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.

Contects
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
5.1.2.1	The convergence of the proposed approach	50
5.1.2.2	The comparisons of different fitness functions	51
5.1.2.3	Evaluation of the derived investment portfolios	54
5.1.2.4	Analyses of parameters	56
5.2	Experimental Results of Method (II)	59
5.2.1   Dataset Descriptions	59
5.2.2 Experimental Evaluations	59
5.2.2.1    The Evolution of the Pareto Fronts of YTMMOPSM	59
5.2.2.2    The ROI of Derived Portfolios	60
Chapter 6	Conclusions and Future Work	63
Reference		65
APPENDIXES	68

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
values  52
Figure 7. Comparison results of three fitness functions in terms of average suitability
values  53
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

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