C this statement is correct 5 substituting the

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c. This statement is correct. 5. Substituting the portfolio returns and betas in the expected return-beta relationship, we obtain two equations with two unknowns, the risk-free rate (r f ) and the factor risk premium (RP): 12 = r f + (1.2 × RP) 9 = r f + (0.8 × RP) Solving these equations, we obtain: r f = 3% and RP = 7.5% 6. a. Shorting an equally-weighted portfolio of the ten negative-alpha stocks and investing the proceeds in an equally-weighted portfolio of the ten positive- alpha stocks eliminates the market exposure and creates a zero-investment portfolio. Denoting the systematic market factor as R M , the expected dollar return is (noting that the expectation of non-systematic risk, e , is zero): $1,000,000 × [0.02 + (1.0 × R M )] $1,000,000 × [(–0.02) + (1.0 × R M )] = $1,000,000 × 0.04 = $40,000 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 cancel out. (Notice that the terms involving R M sum to zero.) Thus, the systematic component of total risk is also 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. 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 stocks is: 20 × [(100,000 × 0.30) 2 ] = 18,000,000,000 The standard deviation of dollar returns is $134,164.
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b. If n = 50 stocks (25 stocks long and 25 stocks short), the investor will have a $40,000 position in each stock, and the variance of dollar returns is: 50 × [(40,000 × 0.30) 2 ] = 7,200,000,000 The standard deviation of dollar returns is $84,853. Similarly, if n = 100 stocks (50 stocks long and 50 stocks short), the investor will have a $20,000 position in each stock, and the variance of dollar returns is: 100 × [(20,000 × 0.30) 2 ] = 3,600,000,000 The standard deviation of dollar returns is $60,000. Notice that, when the number of stocks increases by a factor of 5 (i.e., from 20 to 100), standard deviation decreases by a factor of 5 = 2.23607 (from $134,164 to $60,000). 7 a. ) e ( 2 2 M 2 2 σ + σ β = σ 881 25 ) 20 8 . 0 ( 2 2 2 2 A = + × = σ 500 10 ) 20 0 . 1 ( 2 2 2 2 B = + × = σ 976 20 ) 20 2 . 1 ( 2 2 2 2 C = + × = σ b. If there are an infinite number of assets with identical characteristics, then a
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