Diversification-consistent data envelopment analysis based on directional-distance measures

• DEA models with diversification based on directional distance measures are proposed.
• Coherent risk measures are considered as the inputs and return measures as the outputs.
• Weakly, semi-strongly and strongly Pareto efficient portfolios can be identified.
• An equivalence of the SSD efficiency and the strongest DEA efficiency is discussed.
• Efficiency of 48 representative industry portfolios from US stock markets is investigated.

We propose new diversification-consistent DEA models suitable for assessing efficiency of investment opportunities available on financial markets. The formulations based on directional distance measures enable to use several risk measures as inputs and return measures as outputs, which can take both positive and negative values. We show that various models with different strength can be obtained and strongly, semi-strongly or weakly Pareto–Koopmans efficient investment opportunities can be identified. Moreover, the optimal solutions correspond to efficient investment opportunities and can be used by investors to revise the inefficient ones. If we consider discretely distributed returns, we can prove that under proper choice of the inputs (CVaRs) and outputs (expected return), the strongest model is able to identify efficient investment opportunities with respect to the second-order stochastic dominance. Moreover, the model can be formulated as a linear programming problem. In the numerical study, the proposed DEA models are applied to 48 representative industry portfolios from US stock markets.