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On relations between DEA-risk models and stochastic dominance efficiency tests. (English) Zbl 1339.90229

Summary: 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.

MSC:

90C08 Special problems of linear programming (transportation, multi-index, data envelopment analysis, etc.)
62P05 Applications of statistics to actuarial sciences and financial mathematics
62G10 Nonparametric hypothesis testing
90B50 Management decision making, including multiple objectives
90C35 Programming involving graphs or networks

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[1] 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 · Zbl 0552.90055 · doi:10.1287/mnsc.30.9.1078
[2] Basso A, Funari S (2001) A data envelopment analysis approach to measure the mutual fund performance. Eur J Oper Res 135(3):477-492 · Zbl 0988.90516 · doi:10.1016/S0377-2217(00)00311-8
[3] Basso A, Funari S (2003) Measuring the performance of ethical mutual funds: a DEA approach. J Oper Res Soc 54:521-531 · Zbl 1070.90050 · doi:10.1057/palgrave.jors.2601541
[4] Bawa VS, Bodurtha JN, Rao MR, Suri HL (1985) On determination of stochastic dominance optimal sets. J Financ 40:417-431 · doi:10.1111/j.1540-6261.1985.tb04965.x
[5] Branda M (2012a) Stochastic programming problems with generalized integrated chance constraints. Optimization 61(8):949-968 · Zbl 1252.90056 · doi:10.1080/02331934.2011.587007
[6] Branda M (2012b) Sample approximation technique for mixed-integer stochastic programming problems with several chance constraints. Oper Res Lett 40(3):207-211 · Zbl 1245.90073 · doi:10.1016/j.orl.2012.01.002
[7] 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 · Zbl 1292.91190
[8] Branda M, Dupačová J (2012) Approximations and contamination bounds for probabilistic programs. Ann Oper Res 193(1):3-19 · Zbl 1254.90133 · doi:10.1007/s10479-010-0811-1
[9] Branda M, Kopa M (2012a) DEA-risk efficiency and stochastic dominance efficiency of stock indices. Czech J Econ Financ 62(2):106-124
[10] Branda M, Kopa M (2012b) Equivalence of stochastic dominance and DEA-risk models. Working paper, Oper Res Lett (submitted to) · Zbl 1339.90229
[11] Branda, M.; Kopa, M.; Sakalauskas, L. (ed.); Tomasgard, A. (ed.); Wallace, SW (ed.), From stochastic dominance to DEA-risk models: portfolio efficiency analysis, 13-18 (2012), Vilnius, Lithuania · doi:10.5200/stoprog.2012.03
[12] Charnes A, Cooper W (1962) Programming with linear fractional functionals. Naval Res Logist Q 9:181-196 · Zbl 0127.36901 · doi:10.1002/nav.3800090303
[13] Charnes A, Cooper W, Rhodes E (1978) Measuring the efficiency of decision-making units. Eur J Oper Res 2:429-444 · Zbl 0416.90080 · doi:10.1016/0377-2217(78)90138-8
[14] Chen Z, Lin R (2006) Mutual fund performance evaluation using data envelopment analysis with new risk measures. OR Spectr 28:375-398 · Zbl 1130.90024 · doi:10.1007/s00291-005-0032-1
[15] Cook WD, Seiford LM (2009) Data envelopment analysis (DEA) thirty years on. Eur J Oper Res 192:1-17 · Zbl 1180.90151 · doi:10.1016/j.ejor.2008.01.032
[16] Cooper WW, Seiford LM, Tone K (2007) Data envelopment analysis. Springer, Berlin · Zbl 1111.90001
[17] 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 · Zbl 1137.91442 · doi:10.1016/j.ejor.2005.06.010
[18] Dentcheva D, Ruszczynski A (2006) Portfolio optimization with stochastic dominance constraints. J Banking Financ 30(2):433-451 · doi:10.1016/j.jbankfin.2005.04.024
[19] Dupačová J, Kopa M (2012) Robustness in stochastic programs with risk constraints. Ann Oper Res 200(1):55-74 · Zbl 1255.90088 · doi:10.1007/s10479-010-0824-9
[20] Fábián CI, Mitra G, Roman D (2011) Processing Second-order Stochastic Dominance models using cutting-plane representations. Math Program 130(1):33-57 · Zbl 1229.90108 · doi:10.1007/s10107-009-0326-1
[21] Fishburn PC (1974) Convex stochastic dominance with continuous distribution functions. J Econ Theory 7:143-158 · doi:10.1016/0022-0531(74)90103-3
[22] Galagadera UA, Silvapulle P (2002) Australian mutual fund performance appraisal using data envelopment analysis. Manag Financ 28(9):60-73
