Using MODEA and MODM with Different Risk Measures for Portfolio Optimization

Document Type: Research Paper


1 Department of Mathematics,Faculty of Science, Science and Research Branch, Islamic Azad University, Tehran, Iran

2 Department of Mathematics, Faculty of Science, Science and Research Branch, Islamic Azad University, Tehran, Iran

3 Department of Mathematics, Faculty of Mathematics and Computer Science, Allameh Tabataba'i University, Tehran, Iran.



The purpose of this study is to develop portfolio optimization and assets allocation using our proposed models. The study is based on a non-parametric efficiency analysis tool, namely Data Envelopment Analysis (DEA). Conventional DEA models assume non-negative data for inputs and outputs. However, many of these data take the negative value, therefore we propose the MeanSharp-βRisk (MShβR) model and the Multi-Objective MeanSharp-βRisk (MOMShβR) model base on Range Directional Measure (RDM) that can take positive and negative values. We utilize different risk measures in these models consist of variance, semivariance, Value at Risk (VaR) and Conditional Value at Risk (CVaR) to find the best one as input. After using our proposed models, the efficient stock companies will be selected for making the portfolio. Then, by using Multi-Objective Decision Making (MODM) model we specified the capital allocation to the stock companies that selected for the portfolio. Finally, a numerical example of the Iranian stock companies is presented to demonstrate the usefulness and effectiveness of our models, and compare different risk measures together in our models and allocate assets.


Main Subjects

[1] Artzner P, Eber F, Eber J.M, Heath D, Thinking coherently, Risk, 1997, 10, P. 68–71.


[2] Artzner P, Delbaen F, Eber J.M, Heath D, Coherent measures of risk, Mathematical Finance, 1999, 9, P. 203–28. Doi: 10.1111/1467-9965.00068.


[3] Banihashemi Sh, Navidi S, Portfolio performance evaluation in Mean-CVaR framework: A comparison with non-parametric methods value at risk in Mean-VaR analysis. Operations Research Perspectives, 2017, 4, P. 21–28. Doi: 10.1016/j.orp.2017.02.001.


[4] Banihashemi Sh, Navidi S, Portfolio Optimization By Using MeanSharp-VaR and Multi Objective MeanSharp-VaR Models. Filomat, 2018, 32(3), P. 815–823. Doi: 10.2298/FIL1803815B.


[5] Banker R.D, Charnes A, Coopper W.W, Some models for estimating technical and scale inefficiencies in Data Envelopment Analysis, Journal of Management Science, 1984, 30, P. 1078-1092. Doi: 10.1287/mnsc.30.9.1078.


[6] Baumol W.J, An expected gain confidence limit criterion for portfolio selection, Journal of Management Science, 1963, 10, P. 174-182. Doi: 10.1287/mnsc.10.1.174.


[7] Charnes A, Cooper W.W, Rhodes E, Measuring the efficiency of decision-making units, European Journal of Operational Research, 1978, 2, P. 429–444. Doi: 10.1016/0377-2217(78)90138-8.


[8] Chen S.X, Tang C.Y, Nonparametric inference of Value at Risk for dependent financial returns, Journal of financial econometrics, 2005, 12, P. 227–55. Doi: 10.1093/jjfinec/nbi012.


[9] Claro J, Pinho de Sousa J, A multiobjective metaheuristic for a mean–risk multistage capacity investment problem with process flexibility, Computers and Operations Research, 2012, 39, P. 838–49. Doi: 10.1016/j.cor.2010.08.015.


[10] Duffie D, Pan J, An overview of value at risk, Journal of Derivatives, 1997, 1, P. 7–49. Doi: 10.3905/jod.1997.407971.


[11] Emrouznejad A, A Semi Oriented Radial Measure for measuring the efficiency of decision making units with negative data using DEA, European Journal of Operational Research, 2010, 200:297-304. Doi: 10.1016/j.ejor.2009.01.001.


[12] Farrell M, The measurement of productive efficiency. Journal of the Royal Statistical Society, 1957, 120A, P. 253-281. Doi: 10.2307/2343100.


[13] Glasserman P, Heidelberger P, Shahabuddin P, Portfolio value-at-risk with heavy-tailed risk factors, Mathematical Finance, 2002, 12(3), P. 239–69. Doi: 10.1111/1467-9965.00141.


