, Volume 14, Issue 1, pp 77–99

Mean-value at risk portfolio efficiency: approaches based on data envelopment analysis models with negative data and their empirical behaviour

Research paper


We deal with the problem of an investor who is using a mean-risk model for accessing efficiency of investment opportunities. Our investor employs value at risk on several risk levels at the same time which corresponds to the approach called risk shaping. We review several data envelopment analysis (DEA) models which can deal with negative data. We show that a diversification–consistent extension of the DEA models based on a directional distance measure can be used to identify the Pareto–Koopmans efficient investment opportunities. We derive reformulations as chance constrained, nonlinear and mixed-integer problems under particular assumptions. In the numerical study, we access efficiency of US industry representative portfolios based on empirical distribution of random returns. We employ bootstrap and jackknife to investigate the empirical properties of the efficiency estimators.


Portfolio efficiency Diversification–consistent DEA   Directional distance measure Value at risk Risk-shaping Empirical behaviour 

Mathematics Subject Classification

91B28 90B50 90C15 90C29 

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Probability and Mathematical Statistics, Faculty of Mathematics and PhysicsCharles University in PraguePragueCzech Republic

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