(Im)Possible Frontiers: A Comment

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 an 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. Here, we show that while sample (or randomly drawn) parameter sets almost surely lead to impossible frontiers, slight variations to the parameters, well within their estimation error bounds, lead to frontiers with positive portfolio segments. Parameter sets leading to 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. In reaching a mean/variance equilibrium, asset prices should change slightly from any randomly chosen set to a nearby set that produces a positive segment of the efficient frontier and is thus consistent with the CAPM.