The Econometrics of Financial Markets

The Econometrics of Financial Markets

John Y. CamPgeU AndrewW.Lo I

A. Craig MacKinJay

Princeton University Press Princeton, New Jersey

C"I'yri\l,ht © 1',)97 hy I'rillc elOll Ulliw "ily I'n'" ('fI''' . 41 Willi ,,," SII ,'('1.

Publisht·" hy Princeton Univt"rsily Plince\()II. New Jersey \lilr)~\l

In the linite tl Kingdonl; prinCt.'tlln Unil' t'"ity

We.t

Su~t"X

libra ry of Cong ress Catal oging ·in.Pu blica

1'1'''. Cltid,,·,t.1 Prob lem s with LOllg-1 [oriz on Infc renn :s Test s For Lon g-Ra nge Dep end encc . . . . . . . 2.ti.l Exal llple s of I.on g-Ra ngc lkpc /lclc ncc . 2.6.2 The Hur sl-M ancl clbr ot Resc aled ·Ran gc Stat istic Uni l ROOl Test s . . . . . . . Rec cnt EmpiriC the best forecast one can make or a random variable X is the forecast of the forccast one would make of X if one had superior information J" This can be rewriuen as ~:[X - E[X I J,] I I,J = 0, which has an illtuitive interprctation: Onc cannot lise limited information lito predict thc forecast crror one would make if one had superior information J,. Samuelsou (1965) was the first to show the relevance of the Law of Iterated Expectations ror sccurity market analysis; I.e Roy (l9H9) givcs a lucid review of thc argulIlent. We discuss the point ill detail in Chaptcr 7, hill a brief sllllllllary may be helpful here. Suppose that a security price at lime I, /'" can be wrillen as the rational expectation of some "fundamental value" 1", conditional on information I, available at tillle t. Then we have P,

:=

1':[ V' I I,] = I':, V'.

(\.5.1)

The s;lIne equation holds one period ahead, so 1',+ I = E( V' I I"

I

I =

E'l I \,'.

( 1.5.2)

filiI Ihen Ihe ex(>eclalioll of III(' chan).!;e in the price over the next period is

1-:,1/'1+1-1',1 = E,IE,tIIV'j-F./[V'))

=

0,

Iwcallse I, C l,t I. so E,I 1-:/ + II \"11 = E,I V' J hy Ihe I.aw of Iter;\!ec\ ExpectaliollS. Thlls rl';lli/ell ch;lIIgl's in plin's arc IInlilrcr;lslable ).!;iven information ill the sci 1/.

1.5.2 !., lI/(///U'I/':llifil'llry '/r.\lab!r?

Althollgh the l'mpirirallllt,thodoloh'Y slIllImari7.ed here is wdl-('SI;lhlishcd, Ihere arc sOllie seriolls C\if'firultics ill inllTpreting ils resllJrs, Firsl, allY lest of efficiency nnlst asslln\(' an eqllilibrilllll model thai c\ermes nOflllal secoritv relurns. If dIicicncy is rejeclec\, Ihis could be because the markel is Iruh· incl'lkienl or bccallse all illcorrcci cqllilibrium model has hcen asslIl\lcd, This joillllr:v/ltlllrl'.li,1 pl'OlIl('\1I \\leans 11\;11 lIIarkel crrlciency as slIch em newr he n:je('\I'c\, Second. perli'('\ I'fficiency is an lin realistic benl'hlllark th;1I is uillikely 10 hold in pranicl'. Evell in thcory. as Grossman alld Sliglitz (I !lHO) h;l\c shown. allllonnal \'('tllI'llS will exist iftherl' are costs of ).(.lIherillg and pmcessill)!; ill(III'lIIalioll. Thcse rellirns are necessary 10 compensate illv('siors fill' their infonnalion-gathnill).!; alld informatioll-processing eX)lenses, and are 110 lon).!;er ahllol'mal whell Ihese expellSes arc properly accollnlcd (i,l'. III a large and liqllid mark('I. inlilrlllalion cosls are likely tojllstify (111)' snd} allllOrmall'l'llInlS, hili il is dimcllit 10 say hnw small. evell if sllch costs could he measllred precisc\}', The nolioll or rrialil'f efficicncy-Ihe efficiency of one marketmc;lSlll('d agaillst allothcr. e,g,. tlie N('w York Stock Exchange vs, Ihe Paris BOllrsc. 1',,lures markets vs, SpOl markets, or all('\ion vs, dealer markels-llIay he a mort' IIscrlll rOllccpl Ihall Ihc all-Ol'-llothill).( view lakcn by milch or Ihe tradilional lIIarkel-eflicit'IKY lilnatllre, The adv;llIla).!;es of rclaliw elliciellcy over absolule efliciellcy arc easy 10 SC(' hy way of all ;lIIal0!n'. Physical syslelIIs arc ofIe II ).!;iv(,11 all ('nici('III"}, rat ill).!; has('d Oil the rclalivc propol'.tioll of cllergy or flld ('ollwr!ed to IIsd'lIl work. Therdc)J'(\ a pis Ion ellgille lIIay he rated al n()'){, eflici('lIcy, 1I\('allillg Ihal Oil avcra).!;c 1;0% of tilt" ellnh,)' cOlllailled ill the ell).!;illc'S fllel is IIseli 10 tllm lhe nallkshafr, wilh the remaillill).!; 40'7" lost \0 olhel' forms of work slich as hcal.lighl. or noise. Few ell).!;illeers wOllld e\'eI' t'ollsilln pCfrOrtllill).!; a stalislicaltesllo (11:1('1'lIIille whclhn or 1101 a givclI ('lIgillC i.~ perfcnly d'lirienl-sllch all cllgill!' exists ollly ill 11\l' idt',11 i/.ed frid iOllll'ss world of Ihe imaginatioll, I\lIlnwasll rill).!; I'elalivc eflkiclI('\'-rclalivc 10 Ihe friniolliess idcal-is conHllonplan', hHh-I'I\. wc ha\'\' fllllH' \1) ,"xpcn SlIl'h IIH'aSlln'lI\ellts (i)\' lIIallY hOllsl'!lold prodll('I~: ail' fOlldilioll('JS, hoi wal('J healers. rcfri).(('raIOl's, ('tf. Similarly,

25 market efficiency is an ideali7.ation that is economically unrealizable, but that serves as a useful benchmark for measuring relative efficiency. For these reasons, in this book we do not take a stand on market efficiency iL~e)f, but focus instead on the statistical methods that can be used to test the joint hypothesis of market efllciency and market equilibrium. Although many of the techniques covered ill these pages are central to the market-dflciency debate-tests of variance bounds, Euler equations, th¢ ('APM and the APT-we feel that they can be more profitably applied to measuring efficiency rather than to testing it. And if some markets tuI"it oUI to he particularly inefficient, the diligent reader of this text will be wel~­ prepared to take advantage of the opportunity. ! I

1

f I

I "

2 The Predictability of Asset Returns

and most enduring questiollS of financial econometrics is whether linancial asset prices are forecastable. Perhaps because of the obviolls analogy between financial investments and games of chance, Ill1 dic c box , an d of the die Ihe itself. To the ext ell l to wh ich YOI l dep;1I1 1111111 tha t cqu alit y, if it is in yo ur op po ne nt' s favollr , YOI I arc a fool, all d if ill yo ur ow n, you arc IIn jllst. Th is pas sag e clc arl y con tai ns the no tio n ofa Jai rgt lllle , a gam e wh ich in yo ur favor no r you is ne ith er r op po nen t's, and thi s is the ess enc c of a lIla sto cha stic pro ces s {I',} rtil/ jial l', a wh ich satisfies the fol low ing con dit ion : E[ /"I I I I'/, ["- I, ... J =:= 1'/, (2. 1.2 ) or, equivalently,

E[ I', tl- I'/ 11 ',,1 '/-1 , ... 1 = o. If P, rep rcs cnt s on e's rtll llli lati ve win nin gs or we alth at dat e I fro sOllie galliC of cha nce lll playiflg eac h pcr iod , the ll a fai r gal lic is oll e (l>l' wh exr }'c ted wc alth nex t irh the per iod is sim ply eql lal to thi s pcr iod 's (2.1\.2)), con dit ion ed we alth (se e 011 the his tor y of the gam e. Alt ern ativ ely if tli,c exp ect cd inc rcm , a gal lic is (~Iir ent al win nin gs at any sta ge is zer o wh en con on the his tor y of the gam dit ion ed e (sc c ('2 .1. 3». If 1', is tak cn to be an ass et's pri ce at dat e I, the n the ma rtin gal e hyp oth esis sta tes tha t tom orr ow 's pri ce is exp ect ed to be equ al to tod ay' givch thc ass et's ent ire s pri tT, pri ce history. Alternativ ely, the ass et's exp ect cha lig c is zer o wh en con ed pri ce dit ion ed on the ass et's pri ce his tor y; hen ce iL~ is ju~t as likely to rise pl'i re as it is to fall. Fro m a illr cca stin g per spe cti mar~ingale hyp oth ve, the esi s im pli cs tha t the "be st" for eca st of tom orr sim~y tod ay' s pri ow 's pri ce is cc, wh ere "bc st" me ans mi nim all llc an- sql lar ed Ch al> ter 7). err or (se e An oth er asp cct of the ma rtin gal e hyp oth esi s is tha t no no ver iap pri ed cha ng cs arc un cor pin g rcl ate d at all lea ds and lags, wh ich im pli es the efTccliveness of all line inar for eca stin g rul es for fut ure pri ce cha ng on hIstorical pri ces alo es bas cd ne. Th e hlct tha t so sw eep ing an im pli cat ion cOlin! fro m as sim ple a cou ld mo del as (2.1.2) for esh ado ws the im po rta nt the nla rtin gal e hyp oth rol e Iha l esi s will play in the lIIo del ing of ass et pri ce dyn (se c thc dis cus sio n bel alll ics ow and Ch apt er 7). In fact, the ma rtin gal e was lon g con sid ere d to be a nec ess ary con dit for an 1firienl assct ma ion rke t, on e in wh ich the in( llfI lla tio n con tai ned pri ces is instantly, fully, in pas l an d per pet ual ly ref l(,c ted ill the ass et's cll rre If the ma rke t is eff icie nt pric(':~ nt, th( 'n il slio lll< lno t be pos sib le to pro llt by tra dil lg Oil 'Se e \laI d (1990, Cha pte r 4) III' \",.,

ll
/1, -

/'t-I > 0

if

1'1 -

/'t -

P,--I


.'2 4:>C>':'

4:n.ti 407.2 :nH.I :\47.:'> :~ I :>.:, 2H:t:>

EXPt'{"I('d lotallillmht'f of III"" il1.I ... ;tlHP!c· oln ind(·I)t·,)(It."nl8~rnoulii (rials rt'I)J('s('lIting po,,· iti\'('/lIc'gati\'(' fOlllinlloll.lril\' fOIU}lOlllHlt'd r('lurn" fCu' a GillI.~"iian gromerri(' I\rowllial1 mOlioJl wilh II .. in/1 = Of:k .... ~O'.)'r, allli !'Il.lIUi;lId d('viation n == ~1 %.

(2.2. Hi), anplied to thcse modes of analysis, hut rather that the standards of ('vidence ill this literature have evolved along very different paths. Therefore, we shall present only a cursory review of thl'se techniques. ran~

2.3.1 Fillt'r Rules To test RW2, Alexander (1961, 19(1) applied a filter rule ill which all asset is purchased when its price increases by x%, and (short)sold when its price drops by x%. Such a rule is said to be an x% filter. and was proposed by Alexander (1961) for lhe following reasons: Suppose we tentatively assume the existence of trends in stock market pdccs but believe them to he masked by the jiggling of the market. We might filter out allmovelllellls smaller than a specified sile and examine the remaining movements. The total return of this dynafllil portfolio strateb'Y is then taken to be a mcasllre of the predktability in assel returns. A comparison of the tOlal retllrn to the retllrn from a huy-and-hold stratq..'Y for thc Dow Jones and Standard ,lIld Poor's industrial averages led Alexander to conclude that " ... Ihere liTe trends in siock lIlO, the sample autocprrelation coefficients i>(k) are negatively biased. This negative bias comer from the fact that the autocorrelation coefficient is a scaled sum of cross-p,roducts of deviations of T/ from i\.~ mean, and if the mean is unknown it must be estimated, most commonly by the sample mean (2.4.6). But deviations from the sample mean sum to zero by construction; therefore positive deviations must eventually be followed by negative deviations on average and vice versa, and hence the expected value of cross-produc\.~ of deviations is negative. for smaller samples this eITecl can be significant: The expected value or p(l) ror a sample size of 10 observations is -)0%. Under RWI, fuller (1976) proposes the following bias-corrected estimator p(k):7 -

p(k)

==

_

p(k)

+ --,'{-It --" ( I ( f-I)
ellc\ellt observatiollS-and Ihesl' results lllay he used 10 COllstruct tests or RW~ and RW:~ (sec Sectioll 2.4.3 helow).

2.4.2 Portmall/Pal! .'i/a/istirs Sillce RW I implies that all aUlOcorrclatiolis arc zero, a simple test statistic of RWI that has power against mallY alternative hypothescs is Ihe Q-statislic due to ~ox and Pierce (1970):

~II

'"

-

TLp~(lC).

(2.4.16)

h=1

Ullder the RWI nlill hypothesis, and using (2.4.14), it is ea~y to sec that c1.. = 'I"'£;~I P(lc) is asymptotically distrihuled as X~,. qUllg and Box (197H) provicle the (ollowing finite-sample correctiotl whirh yidds a helll'r (it lO the X~, for slIlall sample sizes: '"

fl' "'-'II

2(k)

== '1'('1'+2) "" -p-

b

T-k

.

(2.4.17)

By summing the squared autocorrclations, the Box-Pierce Q-statistic is designed to detect departures from zero alllOl'orrclatiom in either direction anel al all lags. Therefore, it has power' against a broad r'angt> or alternative hypotheses to the random walk. However, selecting the \l\lInber of autucorrelations III re'l"irc~ SOlllC care-if too /CW arc used, the prcsencc of highcr-orcler autocorrelation lIlay hc IIIbsccl; if too lllallY arc IIsed, the test Illa), lIol have lllllCh power due to insignificant higher-order aUlOcorrelalions. Therefore, while sllth a portlllanteall st,l\islir does have sOllle appeal, IWlLcr tesL~ of thc random walk hypothcscs lIlay he availahle when spedne allcl'Il;lIiVl' h)'I)otheses can be identified. We shall lui'll to slIcll examples ill III

Iii, + Ii" -,i,1

,'\',11 I Ii,

,I\'a'IO" -

,'\';111';" I

Ii,

I

",hefe ;I\',III·J d('IIIJI('~ thl' a~>'llIptolic \',uian("('

I

,'\',III,i" - ,i, I - ,1\';11 \Ii, I,

"pCI ;11111'.

\'~111 p.lIlirlll.tr, apply Ihl" ddt.llIlethod to I({il.,i'.!l:::;(i,/fi',! \\liel"(' fjl=n/~-n,;. fi,;!:=(j,;. aud oh,cl \"(" [h.1I r1f~ ,11111 n-,; ,II (' a.,YIIIIHCJ!i( all~· 1111( 011 (,l.lIc·d IIt'( .111'(' n,! j,;111 dfifi('1I1 {".,lim:.tlol".

-r.-:;

and tll 0Jor whi rhE lklil _rl'l (,H) ) < D. < 00. I nq , ., lim E[i; J = a- < 00. "'1_ 00

nq

L

1:::01

j'llr rlllt, E{il il_; fl (,-.1 = 0 jor (lil)'

/Wl/Z. I'I'O j and k will'l l' j l' k. Con ditio n (Il I) is the unco rrcla ted incr eme nts prop erly of the rand om walk Jhat we wish to test. Con ditio ns (112) and (H3 ) are restr iClio ns on the max illlu m degr ee of dep end ence and hete roge ncit y allow able whil e still perll~illing sO/lle form of the Law of Larg e Num bers and thc Cen trall .illli t TheO rellJ to obta in (sec Whi te [I mH) for the defi nitio ns of cp- and a-m ixin g rand+1IJ sequ ence s). Con ditio n (1!4 ) imp lies that the sam ple aUlO corr elation spf' l are asym ptot icall y IIJlc orrel ate< i; this cond ition may be w('a kene c\ cons iller ably at the expe nse of cOll lput ation al silllp licit y (see !lole I:». 'I~Jis com pou nd null hypo thes is assu mes that jJ, poss esse s lInc one lalc d incr etne nlS but allow s for 'Illit e gene ral form s of hetc rosk edas ticit y, incll ldillg dtte rmin istic chan ges in the vari allce (due . for exam ple, to seas onal factor~) and Eng le's (1~)H 2) ARC II proc esse s (ill whic h the cond ition al variance (~epe~~ on past info nnat ioll) . Sihc e VR( q) still appr oach es one IInd er lit" we need only com pute it~. asym p'tot ic vari ance [call it U(Ij) J to perf orm the stan dard infe renc es. 1.0 and Mac Kin lay (198 8) do this ill two step s. First , reca ll that the follo wing equa lity hold s asym ptot icall y und er 'Illit e gene ral rond ition s: 1

VR(I" j) ==

I

+28 L '~I

(I - ;k) p(h).

