Lecture11Slidesv-4

Lecture11Slidesv-4 - = E ( xR r + (1 x ) R rf ) = xe r + (1...

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Lecture 11: Mean-Variance (cont.) Valentyn Panchenko Certainty Equivalence Deriving expected return-risk frontier: some useful formulae Let X and Y be random variables with expectations E ( X ) and E ( Y ) , variances Var ( X ) and Var ( Y ) and covariance Cov ( X,Y ) . Let a and b be real numbers, then: ° E ( aX + bY )= aE ( X )+ bE ( Y ) ° Var ( aX + bY )= a 2 Var ( X )+ b 2 Var ( Y )+2 ab Cov ( X,Y ) ° Var ( aX + bY )= a 2 Var ( X )+ b 2 Var ( Y )+2 ab ° Var ( X ) ° Var ( Y ) ρ XY , where r XY is the correlation coe cient between X and Y . Deriving expected return-risk frontier Let R 1 and R 2 be random variables for returns of two assets with expectations E ( R 1 )= e 1 and E ( R 2 )= e 2 , variances Var ( R 1 )= v 1 and Var ( R 2 )= v 2 and correlation Corr ( R 1 ,R 2 )= ρ 12 .L e t x denotes the proportion of asset 1 in the portfolio: ° E ( xR 1 +(1 x ) R 2 )= xe 1 +(1 x ) e 2 ° Var ( xR 1 +(1 x ) R 2 )= x 2 v 1 +(1 x ) 2 v 2 +2 x (1 x ) v 1 v 2 ρ 12 ° StDev ( xR 1 +(1 x ) R 2 )= ° Var ( xR 1 +(1 x ) R 2 ) Idea: solve for x
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Deriving expected return-risk frontier, special case: risky ( r )andr isk - free( rf )asset Var ( R rf )=0 and correlation Corr ( R r ,R rf )=0 .
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Unformatted text preview: = E ( xR r + (1 x ) R rf ) = xe r + (1 x ) e rf = e rf + x ( e r e rf ) v p = Var ( xR r + (1 x ) R rf ) = x 2 v r x = v p /v r s p = StDev ( xR r + (1 x ) R rf ) = | x | v r x = s p /s r e p ( v p ) = e rf + e r e rf v r v p , e cient e rf e r e rf v r v p , not e cient e p ( s p ) = e rf + e r e rf s r s p , e cient e rf e r e rf s r s p , not e cient Note: e r e rf is expected excess return e r e rf s r is the slope of the e cient frontier (in this case); it is also called (excess return) Sharpe ratio of risky asset. E cient Frontier: e-v, e-s diagrams e 1 = 10 , s 1 = 5 , e 2 = 5 , s 2 = 0 E cient Frontier: many securities E cient Frontier: capital allocation line...
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Lecture11Slidesv-4 - = E ( xR r + (1 x ) R rf ) = xe r + (1...

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