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August & September  2019, 12(4&5): 761-770. doi: 10.3934/dcdss.2019050

A new approach for worst-case regret portfolio optimization problem

1. 

Business School, University of Shanghai for Science and Technology, Shanghai 200093, China

2. 

School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China

* Corresponding author: Shaojian Qu

Received  June 2017 Revised  October 2017 Published  November 2018

Fund Project: The first author is supported by NSF grant 71571055.

This paper considers the worst-case regret portfolio optimization problem when the distributions of the asset returns are uncertain. In general, the solution to this problem is NP hard and approximation methods that minimise the difference between the maximum return and the sum of each portfolio return are often proposed. Applying the duality of semi-infinite programming, the worst-case regret portfolio optimization problem with uncertain distributions can be equivalently reformulated to a linear optimization problem, and the established solution approaches for linear optimization can then be applied. An example of a portfolio optimization problem is provided to show the efficiency of our method and the results demonstrate that our method can satisfy the portfolio risk diversification property under the uncertain distributions of the returns.

Citation: Ying Ji, Shaojian Qu, Yeming Dai. A new approach for worst-case regret portfolio optimization problem. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 761-770. doi: 10.3934/dcdss.2019050
References:
[1]

A. Ben Tal and A. Nemirovski, Robust convex optimizaion, Math Oper Res, 23 (1998), 769-805.  doi: 10.1287/moor.23.4.769.  Google Scholar

[2]

L. ChenS. He and S. Zhang, Tight bounds for some risk measures with applications to robust portfolio selection, Operations Research, 59 (2011), 847-855.  doi: 10.1287/opre.1110.0950.  Google Scholar

[3]

X. ChenM. Sim and P. Sun, A robust optimization perspective on stochastic programming, Operational Research, 55 (2007), 1058-1071.  doi: 10.1287/opre.1070.0441.  Google Scholar

[4]

E. Delage and Y. Y. Ye, Distributionally robust optimization under moment uncertainty with application data-driven problems, Operational Research, 58 (2010), 595-612.  doi: 10.1287/opre.1090.0741.  Google Scholar

[5]

J. Goh and M. Sim, Distributionally robust optimization and its tractable approximations, Operational Research, 58 (2010), 902-917.  doi: 10.1287/opre.1090.0795.  Google Scholar

[6]

K. Isii, On the sharpness of Chebyshev-type inequalities, Annals of the Institute of Statistical Mathematics, 14 (1962/1963), 185-197.  doi: 10.1007/BF02868641.  Google Scholar

[7]

Y. JiT. N. Wang and M. Goh, The worst-case discounted regret portfolio optimization problem, Applied Mathematics and Computation, 239 (2014), 310-319.  doi: 10.1016/j.amc.2014.04.072.  Google Scholar

[8]

X. LiB. Y. Shou and Z. F. Qin, An expected regret minimization portfolio selection model, European Journal of Operational Research, 218 (2012), 484-492.  doi: 10.1016/j.ejor.2011.11.015.  Google Scholar

[9]

H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91.   Google Scholar

[10]

D. Y. MengQ. Zhao and Z. B. Xu, Improve robustness of sparse PCA by $L_1$-norm maximaization, Pattern Recognit, 45 (2012), 487-497.   Google Scholar

[11]

X. J. Tong and F. Wu, Robust reward-risk ratio optimization with application in allocation of generation asset, Optimization, 63 (2014), 1761-1779.  doi: 10.1080/02331934.2012.672419.  Google Scholar

[12]

M. R. Wagner, Fully distribution-free profit maximization: The inventory management case, Math Operation Reserch, 35 (2010), 728-741.  doi: 10.1287/moor.1100.0468.  Google Scholar

[13]

W. WiesemannD. Kuhn and M. Sim, Distributionally robust convex optimization, Operations Research, 62 (2014), 1358-1376.  doi: 10.1287/opre.2014.1314.  Google Scholar

[14]

S. ZymlerD. Kuhn and B. Rustem, Distributionally robust joint chance constraints with second-order moment information, Mathematical Programming, 137 (2013), 167-198.  doi: 10.1007/s10107-011-0494-7.  Google Scholar

show all references

References:
[1]

