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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 |
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.
Keywords:
Cross-market mixed portfolio,
hierarchical risk integration,
multi-objective robust optimization,
worst-case value at risk.
Mathematics Subject Classification:
91G50, 91G60, 91B30, 62H20, 62E17, 62P99, 65C20.
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:
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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.
|
Figure 2.
Cross-Market Mixed Portfolio
Figure 3.
Index Trend from January 2010 to March 2017
Figure 4.
Standardized Index Trend from January 2010 to March 2017
Figure 5.
Rate of Return from January 2010 to December 2016
Figure 6.
Return Curve of Model 4
Table 1.
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 |
Table 2.
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 |
Table 3.
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 |
Table 4.
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 |
Table 5.
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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