(VI Al)

I

rOIlI~(". second 11I0ll lt'ub an' ",1111 .t....!'IIIIBe d 10 hc' fillite ; other\\,ls(', Iht" \'ari~lIln' lougt"r \\'e11 defin ed. This rull'., r.1l1t1 out (1i:"'l Iibulio lls with IIIlilll tc \·"lri~lIlre. such as thoM ' Ihe st.thlt' P,lrt" lo.. Le\y f.uuil), (with ill flt.n,1 ( It'n:"l. lif eXpO llt'Ill. \ th.1l are Ic~s tll.tn 2) prop ~1.11l(1t·lhr(}t (19tj3 ) al1d o\('d In Failla (I~)I;:)). J1{)WC H'r, 1l1;IIIY olll('r fC)lIII!'o ofl('p tokur t()\is ~,rt· .111(Jwc'd. "I< h 'I' Illal ~enerdled hy ElIgl e', (1 \IH~) ""tol q~ .. ""in · (olldi lioll,l lIv 1t~lel ",~,'d'''li( (AI{( 11IOf~" \'ee ScClill1l 12.2 :111 ill CIt"PI"\' 1'2) l.Jor

j, 110

1,{JI/~-lllJriwll

2, 5,

1&llLm.l

Secolld, II' ,It' Ihal undl'J' 1I~ (conditioll (114» the autocorrelation coelEel,':11 estilll(k) are asym(ltotkally uncorreiatedY' II' the aSYllJptotic \\lri.1J1Ce O. ( " of the p(k)'s call he obtai lied Ul\dn II;,. the asymptotic \'~Iriallre (}(q) oj \'R('1) lIlay he calnilated as the weil-(hted SUIII 01' the Ilk's, whne Ihe weigh Is arc simply the weighL~ ill relalion (~.4.41) squared. Denote by Il. and U(q) the asymptotic variances of Ii(k) and VR(q), respectively. Then ullder Ihe lIull hypothesis II~ Lo and MacKinlay ( I UHH) show that I, The slatistics VD(q), and VR(q)-1 converge almost surely to zero for all q as 11 increases without bound. ') The following is a heleroskedastirity-n)Jlsistenl eSlimator of Il.: (2.4.42)

:t The followillg is a heleroskedastit:ily-rollsisll'nl estimator of O(q): 1- 1

8(q) -

,.

4L (I - ~). 8•. ~~t

(2.4.43)

q

Despite the presence of general heteroskl'dasticity, the standardized test st:11 i,)tic if' (if) if;' (q)

CIII 1)(' used

10

lesl

il,~

J1zij(VR('/) - I)

fii

N(O,i)

(2.4.44)

ill lile usual way.

2.5 Long-Horizon Returns St'\'nal recent sludies have focused on the properties of long-horizon returns to test the random walk hypotheses, in some cases using 5- to 10yt'ar ITtUJ'IlS OWl' a 65-ycar salllple. There are fewer nonovcrlapping longhori/ol1 rcturns for a given tillle span, so samplillg errors are generally l\lthollgb Ihl., 1("~lriClion 011 the fOllrth ('r()!\!\'lIlOlIIl'l1l~ 01 f;, III .. }' ~t'l"Jll sOIllt'whdt lIlIiJlIIl~ it i", :-.ati:-.f"ll'd tor .~ny process with itHkpelHlent illcrelllelits (legaJ(lIess of heterogeneity) .IIHI ~Ibt) luI' Ijllt'~lr (;all:-.~iaJl ARCII processes. This a . . sumptioll !Hay he relaxt'd entirely, req\lir~ IlIg tilt' l"lilll~llioll of the asymptotic rov.uiuHTS of tht' ~IHI()n)IT(·l.ttioll (,!'ttiIllClIOr:'i in order In l'l

ill\(',

\",illl"'" lilt' lilllitillg ';tri"Jl(e f) of VR(q) vi" (~,~,'II), Allhollgh thl' r,'slIilillg estimator of II \\(lIdd he 1110)"(' c.:olllplicatcd than equdliou ('2AA.:-\). ill~ coun'pln.lIly Mlaigilllorw.... d dud lIIay I(,adily IH' forlll('d .d()l1~ the lines of N,·w('y .uHI \t\'(.'~t (I ~JH7). All ('\"('11 III()((~ Kt'llC'lal (and pos~ :-.ihly won.' t'xart) sampling thew), ftH"lht' v,lriillln' r~llio~ m;IY he olHailJ('d IIsing- the result'" uf nll"",r (I ~IH I) alld 1)1I1l1q~

(2.5.8)

IInl'I1("1" tIlUit"f!Iltested to sOllie degree, it is a well,·slablis hcd fact tliatlon g-range depelld ence ell I indel"d he detecte d by the "cLlssical" R/S statistic. Howeve r, perhaps the lIIost illlporta nt shortco llling 01· tile rescaled range is its sensitivi ty to short-ra llge depend ence, implyin g tlLlt allY illcoIllp atibility betweel l the data and the predicte d behavio r of tIle R/S statistic under the null hypothe sis need not COIIIC from long-ra nge depeIld cIIce, but Illay merely he a symptol ll or short-te rIIl Illeillory . III particul ar 1.0 (199 I) shows that under RWI the asympto tic distribution of (I/.[ii) Q,. is given by the randoIll variable V, the range of a firownia ll bridge, but under a stationa ry AR( I) specific ation with autoregressive coeffici ent ¢ the Ilormali Led R/S statistic converg es to ~ V where ~ == .j( I +¢) /( I-¢). For weekly retums of some portfoli os of com IlIon stork, i> is as large as 50%, implyin g that the !Ilcan of C6./.[ii may be biased uI}-

III11S1 he eli",en 10 allow ti)r tlllcillalio ll.' ill Iii .. supply of IVal .. r abm·.. IIie dalll while .Iill Ill~lillt;ljllillg a relatively cunstant flow ofwatc.'r below the dalll. Siufr dam rOJlsrfuctiun costs al to illllllt.'ll~e. IIH~ ililportdn ce of e~[ill1aliIIK Ihe reservoir rapacity Ilt"cessary to meet long-term slOra,;" Ilted., is apparellt. The nUlKe is all eslilllale of ,his quanlity. If JS i. the riverflow (per ,,"it lillle) a/Jove Ihe dam and X n i. Ihe d,,,ired rivedlow below Ihe dalll, Ihe brackeled 'I' .... ltily ill (\!.6.1 0) is Ihe capacilY oflhe reservoir n("~ded 10 tm ....e lhis .Illoolh How Kiven Ihe p.lllnll of Ilows ill p .... iuds I IhrouKh n. For exalllple, Sllpp"S" '"lllllal river How. are ""'"ll1ed to I,,· 100, ;,0, 100. alld :,0 ill years I IhrollKh 4. If a cu,,-,WlIl allllllal flow of 75 below the dam i~ d(,~1I"l'd c.'arli yc:ar, a reservoir must have a minimuJI l total CiJMrity of~:, since it must store 25 111111., ill y.. ars 1 alld:{ 10 provide for Ihe rt"/alivt'ly chy y.. ars:l ,,,"1~. Now slIl'l'ose inslead Ihal Ill .. "'llllrall" 'll'·fIl of riverflow is 100,100, :,0, :,0 ill Y",I" I tilr'"'\'\\i,," H)('fti. i., lib (ill p('rrelll) ;,,1:nificance tests; they are reported onl)' to pro\ide an indication of Ihe ,,,riance ratios' cro.'>Hectional di.'persion. Parenth~ticaJ entries for portfolio \"riance ratio, (panel B) are the heteroskedasticil)'01 it'd

II"

(I."''

H"

O.!l:1H O.H!)':.! O.H:1\) 0.7'2H

H:lI

111

11."

nil II,.,

N~I_'

11:11 -

\

Il;i-\ I~,,-\

l

/(~,-~ n,,_~

/{1I .• l Ilr.,_~

U: I/ . , 1l;'.4 I~.,_,

fI. "

0.171 O.IK':.! 0.1\17 0.201 0.IK7

[{"

0.11:; O.I':.!\) 0.147 0.1:;3 0.147

H.W,li H.\) 14 H.\lti I 1.000 Ilr"

00")

o.o:n

0.0:,:1 0.059 0.057

11:"

Il;,

I~"

0.141 0.1:15 0.121 n.OK4

0.tl:1':.! O.O':.!\) 0.0:12 0.02K 0.111 ':.!

-0.010 '-0.005 -0.006 -0.016

C""

0.141 n.14:1 0.1:17 O.I':.!O

[I,~,-,

1l.~)H

on")

0.0:,7 O.O!) I 0.0:,1 O.lHli 0.02:;

C flit

14

tl.W)t;

0.9-\4 0.\17\) I.oo(l o. \Hi I

Nli

N~,_,\

11,,-4

0.\).\·1

1.(lOO 0.97 1 )

O.OK\) 0.07K 0.07\) 0.071 0.04:,

I55

11."_,,

1l.~)7ti

0.~)7ti

1/,.,

/III

/(,,_:1

Il;,_, //'.,-:1

[{II O.H:\~)

0.3:10 O.3':.!4 0.310 0.':.!ti5

/("

Y"

f(.\, O.HI)~

O.':.!':.!li 0.':.!:1':.! 0.':.!44 0.'242 O.':.!2:1

N,,_~

Y

U::!I

II~,

C'"

/IO,.(/oli(l rf'/unu.

1).lnH 1.\lOIl

/("

11,,- \

Y\

75

Hl'itimfr

rIO'

OJl97 U.UY!'! 0.100 0.094

lIu

11.1,

[1,4,

O.IOli I>.lOO 0.105 0.\04 O.0\)3

0.o7·l 0.071 0.077 0.07\) 0.074

Il.W,O 1),0:,0 0.0:,8 O.Olll 0.061

flll

/(,,,

0.063 0.062 II.I)(iO O.Oti7 0.Oli4

0.0:16 O.O:lti 0.033 O.O:l\) O.03H

n"

0.016 0.017 lUll :, 0.023 0.025

-Iun

0. 11 The intuitioll for this reslilt is simple: Whcn

j exhibils nOlllrading, the relurns to a constantly trading securilY i call forecasl j due to Ihe common f;tctor it prcsent in hoth I·eturns. That j exhibits nontrading implics that future obs('J"ved retllrns 1';;+11 will he a

st~curily

weighted average ofal! past virtllal retllrns r,I+/I_' (with the Xil~lI(k)'s as random weights), of which one lerm will he the l"llncnt virtual returll 1',1' Since the cOlltemporaneous virtual returlls 1"" and 'l ' ;Ire UIITl'iatl'ti (beGl\lse of the common factor). 1':; can forecast ';;4 /I' IloIVevcl", ,.;; is itself unforecastablc bccallse 1';; = Ti' for all t (since 7r, = 0) ,Illt! r,l is II!) by assumption. thus Ij'; is uncolTdatcd with ";;+11 for any 11 > n. The aSyllllllcll)' of (3.1.11) yields all empirically testahle restriction on the crosS-;l\llOcovariallces of returns. Since the only source of asymmetry ill (:\.1.11) is cross-scClional differences in the probabilities of Ilolllrading, :nforlll 0

• al ,

(I-If.)(I-",)

[~] 1-",

2

rr

nq - q+1

b

P.P'''t I-".If,

f3 f3 a 2 n

b

I

for

n == 0

for

n > 0

(3.1.42)

Ii, q > 1, and arbitrary portfolios a, h, and time r.

98

3. Mmlifl Minollrurllln'

Equation (3.1.40) shows thattillle a~~re~ation also affecls Ihe '1II\oCO!"relation of observed portfolio relurns in a highly nonlinear f;lshion. 111 contrast to the alltocorrelation for ti/lle-a!{~reg-ated individual securities, (3.1.40) approaches unity for any fixed I{ as 1(. approaches unity; IIIen,rol"(' the maxim,11 autocorrelation is one. To investig.lle the behavior of the portfolio autocorrelation we plot il as a function of the portlolio nontradin~ probability 1( in Fig-me :t Id Itll' q = !i. 22. G6. and 244. Besides differing in sign. portfolio and individual autocorrclations also differ in absolute magnitude. the former heing much larger than the bner for a givell nOll trading probability. If the nontrading phenomenon is extant. it will be most evident in portfolio returns. Also, portfolio autocorrelations arc 1JI00IOtonically decreasing in if so titat lillie aggregation always decreases nontrading-induced serial dependence ill portfolio relurns. This implies thaI we .Ire lIIostlikely to lind evidcllfc of non trading in short-horizon returns. We exploit both these illlplicatiolls ill Ihe empirical analysis of SeCiioli :1.4.1. 3.1. 2 l~xlensions and GPIleraliwlions

Despite the simplicity of the model of Ilonsynchronous lradill~ in Se(!ion 3.1.1. its implications luI' ohs('I"vcd tillle series arc surprisingly rkh. The framework can bc cxtcnded and gencralized in many directions with lillie dilllculty. It is a simple mailer to relax the assulllption lhat individual virllt.tI rclurn;lare lID by allowing Ihe COIllIltOIl faclOt· 10 be attLOcolTclaled allel the diSH! "bances to be cross-sectionally correlaled. For example, allowill~ ji to he a stationary AR( 1) is conceptll,tlly strai~htr()rward. although the t'alclllatilOns become somcwhat lIIore involved. This specification will yield a dcco61position of observcd ascd llol11r.ulillg. ill what ,,'liS" is this .1ll\o('oIT('latioll SPIlriolls? The prclllis~~ or the extensive literature Ofl flUfls),flcilrouolfs lradiflg is that fI()fltrading is an outCOfllC or institutional features sllch as lagged adjllslIlll'llts anel nonsynchl'()flously reported prices, I\ut if nOJlsYllrhrollicilY is purposeful alld illlorlll 1

2

(3.2.6)

(3.2.7)

Despite the fact that fundamental value P; is fixed, 6.PI exhibits volatility and negative serial correlation as the result of bid-ask bounce. The intuition is clear: If P' is fixed so that prices take on only two values, the bid and the ask, and if the current price is the ask, then the price change between the current price and the previous price must be either 0 or s and the price change between the next price and the current price must be either 0 or -So The sallie argument applies if the current price is the bid, hence the serial correlation between adjacent price changes is non positive. This intuition

102

J. Markd ,HirTU.s/rurturr

applies more generally to cases where the order-type indicator I, is not IID,IH hence the model is considerably Illore general than it may seeJJJ. The larger the spread s, the higher the volatility and the lirsHmll'l' autocovariance, oOlh increasing proportionally so that the first-onkr autocorrelation remains constant at Observe from (3.2.0) that the bid-ask spread docs not induce any higher-order serial correlation. Now let the fundamental value 1',. change through tillie, 11111 slIppose that its increments are serially uncorrclated and independent of 1,.19 Theil (3.2.5) still applies, but the first-order autocorrelation (3.2.7) is no lonp;er - ~ because of the additional variance of 6.P,. in thc denominator. Specifically if a 2 (6.I'.) is the variance of 6.1>,', then

b.

< O.

Although (3.25) shows that a given spread.l implies a first-order alltocovari)nce of _s2 /4, the logic may be reversed so that a givcn autocovariance codlicient and value of /1 imply a particular value for s. Solving for J in (3.2~5) yields

i

s

= '2)-

Cov[M',_I, 6.1',] ,

n·'2·~)

hen(e s may he easily estilllated frolll the sample autocovariances of price cha~ges (see the discussion in Section 3.4.2 regarding the empiric;!1 illlplemcn'lation of (3.2.9) for further details). l:stimating the bid-ask spread lIIay seelll superfluous given the 1;l('t th;lt bid-+k quotes are observable. Ilowever, Roll (1984) argucs that tile (I'loted spre~d may often differ from the 1Jerliue spread, i.e., the spread between the lual market prices of a sell order and a buy order, In many installces, trans clions occur at prices wi/hin the bid-ask spread, perhaps hecause Illarketm kers do nol always update their quotes in a timely fashion, or hecause they ish to rebalance their own inventory and are willing to "beller" their quot s momentarily to achieve this goal, or because they '.

I

= Pal" + Pbh.

(3.2.18)

where I" (lb) is an indicator function that takes on the value one if the transaction occurs at the ask (hid) and zero otherwise. Substituting (3.2;16)(3.2.17) into (3.2.IR) then yields

I>,.

fl"

= -

E[I'·Ir2 U All" + E[P"Ir2 U B1h + Cala - Cbh

(3.~.19)

1'.. + C.. Q,.

(3.2.20)

E[p·lr2 U AlIa + E[P"Ir2 U B1h

(3.2.21 )

ell -

{c,.

if buyer-initiated trade

Cb

if seller-initiated trade

Q,.

{ +1 -I

-

(3.2.22)

if buyer-initiated trade

(3.2.23)

if seller-initiated trade

where A is the event in which the transaction occurs at the ask and B is the event in which the transaction occurs at the bid. Observe that PI! is the common information price lifter the nth transaction. Although (3.2.20) is a decomposition that is frequently used in this literature, Glosten's model adds an important new feature: correlation between 1'" and Q,•. If P is the common information price before the nth transaction and I'" is the common information price afterwards, Glosten shows that

Cov[!'".

Q"IPl

= E[AIPl

where

Aa A:; { Ab

if Q,.

== +1

if Q,.=-l.

(3.2.24)

That I'" ,\Ild Q,. mllst be correlated follows from the existence of adverse selectioll. If Q,.= + I, the possibility that the buyer-initiated trade is informalioll-hased will cause an upward revision in P, and for the same reason,

106

3. Market Microstrurlurr

Q,.=-I will cause a downward revision ill P. There is only one case in which Pn and Q,. are uncorrelated: when the adverse-selection componellt of the spread is zero. Implications fOT Transaction Price Dynamics To derive implications for the dynamics of transactions prices, denote hy f" the revisions in 1'.-1 due to the arrival of new public information between tra(y~s n-I and n. Then the nth transactioll price may be wrillell as

1'" == 1',,_1

\

I

rt~.~:)

+ t" + A"Q".

