Skip to main content
SpringerLink
Log in

On relations between DEA-risk models and stochastic dominance efficiency tests

  • Original Paper
  • Published:
Central European Journal of Operations Research Aims and scope Submit manuscript

Abstract

In this paper, several concepts of portfolio efficiency testing are compared, based either on data envelopment analysis (DEA) or the second-order stochastic dominance (SSD) relation: constant return to scale DEA models, variable return to scale (VRS) DEA models, diversification-consistent 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 representative 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 portfolio 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. The existence of zero value of \(\mathrm CVaR _{\alpha _{s}}(- R_0 )\) can be detected prior to testing and in the positive case, the data matrix can be shifted by an arbitrary constant. This data transformation would be harmless because SSD efficiency is shift equivariant.

  2. The levels belong to the set of values considered in the DEA tests equivalent to SSD tests. This was not the case in Branda (2012c).

References

  • Banker RD, Charnes A, Cooper W (1984) Some models for estimating technical and scale inefficiencies in data envelopment analysis. Manag Sci 30(9):1078–1092

    Article  Google Scholar 

  • Basso A, Funari S (2001) A data envelopment analysis approach to measure the mutual fund performance. Eur J Oper Res 135(3):477–492

    Article  Google Scholar 

  • Basso A, Funari S (2003) Measuring the performance of ethical mutual funds: a DEA approach. J Oper Res Soc 54:521–531

    Article  Google Scholar 

  • Bawa VS, Bodurtha JN, Rao MR, Suri HL (1985) On determination of stochastic dominance optimal sets. J Financ 40:417–431

    Article  Google Scholar 

  • Branda M (2012a) Stochastic programming problems with generalized integrated chance constraints. Optimization 61(8):949–968

    Article  Google Scholar 

  • Branda M (2012b) Sample approximation technique for mixed-integer stochastic programming problems with several chance constraints. Oper Res Lett 40(3):207–211

    Article  Google Scholar 

  • Branda M (2012c) Diversification-consistent data envelopment analysis with general deviation measures. Eur J Oper Res. Available, online 17 November 2012. doi:10.1016/j.ejor.2012.11.007

  • Branda M, Dupačová J (2012) Approximations and contamination bounds for probabilistic programs. Ann Oper Res 193(1):3–19

    Article  Google Scholar 

  • Branda M, Kopa M (2012a) DEA-risk efficiency and stochastic dominance efficiency of stock indices. Czech J Econ Financ 62(2):106–124

    Google Scholar 

  • Branda M, Kopa M (2012b) Equivalence of stochastic dominance and DEA-risk models. Working paper, Oper Res Lett (submitted to)

  • Branda M, Kopa M (2012c) From stochastic dominance to DEA-risk models: portfolio efficiency analysis. In: Sakalauskas L, Tomasgard A, Wallace SW (eds) Proceedings of the international workshop on stochastic programming for implementation and advanced applications. Vilnius Gediminas Technical University, Vilnius, Lithuania, pp 13–18

    Chapter  Google Scholar 

  • Charnes A, Cooper W (1962) Programming with linear fractional functionals. Naval Res Logist Q 9:181–196

    Article  Google Scholar 

  • Charnes A, Cooper W, Rhodes E (1978) Measuring the efficiency of decision-making units. Eur J Oper Res 2:429–444

    Article  Google Scholar 

  • Chen Z, Lin R (2006) Mutual fund performance evaluation using data envelopment analysis with new risk measures. OR Spectr 28:375–398

    Article  Google Scholar 

  • Cook WD, Seiford LM (2009) Data envelopment analysis (DEA) thirty years on. Eur J Oper Res 192:1–17

    Article  Google Scholar 

  • Cooper WW, Seiford LM, Tone K (2007) Data envelopment analysis. Springer, Berlin

    Google Scholar 

  • Daraio C, Simar L (2006) A robust nonparametric approach to evaluate and explain the performance of mutual funds. Eur J Oper Res 175(1):516–542

    Article  Google Scholar 

  • Dentcheva D, Ruszczynski A (2006) Portfolio optimization with stochastic dominance constraints. J Banking Financ 30(2):433–451

    Article  Google Scholar 

  • Dupačová J, Kopa M (2012) Robustness in stochastic programs with risk constraints. Ann Oper Res 200(1):55–74

    Article  Google Scholar 

  • Fábián CI, Mitra G, Roman D (2011) Processing Second-order Stochastic Dominance models using cutting-plane representations. Math Program 130(1):33–57

    Article  Google Scholar 

  • Fishburn PC (1974) Convex stochastic dominance with continuous distribution functions. J Econ Theory 7:143–158

    Article  Google Scholar 

  • Galagadera UA, Silvapulle P (2002) Australian mutual fund performance appraisal using data envelopment analysis. Manag Financ 28(9):60–73

    Google Scholar 

  • Gollmer R, Gotzes U, Schultz R (2011) A note on second-order stochastic dominance constraints induced by mixed-integer linear recourse. Math Program Ser A 126(1):179–190

