Extreme Events: Robust Portfolio Construction in the Presence of Fat Tails (The Wiley Finance Series)

Extreme Events......Page 3
Contents......Page 7
Preface......Page 17
Acknowledgements......Page 19
Abbreviations......Page 21
Notation......Page 23
1.1 Extreme events......Page 25
1.3 Coping with really extreme events......Page 26
1.4 Risk budgeting......Page 27
1.6 Book structure......Page 28
2.2 A fat tail relative to what?......Page 31
2.3.2 Visualising fat tails......Page 35
2.3.4 Behaviour of equity indices......Page 41
2.3.5 Currencies and other asset types......Page 46
2.4.1 Introduction......Page 47
2.4.2 Skew and kurtosis......Page 49
2.4.3 The (fourth-moment) Cornish-Fisher approach......Page 50
2.4.4 Weaknesses of the Cornish-Fisher approach......Page 51
2.4.5 Improving on the Cornish-Fisher approach......Page 52
2.4.6 Statistical tests for non-Normality......Page 53
2.4.7 Higher order moments and the Omega function......Page 55
2.5.1 Introduction......Page 56
2.5.2 The Central Limit Theorem......Page 57
2.5.3 Ways in which the Central Limit Theorem can break down......Page 58
2.6 Lack of diversification......Page 59
2.7.1 Introduction......Page 60
2.7.2 Distributional mixtures......Page 61
2.7.3 Time-varying volatility......Page 62
2.7.4 Regime shifts......Page 64
2.8.1 Introduction......Page 65
2.8.2 Defining characteristics......Page 66
2.8.3 The Generalised Central Limit Theorem......Page 68
2.9.1 Introduction......Page 69
2.9.2 Extreme value distributions......Page 70
2.9.5 Issues with extreme value theory......Page 71
2.10 Parsimony......Page 74
2.11 Combining different possible source mechanisms......Page 76
2.12.3 Crowded trades......Page 77
2.12.5 ‘Rational’ behaviour versus ‘bounded rational’ behaviour......Page 78
2.13.1 Introduction......Page 79
2.13.2 Smoothing of return series......Page 80
2.13.3 Time clocks and non-constant time period lengths......Page 81
2.13.5 Economic sensitivities that change through time......Page 82
3.2 Visualisation of fat tails in multiple return series......Page 85
3.3.1 Introduction......Page 88
3.3.2 Fractile–fractile (i.e., quantile–quantile box) plots......Page 90
3.3.3 Time-varying volatility......Page 96
3.4.1 Introduction......Page 99
3.4.3 The t-copula......Page 100
3.5.1 Introduction......Page 102
3.5.2 Disadvantages of empirically fitting the copula......Page 103
3.5.3 Multi-dimensional quantile–quantile plots......Page 104
3.7 The practitioner perspective......Page 107
3.8.2 Series of different lengths......Page 109
3.8.3 Non-coincidently timed series......Page 110
3.8.4 Cluster analysis......Page 111
3.8.5 Relative entropy and nonlinear cluster analysis......Page 112
4.1 Introduction......Page 115
4.2.1 Introduction......Page 116
4.2.2 Fundamental models......Page 118
4.2.3 Econometric models......Page 119
4.2.5 Similarities and differences between risk models......Page 120
4.3.1 Introduction......Page 121
4.3.2 Principal components analysis......Page 123
4.3.3 The theory behind principal components analysis......Page 124
4.3.4 Weighting schemas......Page 126
4.3.6 Random matrix theory......Page 127
4.4.1 Introduction......Page 130
4.4.2 Practical algorithms......Page 131
4.4.3 Non-Normality and projection pursuit......Page 133
4.4.5 Extracting all the un-mixing weights at the same time......Page 134
4.4.6 Complexity pursuit......Page 135
4.5.1 Introduction......Page 136
4.5.2 Including both variance and kurtosis in the importance criterion......Page 137
4.5.3 Eliminating signals from the remaining dataset......Page 138
4.5.4 Normalising signal strength......Page 139
4.6.2 Quantifying the possible impact of selection effects......Page 140
4.6.3 Decomposition of fat-tailed behaviour......Page 143
4.7.2 Linear regression......Page 144
