## Risk and Asset AllocationThis encyclopedic, detailed exposition spans all the steps of one-period allocation from the foundations to the most advanced developments. Multivariate estimation methods are analyzed in depth, including non-parametric, maximum-likelihood under non-normal hypotheses, shrinkage, robust, and very general Bayesian techniques. Evaluation methods such as stochastic dominance, expected utility, value at risk and coherent measures are thoroughly discussed in a unified setting and applied in a variety of contexts, including prospect theory, total return and benchmark allocation. Portfolio optimization is presented with emphasis on estimation risk, which is tackled by means of Bayesian, resampling and robust optimization techniques. All the statistical and mathematical tools, such as copulas, location-dispersion ellipsoids, matrix-variate distributions, cone programming, are introduced from the basics. Comprehension is supported by a large number of figures and examples, as well as real trading and asset management case studies. At symmys.com the reader will find freely downloadable complementary materials: the Exercise Book; a set of thoroughly documented MATLABŪ applications; and the Technical Appendices with all the proofs. More materials and complete reviews can also be found at symmys.com. |

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Attilio Meucci (One day overlap at RVI.)

### Contents

562 Building coherent indices | 301 |

LXXVII | 302 |

LXXVIII | 303 |

563 Explicit dependence on allocation | 305 |

LXXIX | 306 |

61 The general approach | 311 |

LXXX | 313 |

613 Computing the optimal allocation | 315 |

24 | |

26 | |

28 | |

32 | |

34 | |

XIV | 38 |

Multivariate statistics | 40 |

22 Factorization of a distribution | 45 |

XVI | 48 |

XVIII | 50 |

XIX | 54 |

242 Dispersion | 57 |

XX | 59 |

244 Higherorder statistics | 64 |

XXII | 67 |

XXIII | 70 |

XXV | 72 |

26 Taxonomy of distributions | 77 |

XXVI | 81 |

XXVII | 82 |

263 Student t distribution | 84 |

XXVIII | 87 |

265 Logdistributions | 89 |

266 Wishart distribution | 91 |

268 Order statistics | 96 |

27 Special classes of distributions | 98 |

XXIX | 101 |

XXX | 103 |

XXXI | 105 |

Modeling the market | 109 |

XXXII | 114 |

313 Derivatives | 122 |

XXXIII | 126 |

XXXV | 129 |

XXXVI | 131 |

XXXVII | 133 |

XXXVIII | 138 |

XXXIX | 143 |

XL | 145 |

XLI | 147 |

XLII | 150 |

343 Explicit vs hidden factors | 151 |

XLIV | 160 |

XLV | 162 |

353 The invariants at the investment horizon | 168 |

354 From invariants to prices | 171 |

XLVI | 172 |

XLVII | 173 |

Estimating the distribution of the market invariants | 178 |

411 Definition | 181 |

XLVIII | 184 |

XLIX | 185 |

L | 186 |

421 Location dispersion and hidden factors | 190 |

LI | 192 |

422 Explicit factors | 193 |

LII | 200 |

432 Explicit factors | 201 |

LIII | 204 |

44 Shrinkage estimators | 209 |

LIV | 211 |

LV | 216 |

LVI | 221 |

LVII | 223 |

LIX | 229 |

46 Practical tips | 232 |

LX | 234 |

LXII | 235 |

LXIII | 237 |

LXIV | 239 |

464 Overlapping data | 243 |

LXV | 249 |

LXVI | 260 |

LXVII | 262 |

LXVIII | 270 |

LXIX | 274 |

LXX | 276 |

LXXI | 277 |

LXXII | 278 |

LXXIII | 282 |

544 Sensitivity analysis | 285 |

551 Properties | 287 |

LXXIV | 288 |

LXXV | 292 |

56 Coherent indices expected shortfall | 296 |

LXXVI | 298 |

LXXXI | 316 |

LXXXII | 319 |

62 Constrained optimization | 320 |

LXXXIII | 323 |

LXXXIV | 326 |

LXXXV | 327 |

LXXXVI | 330 |

634 Meanvariance in terms of returns | 332 |

64 Analytical solutions of the meanvariance problem | 335 |

641 Efficient frontier with affine constraints | 336 |

LXXXVII | 338 |

LXXXVIII | 340 |

LXXXIX | 342 |

XC | 343 |

652 MV as an index of satisfaction | 347 |

XCI | 354 |

XCII | 355 |

XCIV | 357 |

XCV | 361 |

XCVI | 364 |

671 Collecting information on the investor | 366 |

XCVIII | 369 |

XCIX | 370 |

CI | 373 |

712 Summarizing the posterior distribution | 377 |

713 Computing the posterior distribution | 380 |

CII | 383 |

CIII | 385 |

CIV | 387 |

CV | 389 |

CVI | 390 |

74 Determining the prior | 394 |

CVIII | 397 |

81 Allocations as decisions | 401 |

CIX | 403 |

CXI | 404 |

CXII | 406 |

CXIII | 407 |

813 Opportunity cost as loss of an estimator | 408 |

814 Evaluation of a generic allocation decision | 412 |

823 Discussion | 417 |

83 Samplebased allocation | 418 |

832 Evaluation | 419 |

CXV | 421 |

CXVI | 422 |

CXVII | 425 |

CXVIII | 426 |

91 Bayesian allocation | 429 |

913 Evaluation | 433 |

914 Discussion | 436 |

92 BlackLitterman allocation | 437 |

CXX | 438 |

linear expertise on normal markets | 440 |

CXXI | 443 |

CXXII | 445 |

CXXV | 450 |

CXXVI | 453 |

933 Evaluation | 454 |

CXXVII | 455 |

CXXVIII | 457 |

CXXIX | 459 |

CXXX | 463 |

951 General definition | 468 |

CXXXII | 469 |

the meanvariance setting | 470 |

CXXXIII | 471 |

953 Discussion | 472 |

CXXXIV | 474 |

CXXXVI | 475 |

Linear algebra | 478 |

CXXXVIII | 480 |

A3 Linear transformations | 482 |

A31 Matrix representation | 483 |

A4 Invariants | 485 |

A42 Trace | 487 |

CXXXIX | 490 |

A6 Matrix operations | 493 |

CXL | 494 |

A63 The vec and vech operators | 496 |

CXLII | 499 |

CXLIII | 501 |

505 | |

CXLV | 514 |

CXLVI | 519 |

525 | |