Lecture11Slidesv-4

# Lecture11Slidesv-4 - = E xR r(1 − x R rf = xe r(1 − x e...

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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 ﬃ cient e rf − e r − e rf √ v r √ v p , not e ﬃ cient ° e p ( s p ) = ± e rf + e r − e rf s r s p , e ﬃ cient e rf − e r − e rf s r s p , not e ﬃ cient Note: e r − e rf is expected excess return e r − e rf s r is the slope of the e ﬃ cient frontier (in this case); it is also called (excess return) Sharpe ratio of risky asset. E ﬃ cient Frontier: e-v, e-s diagrams e 1 = 10 , s 1 = 5 , e 2 = 5 , s 2 = 0 E ﬃ cient Frontier: many securities E ﬃ cient Frontier: capital allocation line...
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## This note was uploaded on 07/13/2011 for the course ECON 3107 at University of New South Wales.

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Lecture11Slidesv-4 - = E xR r(1 − x R rf = xe r(1 − x e...

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