ASSET ALLOCATION IN WEALTH MANAGEMENT USING STOCHASTIC MODELS by STUART ...

217 Pages·2016·3.7 MB·English

ASSET ALLOCATION IN WEALTH MANAGEMENT USING STOCHASTIC
MODELS
by
STUART JACK ROYDEN-TURNER
submitted in accordance with the requirements
for the degree of
MASTER OF SCIENCE
in the subject
OPERATIONS RESEARCH
at the
UNIVERSITY OF SOUTH AFRICA
SUPERVISOR: PROFESSOR P H POTGIETER
FEBRUARY 2016
Declaration
I declare that ASSET ALLOCATION IN WEALTH MANAGEMENT USING STOCHASTIC
MODELS is my own work and that all the sources that I have used or quoted have been indicated
and acknowledged by means of complete references.
I further declare that I have not previously submitted this work, or part of it, for examination at
Unisa for another qualification or at any other higher education institution.
Signature: . . . . . . . . . . . . . . . . . .
STUART J ROYDEN-TURNER
Date: 28th February 2016
Document submitted on Sunday the 28th February 2016 Page 2
Abstract
Modernfinancialassetpricingtheoryisabroad,andattimes,complexfield. Theliteraturereview
in this study covers many of the asset pricing techniques including factor models, random walk
models, correlation models, Bayesian methods, autoregressive models, moment-matching models,
stochastic jumps and mean reversion models. An important topic in finance is portfolio opti-
misation with respect to risk and reward such as the mean variance optimisation introduced by
Markowitz (1952). This study covers optimisation techniques such as single period mean variance
optimisation, optimisation with risk aversion, multi-period stochastic programs, two-fund separa-
tion theory, downside optimisation techniques and multi-period optimisation such as the Bellman
dynamic programming model.
The question asked in this study is, in the context of investing for South African individuals
in a multi-asset portfolio, whether an active investment strategy is significantly different from
a passive investment strategy. The passive strategy is built using stochastic programming with
moment matching methods for non-Gaussian asset class distributions. The strategy is optimised
in a framework using a downside risk metric, the conditional variance at risk. The active strategy
is built with forward forecasts for asset classes using the time-varying transitional-probability
Markov regime switching model. The active portfolio is finalised by a dynamic optimisation using
a two-stage stochastic programme with recourse, which is solved as a large linear program. A
hypothesis test is used to establish whether the results of two strategies are statistically different.
The performance of the strategies are also reviewed relative to multi-asset peer rankings. Lastly,
we consider whether the findings reveal information on the degree of efficiency in the market place
for multi-asset investments for the South African investor.
Key terms: Asset allocation, active and passive investment strategy, multi-period portfolio
optimisation, modern asset pricing, stochastic processes, conditional value at risk, time-varying
transitional-probabilityMarkovregimeswitchingmodel,inter-temporalmean-reversion,integrated
stochastic liability, two-stage problems with recourse, dynamic stochastic programme.
Document submitted on Sunday the 28th February 2016 Page 3
Acknowledgements
I would like to acknowledge and thank the following people:
• Firstly, to Professor Potgieter. I would like to thank you for your very high standards and
patience. These projects take a lot of time in review and in that time our discussions really
deepened my learnings;
• It is very important that I acknowledge the role my wife, Tamryn, played. I can’t thank you
enough, you showed patience and gave me lots of support. Completing this project was only
possible with your support, thank you;
• To my kids, Ethan and Megan, you don’t know yet that Saturday mornings are not usually
reserved for Dad studying. Thanks for your support and patience;
• The FirstRand Investment Management Office which includes Jaco, Andrew and Albert for
his time and willingness to discuss many finance concepts from every angle. I must thank
theFirstRandInvestmentManagementOfficeofficeforallowingmetheuseoftheDMSdata
set and the time away from the office to apply myself;
• Leadership in FirstRand group for their support in making this research possible. This
includes Emil, Francois, Renzi and Jaco.
