Performance evaluation of portfolios with fuzzy returns
RAIRO - Operations Research - Recherche Opérationnelle, Tome 53 (2019) no. 5, pp. 1581-1600.

The existing literature on DEA (Data Envelopment Analysis) for evaluating fuzzy portfolios usually takes risk as an input and return as an output. This assumption is actually not congruent with the real investment process, where the input is the initial wealth and the output is the corresponding terminal wealth. As for the risk and return, which are essentially two indicators derived from the terminal wealth, both should be regarded as outputs. In addition, few studies have employed the diversification model (nonlinear DEA) to estimate the fuzzy portfolio efficiency (PE), despite the fact that there are many studies available within the framework of classical probability theory. Further, the relationship between DEA and diversification models needs to be defined. In this paper, we take the initial wealth as an input, while the return and risk of terminal wealth are taken as desirable and undesirable outputs, respectively. We construct different evaluation models under the fuzzy portfolio framework. The relationships among the evaluation model based on a real frontier, the diversification model and the DEA model are investigated. We show the convergence of the diversification and DEA models under the fuzzy theory framework. Some simulations as well as empirical analysis are presented to further verify the effectiveness of the proposed models. Finally, we check the robustness of the evaluation results by using the bootstrap re-sampling approach.

Reçu le :
Accepté le :
DOI : 10.1051/ro/2019071
Classification : 90B30, 90B50, 90C05, 90C30, 90C70
Mots-clés : Fuzzy portfolio evaluation, possibilistic measures, diversification model, DEA, bootstrap re-sampling
Zhou, Zhongbao 1 ; Chen, Enming 1 ; Xiao, Helu 1 ; Ren, Tiantian 1 ; Jin, Qianying 1

1 
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     author = {Zhou, Zhongbao and Chen, Enming and Xiao, Helu and Ren, Tiantian and Jin, Qianying},
     title = {Performance evaluation of portfolios with fuzzy returns},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {1581--1600},
     publisher = {EDP-Sciences},
     volume = {53},
     number = {5},
     year = {2019},
     doi = {10.1051/ro/2019071},
     mrnumber = {4016080},
     zbl = {1431.90060},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ro/2019071/}
}
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Zhou, Zhongbao; Chen, Enming; Xiao, Helu; Ren, Tiantian; Jin, Qianying. Performance evaluation of portfolios with fuzzy returns. RAIRO - Operations Research - Recherche Opérationnelle, Tome 53 (2019) no. 5, pp. 1581-1600. doi : 10.1051/ro/2019071. http://www.numdam.org/articles/10.1051/ro/2019071/

[1] A. Basso and S. Funari, A data envelopment analysis approach to measure the mutual fund performance. Eur. J. Oper. Res. 135 (2001) 477–492. | DOI | Zbl

[2] M. Branda, Diversification-consistent data envelopment analysis with general deviation measures. Eur. J. Oper. Res. 226 (2013) 626–635. | DOI | MR | Zbl

[3] M. Branda, Reformulations of input–output oriented DEA tests with diversification. Oper. Res. Lett. 41 (2013) 516–520. | DOI | MR | Zbl

[4] M. Branda, Diversification-consistent data envelopment analysis based on directional-distance measures. Omega 52 (2015) 65–76. | DOI

[5] W. Briec and K. Kerstens, Multi-horizon Markowitz portfolio performance appraisals: a general approach. Omega 37 (2009) 50–62. | DOI

[6] W. Briec, K. Kerstens and J.B. Lesourd, Single-period Markowitz portfolio selection, performance gauging, and duality: a variation on the Luenberger shortage function. J. Optim. Theor. App. 120 (2004) 1–27. | DOI | MR | Zbl

[7] W. Briec, K. Kerstens and O. Jokung, Mean-variance-skewness portfolio performance gauging: a general shortage function and dual approach. Manage. Sci. 53 (2007) 135–149. | DOI | Zbl

[8] E. Cao and M. Lai, A hybrid differential evolution algorithm to vehicle routing problem with fuzzy demands. J. Comput. Appl. Math. 231 (2009) 302–310. | DOI | MR | Zbl