4.7.3 Difference equations......Page 145
4.7.4 The potential range of behaviours of linear difference equations......Page 146
4.7.5 Multivariate linear regression......Page 148
4.7.6 ‘Chaotic’ market dynamics......Page 149
4.7.7 Modelling market dynamics using nonlinear methods......Page 151
4.7.8 Locally linear time series analysis......Page 152
4.8.2 Gaussian mixture models and the expectation-maximisation algorithm......Page 153
4.8.3 k-means clustering......Page 155
4.8.5 Regime shifts......Page 156
4.9 The practitioner perspective......Page 157
4.10.1 Introduction......Page 158
4.10.2 Local extrema......Page 159
4.10.3 Global extrema......Page 160
4.10.4 Simulated annealing and genetic portfolio optimisation......Page 161
4.10.6 Run time constraints......Page 162
5.2.1 Introduction......Page 165
5.2.2 Viewing any process through the window of portfolio optimisation......Page 166
5.2.3 Quantitative versus qualitative insights......Page 167
5.2.4 The characteristics of pricing/return anomalies......Page 168
5.3.1 Introduction......Page 169
5.3.2 Mean-variance optimisation......Page 170
5.3.3 Formal mathematical notation......Page 172
5.3.4 The Capital Asset Pricing Model (CAPM)......Page 173
5.4.2 Monotonic changes to risk or return measures......Page 176
5.4.4 Alpha-beta separation......Page 177
5.5 Manager selection......Page 178
5.6.1 Introduction......Page 180
5.6.3 Stochastic optimal control and stochastic programming techniques......Page 181
5.7.2 Different types of transaction costs......Page 182
5.7.3 Impact of transaction costs on portfolio construction......Page 183
5.7.4 Taxes......Page 184
5.8.1 Introduction......Page 185
5.8.2 Information ratios......Page 186
5.9.1 Introduction......Page 187
5.9.2 Different backtest statistics for risk-reward trade-offs......Page 188
5.9.3 In-sample versus out-of-sample backtesting......Page 190
5.9.4 Backtesting risk models......Page 191
5.9.6 Transaction costs......Page 193
5.10.1 Implied alphas......Page 194
5.11 Portfolio optimisation with options......Page 195
5.12 The practitioner perspective......Page 197
5.13.2 Optimisation algorithms......Page 198
5.13.3 Sensitivity to the universe from which ideas are drawn......Page 199
6.1 Introduction......Page 201
6.2.1 Introduction......Page 202
6.2.2 Estimation error......Page 204
6.2.3 Mitigating estimation error in the covariance matrix......Page 205
6.2.4 Exact error estimates......Page 206
6.2.5 Monte Carlo error estimation......Page 209
6.2.6 Sensitivity to the structure of the smaller principal components......Page 210
6.3.1 Introduction......Page 211
6.3.2 Bayesian statistics......Page 212
6.3.3 Applying Bayesian statistics in practice......Page 213
6.4.1 Introduction......Page 214
6.4.2 The Black-Litterman approach......Page 215
6.4.3 Applying the Black-Litterman approach in practice......Page 216
6.4.5 Impact of non-Normality when using more complex risk measures......Page 217
6.5.2 ‘Shrinking’ the sample means......Page 218
6.5.3 ‘Shrinking’ the covariance matrix......Page 220
6.6.1 Introduction......Page 221
6.6.2 The mathematics......Page 222
6.7 The ‘universality’ of Bayesian approaches......Page 224
6.8 Market consistent portfolio construction......Page 225
6.9.1 Introduction......Page 226
6.9.3 Monte Carlo resampling......Page 227
6.9.4 Monte Carlo resampled optimisation without portfolio constraints......Page 230
6.9.5 Monte Carlo resampled optimisation with portfolio constraints......Page 231
6.10 The practitioner perspective......Page 232
6.11.1 Introduction......Page 234
6.11.2 Monte Carlo simulation – changing variables and importance sampling......Page 235
6.11.3 Monte Carlo simulation – stratified sampling......Page 236
6.11.4 Monte Carlo simulation – quasi (i.e., sub-) random sequences......Page 237
