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Unformatted text preview: 2 F11 Modern Portfolio Theory & CAPM – p. 20 Mean-Variance Optimization Formulations (cont’d)
Formulation III: max 2τ µT x − xT Σx subject to eT x = 1
x the parameter τ ≥ 0 is a measure of the risk tolerance
of an investor
what if τ = 0?
what if τ becomes very large?
the efﬁcient portfolio is optimally obtained for all
τ ∈ [0, ∞)
K.S. Tan/Actsc 372 F11 Modern Portfolio Theory & CAPM – p. 21 Method of Lagrangian Multipliers
maximum/minimum of f (x) subject to
the constraint g (x) = k , where xT = (x1 , . . . , xN ).
Optimization Problem: The solution to the above optimization problem can be
obtained via the method of Lagrangian multiplier:
First, set up the Lagrangian function as follows:
L(x, λ) = f (x) + λ[g (x) − k ] Second, derive the following optimality conditions
= 0 i = 1, 2, . . . , N and
∂λ Then x∗ = (x∗ , . . . , x∗ )T that satisﬁes the above
optimality conditions are necessary and sufﬁcient for
the global optimum.
K.S. Tan/Actsc 372 F11 Modern Portfolio Theory & CAPM – p. 22 An Example on Lagrangian Method K.S. Tan/Actsc 372 F11 Modern Portfolio Theory & CAPM – p. 23 Derivation of Markowitz Model K.S. Tan/Actsc 372 F11 Modern Portfolio Theory & CAPM – p. 24 Markowitz M...
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This note was uploaded on 01/04/2012 for the course ACTSC 372 taught by Professor Maryhardy during the Fall '09 term at Waterloo.
- Fall '09