||Constructing the YTM-based Portfolio Selection Model with Constraints by Evolutionary Algorithm
||Department of Computer Science and Information Engineering
multi-objective genetic algorithm
yield to maturity
return on investment
value at risk
||在金融市場中，有多種投資標的與衍生性金融商品，例如：股票、期貨、選擇權等。換句話說，越來越多的投資標的容易讓投資者舉棋不定，使得找出一個合適的投資組合對投資者來說是一件困難的事情。因此，投資組合選擇問題便成為一有趣的議題。根據 Markowitz 所提出的平均變異 (M-V) 模型，有許多演化為基礎的方法被提出用來最佳化投資標的權重。然而，平均變異模型有它的限制。故本論文依據使用者的限制，提出兩個方法來找尋最佳化投資選擇模型，分別為殖利率為基礎的遺傳投資組合選擇模型 (YTMGPSM) 與殖利率為基礎的多目標投資組合選擇模型 (YTMMOPSM)。
在第一個方法中，標的的買與不買及購買張數透過實數將之編入染色體用以建構可能的投資組合。而其染色體的適應值，則透過投資組合的報酬率、風險及其合適性進行評估。其中，投資組合的合適性是由投資組合懲罰值 (PP) 與投資資金懲罰值 (ICP) 所組成，分別用來反應對使用者設定的最大投資資金與最大購買公司數的符合程度。第二個方法中，則透過多目標遺傳演算法，使用兩個目標函數，分別為染色體的合適性與報酬率，探勘柏拉圖集合並提供給投資者不同選擇但都能滿足其限制的投資組合。
實驗利用台灣證券交易所所提供的真實資料來顯示此兩個方法皆是有效用的。YTMGPSM 優點在於能夠以值利率為基礎依照使用者設定的最大投資資金與最大購買公司數來找尋出一個最佳的投資組合，而 YTMMOPSM 的優點，則是透過兩個目標函數且符合投資者的條件下，使用多目標遺傳演算法挖掘柏拉圖集合提供投資者不同的投資組合選擇。
||In financial market, there are many financial instruments and financial derivatives, including stocks, futures, options, etc. In other words, investors have too many choices such that finding an appropriate portfolio is a difficult task. Portfolio selection problem thus becomes an interesting topic in the investment market. Lots approaches focus on optimizing weights of assets by evolutionary algorithms based on Mean-Variance (M-V) model which is proposed by Markowitz. However, M-V model has its limitations. According to user’s constraints, this study thus proposes two methods for optimizing portfolio selection model, namely a yield-to-maturity (YTM)-based genetic portfolio selection model (YTMGPSM) and a YTM-based multi-objective portfolio selection model (YTMMOPSM) with users constraints.
In the first algorithm (YTMGPSM), a set of real numbers are encoded into a chromosome to form a possible portfolio which presents whether buy or not buy and purchased units of assets. The fitness value of a chromosome is evaluated by Return on Investment (ROI), Value at Risk (VaR) and suitability of the respective portfolio. The suitability of a chromosome consists of portfolio penalty (PP) and investment capital penalty (ICP) that are used to reflect the satisfactions of user predefined maximum investment and maximum number of companies, respectively. In the second algorithm (YTMMOPSM), the multi-objective genetic algorithm is utilized with two objective functions, called Suitability and ROI of chromosome, for finding a Pareto set for not only providing investors different choices of portfolios but also filling investors’ preferences.
Experiments on real datasets from Taiwan Stock Exchange (TSE) show two methods are effective. The advantage of YTMGPSM is that it provides investors to find the portfolio according to their preferences, e.g. maximum investment, maximum number of companies constraints. The advantage of YTMMOPSM is that through two objective functions (Suitability and ROI) and investors’ preferences, it could provide Pareto set for investors to select an appropriate investment portfolio.
