# American Institute of Mathematical Sciences

November  2020, 16(6): 3065-3081. doi: 10.3934/jimo.2019094

## Mean-CVaR portfolio selection model with ambiguity in distribution and attitude

 1 Fujian Province University Key Laboratory of Computational Science, School of Mathematical Sciences, Huaqiao University, Fujian, China 2 Sun Yat-sen Business School, Sun Yat-sen University, Guangzhou, China

* Corresponding author: Zhongfei Li

Received  November 2018 Revised  March 2019 Published  November 2020 Early access  July 2019

Fund Project: The first author was supported in part by the Program for Innovative Research Team in Science and Technology in Fujian Province University, and Quanzhou High-Level Talents Support Plan (No. 2017ZT012), and the Scientific Research Foundation of Huaqiao University (No. 18BS311). The third author was supported by the National Natural Science Foundation of China (No. 71721001) and the Natural Science Research Team of Guangdong Province of China (No. 2014A030312003)

In this paper, we develop $\alpha$-robust (maxmin) models, where the Conditional Value-at-Risk (CVaR) is to be optimized under ambiguity in distribution, mean returns, and covariance matrix. Our models allow the investor to distinguish ambiguity and ambiguity attitude with different levels of ambiguity aversion. For the case when there is a risk-free asset and short-selling is allowed, we obtain the analytic solution for the $\alpha$-robust CVaR optimization model subject to a minimum mean return constraint. Moreover, we also derive a closed-form portfolio rule for the $\alpha$-robust mean-CVaR optimization problem in a market without the risk-less asset. The results obtained from solving the numerical example show that if an investor is more ambiguity-averse, his investment strategy will always be more conservative.

