STOCHASTIC

More documents

Recommendations

Info

9.2. Example 2. If (30) u'(x)=(x° + by c with a>0,c>0, then u(x) is strictly decreasingly risk-averse in the region (31) x>[max{0, -b, b(a-l)}y'° . To prove this, note (32) Kx)= --£logu'(x) = aC dx 6 x + bx 1 '' ' RISK AVERSION 133 which is SO and strictly decreasing in the region where the denominator x + bx l ~ a is SO and strictly increasing, which is the region (30). (The condition x£0 is included to insure that x a is defined; for aS 1 it follows from the other conditions.) By Theorem 3, one can obtain a utility function that is strictly decreasingly riskaverse for x>0 by substituting x + d for x above, where d is at least the right-hand side of (31). Multiplying x by a positive factor, as in Theorem 3, is equivalent to multiplying b by a positive factor. Given below are all the strictly decreasingly risk-averse utility functions u(x) on x > 0 which can be obtained by applying Theorem 3 to (30) with the indicated choices of the parameters a and c: (33) (34) (35) (36) (37) a = l,00, c>0, d>0 . (40) u(x)~log(d1+log(x+d2)) with dÔ, d2£0, d^logdjSO . 10. PROPORTIONAL RISK AVERSION So far we have been concerned with risks that remained fixed while assets varied. Let us now view everything as a proportion of assets. Specifically, let Jt*(x,f) be the proportional risk premium corresponding to a proportional risk 2; that is, a PART II. QUALITATIVE ECONOMIC RESULTS
134 JOHN W. PRATT decision maker with assets x and utility function u would be indifferent between receiving a risk xz and receiving the non-random amount E(xz) — xn*(x,z). Then xit*(x,z) equals the risk premium n(x,xz), so (41) n*(x,z) = - n(x,xz) . For a small, actuarially neutral, proportional risk z we have, by (5), (42) 7t*(x,z)=i0 and has negative, strictly increasing risk aversion for c
Page 1 and 2:
STOCHASTIC OPTIMIZATION MODELS IN F
Page 4 and 5:
STOCHASTIC OPTIMIZATION MODELS IN F
Page 6:
Dedicated to the memory of my fathe
Page 9 and 10:
PART II. QUALITATIVE ECONOMIC RESUL
Page 11 and 12:
2. Risk Aversion over Time Implies
Page 14 and 15:
PREFACE AND BRIEF NOTES TO THE 2006
Page 16 and 17:
and shows that current research in
Page 18 and 19:
1990. This area presages the CVaR l
Page 20 and 21:
transaction cost band. Most such mo
Page 22 and 23:
through targets and are less sensit
Page 24 and 25:
ing dates, the papers by Breiman, H
Page 26 and 27:
Carino, D. and W.T. Ziemba (1998).
Page 28 and 29:
Kallberg, J.G. and W.T. Ziemba (198
Page 30 and 31:
Stone, D. and W.T. Ziemba (1993). L
Page 32 and 33:
PREFACE IN 1975 EDITION There is no
Page 34:
particular results are generally st
Page 37:
Part I Mathematical Tools
Page 40 and 41:
wealth, then his utility function m
Page 42 and 43:
cases in Exercise CR-12. The relati
Page 44 and 45:
However, some extensions to infinit
Page 47 and 48:
1. EXPECTED UTILITY THEORY The Anna
Page 49 and 50:
SUBJECTIVE PROBABILITIES ANT EXPECT
Page 51 and 52:
SUBJECTIVE PROBABILITIES AND EXPECT
Page 53 and 54:
Page 55 and 56:
Page 57:
Page 60 and 61:
282 O. L. MANGA3ARIAN" for every X
Page 62 and 63:
284 O. L. MANGASARIAN We have from
Page 64 and 65:
280 O. L. MANQASARIAN For the case
