Question

Assume the return on 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 47%. Suppose an analyst studies 20 stocks and finds that one-half have an alpha of 4.5%, and one-half have an alpha of –4.5%. 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.)

b-1. How does your answer for standard deviation 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.)

b-2. How does your answer for standard deviation change if the analyst examines 100 stocks instead of 20? (Enter your answer in dollars not in millions.)

Answer 1

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ke - Microsoft Excel (Product Activation Failed) Add-Ins File Home Insert Page Layout Formulas Data Review View - 2x % Cut Dr General -11AA = = = D A Wrap Text Merge Merge & Center Center U - SF 4:0 .00 8. Insert Delete Format $ - % % Number Calibri E Copy Paste Format Painter B I Clipboard UT449 I UT Conditional Format Formatting as Table Styles Cell Styles Find & Select Σ AutoSum : A Fill 2 Sort & Clear Filter Editing Font Alignment Cells US uu uv uw ux UV uz VA VB Formula Bar UD VE VE 5 449 450 451 452 453 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 454 455 456 457 458 ALL STOCKS HAVE BETA = 1 459 THE EXPECTED RETURN ON 20 STOCKS = 1500000 x [.045 + 1.0 Rm]- 1500000 x [-.045 + 1.0 Rm] 67500 + 1500000Rm + 67500 - 1500000Rm = 460 135000 461 462 463 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 464 465 466 467 468 469 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 K M convexity DGM betaport future INDEX INTL CAP BUD LEASING PV, FV, ANNUITY DIR right clean YIELD bond portfolio KE CAPM UTILITY, SHARPE, beta ferences: AG470 cap stru WACCRESI el BOJ 120% Rea DOR . * ENG 06-11-2019

ke - Microsoft Excel (Product Activation Failed) Add-Ins File Home Insert Page Layout Formulas Data Review View - 2x % Cut Σ AutoSum : A Fill 2 Sort & Insert Delete Format Cell Styles Clear Filter Editing Find & Select Cells VB VC VD VE VE 5 99405000000 315286 Calibri 11 A A = = = V Wrap Text General E Copy Paste BI U D- DA E E E s FMerce Merge & Center - Center Format Painter $ % % , 4.0 .00 :.: Conditional Format Formatting as Table Clipboard Font Alignment Number Styles VB485 USUT UU UVUW UX UYUZ VA 469 470 The variance from the positions in the 20 firms is : 20 x (150,000 x 0.47)^2 = 471 and the standard deviation is = SQRT (VARIANCE) = 472 473 EXPECTED RETURN = 135000 474 STANDARD DEVIATION = 315286 475 476 b-1 If n = 50 stocks (25 long and 25 short), $60,000 is placed in each position (1500000/25) 477 The variance from the positions in the 50 firms is : 50 X (60000 X 0.47)^2 = 478 And the standard deviation is = 479 480 b-2 Similarly, if n = 100 stocks (50 long and 50 short), $30,000 (1500000/50) is placed in each position 481 The variance is : 100 x ($30,000 x 0.47)^2 = 482 The standard deviation is = 483 484 485 486 487 39762000000 199404 19881000000 141000 488 489 HIM Rea convexity DGM betaport future INDEX INTL CAP BUD LEASING PV, FV, ANNUITY DIR right KE CAPM UTILITY, SHARPE, beta ferences: AG470 clean YIELD bond portfolio cap stru WACCRESI el BO! 120% O ...Rag m * * O ENG 06-11-2019