Citation: Zhilin Kang, Xingyi Li, Zhongfei Li. Mean-CVaR portfolio selection model with ambiguity in distribution and attitude. Journal of Industrial and Management Optimization, 2020, 16 (6) : 3065-3081. doi: 10.3934/jimo.2019094
##### References:
 [1] G. Bayraksan and D. K. Love, Data-driven stochastic programming using phi-divergences, in The Operations Research Revolution, INFORMS, 2015, 1–19. doi: 10.1287/educ.2015.0134. [2] A. Ben-Tal and A. Nemirovski, Robust solutions of uncertain linear programs, Ops. Research Letters, 25 (1999), 1-13.  doi: 10.1016/S0167-6377(99)00016-4. [3] M. J. Best and R. R. Grauer, Sensitivity analysis for mean-variance portfolio problems, Mgmt. Science, 37 (1991), 980-989. [4] P. Bossaerts, P. Ghirardato, S. Guarnaschelli and W. R. Zame, Ambiguity in asset markets: Theory and experiment, The Review of Finan. Studies, 23 (2010), 1325-1359.  doi: 10.1093/rfs/hhp106. [5] J. Cheng, R. Chen, H. Najm, A. Pinar, C. Safta and J. P. Watso, Distributionally robust optimization with principal component analysis, SIAM J. on Optimization, 28 (2018), 1817-1841.  doi: 10.1137/16M1075910. [6] E. Delage and Y. Ye, Distributionally robust optimization under moment uncertainty with application to data-driven problems, Ops. Research, 58 (2010), 595-612.  doi: 10.1287/opre.1090.0741. [7] D. Ellsberg, Risk, ambiguity, and the savage axioms, The Quarterly Journal of Economics, 75 (1961), 643-669.  doi: 10.2307/1884324. [8] P. M. Esfahani and D. Kuhn, Data-driven distributionally robust optimization using the Wasserstein metric: Performance guarantees and tractable reformulations, Math. Programming, 171 (2018), 115-166.  doi: 10.1007/s10107-017-1172-1. [9] C. R. Fox and A. Tversky, Ambiguity aversion and comparative ignorance, The Quarterly Journal of Economics, 110 (1995), 585-603.  doi: 10.2307/2946693. [10] P. Ghirardato, F. Maccheroni and M. Marinacci, Differentiating ambiguity and ambiguity attitude, J. of Econ. Theory, 118 (2004), 133-173.  doi: 10.1016/j.jet.2003.12.004. [11] I. Gilboa and D. Schmeidler, Maxmin expected utility with non-unique prior, J. of Math. Econ., 18 (1989), 141-153.  doi: 10.1016/0304-4068(89)90018-9. [12] C. Heath and A. Tversky, Preference and belief: Ambiguity and competence in choice under uncertainty, J. of Risk and Uncertainty, 4 (1991), 5-28.  doi: 10.1007/BF00057884. [13] R. Jiang and Y. Guan, Data-driven chance constrained stochastic program, Math. Programming, 158 (2016), 291-327.  doi: 10.1007/s10107-015-0929-7. [14] Z. Kang, X. Li, Z. Li and S. Zhu, Data-driven robust mean-CVaR portfolio selection under distribution ambiguity, Quant. Finan., 19 (2019), 105-121.  doi: 10.1080/14697688.2018.1466057. [15] Z. Kang and Z. Li, An exact solution to a robust portfolio choice problem with multiple risk measures under ambiguous distribution, Math. Methods of Ops. Research, 87 (2018), 169-195.  doi: 10.1007/s00186-017-0614-0. [16] B. Li, D. Li and D. Xiong, Alpha-robust mean-variance reinsurance-investment strategy, J. of Econ. Dynamics and Control, 70 (2016), 101-123.  doi: 10.1016/j.jedc.2016.07.001. [17] B. Li, L. Wang and D. Xiong, Robust utility maximization with extremely ambiguity-loving and ambiguity-aversion preferences, Stochastics, 90 (2018), 524-538.  doi: 10.1080/17442508.2017.1371176. [18] J. Liu, Z. Chen, A. Lisser and Z. Xu, Closed-Form optimal portfolios of distributionally robust mean-CVaR problems with unknown mean and variance, Appl. Math. & Optimization, 79 (2019), 671-693.  doi: 10.1007/s00245-017-9452-y. [19] S. Lotfi, M. Salahi and F. Mehrdoust, Adjusted robust mean-value-at-risk model: Less conservative robust portfolios, Optimization and Engineering, 18 (2017), 467-497.  doi: 10.1007/s11081-016-9340-3. [20] S. Lotfi and S. A. Zenios, Robust VaR and CVaR optimization under joint ambiguity in distributions, means, and covariances, European J. of Oper. Research, 269 (2018), 556-576.  doi: 10.1016/j.ejor.2018.02.003. [21] A. B. Paç and M. Ç. Pinar, Robust portfolio choice with CVaR and VaR under distribution and mean return ambiguity, TOP, 22 (2014), 875-891.  doi: 10.1007/s11750-013-0303-y. [22] I. Popescu, Robust mean-covariance solutions for stochastic optimization, Ops. Research, 55 (2007), 98–112. doi: 10.1287/opre.1060.0353. [23] A. G. Quaranta and A. Zaffaroni, Robust optimization of Conditional Value-at-Risk and portfolio selection, J. of Banking & Finance, 32 (2008), 2046-2056.  doi: 10.1016/j.jbankfin.2007.12.025. [24] K. Ruan and M. Fukushima, Robust portfolio selection with a combined WCVaR and factor model, J. of Indust. & Mgmt. Optimization, 8 (2012), 343-362.  doi: 10.3934/jimo.2012.8.343. [25] W. Wiesemann, D. Kuhn and M. Sim, Distributionally robust convex optimization, Ops. Research, 62 (2014), 1358-1376.  doi: 10.1287/opre.2014.1314. [26] S. Zhu and M. Fukushima, Worst-case conditional value-at-risk with application to robust portfolio management, Ops. Research, 57 (2009), 1155-1168.  doi: 10.1287/opre.1080.0684. [27] W. Zhu and H. Shao, Closed-form solutions for extremely-case distortion risk measures and applications to robust portfolio management, 2018. Available from: https://ssrn.com/abstract=3103458.

