Robust Asset Allocation Under Model Ambiguity

219 Pages·2003·2.05 MB·English

Robust Asset Allocation Under
Model Ambiguity
Sandrine Tobelem - Foldvari
Department of Statistics
The London School of Economics and Political Sciences
A thesis submitted for the degree of
Doctor of Philosophy
London, September 2010
I certify that the thesis I have presented for examination for the MPhil/PhD
degree at the London School of Economics and Political Science is solely my own
work other than where I have clearly indicated that it is the work of others (in
which case the extent of any work carried out jointly by me and any other person
is clearly identified in it).
The copyright of this thesis rests with the author. Quotation from it is permitted,
provided that full acknowledgement is made. This thesis may not be reproduced
without the prior written consent of the author.
I warrant that this authorisation does not, to the best of my belief, infringe the
rights of any third party.
i
Acknowledgements
Starting a PhD is like embarking upon a very long and uncertain
journey. Support is a key component for success, and I am extremely
grateful I have been helped so much throughout the long six years I
spent writing this thesis.
I would like to thank the Statistics Department at the London School
of Economics and Political Sciences that allowed me to undertake
an especially difficult part-time PhD. I would like to acknowledge in
particular the academic help I received from Pr Ragnar Norberg, Pr
Qiwei Yao, Pr Umut Cetin and Pr Henry Wynn. I also would like
to thank the administrative staff of the department and in particular
Ian Marshall. Thank to my fellow LSE PhD students; even if I did
not meet with them often, they still made me feel I was part of the
research department.
I would like to acknowledge the financial support of the CREST in
Paris for the grant I received for the first two years of my PhD, and
especially Pr Christian Gourieroux, Pr Laurence Lescourret and Pr
Nizar Touzi.
I also would like to thank my current employer Credit Suisse, which
financed the remaining four years of my research, and gave me enough
flexibility in my job to be able to complete my thesis, despite a very
challenging and demanding environment. I particularly would like to
thank all my colleagues and my managers who were supportive and
allowed me to take my Fridays to work on my PhD for three years.
I especially would like to thank Yves Bentz, my previous manager
at Credit Suisse, who initially offered me to work at the bank while
completing my PhD. He has been a real inspiration and support in
all respects.
I also would like to acknowledge the help and advice I received from
Pr Monique Jean-Blanc, Pr Bernard Sinclair-Desgagne and especially
Pr Leonard Smith and Pr Rudiger Kiesel.
I thank all my friends, and especially David Le Bris and Judith Byd-
lowski who devoted time reading parts of my PhD and giving me
sensible and most welcome advice.
But first and foremost, I would like to thank my PhD supervisor, Pr
Pauline Barrieu. I will never be able to express enough gratitude. She
has been incredibly inspiring, guiding me through every step, never
letting me give up and encouraging me in my most difficult times.
She became a very dear and important person to me and I owe her
much more than the great achievement of finishing my PhD thesis.
Ithankwarmlymyfamily, myparentsandmybrotherandmyparents
in law who gave me confidence and emotional support.
Finally, I would like to thank my husband Peter. He supported me
the most completely and always believed I would finish my PhD thesis
even when I was convinced I could never do it. He gave me an infinite
supply of time, trust and love and I would like to dedicate my research
to him, as I could not have done it without his unconditional support.
I also dedicate my PhD thesis to our two lovely baby daughters Laura
and Emma.
Abstract
A decision maker, when facing a decision problem, often considers
several models to represent the outcomes of the decision variable con-
sidered. More often than not, the decision maker does not trust fully
any of those models and hence displays ambiguity or model uncer-
tainty aversion.
InthisPhDthesis, focusisgiventothespecificcaseofassetallocation
problem under ambiguity faced by financial investors. The aim is not
to find an optimal solution for the investor, but rather come up with
a general methodology that can be applied in particular to the asset
allocation problem and allows the investor to find a tractable, easy to
compute solution for this problem, taking into account ambiguity.
This PhD thesis is structured as follows: First, some classical and
widely used models to represent asset returns are presented. It is
shown that the performance of the asset portfolios built using those
single models is very volatile. No model performs better than the
others consistently over the period considered, which gives empirical
evidence that: no model can be fully trusted over the long run and
that several models are needed to achieve the best asset allocation
possible. Therefore, the classical portfolio theory must be adapted
to take into account ambiguity or model uncertainty. Many authors
have in an early stage attempted to include ambiguity aversion in
the asset allocation problem. A review of the literature is studied
to outline the main models proposed. However, those models often
lack flexibility and tractability. The search for an optimal solution
to the asset allocation problem when considering ambiguity aversion
is often difficult to apply in practice on large dimension problems,
as the ones faced by modern financial investors. This constitutes
the motivation to put forward a novel methodology easily applicable,
robust, flexible and tractable. The Ambiguity Robust Adjustment
(ARA) methodology is theoretically presented and then tested on a
large empirical data set. Several forms of the ARA are considered and
tested. Empirical evidence demonstrates that the ARA methodology
improves portfolio performances greatly.