    Article  Google Scholar 

  • Gotoh J-Y, Konno H (2000) Third-degree stochastic dominance and mean-risk analysis. Manag Sci 46(2):289–301

    Article  Google Scholar 

  • Hardy GH, Littlewood JE, Pólya G (1934) Inequalities, 1st edn. Cambridge University Press, Cambridge

    Google Scholar 

  • Hanoch G, Levy H (1969) The efficient analysis of choices involving risk. Rev Econ Stud 36(3):335–346

    Article  Google Scholar 

  • Jablonsky J (2012) Multicriteria approaches for ranking of efficient units in DEA models. Cent Eur J Oper Res 20(3):435–449

    Article  Google Scholar 

  • Kopa M (2010) Measuring of second-order stochastic dominance portfolio efficiency. Kybernetika 46(3):488–500

    Google Scholar 

  • Kopa M, Chovanec P (2008) A second-order stochastic dominance portfolio efficiency measure. Kybernetika 44(2):243–258

    Google Scholar 

  • Kopa M, Post T (2009) A portfolio efficiency test based on FSD optimality. J Financ Quant Anal 44(5):1103–1124

    Article  Google Scholar 

  • Kuosmanen T (2004) Efficient diversification according to stochastic dominance criteria. Manag Sci 50(10):1390–1406

    Article  Google Scholar 

  • Lamb JD, Tee K-H (2012a) Data envelopment analysis models of investment funds. Eur J Oper Res 216(3):687–696

    Article  Google Scholar 

  • Lamb JD, Tee K-H (2012b) Resampling DEA estimates of investment fund performance. Eur J Oper Res 223(3):834–841

    Article  Google Scholar 

  • Levy H (2006) Stochastic dominance: investment decision making under uncertainty, 2nd edn. Springer, New York

    Google Scholar 

  • Lizyayev A (2012) Stochastic dominance efficiency analysis of diversified portfolios: classification, comparison and refinements. Ann Oper Res 196(1):391–410

    Article  Google Scholar 

  • Lozano S, Gutiérrez E (2008) Data envelopment analysis of mutual funds based on second-order stochastic dominance. Eur J Oper Res 189:230–244

    Article  Google Scholar 

  • Luedtke J (2008) New formulations for optimization under stochastic dominance constraints. SIAM J Optim 19(3):1433–1450

    Article  Google Scholar 

  • Markowitz HM (1952) Portfolio selection. J Financ 7(1):77–91

    Google Scholar 

  • Markowitz HM (1959) Portfolio selection: efficient diversification in investments. Wiley, New York

    Google Scholar 

  • Meskarian R, Xu H, Fliege J (2012) Numerical methods for stochastic programs with second order dominance constraints with applications to portfolio optimization. Eur J Oper Res 216(2):376–385

    Article  Google Scholar 

  • 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):408–418

    Article  Google Scholar 

  • von Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, Princeton

    Google Scholar 

  • Ogryczak W, Ruszczynski A (2001) On consistency of stochastic dominance and mean-semideviation models. Math Program Ser B 89:217–232

    Article  Google Scholar 

  • Ogryczak W, Ruszczynski A (2002) Dual stochastic dominance and related mean-risk models. SIAM J Optim 13:60–78

    Article  Google Scholar 

  • Pflug G (2000) Some remarks on the value-at-risk and the conditional value-at-risk. In: Uryasev SP (ed) Probabilistic constrained optimization. Kluwer, Dordrecht, pp 272–281

    Chapter  Google Scholar 

  • Post T (2003) Empirical tests for stochastic dominance efficiency. J Financ 58:1905–1932

    Article  Google Scholar 

  • Rockafellar RT, Uryasev S (2000) Optimization of conditional value-at-risk. J Risk 2:21–41

    Google Scholar 

  • Rockafellar RT, Uryasev S (2002) Conditional value-at-risk for general loss distributions. J Banking Financ 26:1443–1471

    Article  Google Scholar 

  • Rockafellar RT, Uryasev S, Zabarankin M (2006) Generalized deviations in risk analysis. Financ Stoch 10:51–74

    Article  Google Scholar 

  • Soleimani-damaneh M (2012) On a basic definition of returns to scale. Oper Res Lett 40(2):144–147

    Article  Google Scholar 

  • Whitmore GA (1970) Third-degree stochastic dominance. Am Econ Rev 60(3):457–459

    Google Scholar 

Download references

Acknowledgments

The research was supported by the Czech Science Foundation under the Grants P402/10/1610, P402/12/0558. We would like to express our gratitude to the anonymous referees, whose comments have greatly improved the paper.

Author information

Authors and Affiliations

  1. Department of Probability and Mathematical Statistics, Faculty of Mathematics and Physics, Charles University in Prague, Sokolovská 83, Prague, Czech Republic

    Martin Branda & Miloš Kopa

Authors
  1. Martin Branda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Miloš Kopa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Branda.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Branda, M., Kopa, M. On relations between DEA-risk models and stochastic dominance efficiency tests. Cent Eur J Oper Res 22, 13–35 (2014). https://doi.org/10.1007/s10100-012-0283-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10100-012-0283-2

Keywords

Search

Navigation