Taki/lg the first difference of (:~.~.~O) then yields

==

I \

1\"l,\,lIr-alid Ihis pn'sl'lIls a nllml)('r of prohlems for stalld;tnl ('CllIlOltll'lric Iltockls: oilsl'rvations are IllIlikt'ly to 1)(' idl'lIlically disIrihllll'c\ (sillcl' SOItH' obs( 'rva Iiolts arc vcr)' doselyspaced i II Iillll' wh ilc 01 It crs lIIay Ill' sql:uat('(1 hI' ltoUI SOl' davs), il is dil'flcIIlt to caplllH' scasollal !'Ifl'!'ls (sllch as liItH'-ol:;I'I\· rl')!;lIlarili('s) wilh silllple indicator fllildiolts, all.! «'/'('caslill)!; is 110 loltger a ~Irai)!;hll(.r\\'anl exercise Iwealls(' lite Iransaction t;il\l'S an' ranclolll. I\lso, Ir'lIIs:lI'lioll prin's are always qlloled in dis\Tell' units or lidonIITCn"" $0. I:!:. Ii,,· I'qllilil's, $()': Me"" Anacom,,; API) = Ah Proc.!uc!.O and Chemic.i.; CBS = Columhia Broa(k"~lillg Sy~tl·m:

CC\\

=

C;lpi\al Citic...o;, Al\C; KAB = Kaneb St"rv1(es.

discreten('ss is less problematic for coarser-sampled data, which may be wellapproximated by a continuous-state process. But it becomes more relevant for transaction price changes, since such finely sampled price changes typically take on only a few distinct values. For example, the NYSE Fact Book: J99.J nata reporL~ that in 1994, 97.4% of all transactions on the NYSE occurred with no change or a one-tick price change. Moreover, price changes greater than 4 licks are extremely rare, as documented in Hausman, \.0, and MacKinlay (1992). j)jlrrfif1lfJs and Prias

I

DislTetcnt'ss affects bOlh prices and returns, but in somewhat different ~ays. With respect to prices, several studies have documented the phenome~on of /nla rlus/nin{;, the tendency for prices to fall more frequently on cerp.in valucs than on others.:\\' For example, Figure 3.2a displays the histograms of the fractional part of the daily closing prices of the following five NYSE slo('ks during the three-year period from January 2, 1990, to December,31. 1!l92 (see Tahle 3.1 for sUllImary statistics); Anacomp (AAC), Air Prod~cts and Chcmicals (AP!)), C,oltunbi,l firoaIId n>llnds IIfl. alld the third rounds to the lIearest multiple of d. For simplicity, we shall consider only (3.3.3), although our analysis easily cxtcnds to the othcr two methods. Atthc hcart of the discreteness issuc is thc difli:rence betwecn the retllrn X/ based on continuouS-Slate prices and the return XI" based on discretilelH

O.252!l 0.2524 0.2523 0.2522 O.2!'121

O.2:12!) 0.252H 0.2:127 O.2!'12ti O.2!'12!'1

0.1254 0.1254 0.1254 0.1254 0.1254

0.1256 0.1256 0.1256 0.1256 O.12:,(i

0.1259 O.12:lll O.12511 O.125H (J.1257

0.1261 O.12tiO 0.12fiO O.12(i0 0.1259

O. 12{j~ O.12ti2 0.1262 0.12(;1 O.12til

O.Ofi27 0.0627 0.0627 O.Ofi27 0.0627

O.O(j2H 0.Ofi2H (l.Ofi2H O.()(i2H O.1l62H

0.0629 0.0629 (1.0629 O.t){WI O.O(j211

O.O(i30 0.0630 0.Ofi30 0.01;29 O.OG29

O.0/i31 O.()(i:\\ 0.O(j3\

$1

10% 20% 30% 40% 50%

1',-1

=:

= r,O'Yt,

10%

. T,,_I: - - ' ,

I'(T,,_I)

rt

(1-d d- )} - - - ,I + - 1'(1,,-1) P(T.-Il

(3.3.13)

which reduces to lhe Marsh and Rosenfeld (19Rfi) model in which the incrcments of stopping times are 110t lID. i

,

:\11110\'0'('\'(''', ~t"f: lh.: diSCII.It'\ioll ,at tlw I:lul of Section :t:t2 for ~om(" C3\'eab about (he mod\\uiUIl for tht's(" mudds.

,,. ...

]22

3. Market Microstructure

Limitations Although all of thc prcvious rounding and barricr lIlodcls do capturc pricc discrctcncss and admit cOllsistent cstimators of thc paramctcrs of the IInol)scrvcd continuous-statc price proccss, they suffer from at least three illlportantlimitations. First, ror unobscrved price processes other than geomctric Browniall motion, these models and their correspollding parameter estimators becOllie intractable. Second, the rounding and barrier models focus exclusively 011 prices alld allow no role for other economic variables thatlllight influcnce price behavior, e.g., bid-ask sprcads, volatility, trading volumc, etc. Third, and most importantly, thc distinction between the "true" and obscrved price is artificial at best. and the econoJllic interpretation of the two quantitics is unclear. For example, Ball (lUSH), Cho and Frees (I!IHH), Gottlicb and Kalay (1985), alld lIarris (I U90) all provide methods rO/" estimating the volatility of a continuOUS-lime prke process frpm discrele' 01>selved prices, never questioning the motivation of this arduous task. If lhe continuous-time price process is an approximation to actual market prices, why is the volatility of the approximating process of interest? One lIIight arguc that derivativc pricing models such as the mack-Scholes/Merton formulas depend on thc parameters of stich continuous-time processes, hut thost'i models arc also approximations to market prices, prices which exhibit ~iscreteness as well. Thcrefore, a case must he made for lhe ecollomic ~ rc\cv,!nec of the parameters of continuoUs-slate price processes to properly ~' motiVate the statistical models or discreteness in Section 3.3.2. ,;~ h~ the absencc of a wcll-articulated model of "truc~ pricc. it secms U\Inatur~lto argue that thc "truc" pricc is continuous, implying that ohserved discre~e market priccs are somchow less genuine. After all, the economic dcfini~ion orprice is that quantity oflllllllerairc at which two mutually COliscntif~economiC agents are willing to consummate a tradc. Despite thc f;ICt that if stitutional restrictions llIay rcquirc prices to fall on discrete values, as lon' as both buyers and sellers are aware of this discreteness ill advance and af still willing to engagc in trade, thcn discrete prices correspondi\lg to ma tct trades arc "true" prices ill every sense.

I

J.J.J The OI1/rrl'd l'mbit Modrl

To ad1rcss the limitations of the rOllllding and barrier lIlodels. Hausmall, Lo, anf'! MacKinlay (1992) propose ,11\ altern'ltive ill which price rlulIIgr.\ arc 1lI0dclkd directly using a statisticallllodel known as ordered !)Tobit. a technique used niost frequently in empirical studies ofdepe\ldent variables that take on only a finite mnllbcr of values possessing a \IaluL,1 ()nlering.~7 Heuristically, '7For cxal1\pl~. Ih~ dqwml"111 \'ariahlt- lIlif(hl h,' Ih .. kwl Ill' ,-,Iucllion. as 111'

(u •. ,-x;f3) ",

aq~umellt

.(. , 1 1=

Ill,

whe\·e aA(W.) is written as all ofWk to show how the conditioning varilbles enter the' conditional distribution, and (.) is the standard normal cUlllulative distribution functioll. To develop some inlllilion for the onlcn~d probit lIlodel, ohserve lhat the probability of any particular observed prin~ change is d{'tCl'mined hy where the conditiony a\lowing the data to oetet'minc the partitioll bOlllldaries 0, thc cocnicicnts (3 the conditional JIIl'an, and the cOllditiollal variall(,c ak~' the oJ(kred probit Illodd captures the elllpirical rl'latioll Iwtwl'cn the unobservable continuous st;lle sp;lce S' ;uHllh(' observ('d di"'J'('Il' state space S as a i'lIl1C1ioll of' the economil' variahles X k ;Illd W",

or

M,:xilllllllll.ihrlilwod I~Sli1/llllioll Let hCi) he an indicator variable whirh takes Oil the value one if the ITalizalioll of' the hth observatioll Vk is the ith state ,Ii, and zero otherwise, Theil the log-likelihood function C (il!' the vel'lOl of' price changes Y I 1', y~ l'" j', conditional Oil Ihe expl -./IH .00

.00 .OH .101

10

.O~I

10

-~') -.14 -.09 .00

lir~I~)nll'r;lIl1ororrcLlIi(lIlIJl.HJ"ix i",'ol"tlw (-I x I) slIln'('( lor I ,;' '.~' ,; 1;'111' of ohs('I"\'('" fl'ltlfll:-' lD tell eqllOlI-weig-htcd sile-~orted pOrlfolios w .. illg d~'il)'. week.ly. alld monthly NYSEAMEX (0111111011 stock returns data from th(· CRSP lilt·s for Iht' lime pt:riod .lilly :l. 1962 In Den'mllei" :~(). I!)~H. Storks ;uc assiglled to portfolios 'lT1ll1lall>' lIsing th(~ IH01rket value.1t the cnd urlhe prior }'(·;If. If this market value is missil1K Ihl" cud 01 year markt,t \'.dut' is used. IflxHh mark('1 val lit's ,II"(' IlIis..... ill~ lil(' stork is 1I0t illt. Itlflt'd. ()lIly .\t'nll ilu':-, Wllit nllllpl('I(' daily 1('1(11 II hi!'lloril':-' withiu a gi\'('IJ mOllth arc inrilHh-d ill Ihe d'lily U'WIIiS f.lklll.tlio ll ..... ,;' i. . th(' U'tlil It 10 the portfolio containin g securities with tilt.' smelliest menkel ",,,III('s f~)r thc portfolio of smallest stocks to IlIU = 0.5 for tl!c ponlc>lio of the 11rgcst. The sccond row rcports similar autocorrclatiolls implicd hy 11011l~ading probabilitics cstimated fmlll daily autocorrclations using n.IAI). The largcst implicd !irst-order autocorrelation for the weekly equalw ightcd returns indcx reported in Table 3.6 is only 5.9%. Usillg direct c. timatcsofnontrading via lIegativc sharc priccs yields an autocorrelation of I~' s than 2%. Thcsc magnitudcs arc still considcrably smallcr than the '21 'Yv s· IIlplc autocorrelation of thc e<Jual-wcigllled indcx retUnI. In SIlIllIll,II")', tl e rcccnt cmpirical cvidence provides lillie support for nOlllrad'lIlg as all important source of spllrious correlation in the rcturns of common stock o~er daily and longcr frcquencies. 41

f.

3.4.2 1~,\tiIlUllil/li till' I:J/rrlivc

Bid·A~" .'>/))"('(1/[

In implementing thc model of Scction 3.2.1, Roll (1981) argucs that thc percentage bid-ask spread s, may bc more easily intcrpreted than thc al)solute hid-ask spread s, and he shows that thc Ilrst·urdcr 'lIl1ocovari'lllcc of simple returns is relatcd to .I, in the following way: Cov[ R - l '

s,

.\~1

.\;

--4 Hi

H, I

-

JJ>1I 1',.

~

s,~

4

(:H.'2) (:H.:\)

'

where 5, is defincd as a pcrccntage of thc geometric average of the avcragc bid and ask priccs Po and Ph' Using the approximation in (3.4.2), thc pcrcentage spread may be recovercd as (:H.4)

Notc that (3.4.4) and (3.'2.9) arc only wcll-deflllcd whcn thc returtl al1tocovariance is negalive, sincc by conslruction the hid-ask bOllllce elll only inducc negative !irsl-order serial correlation. Ilowever, in praclice, po~i­ tive scrial correlation in returns is lIot IInCOllllllon, and in thcse cases, Roll simply defines thc spread to he (sc(' footnotcs (l ,lIld b of his Tallk I):

Hnuudollkh, Richa,.dsoll, alld Whit,·law 1199:», M"('h (I!I!I:I) alld Sias alld Sta,b (I!I!I-!) prt·~ClIl

additional empiricfll results on l1onlra ~ (/'k'-,

+ I'~'_I)

if

1'._1

~ U'h'_1

+ 1':'_1)

if

I'k I < W'k'-I -I- 1';_/)'

The spccific ation of X~fJ is then givell by the followillg expressi on: X~fi

=

+ fi~ Yh- I + fi:1 Yh--~ + fi·1 Yk-:I + /I,.SI':,OO._I + fib SI'500.- 2 + fi7SI'500h_:1 + fiXIlISk-1 + fi!,IIIS k- 2 + fiIOIBSh-~ + filiI '/i..(Vk-!l· IBS H I + fil~ 1·I!.(\'k_~) .IIIS._ 2 ) + fil:11 T), (Vk_:l) . IIIS k -:1 ) . fil 61h

The v,lriablt- 61h is illcillde d in .\.

10

allow

fill

clock-lilll!' df(-cls

Oil

Ihe

J. All11kl'l

MiOI lJ/III r·/III l'

rOlu lilio llal lIIeall of I·;. II plic es an' slah le ill Irall saCl ioll lillie ralh er Ihall doc k lillII', Ihis (odf icin ll shou ld 1)(' zel'O. l.a~~l'd prir e chall ~(,s arc illcl ilded 10 aC("OIlIlI lor sni;t 1 dq)( ,lIc1 enci es, and la~~ed reI lints 01 Ill(' SJ(-I':)"/) illd( 'x fllll lln prin ' an' illci llded 10 acco unl for mark ('I-w ide dkC ls 011 prir e challg-(·s. 'Ii) m('asU((' Ihe prir e ililp acl or a Irad e p('r unil \'olll llle, Iht' 1('I"In '1;,(V~_tl is indl lded , whi dl is doll al volll llle Irall sllll' l\l('d ac("ordill~ 10 the Box alld Cox (1!lfi·l) Ilalls ll'I"I lIatio ll '1;,(·); '1;.(\ )

x" I'

whe re I' E 10, II is also ;1 para lll('l er 10 he eSlil llale d. The Box -Cox IrallSf()l"Jllalion allow s doll ,lr voltlll\(' 10 ('n(( 'r into Ihe cond ilion al lIIea ll lIoll lin('arl y,;, parl intla rl), imp orta nt inllo valio n sinc e cOll llllo n intll ilion sugg esls Ihal plic( ' illlp arl ilia), exhi hil ('CO IIOllli('s of scal e with resp eC! to doll ar \'0111111('; i.('., alth oug h IOla ll'ric e imp arl is likely to incr ease with volu me, the lIlar~inal pric( ' illlp;'!'1 proh ahly do('s no!. The Box -Cox trall slim nati capIlIn' s the lil)(' ar sp('c ifica lioll (I' = I) and conc ave spec ifica tion s lipoll to alld illrludin~ till' lo~arilhlllic rlillClioll (I' = 0). The eSli mate d cllrv atur e or Ihis IraIl Sf(.n llalio ll will pl;,y all illlpol"lalll rolt- ill Ihe 1II('''SIl)"{'I I)(,1I1 of pric e illlp act. The Irall sforl ll('d doll ar \'Ollllll(, vari ahk is inle ract ed with IBS k _ I , an indi calo r of ",hel hl'l" Ih(' Irad (' was hll)TI~illitiated (IBS = I), selle r-ini h tiat( ,d IIBS~= - I \. or il)(I( 'I('fl llina ll' (IBS k =/)) . A posi live fill wou ld illlply thaI hlly( 'r-in ilial ed Ilad( 's tl'lId 10 pllsh pri('(~s up and selle r-ini tiate d tlad es t(,lId 10 driv e prir es dow n. Such a rela lion is pred icte d hy seve ral info rm"l .ionhas( 'd lIIod eis of Iradill~, e.g-., Easl ey and O'H ara (I9R 7), Mor eove r, Ihe lIIag llilll de of /ill is II ... p('r-IIl1il \'0111111(' illlp act Oil Ihe cOll ditio llal lIIeall of r~', whic h lila), he f(~adily Ir.lII slale d into the imp act Oil the ('oll dilio llal proh ahil ilics of ohs( 'rv('d pric( ' chan~('s, The sig-n and lIIa~nit\l (-('3.1i4) (- " •. 31i) -O.:l1i!/ -0.~79 (-~I.:,:,) (-:1.:\7)

-0.7·1\) -0501 -0.7!JI -O.HO:l (-7.HI) (-2.H!J) (-17.:{H) (-2:1.01)

-0.174 (-1O.2!J)

tU)79 (1I.!lH)

0.122 (47.37)

(12.97)

-().~!l!l -0.177 -0.022 -0.3111 ( -:l.Ii'l) (-0.17) (-I :,.:17) (-I!/.7HI (l.W,() O.03H tUII.1 0.0:12 ( I.HO) (U.5:,) (~5ti) (·I.rll I

U.!H7 (IH57)

(l.O:lt; (2.H:'»

O.III!/ (7.70)

(l,007 (0.5!1)

0.~17

-0.:\70 -0.:1-10 -O.IH1 -O.IH4 (-:\.fiG) (-0.75) (-I:>.:\H) (-IH.II)

(1.03(i (O.!iS) tUlI " -o.()(/(; (15Ii) (-0.34)

0.01:, (1.:,4 )

0.011

0,1/1·1

(~,:,'I)

('U!1)

11.000, (\1.()9)

n.m!)

