Gray Wolf Optimization Algorithm for MultiConstraints SecondOrder Stochastic Dominance Portfolio Optimization
Abstract
:1. Introduction
2. The MCVSK and MVSK Portfolio Optimization Model
2.1. The Measure of Return and Risk
2.2. The SecondOrder Stochastic Dominance Constraint
2.3. The Skewness and Kurtosis Constraints
2.4. The MCVSK and MVSK Portfolio Optimization Model
3. The GWO Algorithm for the MCVSK and MVSK Portfolio Optimization Model
3.1. Prey Searching
3.2. Prey Encirclement
3.3. Chasing (Hunting)
Algorithm 1 the main procedure of GWO algorithm for the MCVSK and MVSK model. 
Input: problem to solve, problem; number of search agents, SearchAgents no; number of iterations, Max iteration; number of variables, dim; etc 
Output: the best portfolio and the return of the portfolio 

4. Numerical Experiments
4.1. Backtesting and OutofSample Test
4.2. Numerical Analysis
5. Conclusions and Future Research
Author Contributions
Acknowledgments
Conflicts of Interest
Abbreviations
GWO  Gray Wolf Optimization 
PSO  Particle Swarm Optimization 
GA  Genetic Algorithm 
MV  MeanVariance 
MAD  Mean Absolute Deviation 
VaR  Value at Risk 
CVaR  Conditional Value at Risk 
SD  Stochastic Dominance 
FSD  FirstOrder Stochastic Dominance 
SSD  SecondOrder Stochastic Dominance 
ASD  Almost Stochastic Dominance 
MVS  MeanVarianceSkewness 
MOEAs  MultiObjective Evolutionary Algorithms 
NSGAII  NonDominated Sorting Genetic Algorithm II 
SPEAII  Strength Parato Evolutionary Algorithm II 
MCVSK  MeanCVaRskewnesskurtosis 
MVSK  MeanVaRskewnesskurtosis 
IPSO  Immune Particle Swarm Optimization 
ACA  Ant Colony Algrothrim 
BBO  Biogeographybased Optimization 
CNN  Convolutional Neural Network 
LSSVM  Least Squares Support Vector Machine 
Appendix A
Constitution  Index Weight (%)  Constitution  Index Weight (%) 

3i Group  0.38  Admiral Group  0.2 
Anglo American  0.84  Antofagasta  0.13 
Ashtead Group  0.44  Associated British Foods  0.53 
AstraZeneca  3.1  Aviva  1.09 
Babcock International Group  0.27  BAE Systems  1.04 
Barclays  2.09  Barratt Developments  0.26 
BHP Billition  1.53  BP  5.3 
British American Tobacco  4.77  British Land Co  0.36 
BT Group  1.7  Bunzl  0.39 
Burberry Group  0.37  Capita  0.19 
Carnival  0.42  Centrica  0.7 
CocaCola HBC AG  0.19  Compass Group  1.37 
ConvaTec Group  0.08  CRH  1.3 
Croda International  0.23  DCC  0.3 
Diageo  2.94  Direct Line Insurance Group  0.28 
Dixons Carphone  0.2  Easyjet  0.14 
Experian  0.84  Fresnillo  0.11 
GKN  0.31  GlaxoSmithKline  4.21 
Glencore  1.79  Hammerson  0.25 
Hargreaves Lansdown  0.16  Hikma Pharmaceuticals  0.15 
HSBC HIdgs  7.3  Imperial Brands  1.89 
Informa  0.31  InterContinental Hotels Group  0.4 
International Consolidated Airlines Group  0.41  Intertek Group  0.31 
Intu Properties  0.14  ITV  0.43 
Johnson Matthey  0.34  Kingfisher  0.44 
Land Securities Group  0.46  Legal & General Group  0.81 
LIoyds Banking Group  2.22  London Stock Exchange Group  0.51 
Marks & Spencer Group  0.31  Mediclinic International pIc  0.17 
Merlin Entertainments  0.18  Micro Focus International  0.27 
Mondi  0.34  Morrison (Wm) Supermarkets  0.28 
National Grid  1.99  Next  0.39 
Old Mutual  0.56  Paddy Power Betfair  0.4 
Pearson  0.37  Persimmon  0.3 
Provident Financial  0.23  Prudential  2.33 
Randgold Resources  0.33  Reckitt Benckiser Group  2.4 
RELX  0.88  Rio Tinto  2.12 
RollsRoyce Holdings  0.61  Royal Bank Of Scotland Group  0.41 
Royal Dutch Shell A  5.43  Royal Dutch Shell B  4.89 
Royal Mail  0.23  RSA Insurance Group  0.33 
Sage Group  0.39  Sainsbury (J)  0.23 
Schroders  0.19  Severn Trent  0.29 
Shire  2.34  Sky  0.58 
Smith & Nephew  0.6  Smiths Group  0.31 
Smurfit Kappa Group  0.24  SSE  0.87 
St. James’s Place  0.29  Standard Chartered  0.99 
Standard Life  0.41  Taylor Wimpey  0.28 
Tesco  0.93  TUI AG  0.3 
Unilever  2.2  United Utilities Group  0.34 
Vodafone Group  2.94  Whitbread  0.38 
Wolseley  0.69  Worldpay Group  0.25 
WPP  1.29 
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$\mathit{\alpha}$  Algorithm  Skew  Kurt  VaR  CVaR  E [g(x,$\mathit{\xi}$)]  Max  Time (s)  

