CEJOR manuscript No. (will be inserted by the editor) On relations between DEA-risk models and stochastic dominance eﬃciency tests Martin Branda · Miloˇ s Kopa Received: date / Accepted: date Abstract In this paper, several concepts of portfolio eﬃciency testing are com- pared, based either on Data Envelopment Analysis (DEA) or the second-order stochastic dominance (SSD) relation: constant return to scale (CRS) DEA mod- els, variable return to scale (VRS) DEA models, diversiﬁcation-consistent (DC) DEA models, pairwise SSD eﬃciency tests, convex SSD eﬃciency tests and full SSD portfolio eﬃciency tests. Especially, the equivalence between VRS DEA model with binary weights and the SSD pairwise eﬃciency test is proved. DEA models equivalent to convex SSD eﬃciency tests and full SSD portfolio eﬃciency tests are also formulated. In the empirical application, the eﬃciency testing of 48 US rep- resentative industry portfolios using all considered DEA models and SSD tests is presented. The obtained eﬃciency sets are compared. A special attention is paid to the case of small number of the inputs and outputs. It is empirically shown that DEA models equivalent either to the convex SSD test or to the SSD portfo- lio eﬃciency test work well even with quite small number of inputs and outputs. However, the reduced VRS DEA model with binary weights is not able to identify all the pairwise SSD eﬃcient portfolios. Keywords Data Envelopment Analysis · Second-order stochastic dominance · pairwise SSD eﬃciency · convex SSD eﬃciency · SSD portfolio eﬃciency Martin Branda Charles University in Prague, Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics Sokolovsk´ a 83, Czech Republic Tel.: +420 221 913 404 E-mail: branda@karlin.mﬀ.cuni.cz Miloˇ s Kopa Charles University in Prague, Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics Sokolovsk´ a 83, Czech Republic