MPT-II - Motivation for Mean-Variance Optimization A...

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Unformatted text preview: Motivation for Mean-Variance Optimization A mean-variance optimizer can be justified based on either of the following : (i) Investor has quadratic utility: (ii) Asset returns follow a multivariate normal distribution ( R 1 ,R 2 , · · · ,R N ) ∼ MV N ( μ , Σ ) (i) Quadratic utility: u ( x ) = x − αx 2 , x < (2 α ) − 1 ,α > Suppose an initial wealth is invested in a portfolio x with random return R x terminal wealth = w (1 + R x ) utility of the terminal wealth: u ( w (1 + R x )) = w (1 + R x ) − α [ w (1 + R x )] 2 = w (1 − αw ) + (1 − 2 αw ) w R x − αw 2 R 2 x K.S. Tan/Actsc 372 F08 Modern Portfolio Theory & CAPM – p. 25 Motivation for Mean-Variance Optimization (cont’d) Then V ( μ x ,σ x ) = E u ( w (1 + R x )) = w (1 − αw ) + (1 − 2 αw ) w μ x − αw 2 ( σ 2 x + μ 2 x ) Implication? What can you say about the sign of dV dμ x and dV dσ x ? (ii) ( R 1 ,R 2 , · · · ,R N ) ∼ MV N ( μ , Σ ) , then R x ∼ N ( μ x ,σ 2 x ) K.S. Tan/Actsc 372 F08 Modern Portfolio Theory & CAPM – p. 26 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: u [ w (1 + R x )] ≈ u ( w ) + w u ′ ( w ) · R x + 1 2 w 2 u ′′ ( w ) · R 2 x . Ignoring third and higher order terms, E[ u ( w (1 + R x ))] ≈ u ( w ) + w · u ′ ( w ) · μ x + 1 2 w 2 · u ′′ ( w )[ σ 2 x + μ 2 x ] = u ( w ) − 1 2 w 2 · u ′′ ( w ) − 2 u ′ ( w ) w u ′′ ( w ) μ x − ( σ 2 x + μ 2 x ) . K.S. Tan/Actsc 372 F08 Modern Portfolio Theory & CAPM – p. 27 u ( w ) − 1 2 w 2 · u ′′ ( w ) − 2 u ′ ( w ) w u ′′ ( w ) μ x − ( σ 2 x + μ 2 x ) Further assuming μ x ≪ σ 2 x , then E[ u ( w (1+ R x ))] = u ( w ) − 1 2 w 2 · u ′′ ( w ) − 2 u ′ ( w ) w u ′′ ( w ) μ x − σ 2 x . For risk averse agent: max x E[ u ( w (1 + R x ))] ⇔ max x 2 τμ x − σ 2 x ) where τ = − u ′ ( w ) w u ′′ ( w ) which reduces to “formulation III" of the optimization. K.S. Tan/Actsc 372 F08 Modern Portfolio Theory & CAPM – p. 28 Investor Preferences & Distributional Moments Let ˜ W be the terminal wealth with E[ ˜ W ] = μ Applying Taylor expansion (around μ ) to u ( ˜ W ) : u ( ˜ W ) ≈ u ( μ ) + u ′ ( μ )( ˜ W − μ ) + 1 2 u ′′ ( μ )( ˜ W − μ ) 2 + 1 3! u ′′′ ( μ )( ˜ W − μ ) 3 + 1 4! u ′′′′ ( μ )( ˜ W − μ ) 4 + · · · ⇒ E[ u ( ˜ W )] ≈ u ( μ ) + 1 2 u ′′ ( μ ) σ 2 + 1 3! u ′′′ ( μ ) γ + 1 4! u ′′′′ ( μ ) κ + · · · variance: σ 2 = E( ˜ W − μ ) 2 third central moment: γ = E( ˜ W − μ ) 3 , fourth central moment: κ = E( ˜ W − μ ) 4 relate to skewness (i.e....
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This note was uploaded on 10/30/2011 for the course ACTSC 372 taught by Professor Maryhardy during the Fall '09 term at Waterloo.

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MPT-II - Motivation for Mean-Variance Optimization A...

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