[9] J. Cao, G. Lian and T.R.N. Roslan, Pricing variance swaps under stochastic volatility and stochastic interest rate. Appl. Math. Comput. 277 (2016) 72–81. | DOI | MR | Zbl

[10] M.M. Carhart, On persistence in mutual fund performance. J. Finance 52 (1997) 57–82. | DOI

[11] C. Carlsson, R. Fullér and P. Majlender, A possibilistic approach to selecting portfolios with highest utility score. Fuzzy Sets Syst. 1 (2002) 13–21. | DOI | MR | Zbl

[12] Z. Chen and R. Lin, Mutual fund performance evaluation using data envelopment analysis with new risk measures. Or Spectr. 28 (2006) 375–398. | DOI | MR | Zbl

[13] W. Chen, Y. Gai and P. Gupta, Efficiency evaluation of fuzzy portfolio in different risk measures via DEA. Ann. Oper. Res. 269 (2018) 103–127. | DOI | MR | Zbl

[14] W. Chen, S. Li, J. Zhang and M.K. Mehlawat, A comprehensive model for fuzzy multi-objective portfolio selection based on DEA cross-efficiency model. To appear in Soft Comput. DOI: (2018). | DOI

[15] H. Ding, Z. Zhou, H. Xiao, C. Ma and W. Liu, Performance evaluation of portfolios with margin requirements. Math. Prob. Eng. 2014 (2014) 1–8. | DOI | MR | Zbl

[16] D. Dubois and H. Prade, Possibility Theory. Edited by Meyers, R.A. In Encyclopedia of Complexity and Systems Science. Springer, Heidelberg (2009) 6927–6939. | DOI

[17] E.F. Fama and K.R. French, Disagreements, tastes and asset prices. J. Financial Econ. 83 (1993) 667–689. | DOI

[18] X. Huang, Mean-entropy models for fuzzy portfolio selection. IEEE Trans. Fuzzy Syst. 16 (2008) 1096–1101. | DOI

[19] Y. Huang, X. Yang and J. Zhou, Optimal investment and proportional reinsurance for a jump–diffusion risk model with constrained control variables. J. Comput. Appl. Math. 296 (2016) 443–461. | DOI | MR | Zbl

[20] M.C. Jensen, The performance of mutual funds in the period 1945–1964. J. Finance 2 (1968) 389–416.

[21] T. Joro and P. Na, Portfolio performance evaluation in a mean-variance-skewness framework. Eur. J. Oper. Res. 175 (2006) 446–461. | DOI | Zbl

[22] J.S. Kamdem, C.T. Deffo and L.A. Fono, Moments and semi-moments for fuzzy portfolio selection. Insurance Math. Econ. 51 (2012) 517–530. | DOI | MR | Zbl

[23] J.D. Lamb and K.H. Tee, Data envelopment analysis models of investment funds. Eur. J. Oper. Res. 216 (2012) 687–696. | DOI | MR | Zbl

[24] B. Liu and Y.K. Liu, Expected value of fuzzy variable and fuzzy expected value models. IEEE Trans. Fuzzy Syst. 10 (2002) 445–450. | DOI

[25] Y. Liu and W. Zhang, A multi-period fuzzy portfolio optimization model with minimum transaction lots. Eur. J. Oper. Res. 242 (2015) 933–941. | DOI | MR | Zbl

[26] Y. Liu, W. Zhang and W. Xu, Fuzzy multi-period portfolio selection optimization models using multiple criteria. Automatica 48 (2012) 3042–3053. | DOI | MR | Zbl

[27] W. Liu, Z. Zhou, D. Liu and H. Xiao, Estimation of portfolio efficiency via DEA. Omega 52 (2015) 107–118. | DOI

[28] S. Lozano and E. Gutiérrez, Data envelopment analysis of mutual funds based on second-order stochastic dominance. Eur. J. Oper. Res. 189 (2008) 230–244. | DOI | MR | Zbl

[29] H. Markowitz, Portfolio selection. J. Finance 7 (1952) 77–91.

