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 11 N ii i NN iji j ij zw R w z ww = == =+ = =Σ= ∑∑ 2. Mean-variance frontier without a risk-free asset: 1 min 2 ..1 1 st w µ ′Σ = = Forming the Lagrangian: () 2 Lw z w λγ ≡+ + First-order necessary conditions ( ) 1 2 w z w ∇=

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2 [] () ( ) 11 12 13 1 1231 22 32 13 23 33 3 11 1 21 2 31 3 1 2 3 1 12 2 22 3 23 3 2 2 3 33 3 1 1 11 2 12 3 13 2 1 12 2 22 3 23 1 3 2 2 3 3 3 3 w ww w w w w w ww w wwww w w w w w w w w w σσσ σ σσ   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 2 3 2 2 2 2 w w w w w w w w w w w w w w w w w w ∂Σ +++ ∇Σ = = + + + + + + = + 22 3 23 3 3 3 2 w w + A little matrix calculus 11 12 13 1 12 22 23 2 13 23 33 3 1 2 3 2 2 3 3 3 3 11 12 13 3 13 23 33 1 2 3 1 w ww w w w w w w w w w w w σσσσ ′′ = Σ + ′Σ= + 1 2 23 2 1 2 3 3 1 2 3 2 2 3 3 3 3 www w w + +
3 () 1 11 2 12 3 13 1 11 2 12 3 13 1 12 2 22 3 23 1 12 2 22 3 23 1 13 2 23 3 33 1 13 2 23 3 33 11 1 21 2 31 3 2 2 2 2 3 2 3 3 3 33 3 2 2 2 w ww w w ww w w w w w w w w w w w w w σσσ σ σσ ++   ′′ ∇Σ = Σ = + + + + +  =+ + Looking at the constraints [] 1 2 3 3 1 111 w w w w w == + + or in general 1 1232 1 12 23 3 3 w w w w w αα α + + Taking the gradient of the general form 1 2 3 1 22 3 1 1 w w w w w ∇=     ⇒∇ =       =

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

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