A MODIFIED MEAN-VARIANCE-CONDITIONAL VALUE AT RISK MODEL OF MULTI-OBJECTIVE PORTFOLIO OPTIMIZATION WITH AN APPLICATION IN FINANCE YOUNES ELAHI A thesis submitted in fulfilment of the requirements for the award of the degree of Doctor of Philosophy (Mathematics) Faculty of Science UniversitiTeknologi Malaysia DECEMBER 2014
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A MODIFIED MEAN-VARIANCE-CONDITIONAL VALUE AT RISK MODEL
OF MULTI-OBJECTIVE PORTFOLIO OPTIMIZATION WITH AN
APPLICATION IN FINANCE
YOUNES ELAHI
A thesis submitted in fulfilment of the
requirements for the award of the degree of
Doctor of Philosophy (Mathematics)
Faculty of Science
UniversitiTeknologi Malaysia
DECEMBER 2014
iii
To my beloved Parents, Family and my respected Supervisor
iv
ACKNOWLEDGEMENT
I heartily express my gratefulness to Allah s.w.t for His blessing and strength
that He blessed to me during the completion of this research.
My sincere thanks go to my supervisor Prof. Dr. Mohd Ismail Abd Aziz for
his continuous motivation, constant advice, encouragement and support from start to
the completion of my studies.
I am ever grateful to my family, especially my wife, for their continuous
support in term of encouragement and motivation.
Furthermore, very genuine appreciation goes to my father (1952-1990) whom
i owe my very existence to the world, who always gave me the motivation and
courage to look on the bright side every time I felt unmotivated, whom that never let
me down and whom I respect the most in my heart.
This research work has been financially supported by UTM’s International
Doctoral Fellowship (IDF). I would like to thank the members of Universiti
Teknologi Malaysia (UTM) for providing the research facilities.
v
ABSTRACT
This research focuses on the development of a portfolio optimization model
based on the classic optimization method and a meta-heuristic algorithm. The main
goal of a portfolio optimization model is to achieve maximum return with minimum
investment risk by allocating capital based on a set of existing assets. Recently,
mean-variance models have been improved to mean-variance-CVaR (MVC) model
as a multi-objective portfolio optimization (MPO) problem which is difficult to be
solved directly and optimally. In this work, a modified MVC model of portfolio
optimization is constructed using the weighted sum method (WSM). In this method,
each objective function of MVC model is given a weight. The optimization problem
is then minimized as a weighted sum of the objective functions. The implementation
of WSM enables the MVC model to be transformed from a multi-objective function
to one with a single objective function. The modified MVC model is then solved
using ant colony optimization (ACO) algorithm. This algorithm solves the MVC
model by the number of ant colonies and the number of pheromone, a chemical
creating trails for others to follow. The modified MVC model can be used in
managing diverse investment portfolio, including stocks on the stock market and
currency exchange. The applicability and effectiveness of the proposed method are
demonstrated by solving a benchmark problem and a practical investment problem as
examples. The data of practical examples are collected from the foreign currency
exchange of Bank Negara Malaysia for the years 2012 and 2013. In conclusion, this
thesis presented a hybrid optimization algorithm which utilizes a classical approach,
WSM and a meta-heuristic approach, ACO to solve an MVC model of portfolio
optimization.
vi
ABSTRAK
Kajian ini memberi tumpuan kepada pembangunan model pengoptimuman
portfolio berdasarkan kaedah pengoptimuman klasik dan algoritma meta-heuristik.