II.OW,

MOlX;IHll1U likdBu)(){l l·~timat('!\of the ":-.Iopc,''' ("odJiril'Ht!O.ol thl' oH.h.'ICd I)lohit moth'llu .. lIall!'o~

pO". II "~'" ':' """'~"h'...~, ~'.""'~'~ M.,,,,, ...., 0"',, .... ,,," II "". "",."" ....,

"e '.

Chtolin' lilt' 1IH' 0.17 O.I!! 0.20 D.22

1l.24 n.2:, 0.27 0.29 0.31 0.32 0.:14

n.:\(; 0.:\7 n.:I!1 0.47 0.:,4

n.w CUili 0.71 1I.7h 0.8:1 0.1111 1I.!)2 O.!)4 O.!)I; O.!!II O.')!) 1.00 LOt) I.Ot)

0.011 0.11 0.14 0.17 0.20 0.2:1 O.2ti O.2!1 IU2 0.3:, 0.38 nAI 0.44 0.41; 0.49 0,!',2 0.;"1 0.:,1; ()59

IUil lUI 0.711 CUI·l CUI!1 H.92 II.!H 0.!)7 n.!'!) II.!)!) LOll 1.00 1.00 1.00 I.I~I

1.00 1.0\1

pow~r

is reportc(1 for a tcst with a !'lilt" of' 5%. Th(' sample sil.c is the IHlIl1h('1 of t'\'('rH included in the study. Olnd rr is the sqllal't' root of the average \'~tri"I\('(' of the abnormal return ",cro~'" f\nu~.

The

uh~crv"tions

of abno~mal relurn is difficult to delect with the larger variance of 0.00 IIi. In conlraSl, when the CAR is as lar~e as 15% or 2.0% the 5% lCst slill has reasonable power. For example. when the abnormal return is 1.!'J'Yo and

4.6. Analysis ,,/Powl'r

171

0

'" 0

!i'X,. In contrast . when lirms decide to usc straight dcbt linancin! {. Ihe averag'~ ahllorlll al rcturn is doser to I.ero. Mikkels oll alld Partch (1!IH(i) lilld tht' ;IWLlg" ;\I>11orlllal return for debt issues to be -O.~3% for a sample of 171 issues. Finding s such OIS tllesc provide thc fud for the dcvelop ment of new theories . For example . these external financin g results Illotivatc thc pecking order theory of capital Siructur e develop ed by Myers and Majluf (1~IH'I). A major success related to thosc in the corpora le fillancc area is the implicil accepta nce of event-st udy methodo logy hy Ihe lJ .S. Suprem e Court for dctCl'm ining material ity in imider trading cases and li>r determi ning appropr iate disgorge lllent amollnt s in cases of fralld. This implicit aCtTptance in lhc I ~IHH Basic, Incorpo rated v. J .cvinsoll case alld its importa nce for sccuritic s law is discusse d in Mitchell and Neller (1~194). TIH'rc havc also been less slltTt'ssful applicat iolls of cvent-st udy IIlcthod oIOh'Y' An importa nt characte ristic of a slIcccssful e\'t'nt stud), is the ability to idellliry precisely Ihc date of the cvelli. In cascs \\'hne the date is difficult to idelllify or the evcnt is partially antirip'i lcd. eVCIII studics have been less useful. For cxample . thc wealth dferls of n'!{uIator), changes j(>r alkned entilics can hc diHicult to dClect usillg t'\'t'nt-st udy IIlelhodolol-.'Y- The problem is that regulato ry challgc.~ arc often dcbated in Iht' polilic" ,IITlla ovt'J' time alld any aC(,(>lllpanying wealth cf(('((S will he ineol porated graduall y inlO

Ihe vahlt, of a IOIPOl;lIioll as Ih(' probahilily of Ihe change heillg adopled illcreascs. \);11111 alld .lallH's (1~IH:!) discllss Ihis isslle ill Iheir sludy of Ihc illlpaci of t\cposil illll'n'sl ralc t"\'ilillgs Oil Ihrili illSlilllliolls. They look OIl challges ill ralc ('(·jlillgs, hili dccidc 1101 10 cOllsider a changc ill I!173 hecallsc il was lhlt' 10 Icgislalivc aClioli aud 11('11('(' was likely 10 have Iwcll anlicipalcd hy Ihe market. Schipper alld TIIOIllPSOII ( I !IH:{, I !IH5) also ('nCO\llller Ihis prohll'lIl ill a sllidy or lIIergl'l'-I"I'lall'lI rcgulaliolls. They allclIlJlI 10 cirnllllvc:nl Ihe problelll of allticipated reglllatory changes by idenlifying dalf's when the I'rohahilil)' of a IcgUiatol'l' challgc illcrcascs or decreases. Ii owevc 1', lhcy find largely illsigllilicalll n·slllts.lcaving 0JlCJI Ihe possihilily Ihallhc absellce of dislinn evelll datt's accounls {ill' I h(' \;tck or wealth effeclS. Much has I)('clilcal'll('d frolll Ih(' hody ofrf'sf'an:h thaI uses ('V('llt-sludy lJlt'llloC\ology. Mosl gCJlnally. ('venl sllldi('s have showll Ihat. as we would (')(.)1('('( ill a r;lliOlI'.llllIarkt·lp!a('(·. prin's do r('sJlond 10 new inrOI'III;lIioll. We ('''1)('('\ Ihal CV('llI~llIdi('s will (,Olililllll' 10 Ill' a vaillable allel widely IIsed 1001 ill I'UllIOlllics alld lillaIH ....

4,1 Show Ihal whellllsillg till' 1\I;1I·1;.('IIIIOd('IIO nu'aSlllT allllol"lllal retlll"\lS, Ihe salllple ahllol'lllal rellll"JlS frolll eqllalioJl (4.4.7) an' aSYlllplolically illc\ept'llIl('1I1 as Ihe knglh 01 lIlt' ('slilllalioll ",inllow (1. 1) ill('J'('as('s 10 inlinily.

4,2

\ I

\

\

YOII are giv(,11 Ihe li,lIowillg illlimnalioll for an eW1I1. AhllorJllal I'l'IIlI'llS an' salllpkd al all illl("l"val of 0111' day. Th(' (,vcnl-window Ienglh is Ihn'(' days, Th(' IIll'all ahnol'ln,tI relllrn owr Ihe f'V{'nl window is (I.;\'}{, pn da)'. '1'011 ha\'(' a salll)llc oi" :,0 ('VI' II I ohservations. Tht' ahnorJllal r('lllrns an' int!('p(,lld('11l across thl' ('\'('nl observalions as well as across ('Vl'llI days for a giv(,1l CVI'III ohs('I"valion. For :?:, o/Ihe ('\'('nl ohservalions IIII' daily slandalCl d('vialion of Ihl' alllHlllllal 1'('1111'11 is :~'y,) and lilr llle rl'mainillg 2:, OhSI'IT;Iliolls Iht' flail), siandard dl'vi;lIioll is li%. Civclllhis infilrmatioll, whal would 1)(' 111(' pow('\" of Ihe leSI for ;111 ('v('nl sllldy IIsing Ille clIlIllllaliVl' ahllormal rl'tllrn test statistic ill (''1l1alioll (·1.·1.2~)? What would hI' Ihl' pOWl'l' IIsillg lhl' stalldanii/('d C1mlllialiv(' ahllol'lllal n'llIrII lest slalislic in eqllation (4.4. ~·I)) For Ihl' pow('r (,;licubliollS, aSSIlIl\(' lilt' st;IIHbrd dl'viation Ilf II,ll' abnol'lllal rt'llIrllS is knolVlI. 4.3 Whal wOllldl)!' Ih(' answl'I's III qlll'Slioll '1.2 il'lhe IIll'an ahnllrmal r('llIrn is O.li'Y., pl'r dav f(,r Ihe ~:, linll.s willi Ihe larger siandard dcvi;llion?

The Capital Asset Pricing M0ge1

ONE OF TilE IMPORTANT PROBLEMS of modern financial economics is Ihe CluantifIcation of the tradeoff between risk and expected return. A1tho~gh common sense suggests that risky investments such as the stock market will generally yield higher returns than investments free of risk, it was only 'tith the development of the C~pital Asset Pricing Model (CAPM) that economists were able to Cluantify risk and the reward for hearing it. The CAPM implies lhatlhe expected return of an asset Illust be linearly related to the covariance of ils return with the return of the market portfolio. In this chapter we discuss the econometric analysis of this model. The chapter is organized as follows. In Section 5.1 we briefly review the CArM. Section 5.2 presents some results from efficient-set mathematics, including those that are important for understanding the intuition of econometric tests of the CAPM. The methodology for estimation and testing is presenled in Section 5.3. Some tests are based on large-sample statistical theory making the size of the test an issue, as we discuss in Section 5.4. Section 5.5 considers the power of the tests, and Section 5.6 considers testing with weaker distributional assumptions. Implementation issues are covered in Section 5.7, and Section 5.8 considers alternative approaches to testing hased on cross:sectional regressions.

5.1 Review of the CAPM Markowitz (1959) laid the groundwork for the CAPM. In this seminal research, he cast the investor's portfolio selection problem in terms of expeeled return and variance of return. He argued that investors would optimally hold a mean-variance efficient portfolio, that is, a portfolio with the highest expected return for a given level of variance. Sharpe (1964) and l.intner (I !l65b) built on Markowitz's work to develop economy-wide implicatiolls. They showed that if investors have homogeneous expectations IRI

182

5. The Capital And 1'lirillK M{)(M

a~d optimally hold mean-variance ellicient portfolios then, in the ahsclKl' o market frictions, the portfolio of all invested wealth, or the market port~ lio, will itself be a mean-variance ef/lcient portfolio. The IIsllal CAI'M e~uation is a direct implication of the mean-variance efficiency or the mark~t portfolio. . The Sharpe and l.intner derivations of the CAl'M aSSllllle the eXisH~IIlT o~ lending and borrowing at a riskfree rate of interest. for this version or tlie CAPM we have for the expected return of asset i, •

I

E[R;]

~

=:

Ilj

+ fiilll(E[R",]

{Jim

=:

Covlll" Nm ] , Var[Ntn ]

-- IV)

(:;.1.1) (:).I.~)

wi erC Il m is thc relltrn on the market portfolio, and Ilj is the return 011 the ris free asset. The Sharpe-Lintner version can be most compactly expressed in ~erms of returns in excess of this riskfree rate or ill terllls of ,xrru n·lum.l. Le\ lj represent the reUlrn 011 the ilh asset in excess of the riskfree r,lle, Z; H. - Hj . Then for the Sharpe-Lintner CAPM we have

:=

E[l,]

Plln

iJimml",]

([d.:1)

Covl1.j , 1.111 J Var(l,n)

(:>.1.4 )

where l,. is the excess return 011 the market portfolio of assets. Because the riskfree rate is treated as being nonstochastic, equations (5.1.2) alld (!d.4) are equivalent. In empirical implementations, proxies for the riskfree rate arc stochastic and thus the betas can differ. Most empirical work rebting to the Sharpe-Lintner version employs excess returns and thus uses (5.1.4). Empirical tests of the Sharpe-Lintner CAPM have focused on three implications of (5.1.3); (1) The intercept is zero; (2) Beta completely captures the cross-sectional variation of expected excess returns; and (3) The market risk premium, E[z..J is positive. In much of this chapter we will foclls on the first implication; the last two implications will be considered later, in Section S.B. In the absence of a riskfree asset, mack (I ~172) derived a more general version of the CArM. In this version, known as the Black version, tht' expected return of asset i in excess of the l.era-het'l return is linearly related to ito; beta. Specifically, for the expected return of asset i, E[ R,l. wt' havc

E[N,)

=:

Elll"",]

+ fl,,,,(Elll,,,1

- E[ll. .. ).

ll,. is the return on the market portfolio, and It .. is the return on thl' um· portfolio associated with III. This portfolio is defined to he the portfolio that has the minimullI variance of all portfolios nncorrelated with 1/1. (Ally

brla

5.1.

1I1'l/jl'w

oflhl' CAI'M

18;\

other 11l1('olTdatcd ponfolio wOllld have till: same expcctl'd retlll'll, hilt a highl'l'variancl'.) Sincc it is wealth in n~.d tnms th"t is relevant, lill' the Black Illodcl, retllrns arc gener.lily stated on ,III inll.lliOll-"djusted hasis and fI"" is ddilll'(t in tt'nns of real retllrllS, fi,m =

Cov I N,. N", I -----.-. V.u"[l{", I

Econoilletric analysis of the Black version or the CAI'M treats the I.('IO-h('ta ponfolio rellll'll as an u1Iobserved qu.,ntit)', making the 'lll'll),sis more con.plicatcd than that of the Sharpe-Lilll1l('" vl'l'sioll. The Black vlTsion Gill he tested as a restrictioll on the re;!I-retul'll market III(HIl'I. For the real-retltrn m'lrkct model we have E[N;] = u""+/i""E[N,,,I,

"Ild tht' ill'lllicati()11 or the Black vt'l'sioll is

ex"" = E[N"",]

(I -

fI",,)

Vi.

Ud.Hl

III \I'ol'ds, the Black model restricts thl' ;I~set-spt'cilit illtlTt't'j>l of the realretul'll market modcl to be equal to the ('xlH't'tt'd l('lo-Ill'ta portli,lio retul'll tilll('s olle minlls the asset's beta. The CAI'M is a single-period Iliodd; hence (:',.1.:\) and (:,.1.:1) do lIot h'll'c .1 tilllc di mellsion. For ecollometric analysis of till' Illot kl, it is nefessary to add an asslllnption concernin!{ the time-series heh.2.1H)

== 0

where l1al' is the beta of assel (j with respect to portfolio Result 5': For the expecled return of a we have ,

'1wc J1~xt

Ii" = (I - fia/,)Il,,/,

+ /3,,/,/J./,.

I). (!).2.1!1)

introduce a riskf"ree asset into the analysis alld consider portfi>lios Fomposed of a combinatioll of the N risky assets and the riskfree assel. Witl, a riskfrce asset the ponlolio weights of the dsky assets are not COIIslraj~led 10 slim to I, since (I - 'w' L) Gill be invested in the riskfree asst't. I

187

5.2. UI'S II lis from FJjirielll-Sfl Malhl'/lwlirJ I'

I'

" _. - - -' - - _ •• - - - - - -'"

If} (J

Figure 5.1.

MilliJlllllll·\'rllilll/(f'I'orl/oliol lVii/will fli.lk/i,·e A.ufl

Given ;\ riskfree asset with return UJ tilt" minillllllll-\'ariance portfc"fio with expected return ill' will be the solution to the constrained optimization mill w'Dw w

sul~iect to

(5.2.21)

As in the prior problem, we form the Lagrangian fUllctioil I., differentiate it with respect to w, set the resultillg equations to zero, alld thcll solvc for w. For the Lagrallgian function we have I.

==

w'Dw

+ 8 (PI' -

,,1 minimulll·variance portfolio; thai is, lhe lIIarket portfolio is (lIl the c1~\lcient part of the constrained mean-variance frontier. We can suhstitute Y'I into (5.3.62) and (:1.3.63) to obtain (-J' and -t' without resorting to an itclrativc procedure. I We can COllStruct an approximate test of the lIlack version using relllrns in \bx(ess of y as in ([1.3.GH). Ir y is known then the same methodol0h'Y us(·d to ~onstructlhe Sharpe-l.intnC!' version F-test in (5.3.23) applies to testing the null hypothesis that the zero-heta exn'ss-relllnl nl 0.O1l2 0.OH2 0.10:,

0.OH2 0.1:\0 (1.2·17 IUHO

0.(11;1< II.O'lH 0.17·1 0.21;7

0.0:,7 0.077 11.12,1 IUKI

0,101 I).IHO 0,:17-1 (J,r,70

0,07H 0,12H n.2lil O,'I\(i

0.\)(,1 O.O9?> 0.17:, 0.2HO

D.121 o.:!:17 0.:,02 O,nl;

O.OHI) O.IIi:1 o.:\!,,(j 0.:,(;:1

O.(Hi:, 0.1 III 0.2:\1 O.:IH!l

o.on

= II;')(, 0.111:1 0.17'1 (1.:1,10 n.r,OH

= lIi% 0.1:\,1 O.2~>i

O.!,01 0.711 = lIi%

O.lli7 0.:1:12 o.(iD O.H,I:,

---Th"

;llIcllI.tlivc h)'I)('lh('~is i~ rli;u'aru'li/C'cI

h)' lht'

\'.1111(' of llit' ("xllI'(l('d ('X("('!'oS 1«'1111"11

and

Iht' \'alm' of 11i(' ~I.llld.lni .t.uHLu·({ d{'\i.,.ion oi tht.' l'x(T~S l"C.'t\lrn oi

the 1.tngnH·y ptH ltolio. The lHarkcl portfolio is .\\.'';'UIIH'(} to h.nT .m l'Xpt't In} ("x( t's..11 return of H.O'i; . .111-vahl('

Five-year subperiods )/6~)-12/fi!)

2.0~/l

0.019

20./lli7 0.022

1/l.1~2 O.O4/l

n.IU!', n.u I:.

