S tanactsc 372 f11 modern portfolio theory capm p 41

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Unformatted text preview: T = xT Σx ˆ ˆ 0Σ x where 0T = (0, 0, . . . , 0) ∈ ￿N . K.S. Tan/Actsc 372 F11 Modern Portfolio Theory & CAPM – p. 37 Method I: via Standard Optimization Method (cont’d) 2 Recall from Markowitz model: max{2τ µx − σx } In the presence of risk-free asset, the optimization becomes 2 ˆ ˆ ˆ ˆˆ max {2τ µx − σx } = max {2τ µT x − xT Σx} ˆ ˆ ˆ x∈￿N +1 ˆ x∈￿N +1 ˆˆ subject to eT x = 1, The method of Lagrangian can similarly be used to solve the above optimization problem, but need to impose additional assumptions: ˆ (i) Σ is positive definite, but Σ is positive semi-definite, (ii) r ￿= µi for some i ∈ {1, . . . , N }. K.S. Tan/Actsc 372 F11 Modern Portfolio Theory & CAPM – p. 38 Method II: via Capital Allocation Line K.S. Tan/Actsc 372 F11 Modern Portfolio Theory & CAPM – p. 39 Efficient Frontier: Markowitz with Risk-free asset Efficient Frontier: See Figure 11.9 Implications: One-Fund Theorem: There is an unique optimal risky portfolio (or fund) P s...
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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.

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