# American Institute of Mathematical Sciences

March  2020, 16(2): 759-775. doi: 10.3934/jimo.2018177

## Multi-objective robust cross-market mixed portfolio optimization under hierarchical risk integration

 1 Department of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, China 2 Department of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan, 610074, China

* Corresponding author: Nan-jing Huang

Received  August 2017 Revised  August 2018 Published  March 2020 Early access  December 2018

Fund Project: This work was supported by the National Natural Science Foundation of China (11471230, 11671282, 11801462)

In this paper, we consider a multi-objective robust cross-market mixed portfolio optimization model under hierarchical risk integration in the international financial market consisting of finite sub-markets. It is difficult to describe the dependent structure accurately by the traditional copula theory because of the dependent structures of the risk assets in finite sub-markets are different usually. By employing the hierarchical risk integration method, we establish the multi-objective robust cross-market mixed portfolio model in which the worst-case value at risk is used as the risk measurement and the transaction costs, skewness and investment proportion limitation are all considered. We provide a new algorithm to calculate the worst-case value at risk of the cross-market mixed portfolio and give a numerical experiment to show the superiority of the model considered in this paper.

Citation: Han Yang, Jia Yue, Nan-jing Huang. Multi-objective robust cross-market mixed portfolio optimization under hierarchical risk integration. Journal of Industrial and Management Optimization, 2020, 16 (2) : 759-775. doi: 10.3934/jimo.2018177
##### References:
 [1] P. Arbenz, C. Hummel and G. Mainik, Copula based hierarchical risk aggregation through sample reordering, Insurance Mathematics and Economics, 51 (2012), 122-133.  doi: 10.1016/j.insmatheco.2012.03.009. [2] E. D. Arditti and H. Levy, Portfolio efficient analysis in three monents-the multi-period case, Journal of Finance, 30 (1975), 797-809. [3] P. Artzner, F. Delbaen, J. M. Eber and D. Heath, Coherent measures of risk, Mathematical Finance, 9 (1999), 203-228.  doi: 10.1111/1467-9965.00068. [4] I. Bajeux-Besnainou, R. Belhaj, D. Maillard and R. Portait, Portfolio optimization under tracking error and weight constraints, Journal of Financial Research, 34 (2011), 295-330. [5] A. Burcu, D. Fatma and H. Soyuer, Mean-variance-skewness-kurtosis approach to portfolio optimization: an application in Istanbul stock exchange, Ege Academic Review, 11 (2011), 9-17. [6] Y. P. Chang, J. X. Lin and C. T. Yu, Calculating value-at-risk using the granularity adjustment method in the portfolio credit risk model with random loss given default, Journal of Economics and Management, 12 (2016), 157-176. [7] P. Chunhachinda, K. Dandapani, S. Hamid and A. J. Prakash, Portfolio selection and skewness: Evidence from international stock markets, Journal of Banking and Finance, 21 (1997), 143-167. [8] R. Diestel, Graph Theory, Springer-Verlag, New York, 2000. doi: 10.1007/b100033. [9] K. W. Ding, M. H. Wang and N. J. Huang, Distributionally robust chance constrained problem under interval distribution information, Optimlization Letters, 12 (2018), 1315-1328.  doi: 10.1007/s11590-017-1160-7. [10] E. Eberlein and U. Keller, Hyperbolic distributions in finance, Bernoulli, 1 (2000), 281-299. [11] M. R. Fengler and O. Okhrin, Managing risk with a realized copula parameter, Computational Statistics and Data Analysis, 100 (2016), 131-152.  doi: 10.1016/j.csda.2014.07.011. [12] L. Garlappi, R. Uppal and T. Wang, Portfolio selection with parameter and model uncertainty: A multi-prior approach, Review of Financial Studies, 20 (2007), 41-81. [13] L. E. Ghaoui, M. Oks and F. Oustry, Worst-case value-at-risk and robust portfolio optimization: A conic programming approach, Operations Research, 51 (2003), 543-556.  doi: 10.1287/opre.51.4.543.16101. [14] A. Ghorbel and A. Trabelsi, Energy portfolio risk management using time-varying extreme value copula methods, Economic Modelling, 38 (2014), 470-485. [15] C. Giorgio, Tail estimation and mean-VaR portfolio sdeetion in market subject to financial instability, Journal of Banking and Finance, 26 (2002), 1355-1382. [16] D. Goldfarb and G. Iyengar, Robust portfolio selection problems, Mathematics of Operations Research, 28 (2003), 1-38.  doi: 10.1287/moor.28.1.1.14260. [17] Y. Huang and S. H. Fang, Mean-variance analysis of arbitrage portfolios under investment proportion limitations, Chinese Journal of Engineering Mathematics, 24 (2007), 662-668. [18] P. Jana, T. K. Roy and S. K. Mazumder, Multi-objective possibilistic model for portfolio selection with transaction cost, Journal of Computational and Applied Mathematics, 228 (2009), 188-196.  