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Investments 3.4 March 2010 with answers

# Whatisthedollarvalueofthesepositions

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Unformatted text preview: ic source of risk for the three assets: σ2(ei) = 0.01 for all i You can assume that the idiosyncratic sources of risk are uncorrelated, and that rM and Liq are uncorrelated as well. The loadings on the liquidity factor for the three stocks are given by: Stocks γ A 0.7 B 0.9 C 0.3 ‐ Compute the systematic risk of stocks A, B and C. σ2A(syst) = 1.22 x 0.04 + 0.72 x 0.02 = 0.067; σ2B(syst) = 12 x 0.04 + 0.92 x 0.02 = 0.056; σ2C(syst) = 0.82 x 0.04 + 0.32 x 0.02 = 0.027; ‐ Construct an equally weighted portfolio of the three stocks and compute the variance of its non‐systematic risk component. How does it compare to the non‐systematic risk of each individual stock? σ2Portfolio(non‐syst) = 1/3 x 0.01 = 0.0033 (lower than the variance of the non‐systematic component for the individual stocks, equal to 0.01). Part e. (4 points) Taking the same 3 stocks from Part b and d, construct a portfolio that has an exposure of 1 to the liquidity factor and 0 to the market factor. Short sales are allowed. Let wA, wB and wC be the weights of that portfolio. They satisfy the following system of equations: wA + wB + wC = 1 1.2 x wA + 1 x wB + 0.8 x wC = 0 (zero market exposure) 9 0.7 x wA + 0.9 x wB + 0.3 x wC = 1 (unity liquidity risk exposure) So that wA = ‐3.875, wB = 3.75, wC = 1.125. QUESTION 2. (10 points) Portfolio construction and performance measurement Part a. (4 points) A portfolio has an expected rate of return E(r) of 0.15 and a standard deviation s of 0.15. The risk‐free rate is 6 percent. An investor has the following utility function: U = E(r) ‐ (A/2)s2. Which value of A makes this investor indifferent between the risky portfolio and the risk‐free asset? 0.06 = 0.15 ‐ A/2(0.15)2; 0.06 ‐ 0.15 = ‐A/2(0.0225); ‐0.09 = ‐0.01125A; A = 8; U = 0.15 ‐ 8/2(0.15)2 = 6%; U(Rf) = 6%. Part b. (3 points) You are considering investing \$1,000 in a T‐bill that pays 0.05 and a risky portfolio, P, constructed with 2 risky securities, X and Y. The weights of X and Y in P are 0.60 and 0.40, respectively. X has an expected rate of return of 0.14 and variance of 0.01, and Y has an expected rate of return of 0.10 and a variance of 0.0081. ‐ If you want to form a portfolio with an expected rate of retur...
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