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Unformatted text preview: USC - MARSHALL SCHOOL OF BUSINESS FBE 441 Investments – P. Matos – Spring 2010 Homework Assignment #4 - Solutions 1. Consider a market where two factors are sufficient to describe the returns on common stocks. For an asset i, the asset’s expected return is given by E(r i ) = r f + i1 P 1 i2 P 2 , where P 1 and P 2 are the factor premiums (expected return of the factors in excess of the risk-free rate). Both factors are independent. The following table gives the sensitivities of the stocks ABC and PQR to the two factors, as well as the expected returns of each stock: Security i1 i2 E[r i ] ABC 0.6 0.9 18.3 PQR 1.7 1.3 20.2 Riskless 0.0 0.0 5.0 (a) Consider a portfolio, C, made up by selling short $.40 of security PQR and purchasing $1.40 of ABC. How sensitive will this portfolio be to each of the two factors? SOLUTION: This portfolio puts 140% weight in ABC and –40% weight in PQR. Since the sensitivities (betas) are additive, the beta of a portfolio is the weighted average of component betas, sensitivity of this portfolio with respect to the first factor is 1.4(0.6)- 0.4(1.7) = 0.16. Similarly, sensitivity of this portfolio with respect to the second factor is 1.4(0.9)-0.4(1.3) = 0.74. 74 . 3 . 1 1 4 . 9 . 1 4 . 1 β β β 16 . 7 . 1 1 4 . 6 . 1 4 . 1 β β β 2 , 2 , 2 1 , 1 , C1 PRQ PRQ ABC ABC C PRQ PRQ ABC ABC w w w w (b) Consider a portfolio, D, made up by borrowing $1.00 at the risk free rate and investing $1.00 in portfolio C [as in (3.a)]. How sensitive will this portfolio be to each of the factors? SOLUTION: Adding in the risk-free rate does not alter the sensitivities at all. This portfolio return r D = r C- r f . Hence the portfolio sensitivities: 74 . 74 . 1 1 β 16 . 16 . 1 1 β 2 1 D D (c) What combination of securities ABC, PQR and the riskless security will move on a one-to-one basis with factor 1 and be insensitive to factor 2? SOLUTION: To find a portfolio with unit sensitivity to factor one and insensitive to factor two we require: 3 . 1 9 . 1 7 . 1 6 . PRQ ABC PRQ ABC w w w w Solving these we find: 2 . 1 7333 . 1 PRQ ABC w w The remainder of the portfolio position is held in the risk-free asset: 5333 . 1 7333 . 1 2 . 1 1 1 rf rf PRQ ABC w w w w We have created a factor mimicking portfolio for the first factor (and neutral exposure to factor 2)....
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