doi: 10.1016/j.cam.2008.09.008. [19] I. Kakouris and B. Rustem, Robust portfolio optimization with copulas, European Journal of Operational Research, 235 (2014), 28-37.  doi: 10.1016/j.ejor.2013.12.022. [20] C. H. Kimberling, A probabilistic interpretation of complete monotonicity, Aequationes Mathematicae, 10 (1974), 152-164.  doi: 10.1007/BF01832852. [21] H. Konno and K. I. Suzuki, A mean-variance-skewness portfolio optimization model, Journal of the Operations Research Society of Japan, 38 (1995), 173-187. [22] K. K. Lai, L. Yu and S. Wang, Mean-variance-skewness-kurtosis-based portfolio optimization, International Multi-Symposiums on Computer and Computational Sciences, 2 (2006), 292-297. [23] J. Li and J. Xu, Multi-objective portfolio selection model with fuzzy random returns and a compromise approach-based genetic algorithm, Information Sciences, 220 (2013), 507-521.  doi: 10.1016/j.ins.2012.07.005. [24] X. Li, Z. Qin and S. Kar, Mean-variance-skewness model for portfolio selection with fuzzy returns, European Journal of Operational Research, 202 (2010), 239-247. [25] X. H. Li and Y. P. You, A note on allocation of portfolio shares of random assets with Archimedean copula, Annals of Operations Research, 212 (2014), 155-167.  doi: 10.1007/s10479-012-1137-y. [26] M. S. Lobo and S. Boyd, The worst-case risk of a portfolio, Journal of Economic Dynamics and Control, 26 (2000), 1159-1193. [27] H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91. [28] R. B. Nelsen, An Introduction to Copulas, 2$^{nd}$ edition, Springer, Berlin, 2006. [29] M. H. A. Davis and A. R. Norman, Portfolio selection with transaction costs, Mathematics of Operations Research, 15 (1988), 676-713.  doi: 10.1287/moor.15.4.676. [30] H. Sak, W. Hormann and J. Leydold, Efficient risk simulation for liner asset portfolios in the t-copula model, European Journal of Operational Research, 202 (2010), 802-809. [31] A. Sklar, Fonctions de répartition à n dimensions et leurs marges, Publications de l'Institut de Statistique de l'Université de Paris, 8 (1959), 229-231. [32] B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland Publishing, Amsterdam, 1983. [33] H. Soleimani, H. R. Golmakani and M. H. Salimi, Markowitz-based portfolio selection with minimum transaction lots, cardinality constraints and regarding sector capitalization using genetic algorithm, Expert Systems with Applications, 36 (2009), 5058-5063. [34] J. B. Su, Value-at-risk estimates of the stock indices in developed and emerging markets including the spillover effects of currency market, Economic Modelling, 46 (2015), 204-224. [35] G. Szego, Measures of risk, European Journal of Operational Research, 163 (2005), 5-19.  doi: 10.1016/j.ejor.2003.12.016. [36] H. A. L. Thi and M. Moeini, Long-short portfolio optimization under cardinality constraints by difference of convex functions algorithm, Journal of Optimization Theory and Applications, 161 (2014), 199-224.  doi: 10.1007/s10957-012-0197-0. [37] M. H. Wang, J. Yue and N. J. Huang, Robust mean variance portfolio selection model in the jump-diffusion financial market with an intractable claim, Optimization, 66 (2017), 1219-1234.  doi: 10.1080/02331934.2017.1310212. [38] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1941. [39] M. Wu, N. J. Huang and B. S. Lee, Portfolio selection with transaction costs and different borrowing and lending interest rates in varying capital structure, Nonlinear Analysis Forum, 9 (2004), 175-200. [40] H. Yang, W. Zhou and K. W. Ding, Markowitz investment portfolio model with transaction costs and capital budget in varying capital structure, Nonlinear Analysis Forum, 19 (2014), 119-127. [41] H. X. Yao and Z. F. Li, Portfolio model and its explicit expressions of portfolio efficient frontier with minimum investment proportion constraint, OR Transactions, 13 (2009), 119-128. [42] X. L. Zhan and S. Y. Zhang, Risk analysis of financial portfolio based on copula-SV model, Journal of Systems and Management, 16 (2007), 302-306. [43] W. G. Zhang, Y. J. Liu and W. J. Xu, A possibilistic mean-semivariance-entropy model for multi-period portfolio selection with transaction costs, European Journal of Operational Research, 222 (2012), 341-349.  doi: 10.1016/j.ejor.2012.04.023. [44] Q. Zhou, Z. Chen and R. Ming, Copula-based grouped risk aggregation under mixed operation, Applications of Mathematics, 61 (2016), 103-120.  doi: 10.1007/s10492-016-0124-z. [45] S. Zymler, D. Kuhn and B. Rustem, Worst-case Value at Risk of nonlinear portfolios, Management Science, 59 (2013), 172-188.

