Part I The statistics of asset allocation
1Univariatestatistics ....................................... 3
1.1 Buildingblocks.......................................... 3
1.2 Summarystatistics ...................................... 9
1.2.1 Location ......................................... 9
1.2.2 Dispersion........................................ 11
1.2.3 Higher-order statistics ............................. 14
1.2.4 Graphicalrepresentations .......................... 15
1.3 Taxonomy of distributions . . . ............................. 16
1.3.1 Uniform distribution . . ............................. 16
1.3.2 Normaldistribution ............................... 18
1.3.3 Cauchy distribution . . ............................. 20
1.3.4 Student t distribution . ............................. 22
1.3.5 Lognormal distribution ............................. 24
1.3.6 Gammadistribution ............................... 26
1.3.7 Empirical distribution. ............................. 28
1.T Technicalappendix ......................................www
2 Multivariate statistics ..................................... 33
2.1 Buildingblocks.......................................... 34
2.2 Factorization of a distribution ............................. 38
2.2.1 Marginal distribution . ............................. 38
2.2.2 Copulas.......................................... 40
.................................www 1.E Exe rcise s .. . . ... . .. .
2.3 Dependence............................................. 45
2.4 Shapesummarystatistics................................. 48
2.4.1 Location ......................................... 48
2.4.2 Dispersion........................................ 50
2.4.3 Location-dispersionellipsoid ........................ 54
2.4.4 Higher-order statistics ............................. 57
2.5 Dependencesummarystatistics ........................... 59
2.5.1 Measuresofdependence............................ 59
2.5.2 Measuresofconcordance ........................... 64
2.5.3 Correlation....................................... 67
2.6 Taxonomy of distributions . . . ............................. 70
2.6.1 Uniform distribution . . ............................. 70
2.6.2 Normaldistribution ............................... 72
2.6.3 Student t distribution . ............................. 77
2.6.4 Cauchy distribution . . ............................. 81
2.6.5 Log-distributions . ................................. 82
2.6.6 Wishart distribution . . ............................. 84
2.6.7 Empirical distribution. ............................. 87
2.6.8 Order statistics ................................... 89
2.7 Special classes of distributions . . . ......................... 91
2.7.1 Elliptical distributions ............................. 91
2.7.2 Stable distributions . . . ............................. 96
2.7.3 In…nitely divisible distributions ..................... 98
2.T Technicalappendix ......................................www
3 Modeling the market ...................................... 101
3.1 Thequest for invariance..................................103
3.1.1 Equities,commodities, exchangerates................105
3.1.2 Fixed-incomemarket ..............................109
3.1.3 Derivatives .......................................114
3.2 Projectionof theinvariants totheinvestmenthorizon........122
3.3 Frominvariants tomarketprices ..........................126
3.3.1 Rawsecurities ....................................126
3.3.2 Derivatives .......................................129
3.4 Dimensionreduction.....................................131
3.4.1 Explicit factors....................................133
3.4.2 Hiddenfactors ....................................138
3.4.3 Explicitvs.hiddenfactors ..........................143
3.4.4 Notableexamples .................................145
3.4.5 Ausefulroutine...................................147
3.5 Casestudy:modelingtheswapmarket .....................150
3.5.1 Themarket invariants .............................150
3.5.2 Dimensionreduction...............................151
3.5.3 Theinvariantsat the investmenthorizon .............160
3.5.4 Frominvariants toprices ...........................162
.................................www Exercises .. . . ... . .. . .E 2
3.T Technicalappendix ......................................www
Part II Classical asset allocation
4 Estimating the distribution of the market invariants ....... 169
4.1 Estimators .............................................171
4.1.1 De…nition ........................................172
4.1.2 Evaluation .......................................173
4.2 Nonparametricestimators ................................178
4.2.1 Location,dispersionandhiddenfactors ..............181
4.2.2 Explicit factors....................................184
4.2.3 Kernelestimators .................................185
4.3 Maximumlikelihoodestimators ...........................186
4.3.1 Location,dispersionandhiddenfactors ..............190
4.3.2 Explicit factors....................................192
4.3.3 Thenormal case ..................................193
4.4 Shrinkageestimators.....................................200
4.4.1 Location .........................................201
4.4.2 Dispersionandhiddenfactors.......................204
4.4.3 Explicit factors....................................209
4.5 Robustness .............................................209
4.5.1 Measuresofrobustness.............................211
4.5.2 Robustnessofpreviouslyintroducedestimators........216
4.5.3 Robustestimators .................................221
4.6 Practical tips ...........................................223
4.6.1 Detectionofoutliers ...............................223
4.6.2 Missingdata......................................229
4.6.3 Weightedestimates ................................232
4.6.4 Overlappingdata..................................234
4.6.5 Zero-meaninvariants ..............................234
4.6.6 Model-impliedestimation...........................235
4.T Technicalappendix ......................................www
5 Evaluating allocations ..................................... 237
5.1 Investor’sobjectives .....................................239
5.2 Stochasticdominance ....................................243
5.3 Satisfaction.............................................249
5.4 Certainty-equivalent (expectedutility) .....................260
5.4.1 Properties........................................262
5.4.2 Buildingutilityfunctions...........................270
5.4.3 Explicitdependenceonallocation ...................274
5.4.4 Sensitivityanalysis ................................276
5.5 Quantile(valueatrisk) ..................................277
5.5.1 Properties........................................278
.................................www Exercises .. . . ... . .. . .E 3
.................................www .. . . ... . .. . Exercises 4.E
5.5.2 Explicitdependenceonallocation ...................282
5.5.3 Sensitivityanalysis ................................285
5.6 Coherent indices (expectedshortfall).......................287
5.6.1 Properties........................................288
5.6.2 Buildingcoherent indices...........................292
5.6.3 Explicitdependenceonallocation ...................296
5.6.4 Sensitivityanalysis ................................298
5.T Technicalappendix ......................................www