Impossible Frontiers

A key result of the Capital Asset Pricing Model (CAPM) is that the market portfolio---the portfolio of all assets in which each asset's weight is proportional to its total market capitalization---lies on the mean-variance efficient frontier, the set of portfolios having mean-variance characteristics that cannot be improved upon. Therefore, the CAPM cannot be consistent with efficient frontiers for which every frontier portfolio has at least one negative weight or short position. We call such efficient frontiers "impossible", and derive conditions on asset-return means, variances, and covariances that yield impossible frontiers. With the exception of the two-asset case, we show that impossible frontiers are difficult to avoid. Moreover, as the number of assets n grows, we prove that the probability that a generically chosen frontier is impossible tends to one at a geometric rate. In fact, for one natural class of distributions, nearly one-eighth of all assets on a frontier is expected to have negative weights for *every* portfolio on the frontier. We also show that the expected minimum amount of shortselling across frontier portfolios grows linearly with n, and even when shortsales are constrained to some finite level, an impossible frontier remains impossible. Using daily and monthly U.S. stock returns, we document the impossibility of efficient frontiers in the data.