Mean-Variance Optimization

10232012 p kolm 12 global minimum variance portfolio

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Unformatted text preview: ι + γ μ ′Σ μ 2 2 = λ A + 2λγ B + γ C 2 = = = 2 (C − μ0B ) A + 2(C − μ0B )(μ0A − B )B + (μ0A − B ) C 2 Δ 2 2 2 2 2 2 2 3 22 AC − 2μ0ABC + μ0 AB + 2μ0ABC − 2B C − 2μ0 AB + 2μ0B + μ0 A C − 2μ0ABC + B C 2 Δ 2 2 2 2 3 22 AC − μ0 AB − B C + 2μ0B + μ0 A C − 2μ0ABC 2 2 = (AC − B ) Δ 2 Aμ0 − 2B μ0 + C Δ 2 2 = Aμ0 − 2B μ0 + C Δ RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 11 2 Note that σ* (μ0 ) = 2 Aμ0 − 2B μ0 + C Δ is a function μ0 Thus: ( ) 2 • The efficient portfolios w ∗ form a parabola in the σ* (μ0 ), μ0 -plane • The efficient portfolios w ∗ form a hyperbola in the (σ* (μ0 ), μ0 )-plane o This hyperbola we call the efficient frontier RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 10/23/2012. © P. KOLM. 12 Global Minimum Variance Portfolio The global minimum variance portfolio (GMV) is the portfolio that simply minimizes the variance of the portfolio How do we find it? It can be found by solving 2 d σ*...
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