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##### References:
 [1] G. Bayraksan and D. K. Love, Data-driven stochastic programming using phi-divergences, in The Operations Research Revolution, INFORMS, 2015, 1–19. doi: 10.1287/educ.2015.0134. [2] A. Ben-Tal and A. Nemirovski, Robust solutions of uncertain linear programs, Ops. Research Letters, 25 (1999), 1-13.  doi: 10.1016/S0167-6377(99)00016-4. [3] M. J. Best and R. R. Grauer, Sensitivity analysis for mean-variance portfolio problems, Mgmt. Science, 37 (1991), 980-989. [4] P. Bossaerts, P. Ghirardato, S. Guarnaschelli and W. R. Zame, Ambiguity in asset markets: Theory and experiment, The Review of Finan. Studies, 23 (2010), 1325-1359.  doi: 10.1093/rfs/hhp106. [5] J. Cheng, R. Chen, H. Najm, A. Pinar, C. Safta and J. P. Watso, Distributionally robust optimization with principal component analysis, SIAM J. on Optimization, 28 (2018), 1817-1841.  doi: 10.1137/16M1075910. [6] E. Delage and Y. Ye, Distributionally robust optimization under moment uncertainty with application to data-driven problems, Ops. Research, 58 (2010), 595-612.  doi: 10.1287/opre.1090.0741. [7] D. Ellsberg, Risk, ambiguity, and the savage axioms, The Quarterly Journal of Economics, 75 (1961), 643-669.  doi: 10.2307/1884324. [8] P. M. Esfahani and D. Kuhn, Data-driven distributionally robust optimization using the Wasserstein metric: Performance guarantees and tractable reformulations, Math. Programming, 171 (2018), 115-166.  doi: 10.1007/s10107-017-1172-1. [9] C. R. Fox and A. Tversky, Ambiguity aversion and comparative ignorance, The Quarterly Journal of Economics, 110 (1995), 585-603.  doi: 10.2307/2946693. [10] P. Ghirardato, F. Maccheroni and M. Marinacci, Differentiating ambiguity and ambiguity attitude, J. of Econ. Theory, 118 (2004), 133-173.  doi: 10.1016/j.jet.2003.12.004. [11] I. Gilboa and D. Schmeidler, Maxmin expected utility with non-unique prior, J. of Math. Econ., 18 (1989), 141-153.  doi: 10.1016/0304-4068(89)90018-9. [12] C. Heath and A. Tversky, Preference and belief: Ambiguity and competence in choice under uncertainty, J. of Risk and Uncertainty, 4 (1991), 5-28.  doi: 10.1007/BF00057884. [13] R. Jiang and Y. Guan, Data-driven chance constrained stochastic program, Math. Programming, 158 (2016), 291-327.  doi: 10.1007/s10107-015-0929-7. [14] Z. Kang, X. Li, Z. Li and S. Zhu, Data-driven robust mean-CVaR portfolio selection under distribution ambiguity, Quant. Finan., 19 (2019), 105-121.  doi: 10.1080/14697688.2018.1466057. [15] Z. Kang and Z. Li, An exact solution to a robust portfolio choice problem with multiple risk measures under ambiguous distribution, Math. Methods of Ops. Research, 87 (2018), 169-195.  doi: 10.1007/s00186-017-0614-0. [16] B. Li, D. Li and D. Xiong, Alpha-robust mean-variance reinsurance-investment strategy, J. of Econ. Dynamics and Control, 70 (2016), 101-123.  doi: 10.1016/j.jedc.2016.07.001. [17] B. Li, L. Wang and D. Xiong, Robust utility maximization with extremely ambiguity-loving and ambiguity-aversion preferences, Stochastics, 90 (2018), 524-538.  doi: 10.1080/17442508.2017.1371176. [18] J. Liu, Z. Chen, A. Lisser and Z. Xu, Closed-Form optimal portfolios of distributionally robust mean-CVaR problems with unknown mean and variance, Appl. Math. & Optimization, 79 (2019), 671-693.  