Chapter 1 Introduction 1
1.1 Problem Definition and Motivation 1
1.2 Contributions 2
1.3 Reader's Guide 3
Chapter 2 Related Work 4
2.1 Genetic Algorithms 4
2.2 The MOGA-based Optimization Problems 6
2.3 Markowitz’ Portfolio Theory 9
2.4 Portfolio Optimization Approaches 11
Chapter 3 Yield-To-Maturity-Based Genetic Portfolio Selection Model 14
3.1 The Components of the Proposed Approach 14
3.1.1 Chromosome Representation 14
3.1.2 Initial Population 16
3.1.3 Fitness and Selection 17
3.1.4 Genetic operations 21
3.2 The Proposed Algorithm 21
3.3 An Example 24
Chapter 4 Yield-To-Maturity-Based Multi-Objective Portfolio Selection Model 30
4.1 The Components of the Proposed Approach 30
4.1.1 Chromosome Representation 30
4.1.2 Initial Population 32
4.1.3 The Two Objective Functions 33
4.1.4 Fitness Assignment and Selection 35
4.1.5 Genetic operations 36
4.2 The Proposed Algorithm 37
4.3 An Example 41
Chapter 5 Experimental Results 49
5.1 Experimental Results of Method (I) 49
5.1.1 Dataset Descriptions 49
5.1.2 Experimental Evaluations 50
126.96.36.199 The convergence of the proposed approach 50
188.8.131.52 The comparisons of different fitness functions 51
184.108.40.206 Evaluation of the derived investment portfolios 54
220.127.116.11 Analyses of parameters 56
5.2 Experimental Results of Method (II) 59
5.2.1 Dataset Descriptions 59
5.2.2 Experimental Evaluations 59
18.104.22.168 The Evolution of the Pareto Fronts of YTMMOPSM 59
22.214.171.124 The ROI of Derived Portfolios 60
Chapter 6 Conclusions and Future Work 63
List of Figure
Figure 1. The entire GA process 6
Figure 2. An example for the Pareto optimal solutions . 8
Figure 3. Representation of a chromosome Cq . 15
Figure 4. Representation of a chromosome Cq . 31
Figure 5. The convergence of the proposed approach . 50
Figure 6. Comparison results of three fitness functions in terms of average roi + risk
Figure 7. Comparison results of three fitness functions in terms of average suitability
Figure 8. Comparison results of three fitness functions in terms of average designed
fitness values . 54
Figure 9. The proposed approach with and without YTM of companies . 58
Figure 10. The evolution of the Pareto fronts along with generations . 60
List of Tables
Table 1. The yield to maturity (Yi) of companies 16
Table 2. The probabilities of all companies 16
Table 3. The information of the fifteen companies in this example 24
Table 4. The probability of all companies could be selected to the portfolio 25
Table 5. The cumulative probability of all companies 25
Table 6. The ROI values of all chromosomes 27
Table 7. The VaR of all chromosomes with 入 = 99% 27
Table 8. The fitness values of all chromosomes 28
Table 9. The best asset portfolio in this example 29
Table 10. The yield to maturity (Yi) of companies 32
Table 11. The probabilities of all companies 32
Table 12. The information of the fifteen companies in this example 41
Table 13. The probability of all companies could be selected to the portfolio 42
Table 14. The cumulative probability of all companies 42
Table 15. The objective values of all chromosomes 44
Table 16. The ranking results of all the ten chromosomes 45
Table 17. The fitness values of all the ten chromosomes 46
Table 18. The resulting fitness values of the ten chromosomes 47
Table 19. The NDS in this example 48
Table 20. The initial and final portfolios by roi+risk as fitness function 55
Table 21. The initial and final portfolios by suitability as fitness function 55
Table 22. The initial and final portfolios by the designed fitness function 55
Table 23. The average returns of the derived portfolios with different training and testing periods 56
Table 24. The average returns of derived portfolios with different confidence levels 57
Table 25. A portfolio has minimum suitability value 61
Table 26. A portfolio has maximum ROI value 61
Table 27. A portfolio has middle suitability and ROI value 61
Table 28. The investment portfolios of the three portfolios 62
|| T. P. Patalia and G. Kulkarni, "Design of Genetic Algorithm for Knapsack Problem to Perform Stock Portfolio Selection Using Financial Indicators," International Conference on Computational Intelligence and Communication Networks (CICN), pp. 289-292, 2011.
 E. Vercher, "A possibilistic mean-downside risk-skewness model for efficient portfolio selection," IEEE Transactions on Fuzzy Systems, Vol. 21, pp. 585 - 595, 2011.
 R. Kumar and S. Bhattacharya, "Cooperative Search Using Agents for Cardinality Constrained Portfolio Selection Problem," IEEE Transactions on Systems, Man, and Cybernetics, Part C: Applications and Reviews, Vol. 42, pp. 1510 - 1518, 2012.
 E. Wah, Y. Mei and B. W. Wah, "Portfolio Optimization through Data Conditioning and Aggregation," IEEE International Conference on Tools with Artificial Intelligence (ICTAI), pp. 253-260, 2011.
 F. Hassanzadeh, M. Collan and M. Modarres, "A Practical Approach to R&D Portfolio Selection Using the Fuzzy Pay-Off Method," IEEE Transactions on Fuzzy Systems, Vol. 20, pp. 615-622, 2012.
 H. M. Markowitz, "Harry Markowitz: Selected Works," World Scientific Publishing Company, 2009.
 T. J. Chang, S. C. Yang and K. J. Chang, "Portfolio optimization problems in different risk measures using genetic algorithm," Expert Systems with Applications, Vol. 36, pp. 10529-10537, 2009.
