Assume a market index represents the common factor and all stocks in the economy

Question

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 44%. Suppose an analyst studies 20 stocks and finds that one-half have an alpha of 4.4%, and one-half have an alpha of –4.4%. The analyst then buys $1.6 million of an equally weighted portfolio of the positive-alpha stocks and sells short $1.6 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.) Expected Return $ Standard Deviation $ b-1. 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.) Standard Deviation $ b-2. How does your answer change if the analyst examines 100 stocks instead of 20? (Enter your answer in dollars not in millions.) Standard Deviation $

Explanation / Answer

Part a)

The expected return in dollars can be calculated with the use of following formula:

Expected Dollar Return = Investment in Stocks*[Positive Alpha + 1*Market Factor] - Investment in Stocks*[Negative Alpha + 1*Market Factor]

The standard deviation can be calculated with the use of following formula:

Standard Deviation = (Number of Stocks*[(Investment in Each Stock*Standard Deviation of Firm Specific Returns)^2])^(1/2)

___________

Here, Investment in Stock = 1,600,000 and Positive Alpha = 4.4% and Negative Alpha = -4.4%

Using these values in the above formula, we get,

Expected Dollar Return = 1,600,000*(.044 + 1*Market Factor) - 1,600,000*(-.044 + 1*Market Factor)

Expected Dollar Return = 70,400 + 1,600,000*Market Factor - (-70,400) - 1,600,000*Market Factor

Expected Dollar Return = $140,800

___________

Here, Number of Stocks = 20, Investment in Each Stock = (1,600,000*2)/20 = $160,000 and Standard Deviation of Firm Specific Returns = 44%

Standard Deviation = (20*[(160,000*44%)^2])^(1/2) = $314,838

___________

Part b-1

Here, Number of Stocks = 50, Investment in Each Stock = (1,600,000*2)/50 = $64,000 and Standard Deviation of Firm Specific Returns = 44%

Standard Deviation = (50*[(64,000*44%)^2])^(1/2) = $199,121

___________

Part b-2

Here, Number of Stocks = 100, Investment in Each Stock = (1,600,000*2)/100 = $32,000 and Standard Deviation of Firm Specific Returns = 44%

Standard Deviation = (100*[(32,000*44%)^2])^(1/2) = $140,800

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.