1/70-12/74

2.1~1i

O.O:~9

I

1.914

O.O(j(i

19.179 OJnH 17.47ti 0.()(i4

2U~/7

1/75-12/79

21.712 O.UI7 1!/.7H4 0.031

1/80-12/84

1.224

0.:\00

13.:\78 0.20:1

1/85-12/89

1.732

O.JOO

IH.164 OJ/52

I1.HI8 0.297 1(i.()15 0.098

I

1/90-12/94

1.153

0.314

Overall

77.224

0.004

I

\,

12.1i1l0 0.212 1(){i58(i

**

11.200 0.:142 94.151 0.003

n.UIH

27.922 0.002 1:I.(Hi!; 0.22U ](UII :. 0.07(; 12.:\79 (1.2(iO 113.78~,

**

ten-year rubperiods

\ I/r"-12174

2.400

0.013

23.883 O.OOH

22.190 O.oJ:1

24.1i19 O.O()!;

1/75-12/84

2.248

0'

2J.190 0.020

27.192 0.002

1/85-12/94

1.900

0.0,,:1

\9.281 0.037

18.157 0.052

16.373 O.OH'.I

Overall

57.690

O.()(lI

1i[J.{ifi7

fiJ.8:!7 0.001

(iH.215

2.1:,9

0.020

2Uil2 0.017

21.192 0.020

n.17(; O.OJol

**

**

irty-year period 1/65-12/94

\*Le~~ than 0.0005. R~,ult\ are for ten value·weighted portfolios (N = HI) wilh slOe\ wealth. The requirement that the factors be pervasive permits investors to diversify away idiosyncratic risk without restricting their choice offactor risk exposure. Dybvig (1985) and Grinblatt and Titman (1985) take a different approach. They investigate the potential magnitudes of the deviations from exact factor pricing given structure on the preferences of a representative agent. Both papers conclude that given a reasonable specification of the parameters of the economy, theoretical deviations from exact factor pricing arc likely to he negligible. As a consequence empirical work based on the exact pricin~ relation isjustified. Exact factor pricing can also be derived in an intertemporal asset pricing framework. The lntertemporal Capital Asset Pricing Model developed in M(~non ( 1973a) combined with assumptions on the conditional distribution of returns delivers a n1Ultifactor model. In this model, the market portfolio serves as one factor and state variables serve as additional factors.! The additional factors arise from investors' demand to hedge uncertainty'about futllre investment opportunities. Breeden (1979), Campbell (1993a, 1996), and Failla (1993) explore this model, and we discuss it in Chapter 8 .. In this chapter, we will generally not differentiate the APT from the ICAPM. We will analyze models where we have exact factor pri!=ing, t~at is,

IL

= tAo + B>'K.

i (p.1.8)

There is sOllie flexibility in the specification of the factors. Most empiri(';\1 illlpl('IlH'nt,ltions choos(' a proxy for the market portfolio as one factor. Ilow(',,('\', different technic]lles are available for handling the additional facrors. We will consider several cases. In one case, the factors of the Apt anef lhe state variables of the I(,APM need not he traded portfolios. In other rases the factors arc returns on portfolios. These factor portfolios arc called lllillli(kin~ po.rtfolios becallsejointly they are maximally correlated with tIlt' factors. Exact factor pricing will hold with stich portfolios. Huberman, K;llldcl, and Stambaugh (19f\7) and Breeden (1979) discuss this issue ill the context of the APT and ICAPM, respectively.

2 2

6. Multi/actor Pdrill/!. MI}{ft./.1

6.2 Estimation and Testing In this section we consider the estimation and testing of various forms of the ex ICt factor pricing relation. The starting point for the econometric analysis of the model is an assumption about the time-series behavior of retul'lls. W will assume that returns conditional on the factor realizations arc II [) th ough time and joilltly multivariate 1I0rmai. This is a strong assumption, bl~t it docs allow for limited dependence in returns through the time-series behavior of the factors. FunhernlOre, this assllmption can be relaxe(1 by casting the estimation and testing problem in a Generalized Method or Moments framework as outlined in the Appendix. The GMM approach for llIultifactor models isjllst a generali7.ation of the GMM approach to testing the CAPM presented ill Chapter 5. As previously mentioned, the multil;lCtor models specify neither the number of factors nor the identification of the factors. Thus to estimate and test the model we need to detCl'minc the bctors-an issue we will address in Section GA. In this section we will proceed by taking the JIlllnber of brtms and their identification as given. We consider fOllr versions of the exact factor pricing model: (I) F,tctors are portfolios of traded asseL~ amI a riskfree asset exisL~: (2) Factors arc portrolios of traded asseL~ and there is not a riskfree asset; (:{) Factors are 1I0t portfolios of traded assets; and (4) Factors arc portfolios of traded assets and the factor portfolios span the mean-variance frontier of risky asseL~. We lise maximum likelihood estimation to handle all fOllr cases. See Shanken (1992h) for a treatmenl of the same fOllr cases lIsing a cross-sectional n:gression approach. Given thejointllormality assumption for the retnrns condition.1! 011 the factors, we can construct a test of any of the fOllr cases using the likelihood ratio. Since derivation of the test statistic parallels the derivatioll of the likelihood ratio test of the CAI'M presented in Chapter 5, we will not re)leat it here. The likelihood ratio test statistic for all cases takes the sal\le gelll"J"al form. DeflllingJ as the test statistic we have

(ti.2.1)

wlwre t and t" arc the lII'" - J.L/,,) whcre >." is Ihe (Kx I) vcclor of faclor risk premia, for Ihc constrained Illodel, we h,l\'c (G.2.:lS) Thl' constrained model estimators

' . , , olle lIe.:ds lIl.:aSllrcs of the faclOr sellsitivily malrix B, Ihe risk\"ree rate or the zcro-bcla expccted retllrJI Au, alld thc LInnI' risk prcmia >'1:. Ohtainillg mcasures of B and the riskfree ratc or the expectcd I,ero-heta return is stra;~lllforwanl. For the given case the constrained maximllm likelihood estimator S' Gill be IIsed lor B. The observcd riskli'cc 1'.11(' is appropriate li)r the riskl'rec asset or, ill the GIS(~S without a riskCnT asset, the maximum likelihood estim,l\or Yo Can be IIscd for the cxpertctl L('w-lJeta r('tllm. Furthcr estimation is lIeccssary to form ('~Iimalcs of the Linor risk premia. Thc appropri,lIe procedure varics across the [(Jill' cascs of cxact htctor pririllg. [II the case where the faorLfo[ios are /;\('101" I)\n tlln(' is no riskfree asset, the EIClOr risk prelllia (all be estimatcd IIsing Ihe !lith-rCl\cc bctween thc Slll1lplc IIIcall of the factor portl(Jlios and thc estilllalcd {(To-beta rctllrn:

(i.:D)

III this casc, an estilllator of the v;lriall(T or >.1;

IS

(li.:\.1)

().

ill 11//1/1111111 "ril"illg .\lII/It'll

II'hnl' \-;;;-rl)'ul i\ InJlII «(;.:.!.~7). Till' tic! Ihal alld Yu are indl'pI'llIll'nl lias 1)('1'11 Jllili/l'd 10 WI IIII' covariancl' Il'rlll ill (Ii.:!.'!) 10/('('0. In Ihl' C\SI' I\'hert· Ihl' btlors an' 11111 Iradl'd portfolios, an eSlilllator Ill' till' Vl'nor 01 bClo .. ri~k prl'lIIia AA' is Ihl' S\lIn of Ihl' I'sliJllalor of Ihe lIIeall oflhe 1;1('(01' n';tli/;lliolls alld Ihl' I'slimator of' YI,

ii."

I I

All I'slilnalor of II\(' ";lriaJl(,(' of >-i-: is

I

I

. \'arIAi-:1

=

1.-:rn,,+Varl-rd,

(Ii. :l.Ii)

when' \-;;;'[1'11 i~ lrolll (li.~.'I!i). I\l'callse ilfi-: and 1'1 an' illdl'(lI'llIicllt Ihl' lovarianfl' Il'nll ill «;.:U;) is 'l'fIl. Thl' 1IIIIIIh casl', whert' Ihl' faclor portfolios span Ihl' nll'an-variancl' fr(Jnlin, is IIll' sanll' as 1111: firsl casl' exn'pllhal rcal rl'llIJ'llS are SlIilSlilllll'd

\

\ \

I

lill' exn'ss n'llIrtls. 111'1'1' Ai-: is 1111' I'l'l'Ior of !'aclor pOrlldio sampll' 1Ill',\lIS ;\lHI All is 1.1'1'0. For any assel Ihl' t'X(lI't'lI'c\ reillm call bl' ('slimaled by suilsliluliJlg Ihe ('slilllall's 01 R, Au, alld Ai-: illio (li.I.H). Sinn' «(i.I.H) is nOJllillear ill Ihl' par;III1I'ttTS, calculating;\ stalldard ,.\ 1'01' \'-i-: I· T('still~ irindi\,idll;tI Lln"rs ;\1'\' prin'd is sl'nsihll' lill' (,, ('slimalors of B alld D call he fill'llllllated IIsill~ lIIaxillllllll likelihood factor aualysis. Becallse the first-order cOllditiolls ({II' lIIaxilllllllllikelihooc! ar(' highly nOlllincar ill the parameters, solvillg for lhl' estimalors with the IIsual ilerativ(' proccdurl' ('all be slow ancl COnV('IW'IH'l' diflicllIt. Alternative algorithms have hl'en developed by Jiiresko~ (19(i7) and Rubin alld Thayer (I !'H~) which facililate Cluick ('ollvergl'lHT 10 Ihl' l\IaXilll\lII1 likelihood estilllators.

Olle illlnprdortfolios without eliminating the deviations fro III the null hypothesis can ead to substalltial increases in power, because fewer restrictions nlll.\t be ested. The eRR paper focuses all the pricill~ of the factors. They usc a crossectional regressionlllethodo\uh'Y which is similar to the approach presellted II Section 6.3. As previously noted they lind evidence of live priced factors. ~'he factors include the yield spread between long and short interest rates for us government bonds (maturity premium). expected inflation, unexpected inflation, industrial production growth. and the yield ~pread between corporate high- and low-grade bonds (default premium).

~

6.6 Interpreting Deviations from Exact Factor Pricing We have just reviewed empirical evidence which suggests that, while lIIultifaclOr models do a reasonable job of describing the cross section of retunts, deviations frolll the models do exist. Civen this. it is important \0 ('onsiller the possible sources of deviatiolls frolll exact factor pricing. This iss lie is importallt because in a given finite sample it is always possible to lilld all additional factor that will make the deviations vanish. However the procedure of adding an extra factor implicitly assumes that the source of the deviatiolls is a missing risk factor and docs not consider other possible explanatiollS. In this section we allalY/.e the deviations frolll exact factor pricing for a given model with the ol~iective of exploring the source of the deviations. For the analysis the potential sources of deviations are categoriled into two groups-risk-based and non risk-based. The objective is to evaluate the· plausibility of the argument that the deviations from the given factor model call be explained by additional risk factors. The analysis relies on all important distinction between the two categories. namely. a difference in the behavior of the maximum squared Sharpe ratio as the cross sectioll of securities is increased. (Recall that the Sharpe ratio is the ratio of the Jlle,1I\ excess return to the standard deviatioll of the excess relllrn.) For the risk-based alternatives the maximuIII sq\lared Sharpe ratio is bounded and for the nOli risk-based alternatives the lIIaxiJlItJIII squared Sharpe ratio is a less usefid construct and can, in principle, Ire unbounded.

243

6.6. Inll'llmlillg /)l'lIia[ioll.l frollll:xf/d l'ill'lor 1'>i";lIg 6.6. J I:'x(lr[ Far[or l'ririllg Modf'll, M"Ill/-\'arim,,"(' Alla/v.liI, fllld tltp O/llilllll/ Orllwgowi/ I'orlfo/;o

For lhe illitial analysis we drop hack to the level of the primary assets ill the economy. Let N be the number of primary assets. ASSIlIll(' that a riskfree asset exists. l.et Z, represent the (N x I) ve(lor of excess ret mIlS for period [. Assullle Z, is stationary alld ergodic Wilh meall IL alld covariance malrix n that is full r;\Ilk. We also take as giVl'1I ;1 set of 1\ fiKtor I'ollf()lios ;llld allalyze the deviatiolls from exact /it("(or pricillg. For til(" (ilctor model,;ls ill (G.:.!.:n, we have (li.Ii.I) Z, = a + BZ At -I- f" Here B is the (N x I\) 111',11 rix of f~l("\or \o,ld'lllgs, Z/:t 'IS the (l\ x I) vector of lime-llilClor portf()lio excess returns, alld a ,lilt! { t are (N x \ ) venors of asset retum illterccpt.\ ;IIHI dislllrballces, respectivcl)'. The variallce-covariance matrix of the disturballccs is E ;IIHI the variallce-covariallce me zero and that the facl or ITl{ressioli (odI ici(' IlI.' for allY asse t will SlIIll 10 llilily, flu

6.2 COl lside r two ccol loll1 ies, ecollolllY 1\ and ccol lom y II. TIr(' mca ll ('XCl'ss-rcturIi \'('cl or and the cova ri'III (T Illat rix is spn ilicd I)('\o w Ii)!' ('ach of the ecol lom ics, Assu mc ther e exist a riskl n't' 'Isset, N risky ass('ls with IIIcali cxcc ss retu r" f.L alld non sing ular ('()\'ariall(,(' Illat rix n, and a risky \'act or port folio with IIIcali ex(e ss ITIIII'Ii II" alld I'aria llcc The faclo r port folio is lIot a lille ar com billa tion of the N 'Isscls, (Thi s nite riol l Gin he IIl{,thy clim inati n),( one oflh e asse ts whic h is illdl ldcd illlh (' 1;lclor ponf c)lio

or

O. Muftil'll·tor/),.iri,,/!. if III'fl·ss"ry.)

Fill"

,Hot!l'/.~

hllill ('CIIIIO\uil's r\ alld II: It =

a + {-JIJI'

((;.7.1)

n = tWa;; + liIi'al~ + /a/. C:i\'l'lI the OII1 .. \,e IIll"all alld c .. \'ariaIIlT lIlatrix and thl' asslllnplioll Ihal Ihl' brtor pllrlhllio /1 is a traded asset. what is the maximulII sCjuared Sharpe ralio 1'01' Ihe gi\'('11 (·C .. II .. llIil's> 6.:1

Relllrllillg 10 11\l" all .. \,l' proillelll. Ihe I'COIHllllics an' rllrther spl'cili('(1. II\(" ("Ie II It'll IS or a 011'(" .... oss-s("((i .. llalJr illdl"()('llIklll alld idl'lllically

ASSlIlI1l'

distli 1111\('''.

i

=

I ..... N.

The sp"(ilicali"l1 or IIIl' disirilllllioll or Ih,' delllents of {j cOllditional differelltiates 1','Olllllllil's :\ alld II. For econolllY 11.:

;\11111'01'

i

=

I. .... N.

i

=

I ..... N.

Oil

a

((i.7.4)

I"l'OlltIlIlY Ii:

1IIH'Olldiliollal\" III\(o ('xisled ill all ass!'t infinite-lived ag('nlS could selllhe assel short, inv(,st \01\1(' or IIIC IlJ()(,(Tds to pay the divide lid strealll, alld have positi\'(' wealth lerl OWl. This "r"ilr;\gc opportunity rilles Ollt bllbbles.