[14] Hong L.J, Liu G.W, Simulating sensitivities of conditional value at risk, Management Science, 2009, 55(2), P. 281–93. Doi: 10.1287/mnsc.1080.0901.


[15] Huang C.Y, Chiou C.C, Wu T.H, Yang S.C, An integrated DEA-MODM methodology for portfolio optimization, Operational Research International Journal, 2015, 15, P. 115–136. Doi: 10.1007/s12351-014-0164-7.


[16] Huang D.S, Zhu S.S, Fabozzi F.J, Fukushima M, Portfolio selection with uncertain exit time: a robust CVaR approach, Journal of Economic Dynamics and Control, 2008, 32, P. 594–623. Doi: 10.1016/j.jedc.2007.03.003.


[17] Jeong S.O, Kang K.H, Nonparametric estimation of Value at Risk, Journal of Applied Statistics, 2009, 10, P. 1225–38. Doi: 10.1080/02664760802607517.


[18] John M.M, Hafize G.E, Applying CVaR for decentralized risk management of financial companies, Journal of Banking and Finance, 2006, 30, P. 627–44. Doi: 10.1016/j.jbankfin.2005.04.010.


[19] Markowitz H, Portfolio selection, Journal of Finance, 1952, 7(1), P. 77–91. Doi: 10.1111/j.1540-6261.1952.tb01525.x.


[20] Markowitz H, Todd P, Xu G, Yamane Y, Computation of mean-semivariance efficient sets by the critical line algorithm, Annals of Operation Research, 1993, 45(1), P. 307-317. Doi: 10.1007/BF02282055.


[21] Miryekemami S.A, Sadeh E, Amini Sabegh Z,Using Genetic Algorithm in Solving Stochastic Programming for Multi-Objective Portfolio Selection in Tehran Stock Exchange, Advances in Mathematical Finance and Applications, 2017, 2( 4), P. 107-120. Doi: 10.22034/AMFA.2017.536271


[22] Navidi S, Banihashemi Sh, Sanei M, Three steps method for portfolio optimization by using Conditional Value at Risk measure. Journal of New Researches in Mathematics, 2016, 2(5), P. 43–60.


[23] Ogryczak W, Multiple criteria linear programming model for portfolio selection, Annals of Operations Research, 2000, 97, P. 143–162. Doi: 10.1023/A:1018980308807.


[24] Ogryczak W, Ruszczynski A, Dual stochastic dominance and quantile risk measures, International Transactions in Operational Research, 2002, 9, P. 661–680. Doi: 10.1111/1475-3995.00380.


[25] Peracchi F, Tanase A.V, On estimating the conditional expected shortfall, Applied Stochastic Models in Business and Industry, 2008, 24(5), P. 471–93. Doi: 10.1002/asmb.729.


[26] Peykani P; Mohammadi E, Pishvaee M.S, Rostamy-Malkhalifeh M,Jabbarzadeh A,A novel fuzzy data envelopment analysis based on robust possibilistic programming: possibility, necessity and credibility-based approaches, RAIRO-Operations Research, 2018, 52(4), P. 1445-1463. Doi: 10.1051/ro/2018019.


[27] Peykani P; Mohammadi E, Rostamy-Malkhalifeh M, Hosseinzadeh Lotfi F,Fuzzy Data Envelopment Analysis Approach for Ranking of Stocks with an Application to Tehran Stock Exchange, Advances in Mathematical Finance and Applications, 2019, 4(1), P. 31-43. Doi: 10.22034/AMFA.2019.581412.1155.


[28] Peykani P; Mohammadi E, Emrouznejad A, Pishvaee M.S, Rostamy-Malkhalifeh M, Fuzzy data envelopment analysis: An adjustable approach, Expert Systems with Applications, 2019, 136, P. 439-452. Doi: 10.1016/j.eswa.2019.06.039.


[29] Pflug G.Ch, Some remarks on the value-at-risk and the conditional value-at-risk. In: Uryasev S, editor, Probabilistic Constrained Optimization: Methodology and Applications, Dordrecht: Kluwer Academic Publishers, 2000. Doi: 10.1007/978-1-4757-3150-7_15.


[30] Portela M.C, Thanassoulis e, Simpson g, A directional distance approach to deal with negative data in DEA: An application to bank branches, Journal Operational Research Society, 2004, 55(10), P. 1111-1121. Doi: 10.1057/palgrave.jors.2601768.


