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研究生:蘇天恩
研究生(外文):Chester Marvin C. So
論文名稱:投資組合問題之多目標最佳化研究
論文名稱(外文):Multi-objective Evolutionary Approach for Portfolio Optimization
指導教授:陳建宏陳建宏引用關係
指導教授(外文):James Chen
學位類別:碩士
校院名稱:中華大學
系所名稱:資訊工程學系碩士班
學門:工程學門
學類:電資工程學類
論文種類:學術論文
論文出版年:2015
畢業學年度:103
語文別:中文
論文頁數:102
中文關鍵詞:投資組合最佳化遺傳演算法多目標演化式計算多目標遺傳演算法
外文關鍵詞:Portfolio optimizationgenetic algorithmsmulti-objective evolutionary algorithmsmulti-objective genetic algorithm
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在投資組合中最佳化分配自己的資本是每一位投資者的目標。大部分的投資組合優化可以參見 Markowitz 所提出來的理論。最佳化的目標是以得到最大化收益以及最小化風險,一般認為收益越大,風險則越大。投資組合最優化的問題被證明是非常困難的,並且常用啟發式演算法來解決這一類的問題。通常這些方法的目標是以在某一風險水平限制下得到最大化收益。
在本文中提出一個組合優化系統,採用了多目標遺傳演算法 (MOGA) 作為主要的方法來優化本研究的問題。嘗試了幾種實驗以不同的策略以及不同的參數,實驗結果顯示此方法可以盈利,但不一定是最佳組合也不一定是最差的組合。關鍵字:投資組合最佳化;遺傳演算法;多目標演化式計算;多目標遺傳演算法

Allocating one’s assets optimally in a portfolio is the goal of every investor. Much of the theory of portfolio optimization is detailed in Markowitz. Investors need to balance the objective of maximizing the return on their investment with the constraint of minimizing the risk involved. It’s generally accepted that the greater the expected return, the greater the risk. Portfolio optimization problems can be shown to be very hard and heuristic algorithms are a typical way to address them. Typically these approaches aim to maximize the return with the constraint that the risk should not be above a given level.

In this thesis, a portfolio evolutionary optimization system is proposed, which uses a multi-objective genetic algorithm (MOGA) as a major optimization algorithm to optimization the investigated problem. Several experiments are conducted using this approach with different strategies for portfolio optimization.The experimental results indicatedthat the approach can be profitable; it doesn’t necessarily give the optimal portfoliobutdoesn’t give the worst portfolio either.

Keywords: Portfolio optimization, genetic algorithms, multi-objective evolutionary algorithms, multi-objective genetic algorithm.

Table of Contents

摘要 i
Abstract i
Table of Contents ii
List of Tables v
List of Figures vi
Chapter 1 Introduction 1
1.1 Motivation 1
1.2 Objectives 2
1.3 Thesis Organization 3
Chapter 2 Foundations of Portfolio Theory 4
2.1 Asset Prices and Return 4
2.1.1 Stocks 5
2.1.2 Prices and Returns 5
2.2 Mean-Variance Optimaization 7
2.2.1 Risk and Return 9
2.2.2 The Virtues of Diversification 12
2.2.3 Efficient Frontier and Portfolio Selection 12
2.2.4 Closed-form Solutions of the Mean-Variance Potfolio Problem 15
2.2.5 Limits and Problems 15
2.3 Risk Measures 16
2.3.1 Variance 16
2.3.2 Semi-Variance 17
2.3.3 Higher-order Moments 17
2.4 Historical Data and Resampling Methods 18
2.4.1 Historical Approach 18
2.4.2 Fitting Distributions 18
2.4.3 Resampling Methods 18
2.5 Portfolio Optimization Problem 19
2.6 Recent Approaches for Portfolio Optimization 20
2.7 Sumary 22
Chapter 3 Genetic Algorithms and Multi-objective Evolutionary Algorithms 24
3.1 Genetic Algorithms 24
3.2 Principles of Genetic Algorithms 25
3.3 Psuedo-Code 26
3.4 Elements of Genetic Algorithms 27
3.4.1 Fitness Function 27
3.4.2 Operators of Genetic Algorithm 28
3.5 Multi-objective Evolutionary Algorithms 29
3.6 Operation of Multi-objective Evolutionary Algorithm 31
3.7 Multi-objective Evolutionary Algorithms Performance Measures 32
3.8 Summary 32
Chapter 4 Multi-objective Evolutionary Algorithms for Portfolio Optimization 34
4.1 Introduction 34
4.2 Multi-objective Genetic Algorithm Proceedure 35
4.3Multi-objective Genetic Algorithm System 36
4.4 Multi-objective Genetic Algorithm Framework 36
4.5 Chromosome Representation 38
4.5.1 Objective Functions 38
4.5.2 Fitness Assignment Strategy 39
4.5.3 Crossover Operator 41
4.5.4 Mutation Operator 42
4.6 Summary 43
Chapter 5 Data and Results 45
5.1 Data 45
5.2 Investment Universe 45
5.3 Maximum Return and Minimum Risk 45
5.4 Performance Measurement 48
5.5 Experiments and Results 49
5.5.1 Experiment 1 50
5.5.2 Experiment 2 55
5.5.3 Experiment 3 60
5.5.4 Experiment 4 78
5.6 Summary 95
Chapter 6 Conclusion 96
Appendix 97
Reference 98