doi: 10.1007/s00245-017-9452-y. [19] S. Lotfi, M. Salahi and F. Mehrdoust, Adjusted robust mean-value-at-risk model: Less conservative robust portfolios, Optimization and Engineering, 18 (2017), 467-497.  doi: 10.1007/s11081-016-9340-3. [20] S. Lotfi and S. A. Zenios, Robust VaR and CVaR optimization under joint ambiguity in distributions, means, and covariances, European J. of Oper. Research, 269 (2018), 556-576.  doi: 10.1016/j.ejor.2018.02.003. [21] A. B. Paç and M. Ç. Pinar, Robust portfolio choice with CVaR and VaR under distribution and mean return ambiguity, TOP, 22 (2014), 875-891.  doi: 10.1007/s11750-013-0303-y. [22] I. Popescu, Robust mean-covariance solutions for stochastic optimization, Ops. Research, 55 (2007), 98–112. doi: 10.1287/opre.1060.0353. [23] A. G. Quaranta and A. Zaffaroni, Robust optimization of Conditional Value-at-Risk and portfolio selection, J. of Banking & Finance, 32 (2008), 2046-2056.  doi: 10.1016/j.jbankfin.2007.12.025. [24] K. Ruan and M. Fukushima, Robust portfolio selection with a combined WCVaR and factor model, J. of Indust. & Mgmt. Optimization, 8 (2012), 343-362.  doi: 10.3934/jimo.2012.8.343. [25] W. Wiesemann, D. Kuhn and M. Sim, Distributionally robust convex optimization, Ops. Research, 62 (2014), 1358-1376.  doi: 10.1287/opre.2014.1314. [26] S. Zhu and M. Fukushima, Worst-case conditional value-at-risk with application to robust portfolio management, Ops. Research, 57 (2009), 1155-1168.  doi: 10.1287/opre.1080.0684. [27] W. Zhu and H. Shao, Closed-form solutions for extremely-case distortion risk measures and applications to robust portfolio management, 2018. Available from: https://ssrn.com/abstract=3103458.
The left panel shows that $k(\alpha)$ and $b(\alpha)$ are decreasing in $\alpha$. The efficient frontier lines for $\alpha$-robust CVaR model are shown in the right panel. The steepest line (Dash-dot line, black) and flattest line (Solid line, blue) correspond to the cases $\alpha = 0.5$ and $\alpha = 1$, respectively. ($H = 0.4722$, $r_f = 1.01$, $\gamma_1 = 0.0001$, $\gamma_2 = 1.2$, $\beta = 0.95$)
Efficient frontiers of the $\alpha$-maxmin mean-CVaR model with different parameter $\alpha$. The $\alpha$-maxmin portfolio CVaR in the $x$-axis ($\alpha$-maxmin portfolio return in the $y$-axis) is a convex mixture between the worst-case and best-case values of CVaR risk measures (expected return)
Effects of $\alpha$ (the level of ambiguity aversion) on the $\alpha$-maxmin portfolio return and $\alpha$-maxmin portfolio CVaR
Effect of $\alpha$ (the level of ambiguity aversion) on the composition of efficient portfolios from the $\alpha$-maxmin mean-CVaR model. The percentage allocation of assets 1-3 in the optimal allocation $x^*$ have been illustrated in different colors
The variations of optimal portfolio strategies under different levels of ambiguity $\gamma_1$ (for a given $\gamma_2 = 1.2$) and $\gamma_2$ (for a given $\gamma_1 = 0.0001$). The percentage allocation of assets 1-3 in the optimal allocation $x^*$ have been illustrated in different colors
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