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 47%.

Suppose an analyst studies 20 stocks and finds that one-half have an alpha of 4.5%, and one-half have an alpha of –4.5%. 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 for standard deviation 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.)

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

Answer 1

a). excess return R(p) = alpha + (beta*Rm) where Rm is the systematic market factor.

Expected return (in dollars) = dollar value of long portfolio - dollar value of short portfolio

= amount invested*excess return - amount invested*excess return

= 1,500,000*[+4.5% + (1*Rm)] - 1,500,000*[-4.5% + (1*Rm)] = 2*4.5%*1,500,000 = 135,000

Total exposure = amount invested = 2*1,500,000 = 3,000,000

Amount invested in each stock = 3,000,000/20 = 150,000

Dollar variance = number of stocks*[standard deviation*amount invested]^2 = 20*(47%*150,000)^2 = 99.405 billion

Standard deviation = dollar variance^0.5 = (99.405 billion)^0.5 = 315,285.58 or 315,286

b-1). Total exposure is still 3,000,000

Amount invested in each stock = 3,000,000/50 = 60,000

Dollar variance = 50*(60,000*47%)^2 = 39.762 billion

Standard deviation = (39.762 billion)^0.5 = 199,404.11 or 199,404

b-2). Amount invested = 3,000,000/100 = 30,000

Dollar variance = 100*(30,000*47%)^2 = 19.881 billion

Standard deviation = (19.881 billion)^0.5 = 141,000