CAPM_Proof

# CAPM_Proof - Capital Market Line (CML) , thus R P-r P = R...

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CAPM proof Suppose we hold the market portfolio, M , and we wish to invest a small amount α in an asset X , with beta β . Deﬁnition Let R M = return on the market portfolio R X = return on asset X σ M = volatility of the market portfolio σ X = volatility of X r = risk free rate P = M + α X (the new portfolio)

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CAPM proof So R P = R M + α R X 1 + α and σ 2 P = ± 1 1 + α ² 2 σ 2 M + ± α 1 + α ² 2 σ 2 X + 2 ± 1 1 + α ²± α 1 + α ² Cov ( M , X )
CAPM proof So R P = R M + α R X 1 + α and σ 2 P = ± 1 1 + α ² 2 σ 2 M + ± α 1 + α ² 2 σ 2 X + 2 ± 1 1 + α ²± α 1 + α ² Cov ( M , X ) = ± 1 1 + α ² 2 σ 2 M + 2 ± 1 1 + α ²± α 1 + α ² Cov ( M , X ) (since α 2 ± α )

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CAPM proof Thus σ 2 P = 1 ( 1 + α ) 2 ± σ 2 M + 2 α Cov ( M , X ) ² = 1 ( 1 + α ) 2 ³ σ 2 M + 2 α Cov ( M , X ) σ 2 M σ 2 M ! = 1 ( 1 + α ) 2 ± σ 2 M + 2 αβσ 2 M ² (by the deﬁnition of β ) = σ 2 M ( 1 + α ) 2 ( 1 + 2 βα )
CAPM Proof So σ P = σ M ( 1 + α ) p 1 + 2 βα = σ M ( 1 + α ) ( 1 + βα ) (Since 1 + x 1 + 1 2 x )

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CAPM Proof So σ P = σ M ( 1 + α ) p 1 + 2 βα = σ M ( 1 + α ) ( 1 + βα ) (Since 1 + x 1 + 1 2 x ) If the new portfolio is efﬁcient, it must lie on the

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Unformatted text preview: Capital Market Line (CML) , thus R P-r P = R M-r M . CAPM Proof Thus R M + R X 1 + -r M ( 1 + ) 1 + = R M-r M CAPM Proof Thus R M + R X 1 + -r M ( 1 + ) 1 + = R M-r M so R M + R X-r- r 1 + = R M-r CAPM Proof Thus R M + R X 1 + -r M ( 1 + ) 1 + = R M-r M so R M + R X-r- r 1 + = R M-r solving for R X gives R X = r + ( R M-r )...
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## This note was uploaded on 09/18/2011 for the course ACTSC 372 taught by Professor Maryhardy during the Winter '09 term at Waterloo.

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CAPM_Proof - Capital Market Line (CML) , thus R P-r P = R...

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