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 ( , ) (i) Quadratic utility: u ( x ) = x x 2 , x < (2 ) 1 , > 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 ) [ w (1 + R x )] 2 = w (1 w ) + (1 2 w ) w R x w 2 R 2 x K.S. Tan/Actsc 372 F08 Modern Portfolio Theory & CAPM p. 25 Motivation for Mean-Variance Optimization (contd) Then V ( x , x ) = E u ( w (1 + R x )) = w (1 w ) + (1 2 w ) w x w 2 ( 2 x + 2 x ) Implication? What can you say about the sign of dV d x and dV d x ? (ii) ( R 1 ,R 2 , ,R N ) MV N ( , ) , then R x N ( x , 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 ) x + 1 2 w 2 u ( w )[ 2 x + 2 x ] = u ( w ) 1 2 w 2 u ( w ) 2 u ( w ) w u ( w ) x ( 2 x + 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 ) x ( 2 x + 2 x ) Further assuming x 2 x , then E[ u ( w (1+ R x ))] = u ( w ) 1 2 w 2 u ( w ) 2 u ( w ) w u ( w ) x 2 x . For risk averse agent: max x E[ u ( w (1 + R x ))] max x 2 x 2 x ) where = 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 ] = Applying Taylor expansion (around ) to u ( W ) : u ( W ) u ( ) + u ( )( W ) + 1 2 u ( )( W ) 2 + 1 3! u ( )( W ) 3 + 1 4! u ( )( W ) 4 + E[ u ( W )] u ( ) + 1 2 u ( ) 2 + 1 3! u ( ) + 1 4! u ( ) + variance: 2 = E( W ) 2 third central moment: = E( W ) 3 , fourth central moment: = E( W ) 4 relate to skewness (i.e....
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MPT-II - Motivation for Mean-Variance Optimization A...

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