Multi-moment Asset Allocation and Pricing Models

Edited by
Emmanuel Jurczenko and Bertrand Maillet

Wiley Finance 2006

1 Theoretical Foundations of Asset Allocation and Pricing Models with
Higher-order Moments 1
Emmanuel Jurczenko and Bertrand Maillet
1.1 Introduction 1
1.2 Expected utility and higher-order moments 3
1.3 Expected utility as an exact function of the first four moments 10
1.4 Expected utility as an approximating function of the first four moments 16
1.5 Conclusion 22
Appendix A 23
Appendix B 24
Appendix C 25
Appendix D 27
Appendix E 28
Appendix F 30
Acknowledgements 31
References 32
2 On Certain Geometric Aspects of Portfolio Optimisation with Higher
Moments 37
Gustavo M. de Athayde and Renato G. Flôres Jr
2.1 Introduction 37
2.2 Minimal variances and kurtoses subject to the first two odd moments 38
2.2.1 Homothetic properties of the minimum variance set 39
2.2.2 The minimum kurtosis case 41

2.3 Generalising for higher even moments 44
2.4 Further properties and extensions 46
2.5 Concluding remarks 48
Appendix: The matrix notation for the higher-moments arrays 48
Acknowledgements 50
References 50
3 Hedge Fund Portfolio Selection with Higher-order Moments: A
Nonparametric Mean–Variance–Skewness–Kurtosis Efficient Frontier 51
Emmanuel Jurczenko, Bertrand Maillet and Paul Merlin
3.1 Introduction 51
3.2 Portfolio selection with higher-order moments 53
3.3 The shortage function and the mean–variance–skewness–kurtosis
efficient frontier 55
3.4 Data and empirical results 58
3.5 Conclusion 63
Appendix 64
Acknowledgements 65
References 65
4 Higher-order Moments and Beyond 67
Luisa Tibiletti
4.1 Introduction 67
4.2 Higher-order moments and simple algebra 68
4.3 Higher moments: Noncoherent risk measures 71
4.4 One-sided higher moments 72
4.4.1 Portfolio left-sided moment bounds 73
4.4.2 Properties of the upper bound UpS− 74
4.5 Preservation of marginal ordering under portfolios 75
4.5.1 Drawbacks in using higher moments 75
4.5.2 Advantages in using left-sided higher moments 75
4.6 Conclusion 76
Appendix 77
References 77
5 Gram–Charlier Expansions and Portfolio Selection in Non-Gaussian
Universes 79
Fran¸cois Desmoulins-Lebeault
5.1 Introduction 79
5.2 Attempts to extend the CAPM 80
5.2.1 Extensions based on preferences 80
5.2.2 Extensions based on return distributions 83
5.3 An example of portfolio optimisation 85
5.3.1 Portfolio description 86
5.3.2 The various “optimal” portfolios 86

5.4 Extension to any form of distribution 89
5.4.1 Obstacles to distribution-based works 89
5.4.2 Generalised Gram–Charlier expansions 90
5.4.3 Convergence of the fourth-order Gram–Charlier expansion 95
5.5 The Distribution of Portfolio Returns 98
5.5.1 Feasible approaches 98
5.5.2 The moments of the portfolio returns’ distribution 98
5.5.3 Possible portfolio selection methods 100
5.6 Conclusion 105
Appendix A: Additional statistics for the example portfolio 105
A.1 Moments and co-moments 105
A.2 Statistical tests of normality 107
Appendix B: Proofs 108
B.1 Positivity conditions theorem 108
B.2 Approximation of the optimal portfolio density 109
Acknowledgements 110
References 110
6 The Four-moment Capital Asset Pricing Model: Between Asset Pricing and
Asset Allocation 113
Emmanuel Jurczenko and Bertrand Maillet
6.1 Introduction 113
6.2 The four-moment capital asset pricing model 116
6.2.1 Notations and hypotheses 116
6.2.2 Aggregation of the individual asset demands and a two-fund
monetary separation theorem 120
6.2.3 The four-moment CAPM fundamental relation and the security
market hyperplane 125
6.3 An N risky asset four-moment CAPM extension 130
6.3.1 General properties of the mean–variance–skewness–kurtosis
efficient set 131
6.3.2 A zero-beta zero-gamma zero-delta four-moment CAPM 134
6.4 The four-moment CAPM, the cubic market model and the arbitrage asset
pricing model 137
6.4.1 The cubic market model and the four-moment CAPM 137
6.4.2 The arbitrage pricing model and the four-moment CAPM 139
6.5 Conclusion 142
Appendix A 143
Appendix B 145
Appendix C 146
Appendix D 147
Appendix E 150
Appendix F 151
Appendix G 152
Appendix H 154
Appendix I 155
Appendix J 156

Acknowledgements 157
References 157
7 Multi-moment Method for Portfolio Management: Generalised Capital
Asset Pricing Model in Homogeneous and Heterogeneous Markets 165
Yannick Malevergne and Didier Sornette
7.1 Introduction 165
7.2 Measuring large risks of a portfolio 167
7.2.1 Why do higher moments allow us to assess larger risks? 168
7.2.2 Quantifying the fluctuations of an asset 168
7.2.3 Examples 170
7.3 The generalised efficient frontier and some of its properties 172
7.3.1 Efficient frontier without a risk-free asset 173
7.3.2 Efficient frontier with a risk-free asset 175
7.3.3 Two-fund separation theorem 176
7.3.4 Influence of the risk-free interest rate 176
7.4 Classification of the assets and of portfolios 178
7.4.1 The risk-adjustment approach 179
7.4.2 Marginal risk of an asset within a portfolio 181
7.5 A new equilibrium model for asset prices 181
7.5.1 Equilibrium in a homogeneous market 182
7.5.2 Equilibrium in a heterogeneous market 183
7.6 Conclusion 184
Appendix A: Description of the dataset 184
Appendix B: Generalised efficient frontier and two-fund separation theorem 185
B.1 Case of independent assets when the risk is measured by the
cumulants 185
B.2 General case 187
Appendix C: Composition of the market portfolio 188
C.1 Homogeneous case 188
C.2 Heterogeneous case 189
Appendix D: Generalised Capital Asset Pricing Model 190
Acknowledgements 191
References 191
8 Modelling the Dynamics of Conditional Dependency Between Financial
Series 195
Eric Jondeau and Michael Rockinger
8.1 Introduction 195
8.2 A model for the marginal distributions 197
8.2.1 Hansen’s skewed student-t distribution 197
8.2.2 The cdf of the skewed student-t distribution 199
8.2.3 A GARCH model with time-varying skewness and kurtosis 199
8.3 Copula distribution functions 200
8.3.1 Generalities 200
8.3.2 Construction of the estimated copula functions 201

8.4 Modelling dependency and estimation of the model 205
8.4.1 Conditional dependency 205
8.4.2 Estimation in a copula framework 206
8.5 Empirical Results 207
8.5.1 The data 207
8.5.2 Estimation of the marginal model 209
8.5.3 Estimation of the multivariate model 211
8.6 Further research topics 215
Acknowledgements 218
References 219
9 A Test of the Homogeneity of Asset pricing Models 223
Giovanni Barone-Adesi, Patrick Gagliardini and Giovanni Urga
9.1 Introduction 223
9.2 The Quadratic Market Model 224
9.3 Empirical Results 225
9.3.1 Data description 225
9.3.2 Results 226
9.4 Conclusion 229
Acknowledgements 229
References 229
Index 231