show all references

##### References:
 [1] P. Arbenz, C. Hummel and G. Mainik, Copula based hierarchical risk aggregation through sample reordering, Insurance Mathematics and Economics, 51 (2012), 122-133.  doi: 10.1016/j.insmatheco.2012.03.009. [2] E. D. Arditti and H. Levy, Portfolio efficient analysis in three monents-the multi-period case, Journal of Finance, 30 (1975), 797-809. [3] P. Artzner, F. Delbaen, J. M. Eber and D. Heath, Coherent measures of risk, Mathematical Finance, 9 (1999), 203-228.  doi: 10.1111/1467-9965.00068. [4] I. Bajeux-Besnainou, R. Belhaj, D. Maillard and R. Portait, Portfolio optimization under tracking error and weight constraints, Journal of Financial Research, 34 (2011), 295-330. [5] A. Burcu, D. Fatma and H. Soyuer, Mean-variance-skewness-kurtosis approach to portfolio optimization: an application in Istanbul stock exchange, Ege Academic Review, 11 (2011), 9-17. [6] Y. P. Chang, J. X. Lin and C. T. Yu, Calculating value-at-risk using the granularity adjustment method in the portfolio credit risk model with random loss given default, Journal of Economics and Management, 12 (2016), 157-176. [7] P. Chunhachinda, K. Dandapani, S. Hamid and A. J. Prakash, Portfolio selection and skewness: Evidence from international stock markets, Journal of Banking and Finance, 21 (1997), 143-167. [8] R. Diestel, Graph Theory, Springer-Verlag, New York, 2000. doi: 10.1007/b100033. [9] K. W. Ding, M. H. Wang and N. J. Huang, Distributionally robust chance constrained problem under interval distribution information, Optimlization Letters, 12 (2018), 1315-1328.  doi: 10.1007/s11590-017-1160-7. [10] E. Eberlein and U. Keller, Hyperbolic distributions in finance, Bernoulli, 1 (2000), 281-299. [11] M. R. Fengler and O. Okhrin, Managing risk with a realized copula parameter, Computational Statistics and Data Analysis, 100 (2016), 131-152.  doi: 10.1016/j.csda.2014.07.011. [12] L. Garlappi, R. Uppal and T. Wang, Portfolio selection with parameter and model uncertainty: A multi-prior approach, Review of Financial Studies, 20 (2007), 41-81. [13] L. E. Ghaoui, M. Oks and F. Oustry, Worst-case value-at-risk and robust portfolio optimization: A conic programming approach, Operations Research, 51 (2003), 543-556.  doi: 10.1287/opre.51.4.543.16101. [14] A. Ghorbel and A. Trabelsi, Energy portfolio risk management using time-varying extreme value copula methods, Economic Modelling, 38 (2014), 470-485. [15] C. Giorgio, Tail estimation and mean-VaR portfolio sdeetion in market subject to financial instability, Journal of Banking and Finance, 26 (2002), 1355-1382. [16] D. Goldfarb and G. Iyengar, Robust portfolio selection problems, Mathematics of Operations Research, 28 (2003), 1-38.  doi: 10.1287/moor.28.1.1.14260. [17] Y. Huang and S. H. Fang, Mean-variance analysis of arbitrage portfolios under investment proportion limitations, Chinese Journal of Engineering Mathematics, 24 (2007), 662-668. [18] P. Jana, T. K. Roy and S. K. Mazumder, Multi-objective possibilistic model for portfolio selection with transaction cost, Journal of Computational and Applied Mathematics, 228 (2009), 188-196.  doi: 10.1016/j.cam.2008.09.008. [19] I. Kakouris and B. Rustem, Robust portfolio optimization with copulas, European Journal of Operational Research, 235 (2014), 28-37.  