Solving Cardinality Constrained Portfolio Optimisation Problem Using Genetic Algorithms and Ant ...

130 Pages·2015·2.21 MB·English

Solving Cardinality Constrained Portfolio Optimisation Problem
Using Genetic Algorithms and Ant Colony Optimisation

Yibo Li

A thesis submitted for the degree of Doctor of Philosophy
School of Information Systems, Computing and Mathematics,
Brunel University

2015
i

Abstract
In this thesis we consider solution approaches for the index tacking problem, in
which we aim to reproduces the performance of a market index without purchasing
all of the stocks that constitute the index. We solve the problem using three different
solution approaches: Mixed Integer Programming (MIP), Genetic Algorithms (GAs),
and Ant-colony Optimization (ACO) Algorithm by limiting the number of stocks
that can be held. Each index is also assigned with different cardinalities to examine
the change to the solution values.
All of the solution approaches are tested by considering eight market indices. The
smallest data set only consists of 31 stocks whereas the largest data set includes over
2000 stocks. The computational results from the MIP are used as the benchmark to
measure the performance of the other solution approaches. The Computational
results are presented for different solution approaches and conclusions are given.
Finally, we implement post analysis and investigate the best tracking portfolios
achieved from the three solution approaches. We summarise the findings of the
investigation, and in turn, we further improve some of the algorithms.
As the formulations of these problems are mixed-integer linear programs, we use the
solver ‘Cplex’ to solve the problems. All of the programming is coded in AMPL.
ii

Acknowledgement
First, I would like to thank my supervisor Dr. Cormac Lucas for his excellent
guidance, constant support, and great inspiration through my Ph.D. studies. It was
great pleasure to work with him. I would also like to thank my parents and other
family members, without their help and support none of these would have been
possible. I am very lucky to have all of you stand by my side and I am very grateful
for all the things you have done for me.
I would like to thank all the staff in the Mathematics Department for their excellent
work. Finally, I would also like to thank my Ph.D. colleagues and friends: Cristiano,
Xiang, Antonio, Zhenghong, Longhui, Martin, Fei and many others.
iii

Table of Contents

Abstract i
Acknowledgement ii
Table of Contents iii
List of Figures v
List of Tables vii

1. Introduction ........................................................................................................................... 1
1.1 Introduction .................................................................................................................. 1
1.2 Thesis Structure ........................................................................................................... 5
2. Literature Review .................................................................................................................. 7
2.1 History of Portfolio Theory ......................................................................................... 7
2.1.1 Mean-Variance Model ...................................................................................... 7
2.1.2 Capital Asset Pricing Model ........................................................................... 11
2.1.3 Factor Model ................................................................................................... 12
2.2 Index Tracking ........................................................................................................... 14
2.2.1 Genetic Algorithm........................................................................................... 14
2.2.2 Other Approaches ........................................................................................... 19
2.3 Summary .................................................................................................................... 22
3. Mixed Integer Programming ............................................................................................... 24
3.1 Introduction ................................................................................................................ 24
3.2 Research Comparisons ............................................................................................... 25
3.3 Formulation ................................................................................................................ 28
3.4 Computational Results ............................................................................................... 30
4. Genetic Algorithms ............................................................................................................. 32
4.1 Introduction ................................................................................................................ 32
4.2 Application for Index Tracking ................................................................................. 33
iv