7. /'rl'vl//-\'lI/ul' IMII I illl/.! Tirok (I "H:,) has slllllied 1111" possihility ol"hllhhlt's withill the Dialllolld (I!Hi:,) ovnlappillg-gl'III'Lltions lIIolld. In this IIIOtil'! there is an infinite II til III 11'1' of lillitl'-lived agl'lIts, hilt TirolI' shows that evell hne a hllhhle call1lot ;Irisl' \\'hl'll the illtl'll'st \;Itt' excecds the growth raIl' of tht, t'COIlOlllY, hecallsl' the bllhhle I\'ollid I'vl'lltllally hecolIIl' illlillitcly large rdative to the we;t1th of the 1'1'0110111\'. This wOllld violate SOIlW agl'lIt's hlldget cOllstrailll. Thlls hllhhks CIII (111)' exist ill t1.\'I/{/lIIi({//~\' illl,/Jiril'lll overlapping-gl'n!'!'at it illS t'conOlllil's th,lI h,l\'I' O\'nalTlIllllilatl'd private capital, drivin~ till' intl'l cst 1',111' dowlI Iwln\\' Iht, gn)wlh 1',111' of Ihl' I'COlltllllY, Mall), t't'olltllllislS kel Ih,tI lIyn'"llic ill""licil'n ~ lex) + !'(X)(XI+I - x). P,+I -

Substitllting this approximation into (7.1.17), we obtain Y,+I ~ k + P PHI

+ (l

- p)d + 1 - Ph '

(7.1.19)

where p and k are parameters of linearization defined by p == I/(l + exp(cI- jJl), where (d - p) is the average log dividend-price ratio, and k == - log(p) - (I - p) logO I p - I). When the dividend-price ratio is constant, [hclI p = I I( I + D/ P), the reciprocal of one plus the dividend-price ratio. Empirically, in US data over the period 1926 to 1994 the average dividendprice ralio lias been about 4% annually. implying that p should be aboutO.96 ill al1l1l1al data, or ahout 0.997 in monthly data. The Taylor approximation (7.1.1 H) replaces tile log of the slIln of the slock price and the dividend in (7.1.17) with a weighted averar;c of the log stock price and the log dividend in (7.1.19). The log stock price gets a weighl p close to one, while the log dividend v;cts a wcight 1- p close to zero because the dividend is on average

262

7. Prt'st'lli- V(/lut' [{rlatiolls

~lUch smaller than the stock price. so a ~iven proport ional chan~c in the divIdend has a much smaller effect Oll the return than the sallie proport ional in the price. I

~:hange I

ANn"Oximation Accurm), the approxi mation (7.1.19) holds exactly when the lo~ dividend -price ratio ~~ constan t, for then dl+l and /'t+! move togethe r onc-for- one and equatio n ~7.1.19) is equival enlto eqllatio n (7.1.17) . Like any other Taylor expansi on. he approxi mation (7.1.19) will bc accuratc provide d that the variatio n in he log dividend -price ratio is not too great. One can get a sellse for the ,ccuracy of the approxi mation by compar ing the exact return (7.1.17) with tl~le approxi mate return (7.1.19) in actual data. Using monthly nominal ( ividends and prices on the CRSP value-w eighted stock index over the peiod 1926: 1 to 1994: 12. for example . the exact and approxi mate relllrns 1 ave means of 0.78% and 0.72% per month. standar d deviatio ns of 5.55% 'lne! !J56% pcr month. and a correlat ion of 0.!)!)99I. The approxi mation r exalllpk. if expcn,·aynwllls, hut Ille nll"l"t'lIl siock illcl 1..100

jJ(l\)

0.01:)

II~{I\)

().()(l:~

I(jJ(I\)

0.660

24

36

41\

0.1 !}I

O.:Hl:{

O.lHiH

052!! 0.209

2.07'l

0.144 4.113

,t.t;:~1

O.fi!>4 0.267 3.943

(l.W>!) 0.014 0.1\44

0.27'1 0.074 1.677

0.629 11.207 4.!i'21

O.HHO 1I.:m! '2.967

I.O!iO 0.4'24 3.7H3

n.07!) 0.047 3.0:':>

O.:W!) O.I!JO 3.22f\

0.601 0.344 3.'2'25

0.776 O,12H 3.315

0.H63 0.432 3561

1!)27 10 19!H fo(l\)

H"(/o:) I(~( /0:1)

1\)2710 1!):>1

•

1%2to 1\1\14 fo(l\) !t(K) IIjJ(/o:»

n.n24 0.01:> '2.73:~

or

r is II ... 10); ... ·,,1 r~lurn on a valu,,·weighl~d index NYSE, AMEX, and NASDAQ.locks. (d- PI is lilt' log ralio of divideuds o\'er the last year to lhe current price, Regressions are estimated by 01.';, wilh I 1"list· n alld Hodrirk (I!JHO) siandard errors, calculaled rrom equalion (A.3.3) ill Iii,' APP","lix s' ({,,' Ihe expected sturk retllrn at allY horizon, is observable and (';\11 he Ilsnl as a rq~ressor hy Ihe eCOIIOllletririan. Prohlem 7.4 develops ,I strlll'tllr;Il lIIodl'l of SIlKk priccs a!H1 dividends ill which a mllitiple of the log dividend-price ratio has till' pruperties of the variabk X, ill the AR( I) exampk. W" lise the AR( I) example to show that whcll .\'/ is persistellt, the If 01';\ return rcgression on X/ is very small at a short horil.oll; as the horizon increases, the /{~ (irst increases and thcn eventually d\'tTCases. We also discllss (inite-sample difli('uIties with statistil'al inkl'elHT ill long-hot il()n regrcs.'iions. n~ S,(J/i.,/in First cOllSidn regressing the one-period I'ctlll'll 'i+t on the variahle x"~ For simplicity, we will ignore conslant lel'lllS sincc lilese arc lIot tilt, oi>jccts of interest; constan(s l:(l\Ild be included in Ihe regression, or we could simply work with demeaned data. In population, fi(l) = I, so Ihe filled value is jllst x, itself, with variancc while the vari,lIlce or the letlll'll is givcn by e 1/1\. Pllllill~ Ihe 1wo Il'n11S Oil Ihe right-halld side of (7.2.'1) toget Iwl', w,' find Ihal ifex(>"('(l'd slock I'l'l 111'1 IS all' n'l,\, pnsisl"III, Ihe 1lI11ltipl'riod It stalistic grows at lirst approxilllatdv ill plOpor\ioll to Ihe horizo/l 1\. This bl'havior is w('11 illllstrated hy thl' lI'slilts ill '1:1),1" i.l. Illtuitively, it OCClII'S hel'ausl' Ii >recasts or "x pITh'" Il'tlll"llS sl'vl'ral pniods ahead are ollly slightly less variahk thall th,' !I.n·cast of the 11I'xi period's ,~xpl'l'Ied retlll'll, alld they arl' pl'rfenly (,OlTd;ltl''' with it. SIICCl'ssivl' realized retllrllS, Oil Ihe other halld, are slightly /lc)!;ali\'C/y ('orrl'lall'd wilh 011(' allothl'r. Thlls 011 IIrst IIIl' varianI'(' ofthl' IlIl1ltipl'riod liltnl vallie grows more rapidly thall the vari;IIICl' of thc IIIl1ltipniod rl';dilcd rl'tlll'll, illcrl'asillg Ihl' Illllltip('rio
/1; ==

I>l[(l_p)df!lil I-k-

rl.

(7.:!.HI

l="

The perkct-roresighl price g is so namcd becanse frolll Ihl' fX /loJI slock price idcnlily (7.1.:! I) il is Ihe price Ihal would prcvail if ref/liud rCllIrlls were conSlanl at sOllie level T, thal is, if thcre werc 1\0 revisions in expectalions driving unexpected returns. Equivalently. from thc fX alltf slOck price identity (7,1.~':.!) il is Ihc price thai wonld prevail if (,xpcr\cd relnrHs were cunslant and investors had perret:! kllowlcdgc of futllre dividcnds. SubstitUlillg (7.'2.H) inlo (7. I.:! I), we lind Ihal

/I; - /II

'"

= LpJ(li+'+! - rl.

(7.:!.9)

j="

p;

The dilfcrcncc between and III is jllst a dis('olllllcd SIlIlI or Inlml' dl'meaned stuck relllnlS. Irwe now lake expcctalions ;llld usc the definilion givcn in (7. I.':.!:!) ,\II(I (7,1.2:1) oflhe price component/I,,, we lind Ihal (7 .':.!.I 0)

I.

1'11'.11'1I1-\it/1I1'

Udal ilJlIJ

/{I'C alllh al/'" call he illlc rpn' lrcl as tllal CO 1II (>011('11 I oflh (' sloc k priCI' wlli rll is asso cialt 'd with challl!;illl!; ('XP ('cl;u iolis of fulu re sloc k tTlu rns. Thu s Ihe (olld iliol lal ('XI)('CI;lIioli of /': -/" ItIl'aSllreS Ihe t'ffcCI of chal lgill g cxp( 'el('( 1 Sioc k reI urliS Oil lilt' ("1mI'll! SlOck prin '. In I he AR ( I) ('xal llple d('v ('lop ed carl icr, Iht' cond iliol lal I'XI) (Tla lioll of/, : - /I, isjus l x,f(1 -pcp ) froll l (7.1. !!!l). Ifl'x p"(" led sior k n'ltU ns arl' COll Slanl I hrou gh lime , IllI'n Ihl' righ t-ha nd side of (7.~.1O) is /('ro . Tllc rOll slall t-l'x p('C !cd- r('tu nl hypo thl's is imp lies that p; -/" is a fi,,'I' cast ('fro r IlIlrO I"lTl aled with illfo rma tioll kliOWIl attil lll' I. Eqlli val(, lItly , it ililp lil's that thl' stor k prin~ is a ratio llal I'x(l l'cta tioll of the pcrf ",·t-l i,n's il!;h t stoc k pri .. e: (7.~.11

)

Ilow Clli Ihl'S(' iell·a.s hc used 10 !t'SI 111(' hypo lhes is Ihal ('xp ccle d stoc k relm lls are COIISI;iIlI? For silllp lieil) , or I'xp ositi on, w(' heg-ill hy mak ing two llllr( 'alis lic assll mjll iolls : !irsl , Ih:11 IOI!; sloc k pric rs and divi dend s rollo \\' Sl;Ilioll ar), sloc llasl ic pron 'ss,'s , so thai Illey have well -def incd lirsl alld seco lld 1II01lH'lI1S; alld S"COIIII, 111:11 IOI!; divi dl'lI ds are ohse rvah le illio Ihe infil lill' f\llm e, so Ihal IIII' per( i'cl- f(,re sigli l pric c is ohse rvah le 10 IIII' t'con Olll( 'Ilici all. lido \\' \\"1' dis"lI.s~ 110\\ ' II"'SI' assll lllpl iollS arc re\a xed,

g

()rlh"g/lllfllil~ fllld \'flrif lllf'l' -!l"/l ifl/l 'li'lll

E'I" aliol l (7.'2 .11) illip lil's Ihat /,; - I'I is "rlh" K{// w/to infim llali oll vari able s kllow n al lillie I. :\11 mlho ,l\ot lalil ), Il'sl of (7.2.11) rt'l!; ressc s /': - I), 0111 ', infol "lll:t lioll vari ab"' s alld (('SiS for /t'I'( , cod lkie llts. If th,' infil l'lna lio, I v:tri ahlrs ind\ ll'" Ihe Sloc k prilT I'I itsdf , Illis is eq\l ivak llllo a rt'gr ('ssi on of /': onlo /', and olli n \';lri ;lhlc s, II'lIl'rl' III,' hypo llics is 10 be Icsll 'd is 1101,' Ihal /It has a IIl1il cOl'f ficic lII alld lilt, ollll 'r varia bll's have zero cocf ficie nts. The se rt.'l!;r('ssiolls an' "aria llis of Ihe IOIlI!;-II01il.oll n'lu rn regr essio lls Ilisl 'usse d ill Ille pr('v ious sl'fli oll. E'I" alio ll (7.~.!l) show s Ihal g -/1, isjus l a disc oull it'd SIIIII of ftllll l'e 1I1'IIII'alll'd sioc k relUf'IIS, so all orlh ogon alilY I('st of (7.2, II) is a retu rn rl'g-r essio ll wilb all infil lill' Irori llll\, whe re 1\101 '1' lIisla lll relll rns arc I!;co lll('t rical ly dowlI\\'('il!;b ll'd.l~ IlIsl ead ofll'Slill1!; ollll ol!;o llalil Y din' nlr, IIIl1ch oflh e lill'r alllr t· tesls the illlp lical iollS or ol'lh o,l\o llali ty for the ,'ola tilil) , of sloc k pric es. The JIlost f;1I1101lS Slich illlp lic;l lioll . lit-ri ved Ill' I ,I'Ro)' alld POrl ('r (J9H I) alld Shi lln (1!IH I). is Ihl' 1'1111(1/111' illl''i "l/lil \' for IIH' sloc k pric l': (7.'2.I~)

I'.!Thc dOh'II\\'C'igllllllg .dlll\ \\ rill' f('! '1:tli,l ic' ill IIIl' rt'gn. 'ssioll 10 h(' pO!rlilin', WI"'lt ',I" ~hll\\T" ill St'rti oll 7.'.!..11I'.11 \\'(" lilt' N'.! .\',lIi'lil ill ;tlllll l\\'c'i glllt'd lillitt ·-hur i/otl 1('111111 n·gl(· . fOIlH 'lg('\ 10 ,,',0 ,I, lilt, . ~i()11 holil oll illfl(' ;l\('\, 1>1111;1111 alld II.dl (19H~ ••

(:haplt'r I I)

1,,1\(' IlIli It·gn·~,ioll.'

(I!JH~I), SCUll

(If

(IllS .\01 f.

(p.H:,),

.uHI

Shiller

7.2. l'rl'.lt'IIl-\'lliul' Reilliiolls (/lui US Siork l"rire lMulVior

277

The equality in (7.2.12) holds because undcr the null hypothesis (7.2.11) II; - 1'1 must he uncorrelated with PI so no covariance term appears in the variance 0(' I';; the variance inequality follows directly. Equation (7.2.12) can also he understood by noting that an optimal forecast cannot be more variable than the quantity it is forecasting. With constant expected returns the stock price forecasts only the present value of future dividends, so it cannot be more variable than the realized present value of future dividends. TesL~ of this and related propositions are known as variance-bounds lests. A~ Dliriallf and Phillips (1988) point out, variance-bounds tests can be restated as orthogonality tests. To see this, consider a regressiQn of P, on 1'7 - 1'1. This is the reverse of the regression considered above, but it too should have a zero coefficient under the null hypothesis. The reverse regression coefficient is always () == Cov[P; - PI, ptl/Var[p; - pd. It is straightforward to show that Var[p:J - Var[pd Var[p; -

=

PI]

1+28,

(7.2.13)

so the variance inequality (7.2.12) will be satisfied whenever the reverse regression coefficient () > -1/2. This is a weaker restriction than the orthogonality condition () = 0, so the orthogonality test clearly ha~ power in SOIlW situations where the variance-hounds test has none. The justification for using a variance'-hounds test is not increased power; rather it is that a variance-hounds test helps one to descrihe the way in which the null hypothesis fails. lIllil Hool.1

Our analysis so far has assumed that the population variances of log prices and dividcnds exist. This will not bc the case iflog dividends follow a unitroot process; then, as Kleidon (l98!» points out, the sample variances of prices and dividends can be very misleading. Marsh and Merton (1986) provide a particuhlrly neat example. Suppose that expected stock returns an~ constant, so the null hypothesis is true. Suppose also that a firm's managers lise its stock price as an indicator ofMpermanent earnings," selling the firlll's dividcnd equal to a conslant fraction of its stock price last period. In log form, we have (7.2.14) where there is a unique constarlt J that satisfies the null hypothesis (7.2.11). It can Iw shown that both log dividends and log prices follow unit-root processes in this example. Suhstituting (7.2.14) into (7.2.8), we find that the perfert-foresight stock price is related to the actual stock price by 00

p;

== (I -

p)

L )=11

p}Pi+}'

(7.2.15)

~78

7. Presl'1Il- lit/IIII' [{I'llliions

I

"Vhis is just a smoothed version of the actual stock price II" so its variance ([rendS on the variance and aUlOcorrclatioJls of II,. Since autocorrclatiolls c n never be greater than one, g must have a lower variance than flf. The i Ilportance of this result is 1I0tthat it applies to population variances (which a e not well defined in this exalllple because both log prices and log divi ends have unit rooL~), but that it applies to salllple variances in every s;lmple. Thus the variance inequality (7.2.12) will always he violated in the rSh_Merlon example. This unit-root problem is important, but it is also easy to circulllvent. , e variable P; - PI is always stationary provided that stock retuflls arc s tionary, so any test that p; - fll is orthogonal to stationary variables will be wrll-behaved. The problems pointed out by Kleidon (\986) and Marsh and Merton (1986) arise when /'1' - 1'1 is regressed on the stock price 1'1, which has a lunit root. These problems call be avoided by using unit-root I"(~gression tlleory or by choosing a stationary regressor, such as the log divid("nd-pri~e r'llio. SOllie lJlher ways to Ikal with the unit-root problelll arc explored in Problem 7.5. t3

±

Finile-Sample Consideraliuns

So far we have treated the perfect-foresight stock price as if it were .111 observable variable. But as defined in (7.2.8), the perfect-foresight price is unobservable in a finite sample because it is a discounted SUIll of dividends out to the infinite future. The defInition of g implies that T-I-·I

,,; = (1 - p)

L

pj(dH1 +)

+k-

r)

+ pT-I-1 p~.

0.2.16)

)=0

Given data up through tillle T the !irst term 011 the righ t-hand side of (7.2.16) is observable but the second term is not. Following Shiller (1981), olle stalldard response to this dillicu\ty is to replace the unobservable I); hy all ohsCl"vahle proxy 11;:r that IIses ollly illsample information: "/'-1-1 fl;'T

==

(l-fJ)

L

fJ)(dH1'I+h-r)+pr-I-I/,r.

(7.~.17)

1=0

Ilcre the terminal value of the artual stock price, h·, is lIsed ill place of the terminal vallie of tll(· perfect-foresight stork price, fr. Several points arc

'1

"Umlauf and flail (I !JH!/) ;lpply IIl1it·,.oot Il'~,.,·ssio\l Ihl·OI)'. while (;,lIlIpl)("\I ,lIul Shill .. r (I !lRRa,h) replace the 10K Mock price wilh Ihl' lOll () is a hlllClioll oflhl' olhl'r parallH'lI'rs of Ihl' lIIodel. SolI'I' Ii,,' A.

7.2.:-\

Ilisnlss lhl' Slrt'II!41hs alld weakllesses of Ihis 1II00Id 01';1 raliollal hllhhle as 1'I1IIIP;III't\ Wilh Ihl' Blallckml-W;lIsol1 hubhll', (7.1. \(i) ill Ihl" II"X\. NOle: This prohlel\l is

7.3

ha~I'd Oil

Flool alld Ohslldd (I ~191).

COllsidl'r a slock whose I'XIlt'CI"d rl'llIrn ohcys

/':,\/'111

= r

+ x,.

(i.I.2i)

ASSlIlII1' Ihal x, /t.liows all AR( I) pron'ss, ·\·t 1 I

70

"'.