Page 66 and 67:
288 O. L. MANGASAMAN Proof. Conside
Page 68 and 69:
290 O. L. MANGASARIAN Second Berkel
Page 70 and 71:
denotes the /w-dimensional vector o
Page 72 and 73:
Proof (I) We shall prove this part
Page 74 and 75:
(Ill) have not, to the author's kno
Page 76 and 77:
for the case (3); and for the case
Page 79 and 80:
3. DYNAMIC PROGRAMMING Introduction
Page 81 and 82:
INTRODUCTION TO DYNAMIC PROGRAMMING
Page 83 and 84:
Page 85 and 86:
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
COMPUTATIONAL AND REVIEW EXERCISES
Page 95 and 96:
(d) xxx2 on E+ 2 = {x\xt = 0, x2 S
Page 97 and 98:
Show that the Kuhn-Tucker condition
Page 99 and 100:
(a) Show that/, (b, c) = gt (Z>, c)
Page 101:
(c) Show that the monotonicity assu
Page 104 and 105:
An individual's preferences are sai
Page 106 and 107:
12. (Hessians, bordered Hessians, c
Page 108 and 109:
The problem in (b)-(c) hints that t
Page 110 and 111:
(1) Suppose/is twice continuously d
Page 112 and 113: (f) Assume that R(i,a) g 0 and the
Page 114 and 115: (j) Show that an optimal policy Va
Page 117 and 118: INTRODUCTION In the second part of
Page 119 and 120: andom variables when the utility fu
Page 121 and 122: detailed analysis of the qualitativ
Page 123 and 124: separation in a local sense (for al
Page 125 and 126: 1. STOCHASTIC DOMINANCE G. Hanoch a
Page 127 and 128: But now, EFFICIENCY ANALYSIS OF CHO
Page 129 and 130: EFFICIENCY ANALYSIS OF CHOICES INVO
Page 137 and 138: A Unified Approach to Stochastic Do
Page 139 and 140: A UNIFIED APPROACH TO STOCHASTIC DO
Page 149: A UNIFIED APPROACH TO STOCHASTIC DO
Page 152 and 153: RISK AVERSION 123 a given risk the
Page 154 and 155: RISK AVERSION 125 If z is actuarial
Page 156 and 157: RISK AVERSION 127 4. CONCAVITY The
Page 158 and 159: To show that (a) implies (d), note
Page 160 and 161: RISK AVERSION 131 (b') The risk pre
Page 164 and 165: RISK AVERSION 135 12. INCREASING AN
Page 166 and 167: ADDENDUM In retrospect, 1 wish foot
Page 168 and 169: 14 THE REVIEW OF ECONOMICS AND STAT
Page 193 and 194: Separation in Portfolio Analysis R.
Page 195 and 196: SEPARATION IN PORTFOLIO ANALYSIS an
Page 197 and 198: SEPARATION IN PORTFOLIO ANALYSIS II
Page 199 and 200: SEPARATION IN PORTFOLIO ANALYSIS Th
Page 201 and 202: SEPARATION IN PORTFOLIO ANALYSIS Ca
Page 203 and 204: SEPARATION IN PORTFOLIO ANALYSIS It
Page 205 and 206: SEPARATION IN PORTFOLIO ANALYSIS th
Page 207 and 208: COMPUTATIONAL AND REVIEW EXERCISES
Page 209 and 210: 7. Consider an investor having the
Page 211 and 212: Then eliminate p from (2) to obtain
Page 213 and 214:
(a) Show that an indifference curve
Page 215 and 216:
(d) Develop a similar result to (c)
Page 217:
Exercise Source Notes Exercise 1 wa
Page 220 and 221:
(f) Show that k > 0. [Note (Exercis
Page 222 and 223:
(c) Interpret (b). (d) Illustrate s
Page 224 and 225:
9. Let the partial relative risk-av
Page 226 and 227:
Condition (iv) is called the "no-ea
Page 228 and 229:
Then * ( 8xj \ V - s u = \-r-\ -ax,
Page 230 and 231:
exhibiting decreasing absolute and
Page 232 and 233:
(a) Determine necessary and suffici
Page 234 and 235:
Let the random variables Og^Sl and
Page 237:
Part III Static Portfolio Selection