4.3 The Inverse Triangle Genetic Algorithm ................................................................... 38
4.3.1 Introduction ..................................................................................................... 38
4.3.2 Formulation ..................................................................................................... 39
4.3.3 Simulation ....................................................................................................... 40
4.3.4 Computational Results .................................................................................... 46
4.4 Roulette Genetic Algorithms ..................................................................................... 49
4.4.1 Schema Theorem............................................................................................. 49
4.4.2 Application ...................................................................................................... 53
4.4.3 Investigation of population size ...................................................................... 62
4.4.4 Enhancement ................................................................................................... 65
4.4.5 Computational Results .................................................................................... 66
5. Ant Colony Optimization .................................................................................................... 69
5.1 Introduction ................................................................................................................ 69
5.2 Application ................................................................................................................. 73
5.3 Parameters Investigation ............................................................................................ 82
5.4 Computational Results ............................................................................................... 90
6. Post Analysis ....................................................................................................................... 93
6.1 Tracking Portfolios Investigation .............................................................................. 93
6.1.1 Investigation Method ...................................................................................... 93
6.1.2 Investigation Findings ..................................................................................... 98
6.1.3 Further Application ....................................................................................... 100
6.1.4 Computational Results .................................................................................. 101
6.2 Out-of-sample Performance ..................................................................................... 104
7. Conclusion ........................................................................................................................ 113
7.1 Summary .................................................................................................................. 113
7.2 Contribution to Knowledge ...................................................................................... 115
7.3 Future Ideas and Directions ..................................................................................... 116
Reference .............................................................................................................................. 118
v

List of Figures
Figure 1.1: Percentage of active managers outperforming their benchmark over rolling five-
year periods ······························································································································ 3
Figure 1.2: Percentage of active managers outperforming their benchmark during bull/bear
cycles ········································································································································· 3
Figure 2.1: The efficient frontier ······························································································· 9
Figure 2.2: The capital market line ··························································································· 9
Figure 4.1: Crossover ··············································································································34
Figure 4.2: Deformed offspring ·······························································································34
Figure 4.3: Semi-optimization approach ·················································································35
Figure 4.4: Fitness landscape ··································································································36
Figure 4.5: Traditional mutation ·····························································································36
Figure 4.6: Inverse triangle evolution ·····················································································38
Figure 4.7: Evolution with redundant duplicates ····································································44
Figure 4.8: Transformed evolution ·························································································44
Figure 4.9: Portfolio illustration ······························································································54
Figure 4.10: Computing time comparison for f12, f15, and f18 ·············································65
Figure 4.11: Roulette wheel ····································································································66
Figure 5.1: Construction graph ·······························································································70
Figure 5.2: Double bridge experiment ····················································································71
Figure 5.3: Additional experiment ··························································································72
Figure 5.4: Enlargement method ····························································································77
Figure 5.5: Mutation of ACO ···································································································79
Figure 5.6: Extended double bridge experiment ····································································85
Figure 5.7: The results of the extended double bridge experiment ·······································86
Figure 6.1: The return of Hang Seng index and its stocks ·······················································94
Figure 6.2: Half deterministic and half probabilistic searching ············································100
Figure 6.3: Hang Seng, K=10 ·································································································105
Figure 6.4: Hang Seng, K=15 ·································································································105
Figure 6.5: DAX, K=10 ············································································································106
Figure 6.6: DAX, K=15 ············································································································106
vi

Figure 6.7: DAX, K=20 ············································································································107
Figure 6.8: FTSE, K=10 ···········································································································107
Figure 6.9: FTSE, K=15 ···········································································································108
Figure 6.10: FTSE, K=20 ·········································································································108
Figure 6.11: S&P, K=10 ··········································································································109
Figure 6.12: S&P, K=15 ··········································································································109
Figure 6.13: S&P, K=20 ··········································································································110
Figure 6.14: Nikkei, K=10 ······································································································110
Figure 6.15: Nikkei, K=15 ······································································································111
Figure 6.16: Nikkei, K=20 ······································································································111
Figure 6.17: Nikkei, K=25 ······································································································112
vii

