(Im)Possible Frontiers: A Comment*

A totally positive mean-variance efficient aggregate market portfolio — one with no negative weights — is the key equilibrium feature of the Capital Asset Pricing Model (CAPM). Brennan and Lo (2010) designate an efficient frontier as “impossible†when every efficient portfolio has at least one negative weight. For randomly drawn covariance matrices, they prove that the probability of an impossible frontier approaches 1 as the number of assets grows. Impossible frontiers are also invariably found with empirical sample parameters, regardless of the sampling method. These results might seem like a deadly blow to the CAPM. However, we show here that slight variations in sample parameters, well within estimation error bounds, can lead to frontiers with positive portfolio segments. Parameters producing possible frontiers are somewhat like rational numbers on the real line: they occupy a zero-measure of parameter space, but there is always one close by. Thus, starting from an impossible frontier, slight changes in asset prices, as they converge to an economic equilibrium, deliver a possible frontier, consistent with the CAPM.