[31] RahmaniM, Khalili Eraqi M, Nikoomaram H, Portfolio Optimization by Means of Meta Heuristic Algorithms, Advances in Mathematical Finance and Applications,2019, 4(4), P. 83-97, Doi: 10.22034/AMFA.2019.579510.1144


[32] Rockfeller T, Uryasev S, Optimization of conditional value-at-risk, Journal of Risk, 2000, 2(3), P. 21–4. Doi: 10.21314/JOR.2000.038.


[33] Rockfeller T, Uryasev S, Conditional value-at-risk for general loss distribution, Journal of Banking and Finance, 2002, 26(7), P. 1443–71. Doi: 10.1016/S0378-4266(02)00271-6.


[34] Sawik T, Selection of a dynamic supply portfolio in make-to-order environment with risks, Computers and Operations Research, 2011, 38, P. 782–96. Doi: 10.1016/j.cor.2010.09.011.


[35] Scaillet O, Nonparametric estimation and sensitivity analysis of expected shortfall, Mathematical Finance, 2004, 14(1), P. 115–29. Doi: 10.1111/j.0960-1627.2004.00184.x.


[36] Scaillet O, Nonparametric estimation of conditional expected shortfall, Insurance and Risk Management Journal, 2005, 74, P. 639–60. Doi: unige:41796.


[37] Schaumburg J, Predicting extreme Value at Risk: Nonparametric quantile regression with refinements from extreme value theory. Computational Statistics and Data Analysis, 2012, 56, P. 4081-4096. Doi: 10.1016/j.csda.2012.03.016.


[38] Sharpe W.F, A simplifies model for portfolio analysis, Manage Science, 1963, 9, P. 277–293. Doi: 10.1287/mnsc.9.2.277.


[39] Sharpe W.F, A linear programming approximation for the general portfolio analysis problem, Journal Finance Quant Anal, 1971, 6, P. 1263–1275. Doi: 10.2307/2329860.


[40] Sharp J.A, Meng W, Liu W, A modified slacks-based measure model for Data Envelopment Analysis with natural negative outputs and inputs, Journal of the Operational Research Society, 2006, 57, P. 1-6. Doi: 10.1057/palgrave.jors.2602318.


[41] Silvapulle P, Granger C.W, Large returns, conditional correlation and portfolio diversification: A Value at Risk approach, Quantitative Finance, 2001, 10, P. 542–51. Doi: 10.1080/713665877.


[42] Steuer R.E, Paul N, Multiple criteria decision making combined with finance: a categorized bibliographic study, European Journal of Operational Research, 2003, 150(3), P. 496–515. Doi: 10.1016/S0377-2217(02)00774-9.


[43] Subbu R, Bonissone P, Eklund N, Bollapragada S, Chalermkraivuth K, Multiobjective financial portfolio design: a hybrid evolutionary approach, IEEE congress on evolutionary computation, 2005. Doi: 10.1109/CEC.2005.1554896.


[44] Yau S, Kwon R.H, Rogers J.S, Wu D, Financial and operational decisions in the electricity sector: contract portfolio optimization with the conditional value-at-risk criterion, International Journal of Production Economics, 2011, 134, P. 67–77. Doi: 10.1016/j.ijpe.2010.10.007.


[45] Yu K, Allay A, Yang S, Hand D.J, Kernel quantile-based estimation of expected shortfall, The Journal of Risk, 2010, 12(4), P. 15–32. Doi: 2438/14317.


[46] Zeleny M, Multiple criteria decision making, McGraw-Hill, New York, 1982.


[47] Zhu S.S, Fukushima M, Worst-case conditional value-at-risk with application to robust portfolio management, Operations Research, 2009, 57(5), P. 1155–68. Doi: 10.1287/opre.1080.0684.


[48] Zopounidis C, Multi criteria decision aid in financial management, European Journal of Operational Research, 1999, 119, P. 404–415. Doi: 10.1016/S0377-2217(99)00142-3.


[49] Zopounidis C, Doumpos M, Zanakis S, Stock evaluation using a preference disaggregation methodology, Decision Sciences, 1999, 30, P. 313–336. Doi:10.1111/j.1540-5915.1999.tb01612.x.