We develop empirical tests for stochastic dominance efficiency of a given investment portfolio relative to all possible portfolios formed from a given set of assets. Our tests use multivariate statistics, which result in superior statistical power properties compared to existing stochastic dominance efficiency tests and increase the comparability with existing mean-variance efficiency tests. Using our tests, we demonstrate that the mean-variance inefficiency of the CRSP all-share index relative to beta-sorted portfolios can be explained by tail risk not captured by variance.