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CONTENTS
Contents
I Introduction and context 11
1 Introduction 12
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Aims and objectives of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
II Literature survey 16
2 Overview of Modern Portfolio Theory: selected topics 17
2.1 Utility and choice theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 The origins of utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Utility functions and risk aversion coefficients . . . . . . . . . . . . . . . . . . . . . 19
2.4 Graphical illustration of utility functions . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.1 Risk aversion coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Modern asset pricing 21
3.1 Factor models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1.1 APT formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Random walk models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Dependence and correlation in portfolio management . . . . . . . . . . . . . . . . . 25
3.4 Multivariate moment-matching methods . . . . . . . . . . . . . . . . . . . . . . . 29
3.4.1 Fleishman non-normal procedure . . . . . . . . . . . . . . . . . . . . . . . 29
3.4.2 Fleishman, Vale and Maurelli multivariate procedure . . . . . . . . . . . . 30
3.5 Stochastic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5.1 Probability space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5.2 Discrete time stochastic processes . . . . . . . . . . . . . . . . . . . . . . . 31
3.5.3 Geometric random walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5.4 Continuous time stochastic processes . . . . . . . . . . . . . . . . . . . . . . 35
4 Single-period optimisation models 37
4.1 Portfolio optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1.1 Definition of portfolio measures . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1.2 The Markowitz mean variance (MVO) optimisation . . . . . . . . . . . . . 37
4.1.3 The risk aversion approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.1.4 Graphical representation of MVO results . . . . . . . . . . . . . . . . . . . 40
4.2 Tobin’s two fund separation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 41
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4.3 Capital asset pricing model (CAPM) . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4 Stochastic downside approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.4.1 Advances in downside risk measures . . . . . . . . . . . . . . . . . . . . . . 43
4.4.2 Conditional Value at Risk (CVAR) . . . . . . . . . . . . . . . . . . . . . . . 44
4.4.3 CVAR portfolio optimisation formulation . . . . . . . . . . . . . . . . . . . 44
4.5 Advances in econometric time series measures . . . . . . . . . . . . . . . . . . . . . 46
4.5.1 Standard switching models . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.5.2 Markov switching models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.5.3 Hidden Markov model (HMM) . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.5.4 Time-varying Markov-switching model . . . . . . . . . . . . . . . . . . . . 48
4.6 Bayesian portfolio approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.6.1 Black-Litterman model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.6.2 The Black-Litterman optimisation formula . . . . . . . . . . . . . . . . . . 53
5 Multi-period portfolio choice models 54
5.1 Discrete deterministic portfolio models . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.2 Continuous time models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2.1 Continuous deterministic portfolio models . . . . . . . . . . . . . . . . . . . 55
5.2.2 Continuous stochastic portfolio models . . . . . . . . . . . . . . . . . . . . . 55
5.3 Discrete stochastic portfolio models - Stochastic programming . . . . . . . . . . . 56
5.4 Classes of stochastic programming . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.4.1 Two-stage problems with recourse . . . . . . . . . . . . . . . . . . . . . . . 60
5.5 Scenario tree generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6 Discussion: active vs. passive 63
III Portfolio model specification and analysis 64
7 Common portfolio specifications 65
7.1 Asset class universe and proxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.2 Holding costs and trading costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7.3 Use of tools in the modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
IV Passive single-period model 68
8 The passive coherent risk model 69
8.1 Model description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
8.1.1 Return distribution simulation . . . . . . . . . . . . . . . . . . . . . . . . . 69
8.1.2 CVAR portfolio optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . 70
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8.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
8.2.1 Input parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
8.2.2 Results of the moment-matching simulation . . . . . . . . . . . . . . . . . 72
8.2.3 Optimisation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
V Active multi-period framework 76
9 The dynamic multi-stage integrated asset-liability model 77
9.1 Model construct and formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
9.2 Dynamic return forecast: regime switching model . . . . . . . . . . . . . . . . . . 78
9.2.1 Universe of predictive variables . . . . . . . . . . . . . . . . . . . . . . . . . 80
9.2.2 Model selection criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
9.2.3 TVTP BIC test results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
9.2.4 Coefficients of regime-switching model TVTP model . . . . . . . . . . . . . 82
9.2.5 Modelling residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
9.2.6 TVTP model forecasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
9.3 Stochastic asset-pricing model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
9.4 Integrated stochastic liability model . . . . . . . . . . . . . . . . . . . . . . . . . . 91
9.4.1 Liability value model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
9.4.2 Stochastic liability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
9.4.3 Model Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
