Question

Instructions I help < Question 8 (of 10) Save & Ext Submit 10.00 points 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 38% Suppose an analyst studies 20 stocks and finds that one-half have an alpha of 4.2%, and one-half have an alpha of -4,2%. The analyst then buys $10 million of an equally weighted portfolio of the positive-alpha stocks and sells short $1.0 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 207 (Enter your answer in dollars not in millions. Do not round intermediate calculations. Round your answer to the nearest dollar amount.) Standard deviation S b-2. How does your answer change if the analyst examines 100 stocks instead of 207 (Enter your answer in dollars not in millions.) Standard deviation References eBook & Resources Worksheet Learning Objective: 07-04 Take advantage of an arbitrage opportunity with a portfolio that includes mis priced securities

Answer 1

a)
Expected return on 20 stocks = $1,000,000 * (0.042 + ( 1 * Rm)) - $1,000,000 * (-0.042 + (1 * Rm))
= 42,000 + 1,000,000Rm + 42,000 - 1,000,000Rm
= 42,000 + 42,000
= $84,000.

n = 20 stocks ( short 10 stocks and long 10 stocks)
Investor will have $100,000 ($1,000,000/10) in each stock (either long or short)
Market exposure = 0
Firms specific risk has not been fully diversified.

Variance = 20 * (100,000 * 0.38)^2 = 28880000000
Standard deviation = 28880000000 ^0.5 = $169,941

b-1)
n = 50 stocks (25 short and 25 long).
Investor will have$40,000 ($1,000,000/25) in each stock (either long or short).
Variance = 50 * (40,000 * 0.38)^2 = $11552000000
Standard deviation = $11552000000 ^ 0.5 = $107,480.

b-2)
n = 100 stocks (50 short and 50 long).
Investor will have$20,000 ($1,000,000/50) in each stock (either long or short).

Variance = 100 * (20,000 * 0.38)^2 = $5776000000
Standard deviation = $5776000000^0.5 = $76,000.