[30] Z. Mashayekhi and H. Omrani, An integrated multi-objective Markowitz–DEA cross-efficiency model with fuzzy returns for portfolio selection problem. Appl. Soft Comput. 38 (2016) 1–9. | DOI

[31] M.K. Mehlawat, Credibilistic mean-entropy models for multi-period portfolio selection with multi-choice aspiration levels. Inf. Sci. 345 (2016) 9–26. | DOI | Zbl

[32] M.R. Morey and R.C. Morey, Mutual fund performance appraisals: a multi-horizon perspective with endogenous benchmarking. Omega 27 (1999) 241–258. | DOI

[33] B.P.S. Murthi, Y.K. Choi and P. Desai, Efficiency of mutual funds and portfolio performance measurement: a non-parametric approach. Eur. J. Oper. Res. 98 (1997) 408–418. | DOI | Zbl

[34] Z. Qin, X. Li and X. Ji, Portfolio selection based on fuzzy cross-entropy. J. Comput. Appl. Math. 228 (2009) 139–149. | DOI | MR | Zbl

[35] A. Saeidifar and E. Pasha, On the possibilistic moments of fuzzy numbers and their applications. J. Comput. Appl. Math. 223 (2009) 1028–1042. | DOI | MR | Zbl

[36] W.F. Sharpe, Mutual fund performance. J. Bus. 1 (1966) 119–138. | DOI

[37] M. Silva Portela, E. Thanassoulis and G. Simpson, Negative data in DEA: a directional distance approach applied to bank branches. J. Oper. Res. Soc. 55 (2004) 1111–1121. | DOI | Zbl

[38] J.L. Treynor, How to rate management of investment funds. Harvard Bus. Rev. 43 (1965) 63–75.

[39] E. Vercher, J.D. Bermúdez and J.V. Segura, Fuzzy portfolio optimization under downside risk measures. Fuzzy Sets Syst. 158 (2007) 769–782. | DOI | MR | Zbl

[40] L.A. Zadeh, Fuzzy sets. Inf. Control 8 (1965) 338–353. | DOI | MR | Zbl

[41] P. Zhang and W. Zhang, Multiperiod mean absolute deviation fuzzy portfolio selection model with risk control and cardinality constraints. Fuzzy Sets Syst. 255 (2014) 74–91. | DOI | MR | Zbl

[42] W. Zhang, Y. Liu and W. Xu, A possibilistic mean-semivariance-entropy model for multi-period portfolio selection with transaction costs. Eur. J. Oper. Res. 222 (2012) 341–349. | DOI | MR | Zbl

[43] X. Zhao, S. Wang and K.K. Lai, Mutual funds performance evaluation based on endogenous benchmarks. Expert Syst. App. 38 (2011) 3663–3670. | DOI

[44] Z. Zhou, L. Zhao, S. Lui and C. Ma, A generalized fuzzy DEA/AR performance assessment model. Math. Comput. Model. 55 (2012) 2117–2128. | DOI | MR | Zbl

[45] Z. Zhou, H. Xiao, J. Yin, X. Zeng and L. Lin, Pre-commitment vs. time-consistent strategies for the generalized multi-period portfolio optimization with stochastic cash flows. Insurance: Math. Econ. 68 (2016) 187–202. | MR | Zbl

[46] Z. Zhou, Q. Jin, H. Xiao, Q. Wu and W. Liu, Estimation of cardinality constrained portfolio efficiency via segmented DEA. Omega 76 (2018) 28–37. | DOI

[47] Z. Zhou, X. Liu, H. Xiao, T. Ren and W. Liu, Time-consistent strategies for multi-period portfolio optimization with/without the risk-free asset. Math. Prob. Eng. 2018 (2018) 20. | DOI | MR | Zbl

[48] Z. Zhou, H. Xiao, Q. Jin, W. Liu, DEA frontier improvement and portfolio rebalancing: an application of China mutual funds on considering sustainability information disclosure. Eur. J. Oper. Res. 269 (2018) 111–131. | DOI | MR | Zbl

[49] Z. Zhou, X. Zeng, H. Xiao, T. Ren and W. Liu, Multiperiod portfolio optimization for asset-liability management with quadratic transaction costs. J. Ind. Manage. Optim. 15 (2019) 1493–1515. | MR | Zbl

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