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The existence of mean-variance efficient positive portfolios - portfolios with no negative weights - is a key requirement for equilibrium in the Capital Asset Pricing Model (CAPM). Brennan and Lo (2010) define an "Impossible frontier" as a frontier on which all portfolios have at least one negative weight. They prove that for randomly drawn covariance matrices the probability of obtaining impossible frontier approaches 1 as the number of assets grows. Impossible frontiers are also found when the empirical sample parameters are employed, regardless of the specifics of the sampling method. These results seem like a deadly blow to the CAPM.
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