289

/'mh{rlll.1

7.4.2 Now suppose that the Illanagers of the company pay di,,:idends acronling to the rule

rt,

= (+1.11/_1+(1-1.)(//_1+1)"

",line ( and A are constants (0 < A < I), and lit is a white noise error Iln('orreialed wilh ft. Managers partially a(!just dividends towards fundaIIlelllal value, where the speed of a(\justmelll is given by A. Marsh and Merton (I !IHli) have argued for the cmpirical relevance of Ihis dividend polie),. Show that if the price of the stock e'luals its fundamental value, the log cli\'i(kncl-price ratio follows an AR ( I) process. What is the persislence of this process as a function of A allel p? 7.4.3 Now suppose Ihal the slock price docs not eqllal fundamental \'alllc,hlltrathersatisfies/lt = 1I1-y(dl-vtl,wherey > O. That is, price exc('eds fllndamental value whenever fllndamental value is high relative to dividends. Show that the approximate log stock return and the log dividend-price ratio satisfy the AR( I) model (7.1.27) and (7.1.28), where the optiJllal forecaster of the log stock relllrn, XI' is a positive multiple of the log dividend-price ratio. 7.4.4 Show that in this example innovations in stock returns are negatively correhtted with il~novati{)ns in XI' 7.5

Recall the deiinition of the perfect-foresight stock price:

V == 'L}::.o (II [< I

- (I)dl + I +/

+ k - r].

(7.2.8)

Thl' hypOllwsis that ('xpected returns arc constant implies that the actual slOck pricc III is a rational expectation of p;, given investors' information. Now consider forecasting dividends using a smaller information set J,. Deline p, == I~[II; I J,l. 7.5.1 Show that Var(jll) ::: Var(j~I}' Givc somc economic intuition for Ihis n'slllt.

Ptl

~ Var(g - M and that Var(p; - PI) > 7.5.2 Show that Var(IJ; Var(/It - I~I)' Give some economic intuition for these results. Discuss cirClllllstallces where these variance inc'l"alities can be more usefullhan ; Ihe illequality in part 7.:>.1.

I

7.5.3 Nowclcfine I-'tl == k+(I ~'+I+( l-p)d'+I-P" 1-'+1 isthereturn that wOllld prcvail Illlder the constant-expected-return model if dividends were' lim'Casl Ilsing Ihe informatioll set j,. Show that Var(r,+l} ~ Var(r,+l)' i (;iw sollie ecollomic illlllitioll for this result alld discuss circumstances' whert' it can he ilion' Ilserlll than the inl'rallcli 1)(,(,;IIIS(' its posilin' correlatioll with the stochastic discolillt /iH'toi' givcs it a 10\\,('1' 1111',111 gIO." 1'('1111'11 th;1I1 I/M,

Ii. 1111,.,-11'111/'0/111 I:'q II ilibrill III .1/lIdl'l5

VI'I' call 11011' sl;lle 1\\'0 1II0re properties: (1'-1) TIlt" ral in of'siallclarcl devialioll 10 ~ross lII('an for Ihe I)('ndllnark porlll,lio salislies

(H 1.1K)

1\II11he riglll-h;IIHI sid(: oflhis (''1"lioli CAPl\l wilh power Illilily (0 = I) alld th!' Iradiliollal SIalic CAPM (1/ == OJ, It is 1t'lIIpting 10 USt' (H.:I.7) logetlrer with ohsC\'wd dala Oil aggrq~al(' l'IllISlIlIIplion and slock lIIark!'1 relnrns 10 estimale 0111(\ test Ihe EpsleinZill-Wcil IIlw\el. Fps\('ill ;\IId Zill (HI!l1) reporl results Ihis type. In a similar spiril. Ciovallnilli and Wei! (!9H9) lise the 1II()(ld 10 rl'inlerprel the results 01 Mankiw and Shapiro (I !IH(;). who found Ihal h!'las wilL the lIIarkel ha\,,' greal!'r I'xplanalon' pow(')' (1I' Ih!' cross-seclion,1I pailI'm or rellirns Ihan do Iwtas wilh consllinplioll; Ihis is ('onsistent wilh a vahll' or (} dose 10 11'1'0. Ilow("'('r Ihis I' 1'0 ('I' d II 1'1' ignorl's Ihl' (;1('\ Ihal Ihe inlt·rt(·IIIJloral hlldgt'l cOllstr;tinl (H.:I.·I) .dso lillks ('onslllll(lIioll alld llIarkl'l lellirns. \Ve now show lhal lhl' hlU(g('1 (,(JIlsIl'ailll ('all II(' usecllo sllhstilllle consulllption (lut (l11\1(' ;ISSI'I pril'illg 111I)(kl.

or

SItb.\lillllil/~ (.'111/11I1II/J/io/l O/lIIl/liJl' M,"11'1 Call1phell (It}~l:tl) POilllS 0\11 thaI one ('an loglillearize Ihe illl(,),It'll'poral hudg('1 cOllsllaillt (H.:I.·I) arolllld Ihe lIIeall log ('OIlSUIll(llioll-weallh !"tlio loohlaill

t\ "'/I

I

""

r",-, t

I

-1 Ii

I· (I - ~) «(, - "',).

(H,:I.H)

wi II' II' fI == I .-, I the laller dkct dominates, and consulllers requin' a higher return to hold sudl "ssl'ls. There are several possible cirnlmst,llHTS Ilnder which assets Gill he priceriusing only their covariances with the returll Oil the Illarket portfolio, as in the logarithmic version of the static CAI'M. These cases have heen discmsed illllw lileralure Oil inlerll'mporal asset pr'lcing, but (H.:t I 'I) makes it particularly easy to undersland Ihelll. Firsl, if the cod'licicnt of reialiVl' risk aversion y == I. then the opposing efkcts of covariance wilh inveslment opportunities cancel out so thaI only covariance with the market relurn is rclcvalll for asset pricing. Second, if the in\'eSllll('Il! opportunity set is constant, then 0/" is zero for all asseL~, w again assets CUI be priced using only theil covariances wilh Ihe markel reI 11m. Third, if Ihe relurn on Ihe market fo\lO\;s a univariate stochastic process, Ihen news about future rdurns is perICCII), correlated with the currenl return; thus, covariance with the (,\IITl'lIt retlll'Jl is a slIfficiellt statistic fill' covarialll,!, with lICWS ahout luture returns and l',Ul he uSl'd 10 price all assets. Camphell (I !)!)(;a) aq';lIl's Ihat the lirsl Iwo caSl'S do 1101 Ill-scribe US data evell approxim;lIcl)', hilI Ihat Ihe third case is !'Illpiricall)' relevall!. A 'Iliin/ I.oo/i (lllflr 1:'ljuilJ I'rPllliulIIl'uulr

(H.3.11) 111,

(,;,11

he applied to the risk premiulll

Oil

the ,"ark!'t itself. Whell i =

we gel

When the markel relllrn is unforec;tstahlc, Ih(')'c are 110 revisions of ex pet'lalions ill futllre retllrns, so a/I/I, == O. In Ihis case tl,(' l'<Jllity premium wilh theJensen's InequalilY a I, Illis rl'dlln's 111(' equity risk prellliulII associaled willi all}' 1('\'1'1 01 Y alld illCrl';ISes Ih(' risk-aversioll coefficiellt lIeeded to l'xplail\ a givel\ eqllit)' pn·lIIilllll. 1I\IIIitively, whel\ alii" < II Ihe IOllg-rlIn risk of slock Illarkl'l illvcstllll'lIt is less Ihall Ihe short-rull risk hccause the market t('lIds to 1l1eall-i'l·\·(·rl. IlIv('stors wilh high y Gln~ aholll long-mn risk r"lhn Ihall shorHlI1l risk, so Ihl' Frielld alld Hhlllll' calrlliatiol\ oV('rst.llC5 risk and rOIT('spOlulil\gly tJlIt1t-rst;II('s Ihe risk aVl'rsioll Iwededtojllstify lhl' ('(Illity pn·lIIilllll. Camplll'lI (I!)!ltia) sllows Ihal Ihe estimated coefficielll or relative risk aversioll rises hy a bClOr of 1('11 or 1l10re if one allows fill' the empirically cstilll"ted degree of 1I1(';1l1-rt'\'('rsioll ill postwar monthly US dala. III long-run allllllal US data tile dlc('( is less dralllatic hilt still goes ill the same direrlion. (:alllplll'lI also shows Ikl\ risk-awrsioll eSlilllates illcrcasl' if olle allows for hlllllall capilal as a COII1JlOIIt'lIt of I\'(·allh. In Ihis seIlS(~ onl' can clerive Ihe e'lllil), pn'lllilllll JllIlI.k withollt ;IIIY din'('( n:krellcl' 10 cOllslImption data. till J..'quilibriwl/ II luI/ifill/or ..t.l.l~/I),'i(illg Model Wilh a few IllOIT aSSlIllIptiollS, PCI.I 'I) can he IISl'c\ to c\t'rive all t'f)lIilihrilllll IIllllrif;I('lor ;lss('1 pricillg Illodel of tilt' lyP(' discussed in Chapter Ii. We wrile tht' I'('tlll'll 011 lIlt' markt·t as tIll' lirstl·I(·I\lt'lIl or" K-clcllll'1l1 state vecto;' x,+ I. Tht' othn dt'IIH'nts an' v;lriahks that an' knowll to the market hy the ,lid ofpt'riotl/+ I alld arc rt'h'\'allt "H'II,n'casting I'll till"(' r('turlls on tht' lIlarket. W(' asslll\l(' that tht' \'('("\01' XIII (,,!lows a first-ordt'r v('('\or .\Il1on·grt'ssioll (VAR):

Ax,

+ I' ttl.

UU.I()

Tht' asslullplioll that tIll' "AR is lirst-ordn is nol rt'slriniw, sinn' a highnonk!' VAR 1'0111 ;Ih,'al's ht' stack('d into lirst-onll'r form. N('xt 1\'(' ddilll' :1 ,,'·1'1('111('111 v('('tor cl, whost' first ('lellll'lIt is Ollt' alltl whost' olht'r dl'IlH'llls an' :111 /1'1'11. This Vl'l'lor picks OUI Iht' n'al slock return r""/t I frolH lilt' \'('('101' XIII:

"III,1t

I

=::

el'x/+I,

and

r",.I+1 -

E,

rm,l+1

=:::

Th(' li ..,t-lIllln VAR gCllnat('s simple lI\lIltipt'riod fim'l'asts of fu1111'1' n'llIrns:

d'(III.

F,l/m./1

I I

,I

(i·U.li)

8.3. Markrt Frictions

It follows that the discounted sum of revisions in forecast returns can beI wriuen as 00

== el'LpiAiE/+t J=I

(8.3.18) where t.p' is defined to equal e l' pA(I - pAl -I, a nonlinear function of the VAR coefficients. The clements of the vector t.p measure the importance of each state variahle in forecasting future returns on the market. Ifa particular element IPk is large and positive, then a shock to variable k is an important piece of good news about future investment opportunities. We' \lOW define (8.3.19) where ( •. 1+1 is the ktll element of (1+1. Since the first element of the state vector is the return on the market, ail :::= aim. Then (8.3.14) implies that .

E1[rj.I+!l-

2 Tj.I+1

K

= - a~ + yail + (y - I ) LIPkO'ik,

(8:3.20)

k=l

where 'PM is the kth element of t.p. This is a standard K-factor asset pricing model of the type discussed in Chapter 6. The contribution of the intertemporal optimization problem is a set of restrictions on the risk prices of the factors. The first factor (the innovation in the market return) has a risk price of AI :::= Y + (y - I )ipl. The sign of ipl is the sign of the correlation between market return innovations and revisions in expected future market returns. As we have already discussed, this sign affeclS the risk price of the market factor; with a negative !PI, for example, the market factor risk price is reduCt·d if y is greater than one. • The other factors in this model have risk prices of AA (y - 1)rpk for k > I. Factors here are innovations in variables that help to forecast the return on the market, and their risk prices are proportional to their forecasting illiportance as measured by the elements of the vector t.p. If a particular variable has a positive value of IP., this means that innovations in ; that variable arc associated with good news about future investment oppor- ; tunities. Such a variable will have a negative risk price if the coefficient of . relative risk aversion y is less than one, and a positive risk price ify is greater than OIlC. Campbell (I99(b) estimates this model on long-term annual and postWorld War IIl110llthly US stock market data. He estimates CPI to be negative and lar~(' ill absolute value. so that the price of stock market risk At is much

=

32&

H. JlItl'rtelllIJOral/~qllilill1"i1l1ll Models

smaller than the cocfliciclIl of risk aversion y. The other ('"ctors ill the model have imprecisely estimate d risk prices. Althoug h some of these risk prices are substant ial ill magnitu dc, the other filClOrs have minor effects on the lTIean returns of the assets in the SlIIUY, because thcsc asscts typically have slllall covarian ces with the other factors.

8.4 More Genera l Utility Functio ns One str,ligin forward rcspol1se to the difliclIlties of thc stami.~ WI' sh'll! t'llll-;i(\t'r llollparallH'lrir d('ril'ali\'(' pririll!-\ modl'ls ill Chapll'r I~, and (Ints Oil i,SIl('S SllITOUIHlilll-: p;lrallH'lric Illodds ill S('nioll 9,:t Th(' sl'colld aspl'cI ill\,oll'~'s the pricill~ or path-depelld"111 dl'rivalive's hr MOIlIt· Carlo simulalion, A dnil'alil'l' security is saitilO 1)(' Imlll-rl''IH'IIr1m[ ifils payolfdl'lll'lIlis ill SOil\(' 11';1\' Oil Ihl' ('nlin' /mlll oflht'lllHkrl),illg assel's p,.i(T dllrill).!, Ihe dl'ril'alil'l''s lili-, For (·xample. a pllt optioll which giV

It

ILr

->

It

(T

~

r.

(\1. \'(i)

Ji,.

(9.1.7)

and this is accomplished hy selling: rr

n

,

-I

(

2

~

ILfi,) . 1+-(1

(I _fi,) , 11

2

I:;

(T

(1

The adjustlllt'llts (!1.1.7) imply thai the step sil.!: dClTeases as " inCTeases, but at the slower rate of lifo or fit. 'I'll!' probabilities cOllverge 10 ~,also althe I-ate of fit, alld hence we write: I

n = ~+()(fi,),

rr'

=:

~+()(Jh),

I:;

= 0(,/1;).

(!).I.H)

fi,)

where O( denotes terms that are of the same WYllfllOlic Older as fi,." Therefore, as II incrcascs, thc random wall I on a stock with price P(I) at time 1. 10 or course, G also depends on other quantities such as the maturity date T, the strike price X, and other parameters but we shall usually suppress these arguments excep~ when we wish to focus on them specifically. I [owever, expressing G as a function of the currenl stock price pel), and not of past prices, is an important restriction that greatly simplifies the task of fincling r; (in Section 9.4 we shall consider options that do not satisfy

" 110""\'-!T-l 4>((-) is the standarci normal probahility density function. We shan have occasion to consider these l\leaSUres ('1\\'1'1:11 Ihe sampk 1110111('111 concliliolls alld Ihcir popllialion ('Olllllerparis. TIll' pl'o[>cllit's 0(' a CMM eslilllalor dt'lll'lId n ilic:lIl), Oil lilt' choic(' or 11101111'111 COlldilioliS alldthe dislallC'(' 1I\('lril', ;llleilol' sl;1I I< 1:\1'(1 disl'!'('(('-lil\l('

CMt"! ;lppli(;tliOlls, Ihcs(' two isslI(,s

It:!\'('

1)('('11 stlldil'd 1(llit(' tho\\)u).';hly.

i\lollll'lIl COllC!iliollS are I)'pira\ly S\\gg('stl'cl hy thl' illtn 0,

Y > 0,

(!).:~. I~)

This is a cOlltillUOlis-tilllc \'l'l'siol\ of a stalionary AR( I) proccss wilh IlIlconditional Illcall II (SI'(' SC('lioll !).:I.'I for further discussioll), and hCII(", first observed by Merton (1980) for the case of geometric Brownian motion. is true for general dilTusion processes and is an artifact of the nOli-differentiability of diffusion sample paths (see Bertsimas, Kogan, and Lo [19!l6) for further discllssion). In fact, if we observe a continuolls', record of 1'(1) over any finite interval. we can recover without error the diffusion coefficient a(·). even in cases where it is time-varying. Of course, in .

!, 3(j(i

9. lJr'rill((lil".I,,.i"i"~ AlIII/eLI

Table 9.2b.

A.\ym/Jlol;r ,\/flllrlflul,'rmn ./to

?

'!.

I, 1/

- - - - - - _ . _ - .. -'--,

_.. _-I

2 0.0400 4 0.O21l:1 II IUJ200 IIi 11.11141 :12 11.01110 li1 !l.OO71 1211 U.IIO:,O 251i, O.OO:!', ,,12 o.om!:, 1,024 IUlOI!!

O.IHOO O.02H:i 0.0200 11.0141 U.OInO 0.0071 0.00,,0 O.(){I3', Il.OO2', Il.OOl1I

0.0400 O.02H:1 (1.020U 0,0141 O,OIUO 11.0071 O.OW,O O.OO:!', O.OO:.!:, 0.00111

.

