Proof of Mean Variance Tautology

Proof of Mean Variance Tautology - Mean-Variance Efficiency...

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Unformatted text preview: Mean-Variance Efficiency and the Capital Asset Pricing Model Matthew Pollard 28th April 2008 Abstract I provide a simple proof of Roll’s (1977) result that mean-variance efficiency and the capital asset pricing model equation are mathematically equivalent. Proposition Let x p denote any mean-variance efficient portfolio with expected return E [ R p ], except for the minimum variance portfolio. For all assets i, the Capital Asset Pricing Model equation holds: E [ R i ]- E [ R zp ] = β i ( E [ R p ]- E [ R zp ]) where E [ R zp ] the unique mean-variance efficient portfolio satisfying Cov ( R p ,R zp ) = . Furthermore, any pair of portfolios ( x p ,x zp ) that satisfy the above conditions are necessarily both mean-variance efficient. Proof Let Σ = ( σ i,j ) N × N denote the matrix of asset covariances. Suppose x p is efficient; that is, it is the solution to the optimization problem x p = arg min 1 2 x T Σ x subject to x p = ER p , x T p 1 = 1 ....
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This note was uploaded on 04/28/2010 for the course ECON FINC3017 taught by Professor Xelloss during the Spring '10 term at University of St Andrews.

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Proof of Mean Variance Tautology - Mean-Variance Efficiency...

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