28 0 w20u ww0 x u 2 x 2 x 2 further assuming

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Unformatted text preview: w0 Rx − αw0 Rx K.S. Tan/Actsc 372 F11 Modern Portfolio Theory & CAPM – p. 26 Motivation for Mean-Variance Optimization (cont’d) Then V (µx , σx ) = Eu(w0 (1 + Rx )) 22 = w0 (1 − αw0 ) + (1 − 2αw0 )w0 µx − αw0 (σx + µ2 ) x Implication? What can you say about the sign of dV dµx and dV dσx ? (ii) (R1 , R2 , · · · , RN ) ∼ M V N (µ, Σ), then 2 Rx ∼ N (µx , σx ) K.S. Tan/Actsc 372 F11 Modern Portfolio Theory & CAPM – p. 27 Mean-Variance Approximation The assumptions of quadratic utility is counter-intuitive and normally distributed returns are empirically rejected! The mean-variance framework can still be approximately valid for arbitrary preferences and distribution of returns! From the Taylor expansion: 12 2 u[w0 (1 + Rx )] ≈ u(w0 ) + w0 u￿ (w0 ) · Rx + w0 u￿￿ (w0 ) · Rx . 2 Ignoring third and higher order terms, E[u(w0 (1 + Rx ))] 12 2 ≈ u(w0 ) + w0 · u￿ (w0 ) · µx + w0 · u￿￿ (w0 )[σx + µ2 ] x 2 ￿ ￿ 1 2 ￿￿ 2u￿ (w0 ) 2 2 = u(w...
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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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