In this paper, we derive simple, directly computable conditions for minimum-variance portfolios to have all positive weights. We show that either there is no minimum-variance portfolio with all positive weights or there is a single segment of the minimum-variance frontier for which all portfolios have positive weights. Then, we examine the likelihood of observing positively weighted minimum-variance portfolios. Analytical and computational results suggest that: i) even if the mean vector and covariance matrix are compatible with a given positively weighted portfolio being mean-variance efficient, the proportion of the minimum-variance frontier containing positively weighted portfolios is small and decreases as the number of assets in the universe increases, and ii) small perturbations in the means will likely lead to no positively weighted minimum-variance portfolios.

This paper investigates the sensitivity of mean-variance(MV)-efficient portfolios to changes in the means of individual assets. When only a budget constraint is imposed on the investment problem, the analytical results indicate that an MV-efficient portfolio’s weights, mean, and variance can be extremely sensitive to changes in asset means. When nonnegativity constraints are also imposed on the problem, the computational results confirm that a positively weighted MV-efficient portfolio’s weights are extremely sensitive to changes in asset means, but the portfolio’s returns are not. A surprisingly small increase in the mean of just one asset drives half the securities from the portfolio. Yet the portfolio’s expected return and standard deviation are virtually unchanged.

We show that the set of expected return vectors, for which an observed portfolio is mean variance (MV) efficient, is a two-parameter family. We identify ten ways to specify the time series behavior of the two parameters; the result highlights a number of inconsistencies involved in MV modelling. For each of the cases, it permits the inference of the time series of expected return vectors, as well as all the other Capital Asset Pricing Model (CAPM) variables, compatible with a known covariance matrix and the observed time series of market value weights. The empirical work shows that there are substantial case-to-case differences in the time series of mean vectors and many of them are quite different from the constant mean vector envisioned in tests of the CAPM.

We show that the set of expected return vectors, for which an observed portfolio is mean variance (MV) efficient, is a two‐parameter family. We identify ten ways to specify the time series behavior of the two parameters; the result highlights a number of inconsistencies involved in MV modelling. For each of the cases, it permits the inference of the time series of expected return vectors, as well as all the other Capital Asset Pricing Model (CAPM) variables, compatible with a known covariance matrix and the observed time series of market value weights. The empirical work shows that there are substantial case‐to‐case differences in the time series of mean vectors and many of them are quite different from the constant mean vector envisioned in tests of the CAPM.