Problem 7-29 Assume a market index represents the common factor and all stocks i
ID: 2715861 • Letter: P
Question
Problem 7-29
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 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.
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.)
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.)
How does your answer change if the analyst examines 100 stocks instead of 20? (Enter your answer in dollars not in millions.)
References
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 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.
Explanation / Answer
a)
Expected Return = 1400000*(3.4% + 1*Rm) - 1400000*(-3.4% + 1*Rm)
Expected Return = 47600 + 1400000Rm +47600 - 1400000Rm
Expected Return = $ 95200
n= 20 Stock , than 10 Long and 10 short ,
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-1) if n= 50 Stock , than 25 Long and 25 short ,
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
b-2) if n= 100 Stock , than 50 Long and 50 short ,
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