Assume that stock market returns have the market index as a common factor, and t
ID: 2764498 • Letter: A
Question
Assume that stock market returns have the market index as a common factor, and that all stocks in the economy have a beta of 2.0 on the market index. Firm-specific returns all have a standard deviation of 32%.
Suppose that an analyst studies 20 stocks, and finds that one-half have an alpha of +3.0%,and the other half an alpha of –3.0%. Suppose the analyst buys $1.0 million of an equally weighted portfolio of the positive alpha stocks, and shorts $1.0 million of an equally weighted portfolio of the negative alpha stocks.
What is the expected profit (in dollars) and standard deviation of the analyst’s profit?
How does standard deviation change if the analyst examines 50 stocks instead of 20 stocks? 100 stocks?
Assume that stock market returns have the market index as a common factor, and that all stocks in the economy have a beta of 2.0 on the market index. Firm-specific returns all have a standard deviation of 32%.
Suppose that an analyst studies 20 stocks, and finds that one-half have an alpha of +3.0%,and the other half an alpha of –3.0%. Suppose the analyst buys $1.0 million of an equally weighted portfolio of the positive alpha stocks, and shorts $1.0 million of an equally weighted portfolio of the negative alpha stocks.
Explanation / Answer
A.
SHORTING EQUALLY THE 10 NEGATIVE-ALPHA STOCKS AND INVESTING THE PROCEEDS EQUALLY IN THE 10 POSITIVE-ALPHA STOCKSELIMINATES THE MARKET EXPOSURE AND CREATES A ZERO-INVESTMENT PORTFOLIO.USING EQUATION 8.5, AND DENOTING THE MARKETFACTOR AS RM,THE EXPECTED DOLLAR RETURN IS (NOTING THAT THE EXPECTATION OF RESIDUAL RISK,E, IN EQUATION 8.5 IS ZERO)
$1000000´[0.03 + 2.0´RM] – $1000000´[–0.03 + 2.0´RM]
= $1000000 * 0.06 = $60000
THE SENSITIVITY OF THE PAYOFF OF THIS PORTFOLIO TO THE MARKET FACTOR IS ZERO BECAUSE THE EXPOSURES OF THE POSITIVE ALPHA ANDNEGATIVE ALPHA STOCKS CANCELS OUT.(NOTICE THAT THE TERMS INVOLVING RM SUM TO ZERO.)THUS, THE SYSTEMATIC COMPONENT OFTOTAL RISK ALSO IS ZERO.THE VARIANCE OF THE ANALYST'S PROFIT IS NOT ZERO, HOWEVER, SINCE THIS PORTFOLIO IS NOT WELL DIVERSIFIED.
FOR N = 20 STOCKS (I.E., LONG 10 STOCKS AND SHORT 10 STOCKS) THE INVESTOR WILL HAVE A $100000 POSITION(EITHER LONG OR SHORT) IN EACH STOCK. NET MARKET EXPOSURE IS ZERO, BUT FIRM-SPECIFIC RISK HAS NOT BEEN FULLY DIVERSIFIED.THE VARIANCE OF DOLLAR RETURNS FROM THE POSITIONS IN THE 20 FIRMS IS
20´[(100000´0.32)2] = 20480000000
AND THE STANDARD DEVIATION OF DOLLAR RETURNS IS $143108
B.
IF N = 50 STOCKS (25 LONG AND 25 SHORT), $40000 IS PLACED IN EACH POSITION, AND THE VARIANCE OF DOLLAR RETURNS IS
50´[(40000´0.32)2] = 8192000000
THE STANDARD DEVIATION OF DOLLAR RETURNS IS $90509
SIMILARLY, IF N = 100 STOCKS (50 LONG AND 50 SHORT), $20000 IS PLACED IN EACH POSITION, AND THE VARIANCE OF DOLLARRETURNS IS
100´[(20000´0.32)2] = 4096000000
THE STANDARD DEVIATION OF DOLLAR RETURNS IS $64000.
NOTICE THAT WHEN THE NUMBER OF STOCKS INCREASED BY A FACTOR OF 5, FROM 20 TO 100, STANDARD DEVIATION FELL BY A FACTOR OF =2.236, FROM $143108 TO $64000