List of Tables

Table 5.1 Assets name and abbreviation. 46
Table 5.2 Assets picked. 47

List of Figures

Figure 2.1 Daily prices of Starbucks stock (SBUX) from 1st January 2002 to 1st January 2007. 8
Figure 2.2 Weekly prices of Starbucks stock (SBUX) from 1st January 2002 to 1st January 2007. 8
Figure 2.3 Monthly prices of Starbucks stock (SBUX) from 1st January 2002 to 1st January 2007. 9
Figure 2.4 Daily returns of Starbucks stock (SBUX) from 1st January 2002 to 1st January 2007. 10
Figure 2.5 Weekly returns of Starbucks stock (SBUX) from 1st January 2002 to 1st January 2007. 10
Figure 2.6 Monthly returns of Starbucks stock (SBUX) from 1st January 2002 to 1st January 2007. 11
Figure 4.1 MOGA procedure. 35
Figure 4.2 MOGA framework. 37
Figure 4.3 Asset indexes and weights. 38
Figure 4.4 GPSIFF example. 40
Figure 4.5 Example of crossover operator.s 42
Figure 4.6 Example of mutation operator. 43
Figure 5.1 Proves that MOGA willconverge after 100 generations. 49
Figure 5.2 MOGAs with TW asset from 2008-2010 after 30 runs with population size of 100 and mutation rate is 0.01. 51
Figure 5.3 MOGAs with TW asset from 2008-2010 after 30 runs with population size of 100 and mutation rate is 0.10.. 52
Figure 5.4 MOGAs with TW asset from 2008-2010 after 30 runs with population size of 500 and mutation rate is 0.01.. 53
Figure 5.5 MOGAs with TW asset from 2008-2010 after 30 runs with population size of 500 and mutation rate is 0.10. 54
Figure 5.6 MOGAs mean of TW asset from 2008-2010 after 30 runs with different population size and mutation rate.. 55
Figure 5.7 MOGAs with US asset from 2008-2010 after 30 runs with population size of 100 and mutation rate is 0.01. 56
Figure 5.8 MOGAs with US asset from 2008-2010 after 30 runs with population size of 100 and mutation rate is 0.10.. 57
Figure 5.9 MOGAs with US asset from 2008-2010 after 30 runs with population size of 500 and mutation rate is 0.01.. 58
Figure 5.10 MOGAs with US asset from 2008-2010 after 30 runs with population size of 500 and mutation rate is 0.10. 59
Figure 5.11 MOGAs mean of US asset from 2008-2010 after 30 runs with different population size and mutation rate.. 60
Figure 5.12 MOGAs with TW asset from 2010-2012 after 30 runs with population size of 100 and mutation rate is 0.01 choosing the highest solution from their GPSIFF fitness value.. 62
Figure 5.13 MOGAs with TW asset from 2010-2012 after 30 runs with population size of 100 and mutation rate is 0.10 choosing the highest solution from their GPSIFF fitness value.. 63
Figure 5.14 MOGAs with TW asset from 2010-2012 after 30 runs with population size of 500 and mutation rate is 0.01 choosing the highest solution from their GPSIFF fitness value.. 64
Figure 5.15 MOGAs with TW asset from 2010-2012 after 30 runs with population size of 500 and mutation rate is 0.10 choosing the highest solution from their GPSIFF fitness value. 65
Figure 5.16 MOGAs mean of TW asset from 2010-2012 after 30 runs with different population size and mutation ratechoosing the highest solution from their GPSIFF fitness value. 66
Figure 5.17 MOGAs with TW asset from 2010-2012 after 30 runs with population size of 100 and mutation rate is 0.01 choosing the highest solution from their return fitness value.. 67
Figure 5.18 MOGAs with TW asset from 2010-2012 after 30 runs with population size of 100 and mutation rate is 0.10 choosing the highest solution from their return fitness value.. 68
Figure 5.19 MOGAs with TW asset from 2010-2012 after 30 runs with population size of 500 and mutation rate is 0.01 choosing the highest solution from their return fitness value.. 69
Figure 5.20 MOGAs with TW asset from 2010-2012 after 30 runs with population size of 500 and mutation rate is 0.10 choosing the highest solution from their return fitness value. 70
Figure 5.21 MOGAs mean of TW asset from 2010-2012 after 30 runs with different population size and mutation ratechoosing the highest solution from their return fitness value. 71
Figure 5.22 MOGAs with TW asset from 2010-2012 after 30 runs with population size of 100 and mutation rate is 0.01 choosing the highest solution from their risk fitness value.. 72
Figure 5.23 MOGAs with TW asset from 2010-2012 after 30 runs with population size of 100 and mutation rate is 0.10 choosing the highest solution from their risk fitness value.. 73
Figure 5.24 MOGAs with TW asset from 2010-2012 after 30 runs with population size of 500 and mutation rate is 0.01 choosing the highest solution from their risk fitness value.. 74
Figure 5.25 MOGAs with TW asset from 2010-2012 after 30 runs with population size of 500 and mutation rate is 0.10 choosing the highest solution from their risk fitness value. 75
Figure 5.26 MOGAs mean of TW asset from 2010-2012 after 30 runs with different population size and mutation ratechoosing the highest solution from their risk fitness value. 76
Figure 5.27 MOGAs results of TW asset from 2010-2012 with different parameters using different strategies. 77
Figure 5.28 MOGAs with US asset from 2010-2012 after 30 runs with population size of 100 and mutation rate is 0.01 choosing the highest solution from their GPSIFF fitness value.. 79
Figure 5.29 MOGAs with US asset from 2010-2012 after 30 runs with population size of 100 and mutation rate is 0.10 choosing the highest solution from their GPSIFF fitness value.. 80
Figure 5.30 MOGAs with US asset from 2010-2012 after 30 runs with population size of 500 and mutation rate is 0.01 choosing the highest solution from their GPSIFF fitness value.. 81
Figure 5.31 MOGAs with US asset from 2010-2012 after 30 runs with population size of 500 and mutation rate is 0.10 choosing the highest solution from their GPSIFF fitness value. 82
Figure 5.32 MOGAs mean of US asset from 2010-2012 after 30 runs with different population size and mutation ratechoosing the highest solution from their GPSIFF fitness value. 83

