STOCHASTIC

More documents

Recommendations

Info

172 BASIL A. KALYMON more general position compared to the extremes of the Weingartner [6] discounting method which uses the interest rate the firm will be paying on issued bonds, and the Pye [4] discounting scheme in which the precise realized single-period market rate is used. The discount factor as used in our model reflects the time preferences of the decision maker, recognizing the imperfections of the market which do not allow him complete flexibility in borrowing and lending. Such a discount factor would be affected by the market interest rate in as much as the latter contains information on future inflation, but would be principally determined by internal considerations. Note that the importance of this factor is deemphasized in the model in that it is used merely to compare different streams of interest payments and issuing costs and not to evaluate the utility of repayment of the principal. As noted above in this regard, the model assumes that the firm has already decided on the appropriate funding requirement in accordance with its investment opportunities. Current practice employs a single discount rate representing either a cost of capital or some type of single-period or multiperiod market rate. There has been little previous work in bond refunding as a multiperiod decision process. Weingartner [6] is the only one to explicitly formulate the problem as a multiperiod process. His model is completely deterministic, and assumes that a level debt will be maintained, with only a single bond outstanding at a given time. Pye has studied the use of a Markov process to represent interest rate fluctuation in [5], and has used such a representation in [4] to determine the value of the call option by assuming that a single refunding decision will be made in the most favorable period over the term of the issued bond. Also, Bierman in [1] has proposed using a Markovian representation of interest fluctuation to improve short-run timing when refunding a bond, but does not consider the full multiperiod implications. References for work on the term structure and fluctuations of interest rates may be found in Pye [5]. A summary of current practice, (using essentially single decision approaches) as well as a discussion of appropriate discounting rates to be used, can be found in Bowlin [2]. The general model to be discussed is more precisely defined in §1, and the existence of optimal policies of special structure is proved in §2 under differing sets of assumptions. Calculation of optimal debt size after a decision to refund has been made is discussed in §3. In §4, simple necessary conditions for refunding are given for two level debt models. 1. Model Formulation In this section, the basic model to be studied is formally defined. Let Di represent the net requirement for debt financing in period i, for i = 1, 2, • • • , n, with i = n representing the chronologically first period of the n-period planning horizon. The Di'a are assumed to be known. For additional clarification we stress that the financing requirement as used in our model represents the exogenously determined amount of debt financing which the firm envisions requiring. It is based on projected cash flows aggregated over all the major projects/investments of the firm. (See Footnote 1.) Let U , Xi and r, represent, respectively, the time to maturity, the size, and the coupon rate of the bond outstanding in period i. Let p,,m represent the rate of interest that must be paid on a bond issued in period i with time to maturity (or term) m. Defining Pi by pi = pi,*>, we shall define (1) Pi. = Vi(m, in), where Vi(m, p,) is an increasing function in pt. Thus !\(m, p.) represents the term- 564 PART V. DYNAMIC MODELS
BOND REFUNDING WITH <strong>STOCHASTIC</strong> INTEREST RATES 173 curve in period i given that the long-term interest rate is p*. The form of i\(ro, p
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 131 and 132:
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
A Unified Approach to Stochastic Do
Page 139 and 140:
A UNIFIED APPROACH TO STOCHASTIC DO
Page 141 and 142:
Page 143 and 144:
Page 145 and 146:
Page 147 and 148:
Page 149:
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 162 and 163:
9.2. Example 2. If (30) u'(x)=(x°
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 170 and 171:
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
Page 184 and 185:
Page 186 and 187:
Page 188 and 189:
Page 190 and 191:
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:
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:
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:
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 (— U^U^
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 d^Uj
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 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: MINIMAX POLICIES FOR SELLING AN ASS
Page 623 and 624: MINIMAX POLICIES TOR SELLING AN ASS
Page 625 and 626: MINIMAX POLICIES FOB SELLING AN ASS
Page 627: MINIMAX POLICIES FOR SELLING AN ASS
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

Create successful ePaper yourself

Delete template?

Save as template?