, Volume 14, Issue 1, pp 77–99 | Cite as

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

  • Martin Branda
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 



The present work has been supported by the Czech Science Foundation under the Grant P402/12/G097. I would like to express my gratitude to the anonymous referees, whose comments have helped me to improve the paper.


  1. Ahmed S (2014) Convex relaxations of chance constrained optimization problems. Optim Lett 8(1):1–12CrossRefGoogle Scholar
  2. Amaran S, Sahinidis NV, Sharda B, Bury SJ (2014) Simulation optimization: a review of algorithms and applications. 4OR-Q J Oper Res 12(4):301–333CrossRefGoogle Scholar
  3. Artzner P, Delbaen F, Eber J-M, Heath D (1999) Coherent measures of risk. Math Financ 9:203–228CrossRefGoogle Scholar
  4. Banker RD, Charnes A, Cooper W (1984) Some models for estimating technical and scale inefficiencies in data envelopment analysis. Manag Sci 30(9):1078–1092CrossRefGoogle Scholar
  5. Basso A, Funari S (2001) A data envelopment analysis approach to measure the mutual fund performance. Eur J Oper Res 135(3):477–492CrossRefGoogle Scholar
  6. Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  7. Branda M (2013) Diversification–consistent data envelopment analysis with general deviation measures. Eur J Oper Res 226(3):626–635CrossRefGoogle Scholar
  8. Branda M (2013) Reformulations of input–output oriented DEA tests with diversification. Oper Res Lett 41(5):516–520CrossRefGoogle Scholar
  9. Branda M (2014) Sample approximation technique for mixed-integer stochastic programming problems with expected value constraints. Optim Lett 8(3):861–875CrossRefGoogle Scholar
  10. Branda M (2015) Diversification–consistent data envelopment analysis based on directional-distance measures. Omega 52:66–75CrossRefGoogle Scholar
  11. Branda M, Kopa M (2012) DEA-risk efficiency and stochastic dominance efficiency of stock indices. Financ Uver 62(2):106–124Google Scholar
  12. Branda M, Kopa M (2014) On relations between DEA-risk models and stochastic dominance efficiency tests. Cent Eur J Oper Res 22(1):13–35CrossRefGoogle Scholar
  13. Brandouy O, Kerstens K, Van de Woestyne I (2015) Frontier-based vs. traditional mutual fund ratings: a first backtesting analysis. Eur J Oper Res 242(1):332–342CrossRefGoogle Scholar
  14. Briec W, Kerstens K (2009) Multi-horizon Markowitz portfolio performance appraisals: a general approach. Omega 37(1):50–62CrossRefGoogle Scholar
  15. Briec W, Kerstens K, Lesourd J-B (2004) Single period Markowitz portfolio selection, performance gauging and duality: a variation on the Luenberger shortage function. J Optim Theory Appl 120(1):1–27CrossRefGoogle Scholar
  16. Briec W, Kerstens K, Jokung O (2007) Mean–variance–skewness portfolio performance gauging: a general shortage function and dual approach. Manag Sci 53:135–149CrossRefGoogle Scholar
  17. Charnes A, Cooper W, Rhodes E (1978) Measuring the efficiency of decision-making units. Eur J Oper Res 2:429–444CrossRefGoogle Scholar
  18. Chen Z, Lin R (2006) Mutual fund performance evaluation using data envelopment analysis with new risk measures. OR Spectr 28(3):375–398CrossRefGoogle Scholar
  19. Cooper WW, Seiford LM, Zhu J (2011) Handbook on data envelopment analysis. Springer, New YorkCrossRefGoogle Scholar
  20. Cplex solver manual. Visited on 2014-09-23
  21. Dempster MAH (ed) (2010) Risk management: value at risk and beyond. Cambridge University Press, CambridgeGoogle Scholar
  22. Dentcheva D, Martinez G (2012) Augmented Lagrangian method for probabilistic optimization. Ann Oper Res 200(1):109–130CrossRefGoogle Scholar
  23. Ding H, Zhou Z, Xiao H, Ma C, Liu W (2014) Performance evaluation of portfolios with margin requirements, mathematical problems in engineering, vol 2014, Article ID 618706. Hindawi Publishing Corporation, Cairo, p 8 (2014). doi: 10.1155/2014/618706
  24. Dupačová J, Kopa M (2014) Robustness of optimal portfolios under risk and stochastic dominance constraints. Eur J Oper Res 234(2):434–441CrossRefGoogle Scholar
  25. Dupačová J, Hurt J, Štěpán J (2002) Stochastic modeling in economics and finance. In: Applied Optimization, vol 75. Kluwer, DordrechtGoogle Scholar
  26. Efron B (1979) Bootstrap methods: another look at the jackknife. Ann Stat 7:1–26CrossRefGoogle Scholar