Figure 5.33 MOGAs with US asset from 2010-2012 after 30 runs with population size of 100 and mutation rate is 0.01 choosing the highest solution from their return fitness value.. 84
Figure 5.34 MOGAs with US asset from 2010-2012 after 30 runs with population size of 100 and mutation rate is 0.10 choosing the highest solution from their return fitness value.. 85
Figure 5.35 MOGAs with US asset from 2010-2012 after 30 runs with population size of 500 and mutation rate is 0.01 choosing the highest solution from their return fitness value.. 86
Figure 5.36MOGAs with US asset from 2010-2012 after 30 runs with population size of 500 and mutation rate is 0.10 choosing the highest solution from their return fitness value. 87
Figure 5.37MOGAs mean of US asset from 2010-2012 after 30 runs with different population size and mutation ratechoosing the highest solution from their return fitness value. 88
Figure 5.38 MOGAs with US asset from 2010-2012 after 30 runs with population size of 100 and mutation rate is 0.01 choosing the highest solution from their risk fitness value.. 89
Figure 5.39 MOGAs with US asset from 2010-2012 after 30 runs with population size of 100 and mutation rate is 0.10 choosing the highest solution from their risk fitness value.. 90
Figure 5.40 MOGAs with US asset from 2010-2012 after 30 runs with population size of 500 and mutation rate is 0.01 choosing the highest solution from their risk fitness value.. 91
Figure 5.41MOGAs with US asset from 2010-2012 after 30 runs with population size of 500 and mutation rate is 0.10 choosing the highest solution from their risk fitness value. 92
Figure 5.42MOGAs mean of US asset from 2010-2012 after 30 runs with different population size and mutation ratechoosing the highest solution from their risk fitness value. 93
Figure 5.43MOGAs results of US asset from 2010-2012 with different parameters using different strategies. 94



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