Part Ⅰ The statistics of asset allocation 3
1 Univariate statistics 3
1.1 Building blocks 3
1.2 Summary statistics 9
1.2.1 Location 9
1.2.2 Dispersion 11
1.2.3 Higher-order statistics 14
1.2.4 Graphical representations 15
1.3 Taxonomy of distributions 16
1.3.1 Uniform distribution 16
1.3.2 Normal distribution 18
1.3.3 Cauchy distribution 20
1.3.4 Student t distribution 22
1.3.5 Lognormal distribution 24
1.3.6 Gamma distribution 26
1.3.7 Empirical distribution 28
2 Multivariate statistics 33
2.1 Building blocks 34
2.2 Factorization of a distribution 38
2.2.1 Marginal distribution 38
2.2.2 Copulas 40
2.3 Dependence 45
2.4 Shape summary statistics 48
2.4.1 Location 48
2.4.2 Dispersion 50
2.4.3 Location-dispersion ellipsoid 54
2.4.4 Higher-order statistics 57
2.5 Dependence summary statistics 59
2.5.1 Measures of dependence 59
2.5.2 Measures of concordance 64
2.5.3 Correlation 67
2.6 Taxonomy of distributions 70
2.6.1 Uniform distribution 70
2.6.2 Normal distribution 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 Infinitely divisible distributions 98
3 Modeling the market 101
3.1 The quest for invariance 103
3.1.1 Equities,commodities,exchange rates 105
3.1.2 Fixed-income market 109
3.1.3 Derivatives 114
3.2 Projection of the invariants to the investment horizon 122
3.3 From invariants to market prices 126
3.3.1 Raw securities 126
3.3.2 Derivatives 129
3.4 Dimension reduction 131
3.4.1 Explicit factors 133
3.4.2 Hidden factors 138
3.4.3 Explicit vs. hidden factors 143
3.4.4 Notable examples 145
3.4.5 A nseful routine 147
3.5 Case study:modeling the swap market 150
3.5.1 The market invariants 150
3.5.2 Dimension reduction 151
3.5.3 The invariants at the investment horizon 160
3.5.4 From invariants to prices 162
Part Ⅱ Classical asset allocation 169
4 Estimating the distribution of the market invariants 169
4.1 Estimators 171
4.1.1 Definition 172
4.1.2 Evaluation 173
4.2 Nonparametric estimators 178
4.2.1 Location,dispersion and hidden factors 181
4.2.2 Explicit factors 184
4.2.3 Kernel estimators 185
4.3 Maximum likelihood estimators 186
4.3.1 Location,dispersion and hidden factors 190
4.3.2 Explicit factors 192
4.3.3 The normal case 193
4.4 Shrinkage estimators 200
4.4.1 Location 201
4.4.2 Dispersion and hidden factors 204
4.4.3 Explicit factors 209
4.5 Robustness 209
4.5.1 Measures of robustness 211
4.5.2 Robustness of previously introduced estimators 216
4.5.3 Robust estimators 221
4.6 Practical tips 223
4.6.1 Detection of outliers 223
4.6.2 Missing data 229
4.6.3 Weighted estimates 232
4.6.4 Overlapping data 234
4.6.5 Zero-mean invariants 234
4.6.6 Model-implied estimation 235
5 Evaluating allocations 237
5.1 Investor's objectives 239
5.2 Stochastic dominance 243
5.3 Satisfaction 249
5.4 Certainty-equivalent(expected utility) 260
5.4.1 Properties 262
5.4.2 Building utility functions 270
5.4.3 Explicit dependence on allocation 274
5.4.4 Sensitivity analysis 276
5.5 Quantile(value at risk) 277
5.5.1 Properties 278
5.5.2 Explicit dependence on allocation 282
5.5.3 Sensitivity analysis 285
5.6 Coherent indices(expected shortfall) 287
5.6.1 Properties 288
5.6.2 Building coherent indices 292
5.6.3 Explicit dependence on allocation 296
5.6.4 Sensitivity analysis 298
6 Optimizing allocations 301
6.1 The general approach 302
6.1.1 Collecting information on the investor 303
6.1.2 Collecting information on the market 305
6.1.3 Computing the optimal allocation 306
6.2 Constrained optimization 311
6.2.1 Positive orthants:linear programming 313
6.2.2 Ice-cream cones:second-order cone programming 313
6.2.3 Semidefinite cones:semidefinite programming 315
6.3 The mean-variance approach 315
