Implementing mean–variance spanning tests with short-sales constraints

A set of assets is said to span the mean–variance space if the efficient frontier it generates cannot be improved upon with additional assets. Mean–variance spanning is used to determine empirically whether or not particular assets should be included in a given portfolio. Because of typical issues relating to parameter estimation in mean–variance optimization, the results of this empirical approach may differ from those of optimization, which assumes known parameters. In this paper, we show that the Wald tests used to account for short sales are prone to numerical instability. To address this, we exploit the uniqueness of the stochastic discount factor in the presence of a risk-free rate, leading to more robust tests.We also show that the purported Wald tests that have appeared in the literature on retirement plans in the United States do not correspond to mean–variance optimality and that their proper implementation leads to significantly different results.