Page 240 and 241:
given parameter. Exercise ME-30 dev
Page 242 and 243:
utility function. However, many con
Page 244 and 245:
the characteristic exponent is 2 (i
Page 246 and 247:
proportions to maximize expected ut
Page 248 and 249:
investment returns are always nonne
Page 250 and 251:
case to a case of one riskless and
Page 252 and 253:
538 REVIEW OF ECONOMIC STUDIES Howe
Page 254 and 255:
540 REVIEW OF ECONOMIC STUDIES wher
Page 256 and 257:
542 REVIEW OF ECONOMIC STUDIES expa
Page 258 and 259:
JAMES A. OHLSON that quadratic util
Page 260 and 261:
JAMES A. OHLSON Assumptions A1-A3 a
Page 262 and 263:
JAMES A. OHLSON (i) snpl/ieSi\U (3)
Page 264 and 265:
JAMES A. OHLSON A number of interes
Page 266 and 267:
JAMES A. OHLSON for all y,teSx'3C{Q
Page 268 and 269:
Since Z"=o^( w »0 = 0 and/(«,«)
Page 270 and 271:
JAMES A. OHLSON analysis here is "o
Page 272 and 273:
76 THE REVIEW OF ECONOMICS AND STAT
Page 274 and 275:
Page 276 and 277:
Page 279 and 280:
Choosing Investment Portfolios When
Page 281 and 282:
CHOOSING INVESTMENT PORTFOLIOS univ
Page 283 and 284:
CHOOSING INVESTMENT PORTFOLIOS The/
Page 285 and 286:
CHOOSING INVESTMENT PORTFOLIOS Tobi
Page 287 and 288:
CHOOSING INVESTMENT PORTFOLIOS a-di
Page 289 and 290:
CHOOSING INVESTMENT PORTFOLIOS Henc
Page 291 and 292:
CHOOSING INVESTMENT PORTFOLIOS Lemm
Page 293 and 294:
CHOOSING INVESTMENT PORTFOLIOS ID.
Page 295 and 296:
CHOOSING INVESTMENT PORTFOLIOS Proo
Page 297 and 298:
CHOOSING INVESTMENT PORTFOLIOS is c
Page 299 and 300:
CHOOSING INVESTMENT PORTFOLIOS Proo
Page 301 and 302:
CHOOSING INVESTMENT PORTFOLIOS 3. C
Page 303 and 304:
2. EXISTENCE AND DIVERSIFICATION OF
Page 305 and 306:
ON THE EXISTENCE OF OPTIMAL POLICIE
Page 307 and 308:
Page 309 and 310:
Page 311 and 312:
Page 313 and 314:
Reprinted from Journal of Financial
Page 315 and 316:
Remarks: Differentiability assumpti
Page 317 and 318:
distributed from the rest. Then an
Page 319 and 320:
x * "22 "12 1 " (o22- a12) + (0n -
Page 321 and 322:
to decline — as if having more mo
Page 323 and 324:
solved for as a function 8.(y,0"),
Page 325:
FOOTNOTES 1. H. Makower and J. Mars
Page 328 and 329:
264 QUARTERLY JOURNAL OF ECONOMICS
Page 330 and 331:
Page 332 and 333:
Page 334 and 335:
Page 336 and 337:
Page 338 and 339:
274 QUARTERLY JOURNAL OP ECONOMICS
Page 340 and 341:
Page 342 and 343:
Page 344 and 345:
280 QVABTBRLY JOURNAL OF ECONOMICS
Page 346 and 347:
Page 349 and 350:
Reprinted from THE REVIEW OF ECONOM
Page 351 and 352:
(1.1) and (1.2) give SOME EFFECTS O
Page 353 and 354:
SOME EFFECTS OF TAXES ON RISK-TAKIN
Page 355 and 356:
Page 357 and 358:
Page 359 and 360:
Page 361 and 362:
Page 363 and 364:
Page 365 and 366:
REFERENCES SOME EFFECTS OF TAXES ON
Page 367 and 368:
COMPUTATIONAL AND REVIEW EXERCISES
Page 369 and 370:
and a gamble is offered that return
Page 371 and 372:
among two risky assets and a riskle
Page 373 and 374:
Consider problem (1) and suppose th
Page 375 and 376:
21. Let pi,...,p„ be joint-normal
Page 377:
Consider the following four utility
Page 380 and 381:
4. In most stochastic optimizing mo
Page 382 and 383:
(g) Discuss the use of semivariance
Page 384 and 385:
(a) Suppose that the investor alloc