  27. Efron B, Tibshirani RJ (1993) An introduction to the bootstrap. Chapman & Hall CRC, New YorkCrossRefGoogle Scholar
  28. Emrouznejad A, Anouze AL, Thanassoulis E (2010) A semi-oriented radial measure for measuring the efficiency of decision making units with negative data, using DEA. Eur J Oper Res 200(1):297–304CrossRefGoogle Scholar
  29. GAMS—a user’s guide. Visited on 2014-09-23
  30. Gutjahr WJ, Pichler A (2013) Stochastic multi-objective optimization: a survey on non-scalarizing methods. Ann Oper Res 1–25. doi: 10.1007/s10479-013-1369-5
  31. Kenneth French library. Visited on 2015-02-16
  32. Kerstens K, Van de Woestyne I (2011) Negative data in DEA: a simple proportional distance function approach. J Oper Res Soc 62:1413–1419CrossRefGoogle Scholar
  33. Keshavarz E, Toloo M (2015) Efficiency status of a feasible solution in the Multi-Objective Integer Linear Programming problems: a DEA methodology. Appl Math Model 39(12):3236–3247CrossRefGoogle Scholar
  34. Lamb JD, Tee K-H (2012) Data envelopment analysis models of investment funds. Eur J Oper Res 216(3):687–696CrossRefGoogle Scholar
  35. Lamb JD, Tee K-H (2012) Resampling DEA estimates of investment fund performance. Eur J Oper Res 223(3):834–841CrossRefGoogle Scholar
  36. Li X, You Y (2014) A note on allocation of portfolio shares of random assets with Archimedean copula. Ann Oper Res 212:155–167CrossRefGoogle Scholar
  37. Liu W, Zhou Z, Liu D, Xiao H (2015) Estimation of portfolio efficiency via DEA. Omega 52:107–118CrossRefGoogle Scholar
  38. Lozano S, Gutiérrez E (2008) Data envelopment analysis of mutual funds based on second-order stochastic dominance. Eur J Oper Res 189(1):230–244CrossRefGoogle Scholar
  39. Luedtke J, Ahmed S (2008) A sample approximation approach for optimization with probabilistic constraints. SIAM J Optim 19:674–699CrossRefGoogle Scholar
  40. Markowitz HM (1952) Portfolio selection. J Financ 7(1):77–91Google Scholar
  41. Matlab Documentation Center. Visited on 2014-09-23
  42. Morey MR, Morey RC (1999) Mutual fund performance appraisals: a multi-horizon perspective with endogenous benchmarking. Omega 27(2):241–258CrossRefGoogle Scholar
  43. Murthi BPS, Choi YK, Desai P (1997) Efficiency of mutual funds and portfolio performance measurement: a non-parametric approach. Eur J Oper Res 98(2):408–418CrossRefGoogle Scholar
  44. Nemirovski A, Shapiro A (2007) Convex approximations of chance constrained programs. SIAM J Optim 17(4):969–996CrossRefGoogle Scholar
  45. Portela Silva MCA, Thanassoulis E, Simpson G (2004) Negative data in DEA: a directional distance approach applied to bank branches. J Oper Res Soc 55(10):1111–1121CrossRefGoogle Scholar
  46. Prékopa A (2003) Probabilistic programming. In: Ruszczynski A, Shapiro A (eds) Stochastic programming, handbook in operations research and management science, vol 10. Elsevier, Amsterdam, pp 483–554Google Scholar
  47. Raike WM (1970) Dissection methods for solutions in chance constrained programming problems under discrete distributions. Manag Sci 16(11):708–715CrossRefGoogle Scholar
  48. Rockafellar RT, Uryasev S (2002) Conditional value-at-risk for general loss distributions. J Bank Financ 26(7):1443–1471CrossRefGoogle Scholar
  49. Rockafellar RT, Uryasev S, Zabarankin M (2006) Generalized deviations in risk analysis. Financ Stoch 10:51–74CrossRefGoogle Scholar
  50. Roman D, Darby-Dowman K, Mitra G (2007) Mean-risk models using two risk measures: a multi-objective approach. Quant Financ 7(4):443–458CrossRefGoogle Scholar
  51. Shapiro A, Dentcheva D, Ruszczyński A (2009) Lectures on stochastic programming: modeling and theory. SIAM, PhiladelphiaCrossRefGoogle Scholar
  52. Sharp JA, Meng W, Liu W (2007) A modified slacks-based measure model for data envelopment analysis with ‘natural’ negative outputs and inputs. J Oper Res Soc 58(12):1672–1677CrossRefGoogle Scholar
  53. Szegö G (ed) (2004) Risk measures for the 21st century. Wiley, New YorkGoogle Scholar
  54. Tone K (2001) A slacks-based measure of efficiency in data envelopment analysis. Eur J Oper Res 130:498–509CrossRefGoogle Scholar
  55. Wang W, Ahmed S (2008) Sample average approximation of expected value constrained stochastic programs. Oper Res Lett 36(5):515–519CrossRefGoogle Scholar

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

Personalised recommendations