doi: 10.1016/j.ejor.2013.12.022. [20] C. H. Kimberling, A probabilistic interpretation of complete monotonicity, Aequationes Mathematicae, 10 (1974), 152-164.  doi: 10.1007/BF01832852. [21] H. Konno and K. I. Suzuki, A mean-variance-skewness portfolio optimization model, Journal of the Operations Research Society of Japan, 38 (1995), 173-187. [22] K. K. Lai, L. Yu and S. Wang, Mean-variance-skewness-kurtosis-based portfolio optimization, International Multi-Symposiums on Computer and Computational Sciences, 2 (2006), 292-297. [23] J. Li and J. Xu, Multi-objective portfolio selection model with fuzzy random returns and a compromise approach-based genetic algorithm, Information Sciences, 220 (2013), 507-521.  doi: 10.1016/j.ins.2012.07.005. [24] X. Li, Z. Qin and S. Kar, Mean-variance-skewness model for portfolio selection with fuzzy returns, European Journal of Operational Research, 202 (2010), 239-247. [25] X. H. Li and Y. P. You, A note on allocation of portfolio shares of random assets with Archimedean copula, Annals of Operations Research, 212 (2014), 155-167.  doi: 10.1007/s10479-012-1137-y. [26] M. S. Lobo and S. Boyd, The worst-case risk of a portfolio, Journal of Economic Dynamics and Control, 26 (2000), 1159-1193. [27] H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91. [28] R. B. Nelsen, An Introduction to Copulas, 2$^{nd}$ edition, Springer, Berlin, 2006. [29] M. H. A. Davis and A. R. Norman, Portfolio selection with transaction costs, Mathematics of Operations Research, 15 (1988), 676-713.  doi: 10.1287/moor.15.4.676. [30] H. Sak, W. Hormann and J. Leydold, Efficient risk simulation for liner asset portfolios in the t-copula model, European Journal of Operational Research, 202 (2010), 802-809. [31] A. Sklar, Fonctions de répartition à n dimensions et leurs marges, Publications de l'Institut de Statistique de l'Université de Paris, 8 (1959), 229-231. [32] B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland Publishing, Amsterdam, 1983. [33] H. Soleimani, H. R. Golmakani and M. H. Salimi, Markowitz-based portfolio selection with minimum transaction lots, cardinality constraints and regarding sector capitalization using genetic algorithm, Expert Systems with Applications, 36 (2009), 5058-5063. [34] J. B. Su, Value-at-risk estimates of the stock indices in developed and emerging markets including the spillover effects of currency market, Economic Modelling, 46 (2015), 204-224. [35] G. Szego, Measures of risk, European Journal of Operational Research, 163 (2005), 5-19.  doi: 10.1016/j.ejor.2003.12.016. [36] H. A. L. Thi and M. Moeini, Long-short portfolio optimization under cardinality constraints by difference of convex functions algorithm, Journal of Optimization Theory and Applications, 161 (2014), 199-224.  doi: 10.1007/s10957-012-0197-0. [37] M. H. Wang, J. Yue and N. J. Huang, Robust mean variance portfolio selection model in the jump-diffusion financial market with an intractable claim, Optimization, 66 (2017), 1219-1234.  