Matlamat utama model pengoptimuman portfolio adalah untuk mencapai pulangan
maksimum, dengan risiko pelaburan minimum dengan memperuntukkan modal
berdasarkan satu set aset yang ada. Kebelakangan ini, model min-varians telah
diperbaiki kepada model min-varians-CVaR (MVC) sebagai masalah
pengoptimuman portfolio pelbagai-tujuan (MPO) yang sukar untuk diselesaikan
secara langsung dan secara optimum. Dalam penyelidikan ini, satu model MVC
terubahsuai untuk pengoptimuman portfolio dibina menggunakan kaedah
hasiltambah wajaran (WSM). Dalam kaedah ini, setiap fungsi objektif pada model
MVC diberikan satu pemberat. Masalah pengoptimuman tersebut kemudiannya
diminimumkan sebagai hasiltambah wajaran fungsi objektif. Pelaksanaan WSM
membolehkan model MVC ditukarkan dari sebuah fungsi pelbagai-tujuan kepada
satu fungsi objektif tunggal. Model MVC terubahsuai kemudiannya diselesaikan
dengan menggunakan algoritma pengoptimuman koloni semut (ACO). Algoritma ini
menyelesaikan model MVC dengan jumlah koloni semut dan jumlah pheromone,
bahan kimia yang menghasilkan jejak supaya dapat diikuti oleh yang lain. Model
MVC terubahsuai boleh digunakan dalam menguruskan portfolio pelaburan pelbagai,
termasuk saham di pasaran saham dan pertukaran mata wang. Kepenggunaan dan
keberkesanan kaedah yang dicadangkan telah ditunjukkan dengan menyelesaikan
satu masalah penanda aras dan masalah pelaburan praktikal sebagai contoh. Data dari
contoh-contoh praktikal dikumpulkan dari pertukaran mata wang asing Bank Negara
Malaysia bagi tahun 2012 dan 2013. Kesimpulannya, tesis ini membentangkan satu
algoritma pengoptimuman hibrid yang menggunakan pendekatan klasik, WSM dan
pendekatan meta-heuristik, ACO untuk menyelesaikan model MVC pengoptimuman
portfolio.
vii
TABLE OF CONTENTS
CHAPTER TITLE PAGE
DECLARATION ii
DEDICATION iii
ACKNOWLEDGEMENT iv
ABSTRACT v
ABSTRAK vi
TABLE OF CONTENTS vii
LIST OF TABLES x
LIST OF FIGURES xi
LIST OF ABBREVIATIONS xii
LIST OF SYMBOLS xiii
LIST OF APPENDICES xiv
1 INTRODUCTION 1
1.1 Overview 1
1.2 Background of Problem 3
1.3 Problem Statement 5
1.4 Objectives of Study 5
1.5 Scope of the Study 6
1.6 Significance of Study 7
1.6.1 Contribution to Theory 7
1.6.2 Contribution to Practice 7
1.7 Thesis Outline 8
viii
2 LITERATURE REVIEW AND THEORY 9
2.1 Introduction 9
2.2 Preliminary Concepts 10
2.2.1 Multi-Objective Portfolio
Optimization 10
2.2.2 Pareto Optimal Solution 14
2.2.3 Mean-Variance Model 16
2.3 Weighted Sum Method and Permutation-Based
Optimization 18
2.4 Ant Colony Optimization 21
2.4.1 ACO Algorithm 24
2.4.2 ACOR Algorithm 26
2.5 Related Works on Portfolio Optimization 27
2.6 Summary 30
3 RESEARCH METHODOLOGY 31
3.1 Introduction 31
3.2 Research Framework 31
3.3 Overall Research Design 32
3.4 MVC Model Formulation 33
3.5 Summary 40
4 DEVELOPMENT OF MVC MODEL THROUGH WSM 41
4.1 Introduction 41
4.2 Development of Modified MVC Model 42
4.2.1 Conditions for WSM
Implementation 42
4.2.2 Modified MVC Model 47
4.3 Algorithm Procedure 52
4.3.1 Algorithm of MVC Model Based on
PBO 52
4.3.2 Recursive Procedure of PBO 53
4.4 Illustrative Example 54
4.5 Discussion 58
4.6 Summary 59
ix
5 ANT COLONY OPTIMIZATION APPROACH FOR
SOLVING MODIFIED MVC MODEL 60
5.1 Introduction 60
5.2 ACOR Algorithm of MVC Model 62
5.3 Flowchart for Solving a Modified MVC Model
using ACOR 65
5.4 Pseudo Code to Solve a Modified MVC Model
using ACOR 66
5.5 An Illustrative Example 68
5.6 Discussion 73
5.7 Summary 75
6 APPLICATION OF MODIFIED MVC MODEL AND
ACOR 76
6.1 Introduction 76
6.2 Implementation of PBO 76
6.2.1 Investment on USD and EUR in
BNM 77
6.2.2 Investment on USD, GBP and EUR
in BNM 78
6.3 Implementation of ACOR Algorithm 81
6.4 Benchmark Problem and Hybrid Method 81
6.5 Discussion 85
6.6 Summary 86
7 SUMMARY AND FUTURE WORK 87
7.1 Introduction 87
7.2 Main Contribution of Thesis 87
7.3 Limitation of the Study 88
7.4 Direction of Future Researches 89
REFERENCES 91