Through the specific illustration of the asset allocation problem in
finance, this PhD thesis proposes a new general methodology that will
hopefully help decision makers to solve numerous different problems
under ambiguity.
Contents
1 Introduction 1
2 Classical Approaches to the Asset Allocation Problem 7
2.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Notation and setting . . . . . . . . . . . . . . . . . . . . . 10
2.1.2 Trading Strategy . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.3 Decision under risk . . . . . . . . . . . . . . . . . . . . . . 13
2.1.4 The classical portfolio optimisation problem . . . . . . . . 13
2.2 The Efficient Frontier, Markowitz (1952) . . . . . . . . . . . . . . 14
2.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1.1 The mean-variance paradigm . . . . . . . . . . . 15
2.2.1.2 The concept of diversification . . . . . . . . . . . 16
2.2.1.3 Efficient Frontier Definition . . . . . . . . . . . . 17
2.2.2 The Efficient Frontier without a risk-free asset . . . . . . . 18
2.2.2.1 The Minimum Variance portfolio . . . . . . . . . 19
2.2.2.2 The Maximum Sharpe ratio portfolio . . . . . . . 19
2.2.2.3 The Two Fund Theorem. . . . . . . . . . . . . . 21
2.2.3 Introduction of a risk-free asset . . . . . . . . . . . . . . . 22
2.2.3.1 Tangential portfolio . . . . . . . . . . . . . . . . 23
2.2.3.2 The One Fund Theorem. . . . . . . . . . . . . . . 24
2.3 The Capital Asset Pricing Model (CAPM), Sharpe (1964) . . . . 25
2.3.1 The Capital Market Line (CML) . . . . . . . . . . . . . . 26
2.3.1.1 The Market Portfolio . . . . . . . . . . . . . . . . 26
2.3.1.2 The CML Equation . . . . . . . . . . . . . . . . 28
2.3.2 The Security Market Line (SML) . . . . . . . . . . . . . . 28
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CONTENTS
2.3.2.1 The CAPM Betas . . . . . . . . . . . . . . . . . 28
2.3.2.2 The SML Equation . . . . . . . . . . . . . . . . . 29
2.3.3 The CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.3.1 The CAPM Equation . . . . . . . . . . . . . . . 30
2.3.3.2 The Jensen Alpha . . . . . . . . . . . . . . . . . 31
2.4 The Asset Pricing Theory (APT), Ross (1976) . . . . . . . . . . . 33
2.4.1 The APT Framework . . . . . . . . . . . . . . . . . . . . . 34
2.4.2 APT Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4.3 Construction of a factor model portfolio . . . . . . . . . . 37
2.5 Performance measures . . . . . . . . . . . . . . . . . . . . . . . . 38
2.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.6.1 Efficient Frontier without a risk-free asset . . . . . . . . . 41
2.6.2 Efficient Frontier with a risk-free asset . . . . . . . . . . . 43
3 Factor Model Performances: Empirical Evidence 46
3.1 General remarks on factor models . . . . . . . . . . . . . . . . . . 48
3.1.1 Factor models objectives . . . . . . . . . . . . . . . . . . . 48
3.1.2 Types of factor models . . . . . . . . . . . . . . . . . . . . 49
3.2 Presentation of the data . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.1 The raw data . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.2 The cleaning process . . . . . . . . . . . . . . . . . . . . . 53
3.2.3 The returns . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.3.1 Computation and cleaning process . . . . . . . . 54
3.2.3.2 Some basic statistics . . . . . . . . . . . . . . . . 55
3.3 The Capital Asset Pricing Model (CAPM) . . . . . . . . . . . . . 58
3.3.1 Estimation of the CAPM model . . . . . . . . . . . . . . . 58
3.3.1.1 Betas Estimation . . . . . . . . . . . . . . . . . . 58
3.3.1.2 Goodness of fit of the CAPM . . . . . . . . . . . 59
3.3.2 Betas instability . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3.2.1 Betas time series . . . . . . . . . . . . . . . . . . 61
3.3.2.2 Stationary tests . . . . . . . . . . . . . . . . . . . 62
3.3.3 The Jensen statistics . . . . . . . . . . . . . . . . . . . . . 64
3.4 An Exogenous Factor Model (EFM) . . . . . . . . . . . . . . . . . 66
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CONTENTS
3.4.1 The exogenous factors . . . . . . . . . . . . . . . . . . . . 67
3.4.1.1 Expected sensitivities toward the exogenous factors 67
3.4.1.2 Exogenous factor returns and index returns . . . 68
