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eco331-09-MeanVarianceHandout

# eco331-09-MeanVarianceHandout - Overview Mean-Variance...

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Unformatted text preview: Overview Mean-Variance Preferences Portfolio Choice Eco331: Mean-Variance Analysis Marciano Siniscalchi November 2 Overview Mean-Variance Preferences Portfolio Choice Overview When Mean and Variance are All That Matters The portfolio problem in the Mean-Variance Setting Two-asset problem and Mutual-Fund Theorem Overview Mean-Variance Preferences Portfolio Choice Mean Good, Variance Bad Recall Var [ X ] = ∑ n i =1 p i ( x i- E [ X ]) 2 = E [ X 2 ]- ( E [ X ]) 2 ; also StDev [ X ] = p Var [ X ]. Natural measure of dispersion, or “risk” . Formally: recall π ( W , ˜) ≈ 1 2 A ( W ) Var [˜] (small ˜). Now define ˜ W = W + ˜, so E [ ˜ W ] = W , Var h ˜ W i = Var [˜]: then C [ ˜ W ] ≈ E [ ˜ W ]- 1 2 A ( W ) Var [ ˜ W ] . Risk aversion implies A ( W ) ≥ 0, so variance is bad . But this is only an approximation for small risks ! Overview Mean-Variance Preferences Portfolio Choice Mean and Variance are not always enough For general preferences and general risks, EU looks beyond means and variances. X with p.d. (100 , 9 32 ; 200 , 14 32 ; 300 , 9 32 ) Y with p.d. (50 , 1 8 ; 200 , 3 4 ; 350 , 1 8 ) E [ X ] = E [ Y ] = 200 and Var [ X ] = Var [ Y ] = 5625 However, with utility u ( x ) = √ x , X Y : E [ u ( X )] = 9 32 √ 100 + 14 32 √ 200 + 9 32 √ 300 = 13 . 871; E [ u ( Y )] = 1 8 √ 50 + 3 4 √ 200 + 1 8 √ 350 = 13 . 829 ....
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eco331-09-MeanVarianceHandout - Overview Mean-Variance...

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