Question

Assume the return on 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 42%. Suppose an analyst studies 20 stocks and finds that one-half have an alpha of 3.4%, and one-half have an alpha of –3.4%. The analyst then buys $1.4 million of an equally weighted portfolio of the positive-alpha stocks and sells short $1.4 million of an equally weighted portfolio of the negative-alpha stocks.
Required:
a. What is the expected return (in dollars), and what is the standard deviation of the analyst’s profit?
b. How does your answer change if the analyst examines 50 stocks instead of 20?
c. How does your answer change if the analyst examines 100 stocks instead of 20?

Answer 1

Answer:

a. The expected return, and the standard deviation of the analyst’s profit is $95,200 and $262,962.

b. If the analyst examines 50 stocks instead of 20 the Standard deviation would be $ 166,312

c. If the analyst examines 100 stocks instead of 20 the Standard deviation would be $ 117,600

Explanation:

a. In order to calculate the expected return and the standard deviation of the analyst’s profit we would have to make the following calculations:

Expected Return = 1400000*(3.4% + 1*Rm) - 1400000*(-3.4% + 1*Rm)

Expected Return = 47600 + 1400000Rm +47600 - 1400000Rm

Expected Return = $ 95,200

Equal Investment = 1400000/10 = 140000

Variance = 20*((140000*42%)^2) = $ 69,148,800,000

Standard deviation = Variance^(1/2)

Standard deviation = 69,148,800,000^(1/2)

Standard deviation = $ 262,962

b. if n= 50 Stock. then:

Equal Investment = 1400000/25 = 56000

Variance = 50*((56000*42%)^2) = $ 27,659,520,000

Standard deviation = Variance^(1/2)

Standard deviation = 27,659,520,000^(1/2)

Standard deviation = $ 166,312

c. if n= 100 Stock, then:

Equal Investment = 1400000/50 = 28000

Variance = 100*((28000*42%)^2) = $ 13,829,760,000

Standard deviation = Variance^(1/2)

Standard deviation = 13,829,760,000^(1/2)

Standard deviation = $ 117,600