<p>Let me try solving the problem using basic probability concepts directly. (cowrie6’s idea is correct.) </p><p>Suppose that the weights for securities A and B are W<sub>A</sub> and W<sub>B</sub> respectively, with W<sub>A</sub> + W<sub>B</sub> =1. Given that ρ<sub>AB</sub>= -1 (完全负相关), the problem is to find the optimal weight W<sub>A</sub> (and W<sub>B</sub> ) to minimize the variance of the portfolio consisting of A and B: </p><p> Var(W<sub>A</sub> A + W<sub>B</sub> B) = W<sup>2</sup><sub>A</sub><sup><sub> Var </sub></sup>A + W<sup>2</sup><sub>B</sub> Var B + 2 W<sub>A</sub> W<sub>B</sub> Cov(A,B) </p><p> = W<sup>2</sup><sub>A</sub> б<sup>2</sup><sub>A</sub> + W<sup>2</sup><sub>B</sub> б<sup>2</sup><sub>B</sub> + 2 W<sub>A</sub> W<sub>B</sub> ρ<sub>AB</sub> б<sub>A</sub> б<sub>B </sub> </p><p> = 0.16<sup>2</sup> W<sup>2</sup><sub>A</sub> + 0.12<sup>2 </sup>W<sup>2</sup><sub>B</sub> - 2 W<sub>A</sub> W<sub>B</sub> 0.16 0.12 </p><p> = (0.16 W<sub>A</sub> - 0.12(1-W<sub>A</sub> ) ) <sup>2</sup> = (0.28 W<sub>A</sub> - 0.12) <sup>2</sup> </p><p> Therefore Var(W<sub>A</sub> A + W<sub>B</sub> B) reaches its minimal if 0.28 W<sub>A</sub> - 0.12 = 0 or W<sub>A</sub> = 3/7. (W<sub>B</sub> = 1 - W<sub>A</sub> = 4/7, accordingly) Note that ρ<sub>AB</sub>= -1 is a known fact given in the problem. </p>
[此贴子已经被作者于2009-5-30 7:52:59编辑过]
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