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 1.0 on the market index. Firm-specific returns all have a standard deviation of 22%. Suppose that an analyst studies 20 stocks and finds that one-half of them have an alpha of +1.5%, and the other half have an alpha of -1.5%. Suppose the analyst invests $1.0 million in an equally weighted portfolio of the positive alpha stocks, and shorts $1 million of an equally weighted portfolio of the negative alpha stocks. a. What is the expected profit (in dollars) and standard deviation of the analyst's profit? (Do not round intermediate calculations. Round your answers to the nearest whole dollar amount.) Expected profit (in dollars) Standard deviation b. How does your answer change if the analyst examines 40 stocks instead of 20 stocks? 100 stocks? (Do not round intermediate calculations. Round your answers to the nearest whole dollar amount.) 40 stocks Standard deviation 100 stocks 40 stocks 100 stocks Standard deviation

Answer 1

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w v * 09-10 ENG 16-03-2020 ENG 16 22 VB741 : X URUS fax UT UU UVUW UX UY UZ VA VB VC VD 741 743 If alpha = a(p) then for the long component we have R(p) = a(p) + B(p)*R(m) Variance = no of stocks x (position in each stock x standard deviation) Shorting equally the 10 negative-alpha stocks and investing the proceeds equally in the 10 positive-alpha stocks eliminates the market exposure and creates a zero-investment portfolio 745 746 747 ALL STOCKS HAVE BETA = 1 748 749 THE EXPECTED RETURN ON 20 STOCKS = 750 1000000 x [.015 + 1.0 Rm]– 1000000 x [-015 + 1.0 Rm) 15000 + 1000000Rm + 15000 - 1000000Rm = 30000 751 752 753 The sensitivity of the payoff of this portfolio to the market factor is zero because the exposures of the positive alpha and negative alpha stocks cancels out 754 755 756 757 758 Thus, the systematic component of total 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 $100,000 position (either long or short) in each stock (1000000/10). Net market exposure is zero, but firm-specific risk has not been fully diversified 759 The variance from the positions in the 20 firms is : 20 x (100,000 x 0.22)^2 = and the standard deviation is = SQRT (VARIANCE) = 9680000000 98387 761 762 763 764 EXPECTED RETURN = STANDARD DEVIATION = ... UTILITY, SHARPE, beta convexity DGM beta 30000 98387 port future INDEX INTL CAP BUD SANDRINE LEASING PORTFOLOQ DIR right clean YIELD bond portfolio ... ►

w Q v 1 ENG 09:11 16-03-2020 22 VB782 -X UR us UT W UV UW Ux ur uz VA VB vc VD - The variance from the positions in the 20 firms is : 20 x (100,000 x 0.22)^2 = and the standard deviation is = SQRT (VARIANCE) = 9680000000 98387 759 760 761 762 763 764 765 766 767 768 EXPECTED RETURN = STANDARD DEVIATION = 30000 98387 b-1 If n = 40 stocks (20 long and 20 short), $50000 is placed in each position (1000000/20) The variance from the positions in the 40 firms is: 40 X (50000 X 0.22)^2 = And the standard deviation is = 4840000000 69570 769 b-2 770 771 Similarly, if n = 100 stocks (50 long and 50 short), $20000 (1000000/50) is placed in each position The variance is : 100 x ($20000 x 0.22)^2 = The standard deviation is = 772 1936000000 44000 773 774 775 776 777 778 779 780 781 782 ... UTILITY, SHARPE, beta coNvexity DGM beta port future INDEX INTL CAP BUD SANDRINE LEASING PORTFOLOQ DIR right clean YIELD bond portfolio ... :