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.

This paper theoretically evaluates the robustness of the Security Market Line relationship when the market proxy employed is not mean‐variance efficient. The analysis focuses on the behavior of the “benchmark errors,” the deviations of assets and portfolios from the Security Market Line. First, we characterize how the location of an asset in mean‐variance space determines its benchmark error. Then the continuity properties of the benchmark errors are studied. The results indicate that the magnitudes of the errors exhibit continuous but not uniformly continuous behaviors. The relative rankings based on deviations from the Security Market Line, however, exhibit some severe discontinuities. In fact, these can be exactly reversed for two proxies arbitrarily close in mean‐variance space.

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.