 J. S. Chen, J. L. Hou, S. M. Wu and Y. W. Chang Chien, "Constructing investment strategy portfolios by combination genetic algorithms," Expert Systems with Applications, Vol. 36, pp. 3824-3828, 2009.
 L. R. Z. Hoklie, "Resolving multi objective stock portfolio optimization problem using genetic algorithm," The International Conference on Computer and Automation Engineering, pp. 40-44, 2010.
 V. Bevilacqua, V. Pacelli and S. Saladino, "A novel multi objective genetic algorithm for the portfolio optimization," Advanced Intelligent Computing, pp. 186-193, 2012.
 P. C. Lin, "Portfolio Optimization and Risk Measurement based on Non-Dominated Sorting Genetic Algorithm," Journal of Industrial and Management optimization, Vol. 8, pp. 549-564, 2012.
 R. Mansini and M. G. Speranza, "Heuristic algorithms for the portfolio selection problem with minimum transaction lots," European Journal of Operational Research, Vol. 114, pp. 219-233, 1999.
 H. Kellerer, R. Mansini and S. M. Grazia, "Selecting portfolios with fixed costs and minimum transaction lots," Annals of Operations Research, Vol. 99, pp. 287-304, 2000.
 H. Konno and A. Wijayanayake, "Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints," Mathematical Programming, Vol. 89, pp. 233-250, 2001.
 R. Mansini and M. G. Speranza, "An exact approach for portfolio selection with transaction costs and rounds," IIE transactions, Vol. 37, pp. 919-929, 2005.
 C. C. Lin and Y. T. Liu, "Genetic algorithms for portfolio selection problems with minimum transaction lots," European Journal of Operational Research, Vol. 185, pp. 393-404, 2008.
 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, Vol. 36, pp. 5058-5063, 2009.
 M. H. Babaei, M. Hamidi, E. Jahani and H. Pasha, "A New Approach to Solve an Extended Portfolio Selection Problem," presented at the Proceedings of the 2012 International Conference on Industrial Engineering and Operations Management, Istanbul, Turkey, 2012.
 P. Gupta, M. K. Mehlawat and G. Mittal, "Asset portfolio optimization using support vector machines and real-coded genetic algorithm," Journal of Global Optimization, Vol. 53, pp. 297-315, 2012.
 P. Gupta, M. K. Mehlawat and A. Saxena, "Hybrid optimization models of portfolio selection involving financial and ethical considerations," Knowledge-Based Systems, Vol. 37, pp. 318-337, 2012.
 J. H. Holland, "Adaptation in Natural and Artificial Systems," University of Michigan Press, 1975.
 D. E. Goldberg, "Genetic algorithms in search, optimization, and machine learning," Addison Wesley, 1989.
 J. J. Grefenstette, "Optimization of control parameters for genetic algorithms," IEEE Trans System Man, and Cybernetics, Vol. 16, pp. 122-128, 1986.
 J. D. Schaffer, "Multiple objective optimization with vector evaluated genetic algorithms," Proceedings of the 1st international Conference on Genetic Algorithms, pp. 93-100, 1985.
 C. M. Fonseca and P. J. Fleming, "Genetic algorithms for multiobjective optimization: Formulation, discussion and generalization," The international conference on genetic algorithms, pp. 416-423, 1993.
 K. Deb, A. Pratap, S. Agarwal and T. Meyarivan, "A fast and elitist multiobjective genetic algorithm: NSGA-II," IEEE Transactions on Evolutionary Computation, Vol. 6, pp. 182-197, 2002.
 E. Zitzler, M. Laumanns and L. Thiele, "SPEA2: Improving the strength Pareto evolutionary algorithm," Evolutionary Methods for Design, Optimization and Control With Applications to Industrial Problems, pp. 95-100, 2001.
 Z. G. M. El hachloufi, F. Hamza, "Stocks Portfolio Optimization Using Classification and Genetic Algorithms," Applied Mathematical Sciences, Vol. 6, pp. 4673-4684, 2012.
 J. Bermudez, J. Segura and E. Vercher, "A multi-objective genetic algorithm for cardinality constrained fuzzy portfolio selection," Fuzzy Sets and Systems, Vol. 188, pp. 16-26, 2012.
 J. Li and J. Xu, "Multi-objective portfolio selection model with fuzzy random returns and a compromise approach-based genetic algorithm," Information Sciences, Vol. 220, pp. 507-521, 2012.
 A. D. Roy, "Safety first and the holding of assets," Econometrica, Vol. 20, pp. 431-449, Jul. 1952.
 F. Herrera, M. Lozano and J. L. Verdegay, "Fuzzy connectives based crossover operators to model genetic algorithms population diversity," Fuzzy Sets and Systems, Vol. 92, pp. 21-30, 1997.