6.11.6 Monte Carlo simulation of fat tails......Page 238
7.2.1 Introduction......Page 241
7.2.2 An example of regime switching model......Page 242
7.2.3 The operation of regime switching model......Page 243
7.2.4 Complications......Page 244
7.3.1 Introduction......Page 245
7.3.2 Expected utility theory......Page 246
7.3.3 Constant relative risk aversion (CRRA)......Page 247
7.3.4 The need for monetary value to be well defined......Page 248
7.4.2 Specification of problem......Page 250
7.4.4 Introducing regime switching......Page 251
7.4.5 Dependency on ability to identify prevailing regime......Page 252
7.4.7 Incorporating constraints on efficient portfolios......Page 253
7.4.9 General form of RS optimal portfolio behaviour......Page 254
7.5 Links with derivative pricing theory......Page 255
7.6 Transaction costs......Page 256
7.7.1 Introduction......Page 258
7.7.3 Threshold autoregressive models......Page 259
7.8 Incorporating more intrinsically fat-tailed behaviour......Page 260
7.9.2 Straightforward ways of adjusting for time-varying volatility......Page 261
7.9.3 Lower partial moments......Page 262
7.10 The practitioner perspective......Page 264
7.11.2 Solving problems with time-varying parameters......Page 265
7.11.4 Catering for transaction costs and liquidity risk......Page 266
7.11.5 Need for coherent risk measures......Page 267
8.1 Introduction......Page 269
8.2 Limitations of current stress testing methodologies......Page 270
8.3.1 Introduction......Page 271
8.3.3 Use for standardised regulatory capital computations......Page 272
8.3.4 A greater focus on what might lead to large losses......Page 274
8.3.5 Further comments......Page 275
8.4 Reverse stress testing......Page 276
8.5.3 Idiosyncratic risk......Page 277
8.5.4 Key exposures......Page 278
8.6 Designing stress tests statistically......Page 279
8.7 The practitioner perspective......Page 281
8.8.2 Being in a position to ‘stress’ the portfolio......Page 282
8.8.3 Calculating the impact of a particular stress on each position......Page 283
8.8.4 Calculating the impact of a particular stress on the whole portfolio......Page 284
8.8.5 Engaging with boards, regulators and other relevant third parties......Page 285
9.2 Thinking outside the box......Page 287
9.3 Portfolio purpose......Page 289
9.4.2 Drawing inferences about Knightian uncertainty......Page 290
9.4.3 Reacting to really extreme events......Page 292
9.4.4 Non-discoverable processes......Page 293
9.5.1 Introduction......Page 294
9.5.3 Knightian uncertainty again......Page 295
9.6.1 Introduction......Page 296
9.6.2 Governance models......Page 297
9.6.3 Enterprise risk management......Page 298
9.6.4 Formulating a risk appetite......Page 299
9.6.5 Management structures......Page 300
9.6.6 Operational risk......Page 301
9.7 The practitioner perspective......Page 302
9.8.3 Implementing enterprise risk management......Page 303
10.2.1 Chapter 2: Fat tails – in single (i.e., univariate) return series......Page 305
10.2.2 Chapter 3: Fat tails – in joint (i.e., multivariate) return series......Page 306
10.2.4 Chapter 5: Traditional portfolio construction techniques......Page 307
10.2.5 Chapter 6: Robust mean-variance portfolio construction......Page 308
10.2.8 Chapter 9: Really extreme events......Page 309
A.2 Fat tails – In single (i.e., univariate) return series......Page 311
A.3 Fat tails – In joint (i.e., multivariate) return series......Page 312
A.5 Traditional portfolio construction techniques......Page 313
A.6 Robust mean-variance portfolio construction......Page 314
A.7 Regime switching and time-varying risk and return parameters......Page 315
A.8 Stress testing......Page 316
A.9 Really extreme events......Page 317
References......Page 319
Index......Page 325