MCVSK  0.01  GWO  0.1726  4.0974  −2.2535  −2.7743  0.1430  0.0484  30.77 
PSO  0.0878  3.9355  −2.3300  −2.7368  0.1107  0.0389  27.77  
GA  0.1307  3.8451  −2.2777  −2.7744  0.1215  0.0378  28.59  
0.05  GWO  0.0734  3.8483  −1.7844  −2.1648  0.1478  0.0488  24.94  
PSO  0.0292  4.0188  −1.5916  −2.2091  0.1109  0.0369  32.45  
GA  0.1362  4.0364  −1.7264  −2.1502  0.1282  0.0386  30.20  
0.10  GWO  0.2033  3.8933  −1.1009  −1.7413  0.1512  0.0479  32.12  
PSO  0.0346  3.9227  −1.0638  −1.7982  0.1110  0.0349  30.07  
GA  0.2087  3.9117  −1.1509  −1.7554  0.1210  0.0356  27.47  
MVSK  0.01  GWO  0.3197  3.9699  −2.1251  −2.6291  0.1402  0.0494  30.12 
PSO  0.0934  3.8752  −2.4055  −2.8165  0.1117  0.0353  31.21  
GA  0.0898  3.9745  −2.2682  −2.7807  0.1228  0.0366  27.47  
0.05  GWO  0.2464  4.1137  −1.6823  −2.0712  0.1451  0.0487  28.60  
PSO  0.1575  3.9913  −1.7283  −2.1363  0.1347  0.0423  27.77  
GA  0.1324  3.9691  −1.7344  −2.1379  0.1215  0.0359  23.86  
0.10  GWO  0.1880  4.1003  −1.1198  −1.7461  0.1393  0.0479  26.73  
PSO  0.0440  3.9591  −0.9915  −1.7541  0.1064  0.0333  26.81  
GA  0.1612  3.9245  1.1932  1.7606  0.1261  0.0341  26.02  
FTSE100  0.01  ×  0.0151  4.3361  −2.6739  −3.2503  0.1017  0.0730  × 
0.05  −1.5442  −2.3205  
0.10  −1.1953  −1.8086  
y  0.01  ×  −0.1992  4.6663  −3.1508  −3.2600  0.0612  0.0099  × 
0.05  −1.6022  −2.4337  
0.10  −1.0599  −1.8505 
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Ren, Y.; Ye, T.; Huang, M.; Feng, S. Gray Wolf Optimization Algorithm for MultiConstraints SecondOrder Stochastic Dominance Portfolio Optimization. Algorithms 2018, 11, 72. https://doi.org/10.3390/a11050072
Ren Y, Ye T, Huang M, Feng S. Gray Wolf Optimization Algorithm for MultiConstraints SecondOrder Stochastic Dominance Portfolio Optimization. Algorithms. 2018; 11(5):72. https://doi.org/10.3390/a11050072
Chicago/Turabian StyleRen, Yixuan, Tao Ye, Mengxing Huang, and Siling Feng. 2018. "Gray Wolf Optimization Algorithm for MultiConstraints SecondOrder Stochastic Dominance Portfolio Optimization" Algorithms 11, no. 5: 72. https://doi.org/10.3390/a11050072