Appendices A-F 100-115
x
LIST OF TABLES
TABLE NO. TITLE PAGE
2.1 Summarized review of multi-objective portfolio optimization 28
3.1 Research design 33
4.1 Algorithm of MVC model based on PBO 52
4.2 Constants used by WSM for MVC model (USD and GBP) 55
4.3 Summary of result of modified MVC model for USD and GBP 57
4.4 Comparison between PBO and LINGO for USD and GBP 57
4.5 Conditions and their related mechanism 58
5.1 Comparison between PBO and ACOR 74
5.2 Comparison between ACOR and LINGO for USD and GBP 74
6.1 Results of min MVC model for USD and EUR 78
6.2 Solution of the MVC model for USD, GBP and EUR (case a) 80
6.3 Solution of the MVC model for USD, GBP and EUR (case b) 80
6.4 Comparison between PBO and LINGO for USD, GBP and
EUR 81
6.5 Results of MV Model for Handan and Baidu 83
6.6 Summary of comparison for benchmark problem 84
xi
LIST OF FIGURES
FIGURE NO. TITLE PAGE
2.1 The global objectives hierarchy 11
2.2 The chart of the ACO as a meta-heuristic method 22
2.3 The probability density function (PDF) 26
3.1 Diagram of research framework 32
4.1 Recursive procedure of PBO 53
4.2 Historical chart of rate of USD and GBP against RM 55
5.1 The PDF to determine the solutions in ACOR 62
5.2 Solutions archive in ACOR 63
5.3 Procedure flowchart of ACOR 65
5.4 Simulation pseudo code of ACOR 67
5.5 Initial population in the ACOR for USD and GBP 69
5.6 Sorted initial population of MVC for USD and GBP 70
5.7 Solutions archive for USD and GBP with normal weights 70
5.8 Mean vector of USD and GBP in the ACOR 71
5.9 Standard deviations of USD and GBP in ACOR 71
5.10 New generated rows in ACOR for USD and GBP 71
5.11 Unsorted merged archive in ACOR for USD and GBP 72
5.12 Sorted merged archive in ACOR for USD and GBP 72
5.13 Final solutions archive of first iteration in ACOR for USD and
GBP 73
5.14 Solutions archive in ACOR for 50 iterations 73
6.1 Historical chart of fluctuation of USD and EUR against RM 77
6.2 Historical chart of fluctuation of USD, GBP and EUR to RM 79
6.3 Historical price fluctuation of Baidu and Handan stocks 82
xii
LIST OF ABBREVIATIONS
ACO - Ant Colony Optimization
ACOR - Ant Colony Optimization in continues space
MV - Mean Variance
WSM - Weighted Sum Method
MPO - Multi-Objective Portfolio Optimization
VaR - Value at Risk
CVaR - Conditional Value at Risk
DM
PBO
-
-
Decision Maker
Permutation Based Optimization
AWS - Adaptive Weighted Sum
BNM - Bank Negara Malaysia
RM - Ringgit Malaysia
MVSK - Mean-Variance-Skewness-Kurtosis
MVS - Mean-Variance-Skewness
MVC - Mean-Variance-CVaR
EDA - Estimation of Distribution Algorithm
xiii
LIST OF SYMBOLS
- The number of assets that are available
- Return depending on a decision vector that belongs to
a feasible set
- Proportion of investment in asset
- The feasible set of solutions
- A solution of problem
- The expected mean of the asset
( ) - The variance belonging to
- The number of assets to invest
- Covariance among the return of assets
- Covariance matrix
- Confidence level
- Overall maximum
- Best solution
( ) - Pareto optimal
- Weight of objective functions
xiv
LIST OF APPENDICES
APPENDIX NO. TITLE PAGE
A The dataset of USD and GBP to RM from BNM 99
B The results of MVC model based on WSM for USD and
GBP 103
C Data for USD and EUR against RN from BNM (2012,
2013) 105
D The results of portfolio included USD and EUR against
RM (2012, 2013) 109
E Some results of portfolio included USD, GBP and EUR
against RM (2012, 2013) 112
F List of publications 117
CHAPTER 1
1 INTRODUCTION
1.1 Overview
In finance, a portfolio refers to a collection of investments. Usually a person
with a certain amount of fund wants to obtain higher income than interests paid by
saving accounts or fixed deposits. The investor tries to choose various assets to
purchase with the hope of getting a better return. These assets are commonly shares
on the stocks markets. They may also be commodities (gold, iron, aluminum,