3.4.2 The EFM model . . . . . . . . . . . . . . . . . . . . . . . 71
3.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.4.3.1 Betas instability . . . . . . . . . . . . . . . . . . 72
3.4.3.2 Jensen statistics . . . . . . . . . . . . . . . . . . 73
3.5 A Fundamental Factor Model (FFM) . . . . . . . . . . . . . . . . 75
3.5.1 The FFM model . . . . . . . . . . . . . . . . . . . . . . . 75
3.5.1.1 The FFM factors . . . . . . . . . . . . . . . . . . 75
3.5.1.2 FFM equation . . . . . . . . . . . . . . . . . . . 76
3.5.2 Estimation of the FFM model . . . . . . . . . . . . . . . . 76
3.5.3 Comments on the FFM . . . . . . . . . . . . . . . . . . . 78
3.6 Principal Component Analysis (PCA) . . . . . . . . . . . . . . . . 79
3.6.1 The PCA model . . . . . . . . . . . . . . . . . . . . . . . . 79
3.6.1.1 The PCA principle . . . . . . . . . . . . . . . . . 79
3.6.1.2 Factors selection and Betas calibration . . . . . . 80
3.6.2 PCA Estimation and Results . . . . . . . . . . . . . . . . 81
3.6.2.1 Explanatory power of the PCA . . . . . . . . . . 81
3.6.2.2 PCA Factors Identification . . . . . . . . . . . . 82
3.6.2.3 The Jensen Statistics . . . . . . . . . . . . . . . . 84
3.6.3 Comment on the PCA model . . . . . . . . . . . . . . . . 85
3.7 Independent Component Analysis (ICA) . . . . . . . . . . . . . . 86
3.7.1 ICA principle . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.7.2 ICA Fast Algorithm . . . . . . . . . . . . . . . . . . . . . 87
3.7.2.1 Negentropy: a measure of non-Gaussianity . . . . 88
3.7.2.2 Mutual information . . . . . . . . . . . . . . . . 88
3.7.3 Estimation of the model . . . . . . . . . . . . . . . . . . . 89
3.8 Clusters Analysis (CA) . . . . . . . . . . . . . . . . . . . . . . . . 91
3.8.1 Cluster Analysis principle . . . . . . . . . . . . . . . . . . 92
3.8.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.8.3 Estimation of the model . . . . . . . . . . . . . . . . . . . 94
3.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
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CONTENTS
4 Decision Under Ambiguity: Literature Review 99
4.1 The concept of Ambiguity . . . . . . . . . . . . . . . . . . . . . . 100
4.1.1 The Knight Uncertainty . . . . . . . . . . . . . . . . . . . 100
4.1.2 Savage Subjective Expected Utility (1954) . . . . . . . . . 101
4.1.3 The Ellsberg Paradox . . . . . . . . . . . . . . . . . . . . . 102
4.1.4 Kahneman and Tversky Prospect Theory (1979) . . . . . . 105
4.2 Decision under Ambiguity . . . . . . . . . . . . . . . . . . . . . . 106
4.2.1 Gilboa and Schmeidler Max-Min Expected Utility (1989) . 106
4.2.2 Hansen and Sargent Robust Control (2001) . . . . . . . . . 107
4.3 Portfolio Allocation and Model Risk . . . . . . . . . . . . . . . . . 108
4.3.1 The classical Markowitz settings . . . . . . . . . . . . . . . 109
4.3.2 Learning and Filtering . . . . . . . . . . . . . . . . . . . . 110
4.3.2.1 Classical Learning: the Bayesian approach . . . . 110
4.3.2.2 Learning Under Ambiguity . . . . . . . . . . . . 113
4.3.2.3 The shrinkage approach . . . . . . . . . . . . . . 115
4.3.2.4 The Multiple Prior Approach . . . . . . . . . . . 115
4.3.3 Generalised framework to model ambiguity in the asset al-
location problem . . . . . . . . . . . . . . . . . . . . . . . 116
4.3.3.1 Asmoothclassofpreferencesthatmodelsambiguity117
4.3.3.2 A Generalised Model . . . . . . . . . . . . . . . . 117
4.4 The Poor Performance of Ambiguity Models on Empirical Data. . 118
5 A Robust Alternative Approach to Model Ambiguity 121
5.1 Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.2 An alternative robust approach to model uncertainty: the Ambi-
guity Robust Adjustment (ARA) . . . . . . . . . . . . . . . . . . 125
5.2.1 The Absolute Ambiguity Robust Adjustment (AARA) . . 127
5.2.2 The Relative Ambiguity Robust Adjustment (RARA) . . . 131
5.2.3 The role of the risk free asset . . . . . . . . . . . . . . . . 132
5.2.4 ARA parameterisation . . . . . . . . . . . . . . . . . . . . 133
5.3 Some definitions relative to the ARA asset allocation . . . . . . . 134
5.4 Comparison with the Klibanoff, Marinacci, Mukerji model (KMM) 136
5.4.1 Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
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