Page 386 and 387:
The linear programming subproblems
Page 388 and 389:
(f) Show that (j> is proportional t
Page 390 and 391:
(e) Show that X has mean and varian
Page 392 and 393:
(i) Show that for all Aru ...,km, t
Page 394 and 395:
Some properties of the expected val
Page 396 and 397:
(i) Suppose that V0 is positive def
Page 398 and 399:
where V is the gradient operator an
Page 400 and 401:
and where Z, Ylt...,Ym are independ
Page 403 and 404:
INTRODUCTION This part of the book
Page 405 and 406:
some qualitative characteristics of
Page 407:
function of wealth. Hence the optim
Page 410 and 411:
INVESTMENT ANALYSIS UNDER UNCERTAIN
Page 412 and 413:
Page 414 and 415:
P if and only if INVESTMENT ANALYSI
Page 416 and 417:
INVESTMENT ANALYSIS UNDEB UNCERTAIN
Page 418 and 419:
Page 420 and 421:
Page 422 and 423:
Page 425 and 426:
2. RISK AVERSION OVER TIME IMPLIES
Page 427 and 428:
ables at t, and the primed variable
Page 429 and 430:
ehavior in choosing among timeless
Page 431 and 432:
is to asume that the consumer behav
Page 433 and 434:
The function t/,(C,_,, wt|a,, ft) i
Page 435 and 436:
sumption, given state j3,+i at r+l,
Page 437 and 438:
3. MYOPIC PORTFOLIO POLICIES Reprin
Page 439 and 440:
326 THE JOURNAL OF BUSINESS this op
Page 441 and 442:
328 THE JOURNAL OF BUSINESS with th
Page 443 and 444:
330 THE JOURNAL OF BUSINESS In sum
Page 445 and 446:
332 THE JOURNAL OF BUSINESS subject
Page 447:
334 THE JOURNAL OF BUSINESS myopic
Page 450 and 451:
where and k" 1 f (Mt-Ui) 2 4k {Oi 1
Page 453 and 454:
MIND-EXPANDING EXERCISES 1. Conside
Page 455 and 456:
(i.e., partial separability) and #,
Page 457 and 458:
(c) Discuss the solution properties
Page 459 and 460:
(d) By differentiating (cl) and (c2
Page 461:
Let U be the utility function for t
Page 465 and 466:
INTRODUCTION I. Two-Period Consumpt
Page 467 and 468:
asked to develop explicit solutions
Page 469 and 470:
that the property of nonincreasing
Page 471 and 472:
The Hakansson paper studies a gener
Page 473 and 474:
In the Samuelson-Hakansson additive
Page 475 and 476:
techniques. The article by Breiman
Page 477 and 478:
enously). These requirements are to
Page 479 and 480:
in general, a monotonic function of
Page 481 and 482:
ergodic theory, the book by Breiman
Page 483 and 484:
or less than u; either do nothing o
Page 485 and 486:
ME-24 the reader is invited to cons
Page 487:
In discrete time, a nonnegative lin
Page 490 and 491:
R. G. VICKSON This must not be inte
Page 492 and 493:
This gives with R. G. VICKSON t(t)
Page 495 and 496:
TWO-PERIOD CONSUMPTION MODELS AND P
Page 497 and 498:
310 DRfeZE AND MODIGLIANI second or
Page 499 and 500:
312 DREZE AND MODIGLIANI Had the sa
Page 501 and 502:
314 DRfezE AND MODIGLIANI (— UÛ^
Page 503 and 504:
316 DREZE AND MODIGLIANI ure for te
Page 505 and 506:
318 DREZE AND MODIGLIANI income sid
Page 507 and 508:
320 DREZE AND MODIGLIANI (3.5) The
Page 509 and 510:
322 DREZE AND MODIGLIANI Consumptio
Page 511 and 512:
324 DREZE AND MODIGLIANI in (cj, c2
Page 513 and 514:
326 DREZE AND MODIGLIANI Since dÛj
Page 515 and 516:
328 DREZE AND MODIGLIANI On the rig
Page 517 and 518:
330 DRfeZE AND MODIGLIANI and its p
Page 519 and 520:
332 DREZE AND M0DIGLIAN1 LEMMA C.2.
Page 521 and 522:
334 DRfeZE AND MOD1GLIANI if the sm
Page 523 and 524:
MANAGEMENT SCIENCE Vol. 19, No. 2,
Page 525 and 526:
A DYNAMIC MODEL FOR BOND PORTFOLIO
Page 527 and 528:
Page 529 and 530:
Page 531 and 532:
Maximize A DYNAMIC MODEL FOE BOND P
Page 533 and 534:
Page 535:
Page 538 and 539:
MULTIPERIOD RISK PREFERENCE 41 mode
Page 540 and 541:
MULTIPERIOD RISK PREFERENCE 43 The
Page 542 and 543:
MULT1PERI0D RISK PREFERENCE between
Page 544 and 545:
MULTIPERIOD RISK PREFERENCE 47 (ii)
Page 546 and 547:
MULTIPERIOD RISK PREFERENCE 49 that
Page 548 and 549:
or if s"(s' MULTIPERIOD RISK PREFER
Page 550 and 551:
MULTIPERIOD RISK PREFERENCE 53 5. C
Page 553 and 554:
Reprinted from The Review of Econom
Page 555 and 556:
the risky asset, with 1 — w, goin
Page 557 and 558:
w*t = glW,;Z,-1,...,Z0] = gT_i[W>]
Page 559 and 560:
Since proof of the theorem is strai
Page 561 and 562:
E C O N O M E T R I C A VOLUME 38 S
Page 563 and 564:
OPTIMAL INVESTMENT 589 M: the numbe
Page 565 and 566:
OPTIMAL INVESTMENT 591 For comparis
Page 567 and 568:
(ii) in the case of Model III, 1 1
Page 569 and 570:
Let Then OPTIMAL INVESTMENT 595 *.=
Page 571 and 572:
where OPTIMAL INVESTMENT 597 (41) T
Page 573 and 574:
where (54) T(x) = sup \ log c + log
Page 575 and 576:
where b(v*) is the greatest lower b
Page 577 and 578:
OPTIMAL INVESTMENT 603 noncapital i
Page 579 and 580:
OPTIMAL INVESTMENT 605 Let us now t
Page 581:
OPTIMAL INVESTMENT 607 REFERENCES [
Page 584 and 585:
present value. Let ( be time, r the
Page 586 and 587:
constant or decrease at a constant
Page 588 and 589:
Denote the vector in the parenthese
Page 590 and 591:
112 HOWARD M. TAYLOB easy to pictur
Page 592 and 593:
114 HOWARD M. TAYLOR market and ign
Page 594 and 595:
116 HOWARD M. TAYLOR the investor w
Page 596 and 597:
118 HOWARD M. TAYLOR option which a
Page 598 and 599:
120 HOWARD M. TAYLOR 3. BR.VDA, J.
Page 600 and 601:
172 BASIL A. KALYMON more general p
Page 602 and 603:
174 BASIL A. KALYMON The three case
Page 604 and 605:
176 BASIL A. KALYMON an additional
Page 606 and 607:
178 BASIL A. KALYMON (24) d\ = r -
Page 608 and 609:
180 BASIL A. KALYMON The above resu
Page 610 and 611:
182 BASIL A. KALYMON PROOF. First w
Page 613 and 614:
MANAGEMENT SCIENCE Vol. 17, No. 7,
Page 615 and 616:
M1NIMAX POLICIES FOB SELLING AN ASS
Page 617 and 618:
MINIMAX POLICIES FOB SELLING AN ASS
Page 619 and 620:
MINIMAX POLICIES FOR SELLING AN ASS
Page 621 and 622:
Page 623 and 624:
MINIMAX POLICIES TOR SELLING AN ASS
Page 625 and 626:
MINIMAX POLICIES FOB SELLING AN ASS
Page 627:
Page 630 and 631:
648 L. BREIMAN We have discussed th
Page 632 and 633:
650 L. BREIMAN Hence, it is suffici
Page 634 and 635:
(a.s.) on the set lim XN < oo > u <
Page 636 and 637:
EDWARD O. THORP method to this arti
Page 638 and 639:
EDWARD O. THORP drastic simplificat
Page 640 and 641:
EDWARD O. THORP us is that the prop
Page 642 and 643:
EDWARD O. THORP Let the range of Rj
Page 644 and 645:
EDWARD O. THORP is approximately on
Page 646 and 647:
EDWARD O. THORP choosing P in repea
Page 648 and 649:
EDWARD O. THORP return from the hed
Page 650 and 651:
EDWARD O. THORP it is plausible to
Page 652 and 653:
EDWARD O. THORP is maximization of
Page 654 and 655:
EDWARD O. THORP 6. BREIMAN, L., "Op
Page 657 and 658:
Reprinted from JOURNAL OF ECONOMIC
Page 659 and 660:
CONSUMPTION AND PORTFOLIO RULES 375
Page 661 and 662:
Page 663 and 664:
and CONSUMPTION AND PORTFOLIO RULES
Page 665 and 666:
CONSUMPTION AND PORTFOLIO RULES giv
Page 667 and 668:
Page 669 and 670:
Page 671 and 672:
Page 673 and 674:
Page 675 and 676:
Page 677 and 678:
Page 679 and 680:
Page 681 and 682:
Page 683 and 684:
Page 685 and 686:
Page 687 and 688:
Page 689 and 690:
Yin CONSUMPTION AND PORTFOLIO RULES
Page 691 and 692:
Page 693 and 694:
Y(t) Y CONSUMPTION AND PORTFOLIO RU
Page 695 and 696:
Page 697:
Page 700 and 701:
(d) Suppose n > r2. Formulate the o
Page 702 and 703:
(j) Show that the absolute risk-ave
Page 704 and 705:
The optimal first-period decision a
Page 706 and 707:
(b) Show that the optimal policy in
Page 708 and 709:
the game is stationary and favorabl
Page 710 and 711:
(v) UHEV-E^CV-Af- 1 )^ w, V&0, w>0,
Page 713 and 714:
MIND-EXPANDING EXERCISES 1. Conside
Page 715 and 716:
Note the following distinction betw
Page 717 and 718:
(e) Discuss the computational aspec
Page 719 and 720:
(a) Show that U obeys the functiona
Page 721 and 722:
(g) If ;ij(0) > 0 (1 g; g N), show
Page 723 and 724:
Note that problem P2 has 3(N- 1)+1
Page 725 and 726:
Note that (f) and (g) imply the exi
Page 727 and 728:
P = (Pu), where pu = P[X„+l =j\ X
Page 729 and 730:
(a) Show that {X„} is a stationar
Page 731 and 732:
(b) Show that T T u = a + b Y[ c\'
Page 733 and 734:
(b) Interpret this assumption. Is i
Page 735 and 736:
(d) Show that (c) may be written as
Page 737 and 738:
Bibliography Journal Abbreviations
Page 739 and 740:
BHARUCHA-REID, A. T. (1960). Elemen
Page 741 and 742:
FIACCO, A. V., and MCCORMICK, G. P.
Page 743 and 744:
IGLEHART, D. (1965). "Capital accum
Page 745 and 746:
MANGASARIAN, O. L. (1966). "Suffici
Page 747 and 748:
PORTEUS, E. L. (1972). "Equivalent
Page 749 and 750:
THOMASION, A. J. (1969). The Struct
Page 751 and 752:
A Absolute risk aversion, 84, 85, 9
Page 753 and 754:
Logarithmic utility functions, see
Page 755 and 756:
V Value of information, 462-467, 68
show all

STOCHASTIC

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?