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Risk-balanced territory design optimization for a Micro finance institution
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.
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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[27] |
H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[27] |
H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), 77-91. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |






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 |
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 |
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 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 |
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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