )  (either long or short) in each stock where the number of stock is 20 i.e. 10 short and 10 long positions in different stocks.

The variance of dollar returns from the positions in the 20 firms is:

The standard deviation of dollar returns is calculated as a square root of variance at  ~ $89,443

Part II

Evaluate the change to 20 stocks and 100 stocks as follows :

If the number of stocks are 50 (i.e., 25 long and 25 short), $40,000 is placed in each position, and the variance of dollar returns is:

The standard deviation of dollar returns is $56,569

The standard deviation of dollar returns is calculated as a square root of variance at  ~ $56,569

If the number of stock are 100 (i.e., 50 long and 50 short), $20,000 is placed in each position, and the variance of dollar returns is:

The standard deviation of dollar returns is calculated as a square root of variance at  ~ $40,000