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Lecture04-1996

# Lecture04-1996 - Lecture V Derivation of the...

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1 Lecture V: Derivation of the Minimum-Variance Frontier, Minimum- Variance Portfolio, and the Efficient Set with a Risk-Free Asset I. Derivation of the Minimum-Variance Frontier A. The minimum-variance frontier is the locus of points that represent the minimum variance for a given expected return. z σ B. Mathematical derivation 1. General Formulation 0 0 2 1 1 N i i i N N i j ij i j z w z w R w z w w w w σ σ = = = = + = = Σ = ∑∑ 2. Mean-variance frontier without a risk-free asset: 1 min 2 . . 1 1 w w s t w z w µ ′Σ = = Forming the Lagrangian: ( ) ( ) 1 1 2 w w L w z w λ γ µ ′Σ + + First-order necessary conditions ( ) ( ) ( ) 1 2 w w w w w L w z w λ γ ′Σ = ∇

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2 [ ] [ ] ( ) ( ) ( ) 11 12 13 1 1 2 3 12 22 23 2 13 23 33 3 1 11 2 12 3 13 1 2 3 1 12 2 22 3 23 1 13 2 23 3 33 1 1 11 2 12 3 13 2 1 12 2 22 3 23 3 1 13 2 23 3 33 w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ     ′Σ =       + + = + + + + = + + + + + + + + ( ) ( ) ( ) ( ) 1 1 11 2 12 3 13 1 11 2 12 3 13 1 12 2 22 3 23 1 12 2 22 3 23 2 1 13 2 23 3 33 1 13 2 23 3 33 3 1 11 2 12 3 13 1 12 2 2 2 w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ Σ + + + + + Σ Σ = = + + + + + + + + + + Σ + + Σ = + ( ) ( ) 22 3 23 1 13 2 23 3 33 2 w w w w σ σ σ σ + + + A little matrix calculus ( ) [ ] 11 12 13 1 12 22 23 2 13 23 33 3 1 11 2 12 3 13 1 12 2 22 3 23 1 13 2 23 3 33 11 12 13 1 2 3 12 22 23 13 23 33 1 11 2 12 3 13 1 w w w w w w w w w w w w w w w w w w w w w w w w w w σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ Σ = Σ + Σ     Σ =       + + = + + + +
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