A. Ben Tal and A. Nemirovski, Robust convex optimizaion, Math Oper Res, 23 (1998), 769-805.  doi: 10.1287/moor.23.4.769.  Google Scholar

[2]

L. ChenS. He and S. Zhang, Tight bounds for some risk measures with applications to robust portfolio selection, Operations Research, 59 (2011), 847-855.  doi: 10.1287/opre.1110.0950.  Google Scholar

[3]

X. ChenM. Sim and P. Sun, A robust optimization perspective on stochastic programming, Operational Research, 55 (2007), 1058-1071.  doi: 10.1287/opre.1070.0441.  Google Scholar

[4]

E. Delage and Y. Y. Ye, Distributionally robust optimization under moment uncertainty with application data-driven problems, Operational Research, 58 (2010), 595-612.  doi: 10.1287/opre.1090.0741.  Google Scholar

[5]

J. Goh and M. Sim, Distributionally robust optimization and its tractable approximations, Operational Research, 58 (2010), 902-917.  doi: 10.1287/opre.1090.0795.  Google Scholar

[6]

K. Isii, On the sharpness of Chebyshev-type inequalities, Annals of the Institute of Statistical Mathematics, 14 (1962/1963), 185-197.  doi: 10.1007/BF02868641.  Google Scholar

[7]

Y. JiT. N. Wang and M. Goh, The worst-case discounted regret portfolio optimization problem, Applied Mathematics and Computation, 239 (2014), 310-319.  doi: 10.1016/j.amc.2014.04.072.  Google Scholar

[8]

X. LiB. Y. Shou and Z. F. Qin, An expected regret minimization portfolio selection model, European Journal of Operational Research, 218 (2012), 484-492.  doi: 10.1016/j.ejor.2011.11.015.  Google Scholar

[9]

H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91.   Google Scholar

[10]

D. Y. MengQ. Zhao and Z. B. Xu, Improve robustness of sparse PCA by $L_1$-norm maximaization, Pattern Recognit, 45 (2012), 487-497.   Google Scholar

[11]

X. J. Tong and F. Wu, Robust reward-risk ratio optimization with application in allocation of generation asset, Optimization, 63 (2014), 1761-1779.  doi: 10.1080/02331934.2012.672419.  Google Scholar

[12]

M. R. Wagner, Fully distribution-free profit maximization: The inventory management case, Math Operation Reserch, 35 (2010), 728-741.  doi: 10.1287/moor.1100.0468.  Google Scholar

[13]

W. WiesemannD. Kuhn and M. Sim, Distributionally robust convex optimization, Operations Research, 62 (2014), 1358-1376.  doi: 10.1287/opre.2014.1314.  Google Scholar

[14]

S. ZymlerD. Kuhn and B. Rustem, Distributionally robust joint chance constraints with second-order moment information, Mathematical Programming, 137 (2013), 167-198.  doi: 10.1007/s10107-011-0494-7.  Google Scholar

Table 1.  Numerical Results
This paper[4]Ix
$D^1$ $D^2$IterTimeWrvTimeIterWdrvIx
501060.0120.20240.01770.22550.1024
502560.0130.21330.01970.23710.1003
10025210.110.29780.174270.2494-0.1941
10050230.1290.23470.285420.29360.1914
20050370.9300.54841.708460.5195-0.0556
200100371.2880.39222.502370.3184-0.2318
300100493.5770.52664.81590.70260.2505
300200789.5630.448611.84810.48870.0821
This paper[4]Ix
$D^1$ $D^2$IterTimeWrvTimeIterWdrvIx
501060.0120.20240.01770.22550.1024
502560.0130.21330.01970.23710.1003
10025210.110.29780.174270.2494-0.1941
10050230.1290.23470.285420.29360.1914
20050370.9300.54841.708460.5195-0.0556
200100371.2880.39222.502370.3184-0.2318
300100493.5770.52664.81590.70260.2505
300200789.5630.448611.84810.48870.0821
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