xg2185 HW4 - IEOR4721 Global Capital Markets Assignment#4...

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1 / 6 IEOR4721 Global Capital Markets Assignment #4 Uni: xg2185, Name: Xiao Guo Q1: (a) var(r ? ) = 𝜎 ? 2 = 0.1 2 = 0.01 var(r ? ) = 𝜎 ? 2 = 0.2 2 = 0.04 var(r ? ) = 𝜎 ? 2 = 0.3 2 = 0.09 cov(r ? ,r ? ) = 𝜌 ?? ∗ 𝜎 ? 𝜎 ? = 0 cov(r ? ,r ? ) = 𝜌 ?? ∗ 𝜎 ? 𝜎 ? = 0.5 ∗ 0.1 ∗ 0.3 = 0.015 cov(r ? , r ? ) = 𝜌 ?? ∗ 𝜎 ? 𝜎 ? = 0.5 ∗ 0.2 ∗ 0.3 = 0.03 Cov(r ?? , r 1 ) = w ? ∗ w ? ∗ var(r ? ) + w ? ∗ w ? ∗ var(r ? ) + w ? ∗ w ? ∗ var(r ? ) + (w ? ∗ w ? + w ? ∗ w ? ) ∗ cov(r ? , r ? )+(w ? ∗ w ? + w ? ∗ w ? ) ∗ cov(r ? ,r ? )+(w ? ∗ w ? + w ? ∗ w ? ) ∗ cov(r ? , r ? ) = (0.887 ∗ 0.294) ∗ 0.001 + 0.287 ∗ 0.412 ∗ 0.04 − 0.174 ∗ 0.294 ∗ 0.09 + (0.887 ∗ 0.412 + 0.287 ∗ 0.294) ∗ 0 + (0.887 ∗ 0.294 − 0.174 ∗ 0.294) ∗ 0.015 + (0.287 ∗ 0.294 − 0.174 ∗ 0.412) ∗ 0.03 = 0.00626 (b) Let w be the weight of GMV, 1-w be the weight of Portfolio 1. The expect return of the portfolio is: E(r ? ) = w ∗ 0.047 + (1 − w) ∗ 0.1 The variance of the portfolio return is: Var(r ? ) = w 2 ∗ 𝜎 ?? 2 + (1 − w) 2 ∗ 𝜎 1 2 + 2 ∗ w ∗ (1 − w) ∗ Cov(r ?? , r 1 ) = w 2 ∗ 0.079 2 + (1 − w) 2 ∗ 0.159 2 + 2 ∗ w ∗ (1 − w) ∗ 0.00626 Then we can plot the mean variance frontier with Excel: 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.005 0.01 0.015 0.02 0.025 0.03 mean mean This study resource was shared via CourseHero.com
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2 / 6 (c) By maximizing the sharp ratio, we have the linear programming problem: Max Sharp Ratio = ?(? ? ) σ ? E(r ? ) = w ∗ 0.047 + (1 − w) ∗ 0.1 σ ? = w 2 ∗ 𝜎 ?? 2 + (1 − w) 2 ∗ 𝜎 1 2 + 2 ∗ w ∗ (1 − w) ∗ Cov(r ?? ,r 1 ) ∗ 𝜎 ?? ∗ 𝜎 1 The answer is: w = ?[? ?? − ? 𝑓 ] ∗ 𝜎 1 2 − ?[? 1 − ? 𝑓 ] ∗ 𝜎 1,?? ?[? ?? − ? 𝑓 ] ∗ 𝜎 1 2 + ?[? 1 − ? 𝑓 ] ∗ 𝜎 ?? 2 − (?[? ?? − ? 𝑓 ] + ?[? 1 − ? 𝑓 ]) ∗ 𝜎 1,?? = −45.539% Therefore the weight of the portfolio 1 is 1-w=1.45539 So , the structure of the MVE portfolio is: A: −0.45539 ∗ 0.887 + 1.45539 ∗ 0.294 = 2.395% B: −0.45539 ∗ 0.287 + 1.45539 ∗ 0.412 = 46.892% C: −0.45539 ∗ −0.174 + 1.45539 ∗ 0.294 = 50.712% (d) The best CAL will combine risk-free asset with the MVE portfolio.
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