Positively Weighted Portfolios on the Minimum-Variance Frontier

Duality theory is employed to provide necessary and sufficient conditions for portfolios on the minimum-variance frontier to have positive investment proportions in all assets. These conditions involve the feasibility of portfolios that have non-negative correlation with all assets and positive correlation with at least one. Using these results, several "qualitative" results concerning the signs of investment proportions in efficient portfolios are proved. It is argued that the conditions that ensure all-positive weights in efficient portfolios are intuitively compelling and are not unique to the CAPM. With large numbers of assets, however, the signs of weights in minimum-variance portfolios can be very sensitive to slight departures from these conditions due to, for example, sampling error. THE BEHAVIOR OF THE minimum-variance frontier is of general concern to portfolio theory and plays a central role in the study of asset pricing. An unresolved question in the theory of the minimum-variance frontier involves when it will contain portfolios with positive weights on all assets. This paper addresses that question. We show that the conditions that ensure positive investment proportions in efficient portfolios have, by Farkas' Lemma, a simple dual. This dual, or alternative, system of inequalities puts conditions directly on the variances, covariances, and means of the returns, and these conditions can be interpreted intuitively. This contrasts with simply considering the first-order conditions for variance minimization, since these involve the inverted covariance matrix. With a large number of assets, it can be difficult to translate one's intuitive sense about how assets covary into statements about the inverse of the covariance matrix. The duality conditions we derive involve the existence of portfolios that are positively (or negatively) correlated with all assets. Indeed, we show that a necessary and sufficient condition for no portfolio on the frontier to have all-positive weights is the feasibility of an "arbitrage" or "hedge" portfolio, with weights that sum to zero, that has an expected payoff of zero and has non-negative correlation with all assets. The duality conditions also make it easy to construct examples of the types of covariability that permit or preclude positive investment proportions on all assets in frontier portfolios. There are at least three apparent reasons for interest in the signs of weights