List of Tables
Table 3-1: BMC’s MIP data sets ······························································································27
Table 3-2: MIP data sets of our research ················································································27
Table 3-3: Computational results of MIP ················································································31
Table 4-1: IT-GA simulation ·····································································································41
Table 4-2: Initial population size for each index ·····································································46
Table 4-3: Computing results of MIP, IT-GA, and EIT-GA for the first five indices ·················47
Table 4-4: Computational results of EIT-GA and MIP for the larger indices ···························48
Table 4-5 (a): Semi-crossover operator simulation – portfolio processing ····························56
Table 4-6 (a): Semi-optimization operator simulation – portfolio processing ·······················59
Table 4-7: Initial population size under different frequencies ···············································62
Table 4-8: Computational results of R-GA for smaller size indices under different input
frequencies ·····························································································································63
Table 4-9: Average Tracking Error ···························································································64
Table 4-10: EIT-GA vs ER-GA ···································································································67
Table 5-1: Simulation ··············································································································76
Table 5-2: Attraction table before amplification ····································································77
Table 5-3: Attraction table after amplification ·······································································77
Table 5-4: Stopping criteria illustration ··················································································81
Table 5-5: Colony size for each index under different frequencies ········································83
Table 5-6: Computational results under different input frequencies ·····································84
Table 5-7: Computational results under different evaporation rates ····································87
Table 5-8: ER-GA vs ACO ·········································································································91
Table 6-1: Close stocks of the Hang Seng index ······································································96
Table 6-2: Optimal portfolio of the Hang Seng index with K=10 and K=15 ····························97
Table 6-3: Tracking portfolios investigation ···········································································98
Table 6-4: Results comparison for GA ···················································································102
Table 6-5: Results comparison for ACO ················································································103
1

Chapter 1
.
1 Introduction
1.1 Introduction
Fund managers are often hired to control a large amount of money which is invested
in various securities and other assets such as: real estate. The investor can be
categorized into two groups: institutions and private investors. Fund managers
usually provide investment management services that include financial statement
analysis, asset/stock selection, and direction of management plans and ongoing
monitoring of investments. The aim for fund managers is to make capital growth and
income over the short-term or the long-term. The basic investment strategies adopted
by fund managers can be broadly classified into two types: active management and
passive management.
 Active management
For active management, the fund managers have more confidence in their
own ability to estimate cash flows, growth rates, and discount rates. Based on
these estimates, they value assets and determine whether an asset is fairly
valued. In an actively managed portfolio, assets that are undervalued, or have
a chance of offering above-normal returns, will have a higher weight than
that in the index, whereas other assets will have a zero weight, or even
2

negative weight if short selling is permitted. This style of investing is called
active management, and the portfolios are referred to as active portfolios.
Most open-end mutual funds and hedge funds practice active management,
and most analysts believe that active investment can provide capital growth.
Active management often has high fixed cost (payments to the management
team) and its high frequency of trading often incurs high transaction costs. If
the market performs well these fees will be offset by the returns.
 Passive management
This is based on the assumption of an efficient market. If the markets are
efficient, the price in the market is an unbiased estimate of all future
discounted cash flows. In other words, the price aggregates and reflects all
information that is publicly available, and investors cannot expect to earn a
return that is greater than the required rate of return for that asset. So there is
no way to outperform the market. In that case, a simple and convenient
approach to investing is to rely on the prices set by the market. Portfolios that
are based on the assumption of unbiased market prices are referred to as
passive portfolios. Passive portfolios most commonly replicate and track
market indices, which are passively constructed on the basis of market prices
and market capitalizations. Examples of market indices are the FTSE 100, the
Nikkei 225, and the S&P 500. Passive portfolios based on market indices are
called index funds and generally have low running costs because no
significant effort is expended in valuing securities that are included in an
index. By comparison, passive management has lower fixed costs and lower
transaction costs, but once the market falls it will inevitably affect the index
fund.
Historical evidence reveals that active fund managers averagely underperform their
corresponding benchmarks. Researchers also found that some of the best active fund
managers did perform reasonably well in some periods, but most of them failed to
carry their success over a long-term period. The following two figures show the
evidence found by researchers from ‘Vanguard’ (Charles Thomas, Peter Westaway
and Todd Schlanger, n.d.):