9.4.4 Liability target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
9.5 Multi-stage portfolio choice model . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
9.5.1 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
9.5.2 Multi-period asset allocation . . . . . . . . . . . . . . . . . . . . . . . . . . 103
VI Results and Analysis 104
10 Results of the investment strategies 105
10.1 In-sample analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
10.2 Out-of-sample analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
10.3 Hypothesis testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
VII Conclusion 110
11 Conclusion 111
11.1 Areas for further research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
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VIII Appendix 123
A Asset class simulation testing 123
A.1 t-test and histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
A.2 TVTP data parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
IX Addendum 126
B Single-period model 127
B.1 SAS Code: inferential statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
B.2 SAS Code: Fleishman simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
B.3 Matlab code: portfolio optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . 169
C Multi-period model 176
C.1 Eviews Code: regime-switching model . . . . . . . . . . . . . . . . . . . . . . . . . 176
C.2 SAS Code: asset price simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
C.3 SAS Code: liability model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
C.4 AMPL DEV code: multi-period optimisation . . . . . . . . . . . . . . . . . . . . . 216
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LIST OF FIGURES
List of Figures
1 Utility Functions: Investor attitudes toward risk . . . . . . . . . . . . . . . . . . . 19
2 Mean-Variance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3 The security market line (SML) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 Riskmeasuresillustratedonasimulatedequityreturndistribution(SASchartoutput) 46
5 Derivingthecombinedreturnvector. ThisillustrationissourcedfromIdzorek(2007,
page 27). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6 Categorisation of stochastic programs . . . . . . . . . . . . . . . . . . . . . . . . . 58
7 A scenario tree with three stages and eight scenarios . . . . . . . . . . . . . . . . . 61
8 CVAR Efficient Frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
9 Optimal CVAR portfolios (CPI is in ZAR). . . . . . . . . . . . . . . . . . . . . . . 74
10 Aggregate asset allocation of over time. . . . . . . . . . . . . . . . . . . . . . . . . 75
11 Model schema: multi-period dynamic stochastic optimal ALM portfolio model . . 78
12 Regime-switching model construct . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
13 Actuals, model results and residual error (charts are an Eviews output). . . . . . . 84
14 Actuals, model results and residual errors (charts are an Eviews output). . . . . . 85
15 Assetreturnsimulationforcommodities(left)andRSAequity(right)inZAR,based
on the 2007 forecast (SAS output) . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
16 AssetreturnsimulationforcommoditiesinZAR(left)andRSAequity(right),based
on the 2008 forecast (SAS output) . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
17 Actuarial life expectancy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
18 The liability values for an affluent young client. . . . . . . . . . . . . . . . . . . . . 95
19 Liability profile for an affluent middle client.. . . . . . . . . . . . . . . . . . . . . . 95
20 HNWclientliabilityvalues. Thisclientretiresin2011andstartsmakingwithdrawals. 96
21 UHNW client liability values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
22 IRR calculation illustrated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
23 Index value over investment horizon per each segment. . . . . . . . . . . . . . . . . 97
24 Scenario tree scope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
25 Asset returns in a dynamic stochastic setting. . . . . . . . . . . . . . . . . . . . . . 99
26 Asset allocation of the ALM stochastic linear program over time . . . . . . . . . . 103
27 attribution of returns for CVAR portfolio . . . . . . . . . . . . . . . . . . . . . . . 106
28 Attribution of returns for ALM portfolio . . . . . . . . . . . . . . . . . . . . . . . . 106
29 In-sample performance tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
30 Out-of-sample performance tracking . . . . . . . . . . . . . . . . . . . . . . . . . . 108
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LIST OF TABLES
List of Tables
1 A South African investor’s passive costs . . . . . . . . . . . . . . . . . . . . . . . . 66
2 A summary of the use of tools in modelling and data analysis . . . . . . . . . . . . 67
3 Historical descriptive statistics: key inputs for the moment-matching simulation . . 71
4 Correlationinputsforsimulationbasedonhistoricaldata-Pearsoncorrelationmatrix 71
5 Descriptive statistics of simulated data: key inputs for the moment-matching simu-
lation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6 Pearson correlation matrix of the simulated data . . . . . . . . . . . . . . . . . . . 72
7 Comparison of historical and simulated data . . . . . . . . . . . . . . . . . . . . . . 73
8 Aggregate allocation to risk levels over time.. . . . . . . . . . . . . . . . . . . . . . 75
9 TVTP BIC testing results.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
10 Comparison of standard deviation of the sample and standard deviation of residuals. 82
11 Regime-switching model TVTP model coefficients (modelling outputs from Eviews) 83
12 In-sample regime-switching TVTP model return forecast . . . . . . . . . . . . . . . 86
13 Out-of-sample regime-switching TVTP model return forecast . . . . . . . . . . . . 87
14 Liability book breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
15 Liability model assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
16 In-sample (2005-2009) performance results . . . . . . . . . . . . . . . . . . . . . . . 107
17 Out-of-sample (2010-2014) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
18 Add caption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
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