6.3.1 The geometry of allocation optimization 316
6.3.2 Dimension reduction:the mean-variance framework 319
6.3.3 Setting up the mean-variance optimization 320
6.3.4 Mean-variance in terms of returns 323
6.4 Analytical solutions of the mean-variance problem 326
6.4.1 Efficient frontier with affine constraints 327
6.4.2 Efficient frontier with linear constraints 330
6.4.3 Effects of correlations and other parameters 332
6.4.4 Effects of the market dimension 335
6.5 Pitfalls of the mean-variance framework 336
6.5.1 MV as an approximation 336
6.5.2 MV as an index of satisfaction 338
6.5.3 Quadratic programming and dual formulation 340
6.5.4 MV on returns:estimation versus optimization 342
6.5.5 MV on returns:investment at different horizons 343
6.6 Total-return versus benchmark allocation 347
6.7 Case study:allocation in stocks 354
6.7.1 Collecting information on the investor 355
6.7.2 Collecting information on the market 355
6.7.3 Computing the optimal allocation 357
Part Ⅲ Accounting for estimation risk 363
7 Estimating the distribution of the market invariants 363
7.1 Bayesian estimation 364
7.1.1 Bayesian posterior distribution 364
7.1.2 Summarizing the posterior distribution 366
7.1.3 Computing the posterior distribution 369
7.2 Location and dispersion parameters 370
7.2.1 Computing the posterior distribution 370
7.2.2 Summarizing the posterior distribution 373
7.3 Explicit factors 377
7.3.1 Computing the posterior distribution 377
7.3.2 Summarizing the posterior distribution 380
7.4 Determining the prior 383
7.4.1 Allocation-implied parameters 385
7.4.2 Likelihood maximization 387
8 Evaluating allocations 389
8.1 Allocations as decisions 390
8.1.1 Opportunity cost of a sub-optimal allocation 390
8.1.2 Opportunity cost as function of the market parameters 394
8.1.3 Opportunity cost as loss of an estimator 397
8.1.4 Evaluation of a generic allocation decision 401
8.2 Prior allocation 403
8.2.1 Definition 403
8.2.2 Evaluation 404
8.2.3 Discussion 406
8.3 Sample-based allocation 407
8.3.1 Definition 407
8.3.2 Evaluation 408
8.3.3 Discussion 412
9 Optimizing allocations 417
9.1 Bayesian allocation 418
9.1.1 Utility maximization 419
9.1.2 Classical-equivalent maximization 421
9.1.3 Evaluation 422
9.1.4 Discussion 425
9.2 Black-Litterman allocation 426
9.2.1 General definition 426
9.2.2 Practicable definition:linear expertise on normal markets 429
9.2.3 Evaluation 433
9.2.4 Discussion 436
9.3 Resampled allocation 437
9.3.1 Practicable definition:the mean-variance setting 438
9.3.2 General definition 440
9.3.3 Evaluation 443
9.3.4 Discussion 445
9.4 Robust allocation 445
9.4.1 General definition 445
9.4.2 Practicable definition:the mean-variance setting 450
9.4.3 Discussion 453
9.5 Robust Bayesian allocation 454
9.5.1 General definition 455
9.5.2 Practicable definition:the mean-variance setting 457
9.5.3 Discussion 459
Part Ⅳ Appendices 465
A Linear algebra 465
A.1 Vector space 465
A.2 Basis 468
A.3 Linear transformations 469
A.3.1 Matrix representation 470
A.3.2 Rotations 471
A.4 Invariants 472
A.4.1 Determinant 472
A.4.2 Trace 474
A.4.3 Eigenvalues 474
A.5 Spectral theorem 475
A.5.1 Analytical result 475
A.5.2 Geometrical interpretation 478
A.6 Matrix operations 480
A.6.1 Useful identities 480
A.6.2 Tensors and Kronecker product 482
A.6.3 The"vec"and"vech"operators 483
A.6.4 Matrix calculus 485
B Functional Analysis 487
B.1 Vector space 487
B.2 Basis 490
B.3 Linear operators 493
B.3.1 Kernel representations 494
B.3.2 Unitary operators 494
B.4 Regularization 496
B.5 Expectation operator 499
B.6 Some special functions 501
References 505
List of figures 515
Notation 519
Index 525