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 31%. Suppose an analyst studies 20 stocks and finds that one-half have an alpha of 2.0%, and one-half have an alpha of –2.0%. The analyst then buys $1.1 million of an equally weighted portfolio of the positive-alpha stocks and sells short $1.1 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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Home ert Page Layout Formulas Data Review V 義Cut a copy ▼ Add-Ins Σ AutoSum , A -Ξ: 'ila Formattinnalasonnatst el" insert Delete Format Sort &Find & /Format Pan ter B 1 u. 田 Merge & Center. $. % , 2 ClearFlter Select Clipboard Font Cells dting Alignment UR12 UR US UT If alpha-a(p) then for the long component we have R(p) = a(p) + β(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 UV UW 1138 UY UZ VA VB VD 6 ALL STOCKS HAVE BETA THE EXPECTED RETURN ON 20 STOCKS 9 $ 1,100.000 × [.02 + 1.0 ×Rm]_ $1,100,000 x[-02 + 1.0 ×Rm] -$22000$1100000Rm S22000 - $1100000Rm S44000 10 12 13 1.1 15 16 17 18 19 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 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 $110,000 position (either long or short) in each stock (1100000/10). Net market exposure is zero, but firm-specific risk has not been fully diversified covexty DGM port future INDEX INTL CAP BUD LEASING PV,FV, ANNUITY DIR nght dlean YIELD bond portfolocap stru WACC RESI ex clD 4HKE CAPM UTILITY, SHARPE, beta bond erences: AG470 03:50 06-05-2019

Home ert Page Layout Formulas Data Review V Add-Ins % Cut Σ AutoSum , Copy Merge & Center. $, % , 'ila Conditional Format Cell Insert Delete Format Sort &Find & Filter Select ▼ е Format Painter B 1 u. ㄧ˙ タ.A. E- 袖 2 Clear , Formatting ▼ as Table* Styles ▼ Styles ▼ ▼ ▼ Clipboard Font Cells dting Alignment US20 US UT Thus, the systematic component of total risk also is zero. The variance of the analyst's profit is not ero, 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 $110,000 position (either long or short) in each stock (1100000/10). Net market exposure is zero, but firm-specific risk has not been fully diversified UV UW 1138 UY UZ VA VB VC VD 183 17 18 19 20 21 The variance from the positions in the 20 firms is : 20 ×($110,000 x.31)| and the standard deviation is $2888000000. 23256200000 152500 EXPECTED RETURN-$44000 STANDARD DEVIATION $152500 23 24 25 26 27 28 29 30 31 32 (b-1) Ifn- 50 stocks (25 long and 25 short), $44,000 is placed in each position (1100000/25) The variance from the positions in the 50 firms is And the standard deviation is 50 x[(S44,000x.31)9302480000 96449 (b-2) Similarly, ifn-100 stocks (50 long and 50 short), $22,000 (1100000/50) is placed in each position The variance is 100 xI(S22,000 x31)1 The standard deviation is 4651240000 68200 PM UTILITY, SHARPE, beta-bond cove ty DGM port future INDEX IN TL CAP BUD LEASING PV FV ANNUITY DIR ght dean YELD bond porti i ap tru WACC RESI ex d d , erences: AG470 03:51 06-05-2019