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CAPM_Proof - Capital Market Line(CML thus R P-r σ 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 σ 2 M + α 1 + α 2 σ 2 X + 2 1 1 + α α 1 + α Cov ( M , X )
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CAPM proof So R P = R M + α R X 1 + α and σ 2 P = 1 1 + α 2 σ 2 M + α 1 + α 2 σ 2 X + 2 1 1 + α α 1 + α Cov ( M , X ) = 1 1 + α 2 σ 2 M + 2 1 1 + α α 1 + α 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 ) = 1 ( 1 + α ) 2 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 (by the definition of β ) = σ 2 M ( 1 + α ) 2 ( 1 + 2 βα )
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CAPM Proof So σ P = σ M ( 1 + α ) 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 ) 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 σ P = R M-r σ M . CAPM Proof Thus R M + α R X 1 + α-r σ M ( 1 + βα ) 1 + α = R M-r σ M CAPM Proof Thus R M + α R X 1 + α-r σ M ( 1 + βα ) 1 + α = R M-r σ M so R M + α R X-r-α r 1 + βα = R M-r CAPM Proof Thus R M + α R X 1 + α-r σ M ( 1 + βα ) 1 + α = R M-r σ M so R M + α R X-r-α r 1 + βα = R M-r solving for R X gives R X = r + β ( R M-r )...
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