We conduct a broad study of stochastic dominance efficiency on financial markets. We show that in the long run the vast majority of 17 equity market indices across the globe are inefficient at order two relative to their industry components. In the short run, the past stochastic dominance relation between the index and sub-indices predicts future dominance. Trading rules accounting for the predictability of stochastic dominance improve the out-of-sample certainty equivalents of risk-averse investors. The gains are especially pronounced for European and developing markets, while no consistent outperformance of alternative strategies is found for the S&P 100 and Nikkei 225 indices.