doi: 10.1080/02331934.2017.1310212. [38] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1941. [39] M. Wu, N. J. Huang and B. S. Lee, Portfolio selection with transaction costs and different borrowing and lending interest rates in varying capital structure, Nonlinear Analysis Forum, 9 (2004), 175-200. [40] H. Yang, W. Zhou and K. W. Ding, Markowitz investment portfolio model with transaction costs and capital budget in varying capital structure, Nonlinear Analysis Forum, 19 (2014), 119-127. [41] H. X. Yao and Z. F. Li, Portfolio model and its explicit expressions of portfolio efficient frontier with minimum investment proportion constraint, OR Transactions, 13 (2009), 119-128. [42] X. L. Zhan and S. Y. Zhang, Risk analysis of financial portfolio based on copula-SV model, Journal of Systems and Management, 16 (2007), 302-306. [43] W. G. Zhang, Y. J. Liu and W. J. Xu, A possibilistic mean-semivariance-entropy model for multi-period portfolio selection with transaction costs, European Journal of Operational Research, 222 (2012), 341-349.  doi: 10.1016/j.ejor.2012.04.023. [44] Q. Zhou, Z. Chen and R. Ming, Copula-based grouped risk aggregation under mixed operation, Applications of Mathematics, 61 (2016), 103-120.  doi: 10.1007/s10492-016-0124-z. [45] S. Zymler, D. Kuhn and B. Rustem, Worst-case Value at Risk of nonlinear portfolios, Management Science, 59 (2013), 172-188.
The Structure of Financial Market
Cross-Market Mixed Portfolio
Index Trend from January 2010 to March 2017
Standardized Index Trend from January 2010 to March 2017
Rate of Return from January 2010 to December 2016
Return Curve of Model 4
Moment information of daily return rate from January 2010 to December 2016
 Name Mean(%) Variance Skewness Kurtosis SZZS 0.0170 2.1816 -0.3702 7.6127 HSI 0.0072 1.3876 -0.0621 6.0437 DAX 0.0483 1.7579 -0.1611 5.2904 AEX 0.0243 1.3845 -0.1214 5.8596
 Name Mean(%) Variance Skewness Kurtosis SZZS 0.0170 2.1816 -0.3702 7.6127 HSI 0.0072 1.3876 -0.0621 6.0437 DAX 0.0483 1.7579 -0.1611 5.2904 AEX 0.0243 1.3845 -0.1214 5.8596
Correlation Coefficient Matrix
 SZZS HSI DAX AEX SZZS 1.0000 0.5265 0.1463 0.1700 HSI 1.0000 0.3596 0.3900 DAX 1.0000 0.9146 AEX 1.0000
 SZZS HSI DAX AEX SZZS 1.0000 0.5265 0.1463 0.1700 HSI 1.0000 0.3596 0.3900 DAX 1.0000 0.9146 AEX 1.0000
Rank Correlation Coefficient Kendall's $\tau$
 DJIA DAX SZZS N225 DJIA 1.0000 0.3575 0.0865 0.1060 DAX 1.0000 0.2321 0.2458 SZZS 1.0000 0.7144 N225 1.0000
 DJIA DAX SZZS N225 DJIA 1.0000 0.3575 0.0865 0.1060 DAX 1.0000 0.2321 0.2458 SZZS 1.0000 0.7144 N225 1.0000
Model Settings
 Model 1 Model 2 Model 3 Model 4 Risk Measurement Variance WCVaR WCVaR WCVaR Hierarchical Risk Integration NO NO YES YES Skewness NO NO NO YES Transaction Cost YES YES YES YES Investment Proportion Limitation YES YES YES YES
 Model 1 Model 2 Model 3 Model 4 Risk Measurement Variance WCVaR WCVaR WCVaR Hierarchical Risk Integration NO NO YES YES Skewness NO NO NO YES Transaction Cost YES YES YES YES Investment Proportion Limitation YES YES YES YES
Statistic Comparison
 Item Model 1 Model 2 Model 3 Model 4 WCVaR of optimal portfolio - 1.7816 1.7254 1.6914 Total Rate of Return(%) 15.4084 16.1731 17.6015 18.3447 Daily Rate of Return(%) 0.0518 0.0541 0.0586 0.0609 Variance 0.8661 0.8876 0.9833 1.0135 Daily Maximum Loss(%) -5.6033 -5.3498 -5.1381 -4.9078
 Item Model 1 Model 2 Model 3 Model 4 WCVaR of optimal portfolio - 1.7816 1.7254 1.6914 Total Rate of Return(%) 15.4084 16.1731 17.6015 18.3447 Daily Rate of Return(%) 0.0518 0.0541 0.0586 0.0609 Variance 0.8661 0.8876 0.9833 1.0135 Daily Maximum Loss(%) -5.6033 -5.3498 -5.1381 -4.9078
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