[23] 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 · Zbl 1229.90109 · doi:10.1007/s10107-009-0270-0
[24] Gotoh J-Y, Konno H (2000) Third-degree stochastic dominance and mean-risk analysis. Manag Sci 46(2):289-301 · Zbl 1231.91192 · doi:10.1287/mnsc.46.2.289.11928
[25] Hardy GH, Littlewood JE, Pólya G (1934) Inequalities, 1st edn. Cambridge University Press, Cambridge · Zbl 0010.10703
[26] Hanoch G, Levy H (1969) The efficient analysis of choices involving risk. Rev Econ Stud 36(3):335-346 · Zbl 0184.45202 · doi:10.2307/2296431
[27] Jablonsky J (2012) Multicriteria approaches for ranking of efficient units in DEA models. Cent Eur J Oper Res 20(3):435-449 · Zbl 1339.90231 · doi:10.1007/s10100-011-0223-6
[28] Kopa M (2010) Measuring of second-order stochastic dominance portfolio efficiency. Kybernetika 46(3):488-500 · Zbl 1193.91140
[29] Kopa M, Chovanec P (2008) A second-order stochastic dominance portfolio efficiency measure. Kybernetika 44(2):243-258 · Zbl 1154.91456
[30] Kopa M, Post T (2009) A portfolio efficiency test based on FSD optimality. J Financ Quant Anal 44(5):1103-1124 · doi:10.1017/S0022109009990251
[31] Kuosmanen T (2004) Efficient diversification according to stochastic dominance criteria. Manag Sci 50(10):1390-1406 · doi:10.1287/mnsc.1040.0284
[32] Lamb JD, Tee K-H (2012a) Data envelopment analysis models of investment funds. Eur J Oper Res 216(3):687-696 · Zbl 1237.91239 · doi:10.1016/j.ejor.2011.08.019
[33] Lamb JD, Tee K-H (2012b) Resampling DEA estimates of investment fund performance. Eur J Oper Res 223(3):834-841 · Zbl 1292.91194 · doi:10.1016/j.ejor.2012.07.015
[34] Levy H (2006) Stochastic dominance: investment decision making under uncertainty, 2nd edn. Springer, New York · Zbl 1109.91037
[35] Lizyayev A (2012) Stochastic dominance efficiency analysis of diversified portfolios: classification, comparison and refinements. Ann Oper Res 196(1):391-410 · Zbl 1259.91079 · doi:10.1007/s10479-012-1123-4
[36] Lozano S, Gutiérrez E (2008) Data envelopment analysis of mutual funds based on second-order stochastic dominance. Eur J Oper Res 189:230-244 · Zbl 1147.90016 · doi:10.1016/j.ejor.2007.04.014
[37] Luedtke J (2008) New formulations for optimization under stochastic dominance constraints. SIAM J Optim 19(3):1433-1450 · Zbl 1180.90215 · doi:10.1137/070707956
[38] Markowitz HM (1952) Portfolio selection. J Financ 7(1):77-91
[39] Markowitz HM (1959) Portfolio selection: efficient diversification in investments. Wiley, New York
[40] 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 · Zbl 1242.90146 · doi:10.1016/j.ejor.2011.07.044
[41] 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 · Zbl 0930.91020 · doi:10.1016/S0377-2217(96)00356-6
[42] von Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, Princeton · Zbl 0063.05930
[43] Ogryczak W, Ruszczynski A (2001) On consistency of stochastic dominance and mean-semideviation models. Math Program Ser B 89:217-232 · Zbl 1014.91021 · doi:10.1007/PL00011396
[44] Ogryczak W, Ruszczynski A (2002) Dual stochastic dominance and related mean-risk models. SIAM J Optim 13:60-78 · Zbl 1022.91017 · doi:10.1137/S1052623400375075
[45] Pflug, G.; Uryasev, SP (ed.), Some remarks on the value-at-risk and the conditional value-at-risk, 272-281 (2000), Dordrecht · Zbl 0994.91031 · doi:10.1007/978-1-4757-3150-7_15
[46] Post T (2003) Empirical tests for stochastic dominance efficiency. J Financ 58:1905-1932 · doi:10.1111/1540-6261.00592
[47] Rockafellar RT, Uryasev S (2000) Optimization of conditional value-at-risk. J Risk 2:21-41
[48] Rockafellar RT, Uryasev S (2002) Conditional value-at-risk for general loss distributions. J Banking Financ 26:1443-1471 · doi:10.1016/S0378-4266(02)00271-6
[49] Rockafellar RT, Uryasev S, Zabarankin M (2006) Generalized deviations in risk analysis. Financ Stoch 10:51-74 · Zbl 1150.90006 · doi:10.1007/s00780-005-0165-8
[50] Soleimani-damaneh M (2012) On a basic definition of returns to scale. Oper Res Lett 40(2):144-147 · Zbl 1242.90096 · doi:10.1016/j.orl.2011.11.005
[51] Whitmore GA (1970) Third-degree stochastic dominance. Am Econ Rev 60(3):457-459
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