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 43%.

Suppose an analyst studies 20 stocks and finds that one-half have an alpha of 2.4%, and one-half have an alpha of –2.4%. The analyst then buys $1.5 million of an equally weighted portfolio of the positive-alpha stocks and sells short $1.5 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            $

Answer 1

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v 4* - ENG 07-26 18-03-2070 23 VB866 - X UT fix uu us uv UW UX UY UZ VA VB VC VD VE - 844 845 846 847 848 849 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 850 ALL STOCKS HAVE BETA = 1 THE EXPECTED RETURN ON 20 STOCKS = 851 852 853 854 1500000 x 1.024 + 1.0 Rm]- 1500000 x (-024 + 1.0 Rm] 36000 + 1500000Rm + 36000 - 1500000Rm = 72000 855 856 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 857 858 859 860 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 $150,000 position (either long or short) in each stock (1500000/10). Net market exposure is zero, but firm-specific risk has not been fully diversified 861 862 863 The variance from the positions in the 20 firms is : 20 x (150,000 x 0.43)^2 = and the standard deviation is = SQRT (VARIANCE) = 83205000000 288453 864 865 866 a 867 EXPECTED RETURN = STANDARD DEVIATION = ... UTILITY, SHARPE, beta convexity 72000 288453 DGM beta port future INDEX INTL CAP BUD SANDRINE LEASING PORTFOLOQ DIR right clean YIELD bond portfolio ... ►

11 Ale $ v el 4* - ENG 07-26 18-03-2070 23 VB888 X UT fix U us UV UW ux ur uz VA VB VC VO VE 865 866 a 867 868 869 EXPECTED RETURN = STANDARD DEVIATION = 72000 288453 269 b-1 870 If n = 50 stocks (25 long and 25 short), $60,000 is placed in each position (1500000/25) The variance from the positions in the 50 firms is : 50 X (60000 X 0.43)^2 = And the standard deviation is = 33282000000 182434 871 872 873 b- 874 Similarly, if n = 100 stocks (50 long and 50 short), $30,000 (1500000/50) is placed in each position The variance is : 100 x (30000 x 0.43)^2 = The standard deviation is = 875 16641000000 129000 876 877 878 879 880 881 882 883 884 885 886 887 888 ... UTILITY, SHARPE, beta coNvexity DGM beta port future INDEX INTL CAP BUD SANDRINE LEASING PORTFOLOQ DIR right clean YIELD bond portfolio ... :