I

I

iii

1:!

r.:i

O.(HOO O.O:lH:1 (J.0200 11.111·11 O,OIUU I!.OO71 O.OW,O 0,00:1" 0,002" O,OOIH

0.0100 O.O:lH:1 O.O:lOO 0,0141 0,11100 0,0071 0.00,,0 O,OO:!!", 0.002" O.OOIH

0.11100 O.O:lH:1 0.0200 0.0141 0.0100 0,0071 O.OW,O

I

I

l:!x

'!"h

O.(HOO U.O:lH:1 O.O:lUO (1.11141 0.01U0 0.0071 0,00:,0 O,OO~" O.OIJ:i', 0,002" O,OO:.!', o,OOIH O,OOIH

0.11-100 O.O:lH:1 0.0200 11.(11,11 0.0100 0,0071 O.IIW,O 0.00:1:, 0,002" O.OOIH

A'\ymp\otir slc\1\(\anl ('nor of;' '! for \',W11)\\:o\ ""hu':" of n .Hul h, ass\\min~

year and

(1

= 0.20.

of 64 ()hs(~lYJ.\i()ns

Recall Ihal '/' '"

I"'n('" II ... \'alll(,s t'q\lc,\ly :'olMfl'd o\'(.'r .\ Yl'ars. 1/1,;

1/ ' "

I

li4 .111

0

if

{:\ < I.

I,

ax ill; i)(T-tJ

I'

()

ill'

X>

rf

iJI'(

ilU'- tJ ",h(' 1'('

" "

.(. cc)II .. /I .

. (.! ~)i'r./I .

IT::j (T'!) - - ( r+. n

~

IUl'qualilY «1.:1.·10) shows liI;11 Ihl' ;lccllracy of i; dt'creases witli Ihe kl'd of Ihe Siock plin' ;IS long ;IS IllI

r > 0

(~),:UO)

.,

( :O\' f Ii, ( r

). li~ ( r ) I

(1'

- - I ' -y(l,-/,- r)

2y

II -/ r

Conl'i( r),

III'

(r) I

[I - I-Y']~ '.

< I',!

(~),::l,:l

I)

I

-- [I - I'-Y'l, 2

Sillcl' (~),:\..J(;) is ;1 (;allssia ll process, Ihe 1Il01l}('nts (~1.:1.·19)-(~),3,:1 I) (Olllpletely charaClt ·rilt· Ihe liJlitc-di lllt'lIsiol lal distribu tiolls of I, (I) (see Sectioll ~),l.l Itlr the defillitio ll of a fillilt'-di lll(,lIsio nal distrihu tion), Ullli,'c the arilhlllc tic Browllia ll IIlOlioll or ralldolll walk which is lIonst,ll iollary 'md ortell said 10 hc tli/li'Il'wl'-I/(/liol/f/}~v or ,I Jlor/uUli c Irnu/, the trclldin g O-U pn)~'('SS is said 10 I~,I' Irl'lltl-,l/l Iliol/(u)' siJlce its dcviatio lls frolll trcnd filII ow a slallona ry proccss, ·1 fOlulitiollC 'cI 011 lIfO) = /1" in dt'lil1in~ Iht, dellc'lule d I()~~prift· pro("('."i.s, ahus(' 01 h'llIIillCilo gy to (all 11It'~t· IIHJlUt'lIts "uJlcullcl irioIMI", Ilo\\'t'\'('r, in this

:.'IISilll (' Wt' 1i;1\'(,

it is a

~Iiglll

fit'" Ih,' di,tillnio ll i.1ri prilllal il)' ~c'JII;lIl1i( ~illn' lilt' c:nllclili()lIin~ v.lriahk is lIlort' of all initial

nmdilioll lhan.1I1 illl"ollll"li oll \,;II·jahlC'-il"\\'(' dt'fillt' Iht' hq~illllillg : oirilllt" as I = () and Ihe flilly oh"'rv"hlt - >1.11 lill~ \',tI"" 01/,(0) '" I'". d"',, ('I,:l,1~1)-(~I,:\,r.~) ;Ir!' 11I"'lIlldil illll.tI IIIOIll('lll' rdativt, 10 ,!I"M' illil;;11 ("OIHlilioll!'oo, \\'e' ~";tli ,uloptlliis definition 01 all1llH'OIH litiol1;IIIII IIIIH'ut tilioligho ul liIc' rc"1I1ailHic-r of llai~, ".IPlt'l. :!I All illlplicllio ll uf 11 "IHI",t;"io ll;U it), is Ihal lilt' variau('(' of r-pt"riod returlls ha.1ri a finite limit as r iIHT(,;IM" \\"illwHI !,Olllld, ill Ilti.1ri ( ... ~" n '.! /y. wht'J"t'as Ihis variallct' increas(,s linearly with r 1IJ1(kr a r;lIIdom \ ....all. \\,I!ik IH'lId-st;lI ioliary prof('sst's art' oht'll simplt·J' 10 (·~tililate. "H'Y Ita\'(' ht'('11 n ili,·i/C'd tI~ 1I111e;lli.lriti( lIuldds of financial ;1~S('t prict's sinn' they do ltol "(ford w('11 with 11i(' (0111111011 illtlliliOlllh~lll()lIg('I-hol i/oll ;,ts:o.t't rt'IIII"J1S ('xhihit llIol"(' .. bk 01 th~lt prirc IOI"l'(";lSt'\ ('xhibillll on' 1IIIr(' I laillt\';ts tlu' 1(11('( ;I.,t hOI i/llU grows. Ilo\\,t'\,(,,,, if lIu' ~ollr('t· of stich i~ ('lIIpiri,.1I 1111,("1 \·~lli'lil. it III.IY wl'll Ill' ({llIsistt'!}1 wilh Irt'lul-M.u ioll;.lrily silln' it is IIOW \\'('II-llloW lIlkll fiu .IIIY fi II Itt' .'1'1 Clf cI.lla, (rc'IItI-M~lIioll~lrily :.uul t1i1f"t'IC·lIft'·!'\I~uiollarity an' \"ittllall\' illtii,lillglli!'ooh;lhl(' (,t"", fUI (,,\.lIllpl,,, St·, lion '2.7 in Ch~lp(('r~.
n'sulls Ic,r Ilc"HOIli\"dy alllnccnrc'I,II('(1 .llld pc)sili\'('ly alllclftJlTt'l.llc.'cl asst't (I!N:.) for fllnller d,-Iai"-

W;III~

9. .>.

jllll'/l'IIlI'nli ng j'fl/w//('/lir 0l'lion I'rici ng Mod!'/.,

.\s Ihe l'ellll'lI intcrval r derreases, il call 1)(' showlI Ihal Ihl' adjllslllwlll faclor 10 .\~lli(r)l/r in (!l.:t(il) appro;H'h('s IIl1ill' (IIS(' 1:llt/pilal's 1'1111'). III Ihe cOlltilluous-lillW limit, thc stalldard devi;llioll of cOlllillllollSly compOll IHlt-d ret u rtIS is a cOllsistellt esl iilia Ior Ii 'I' a ;1I H II he clkcls of predictahilill' OIl a vanish. The illtllitioll (,(III1CS frolll Ihe raI'l th,ll a is a IIwaSllre of /()((J/1/("lIlilil)~lhe volalility of illfillilesilllal pricc challges-alld Ih('l"(' is 110 prcdictahilily OV('I" allY illfillitesimallilllc illll'l'val hy cOllslructioll (see Seclioll 9.1.1). TIH'rd()re, Ihe illfhlcll(,(, of prl'llitlahilily 011 cslimalors tI)r
9.3.5 Implied Volatility Estimators Suppose the current market price of a one-year European call option on a nondividend-paying stock is $7.382. Suppose further that its strike price is $35, the current stock price is $40, and the annual simple riskfree interest rate is 5%. If the Black-Scholes model holds, then the volatility (1 implied by the values given above can only take on one value--{).200-because' the Black-Scholes formula (9.2.16) yields a one-tCK>ne relation between the option's price and a, holding all other parameters fixed. Therefore. the optioll describen above is sain to have an implied Il()lfllilily of 0.200 or 20%. So COlllmOll is this notion of implied volatility that options traders often quote prices lIot in dollars but in units of volatility. e.g., "The one-year European call with $35.000 strike is trading at 20%.n i . Because implied volatilities are linked directly to current market pri~es (via the Black-Scholes formula). some investment professionals have argtled that they arc betlere~timator!lofv()latility than f'~timlllf(yfll tm!W"tf NNM41ltm·1f1, llala '!leh .... lit . '",...... ,,." 1I(.1·~,.,r,n"( "fI' I~'i'(, '/.011 t"

If( '/,.

"'(fff/(#fi4 "II/hUtll' \

9. /)1'I"illlllitll'

I),.i(il/~

Mlltld,

SiJ1(1they arc based on nllH'lIt prices which presumably have expectations of tl e future impounded in them. -!owevcr, sllch an argullIellt ovcrlooks tlte fact that an illlplied volatility is i11~imately related to a specific /ml'flllll'lrir option·pricing lIIodel-typically thc B1ack:Scholcs model-which, in turn, is intimately related to a particular set ~If dynamics for thc undcrlying stock price (geollletric Brownian motion ill tl e B1ack·Scholes case). llcrein lies tlw prohlem Wilh illlplicd volatilities: If th B1ack-Scholcs forllluia holds, then the parallleter a can he ret'overed with III m-or by inverting the Black-Scholes fOnJlllinllllia provides no new information. Of course, in this case the historical volatility estimator is elJuall), IIseless, since it need not enter into the correct stochastic volatility optioll-pricing /(lJ'IlIltla (in ract it does not, as shown by Ilull and White ll9H7 J, Wiggins 119H7J and others-sec Section 9.3.(j). The correct approach is to IIS(' a historical estimator of the unknown parameters clllering into the pricing formula-in the Black-Scholes case, the parallleter a is related to the historical volatility estimator of continuously compounded retllrns, bllt IInder other assllmptions for the stock price dynamics, historical v~)latility need 1I0t phI)' such a celllral role. This raises all interestill!{ iSSllt' n~!{anlill!{ the validit}' of the Black-Scholes /i)flllula. If lhe Black-Scholes formula is indeed t'(lI'rect, thell tht~ implied volatilities orany set of optiolls ont he sallie stock lIIust be 1I11/1l1'rimll)' ilil'lllil'lli. ()f course, in practilT til!'y n('ver ar(,; thlls til(' assllmpliolls of lile IlhH'kScholes lIIodel canllotlitnally be true. This should not cOllie as a cOlllpletc sltrpriS('; aha all the assumptiolls or the Black-Scholes IlIlHld impl)' Ihal options are redundant securities, which e1iminatcs the nccd for organil.cd optiolls markets altog(,ther. The ciifficlllty lies ill d('tnminillg which of the man)' Black-Scholes assmll)ltions arc violatcd. If, for example. tltc Black-Scholes model /;Iib ('111-

(/ J. IlIIjill'llll'lIlillgl'f/rfil/lI'lnr (Jjllillll I',irillg ;\/"r/,./,

piritidl\' I)('callse stock prices do not 1()lIow ,I IOf.:1I01lJl.t1 dilfllsion, we lIIay Ill' ilhk 10 specil)' all alterllate pritT pron'ss Ih.1l lils ill(' data heller, ill which case the "illlplit'll" parallwt('r(s) of options Oil tite SilllH' stod;, may illdel'd he nUlllerically identical. Altenratiq'ly, if Ihe Black-Scitoles model bib l'lllpiricillly hecause ill practic(, it is illlpllssihk to tl;uk cOlltilllHHlsly dUI' 10 transactions costs and other illstitlllioll,t1 cOllstrilillls, thell 111,11 kets arc n('v('r dYllalllically complete, optiolls an' lIeH'I" rnirnHl;llIt serllritit,s, and we sholiid never cxpect "illlplied" parailleters of optiolls on the sallie slock 10 he 1IIIIIH'rically idelltical I()J' any oplion-pricillg I()rlllula. In this case, Ihe degrce to which implied volatilities disagtn' 111,1)" 1)(' iUI indicatioll of h,lI\' "redIIlHLIIII" options really arc. The 1;lcl thill opliolls Iradns quo\(' pri, es in 1"1 IllS olBI,llio alld if this 11lH·('rtainty (lIn) is lIot perIc'('(11' cOl'relall'd with Ihl' IlIlcertainl)' illhel'ellt ill th(' stock pl'ic(' process (II,.), Ihl' rl'plicalillg portlillio will lIot 1)(' ahle to "spall" the possihll' O\ltCOIIJ(', Ihal all oplioll 111;1\' rl';t1i/(';11 Illallll'il), (WI' Ilarrison ;lnd Kn'ps II D7~) I alld Ililllie ;In
5%

Annual Expected Stock Return

15%

Annual Standard Deviation of Stock Return

20%

Initial Stock Price 1'(0) Time to Maturity 'I'

$40 I Year.

Frolll the entries in Table 9.G, we see that large differences between the con tinuous-time price fI (0) = $4.7937 and the crude Monte Carlo estimator if (0) can arise, even when III and n are relatively large (the antithetic estimator is defined and discussed in the next section). For example. H(O) and H(O) differ by 30 cenlS when n = 250, a nontrivial discrepancy given the typical sizes of options portfolios. ' The difference between /itO) and 11(0) arises from two sources: sampling variatioll ill li(O) and the discreteness of the simulated sample path~ of prices. The former source of discrepancy is controlled by the number of replications m, while the latter source is controlled by the number of ohservations 11 in each simulated sample path. Increasing m will allow us to estimate r:" [Ii (0) 1with arbitrary accuracy, but if n is fixed then E" [H(O)] llced not converge to the continuous-time price H(O). Does this discrepallc), illlply that MOllte Carlo estilllators are inferior to closed-form solutions whell such solutions are available? Not necessarily. This difference hig-hlights the importance of discretization in the pricillg of path-dependent securities. Since ''ie are selecting the maximum ove' /( ('xpollt'lltials of the (discrete) partial slim L~=I where k ranges from 0 to II, as II il1creases the maxinllllll is likely to increase as well. 2 t; Heuristically.

r,·,

~h/\hhllllgh

it is prohahle that the.' maximulII of the partial sum will inrrt"a~ with n. it A. . We illfl('OIS(, II in 'l';,bk ~Ui. Wt" gt'IWfClIt";" llt"W indt")>t"nc\f'lll t-andmu S("'1I11'II(T II; 1;'=1' and ,heft' is ;,dway~sc))nt' challct" IIla1this I)("W sequenre with mure tenns will 1H"\·(·r1h('h-~'\ yic.·}(\ ~ll1al!("J' panial Slims, i...

1I0t gll;U";I1H(·(·(l.

I).

Jah/,' 9.6.

f)rri!lfIti!,,·f'Ii,.;lIg .\I1I11t·11

,\101111' t:llt/" "\/illlll/IOIIII/ /ouiduII!.- o/IIUJII

(:,11I"III;II"T "r 1I101l\(' (:arlo l'Slilll;1I0rS is \0 replan' estilllates hy their popilialioll ('('"lllnp"nS Wh('IJeI'tT possihle, fill' Ihis rl'dllces salliplill~ variatioll ill Ihl' ('slilllaIOl, For l'X;1I II pie. whell sillllliatill~ risk-liculralized ass('1 rl'lul"lls, III-pr('ssillg E" [.I (X) I as lite sllill or E'I g( Yl J alld E'U( X)-- g( }') I, Ih .. n'l'('(t;r[ioll 10 he ('sti11I;t[ed is de('ollll'osed illto two t(,rllIS wherc [hc firs[ [nlll is kllown alld thc S(,(,(IlH\ 1ol1l agellts' prereu~ul'l's auel their il\\'{'stllll'lIl opportullity Sl'" CIS, rill' {'''ClIlll'k, in Ihl' slOeltastic,,,"I"Iilill' Illlldd .. rSl'l'Iioll !l,:\'(i,

9,5 ConcllLo;ion Thl' pricing or dnil',lIil'l' s(,curities is lIue 0(' Iltl' ullqualiliec\ SUllTSSCS 0(' 1110dlTil {,(,OIlOl1lics, It Itas cltangeeltltc \\'a)' e{'t'lloJlli~Is view d)'llamic modcis o('securilics prin's, and itltas had all C'lHlI'IlIOllS ill1P;ICI 011 lite invest11lcllt COllllllllllilY, TIll' ('('eatioll of' ever moJ'(' COlllPlcx lill;111l i;t\ illStl'l111ll'lItS has I>e(,11 all iJ1lport;1I11 stim11l11s Ii.r a(,;1
dillll,itlll,'jllllll'

~1t-nuu (1~17t}h}

PICHt',,,,,,,,,,

Sc'('

393

l'rub/ellu

that volatilities do shift over time ill random fashion, it is clear that issues regarding market incompleteness are central to the pricing of derivative securities. In this chapter we have only touched upon a small set of issues that smrollnd derivatives research, those that have received the least attention in the extant literature, with the hope that a wider group of academics and investment professionals will be encouraged to join in the fray and quicken the progress in this exciting area.

Problems-Chapter 9 9.1 Show that the continuous-time process /JII(t) of Section 9.1.1 converges in distribution to a normally distributed continuous-time process PCt) by calculating tile the moment-generating function of pn(t) and taking limits. Derive (9.3.30) and (9.3.31) explicitly by evaluating and inverting ~e Fisher information matrix in (9.3.7) for the maximum likelihood estimators I'- and ~ of the parameters of a geometric Brownian motion based pn regularly sampled data. !

9.2

a

a

9.3 Derive tile maximum likelhood estimators 1'-, 2 , and y of the parabeters of the trending Ornstein-Uhlenbeck process (9.3.46), and calculate their asymptotic distribution explicitly using (9.3.7). How do these three estimators differ in their asymptotic properties under standard asympLOtiCS and under continuous-record asymptotics? 9.4 You are currently managing a large pension fund and have invested most (lfit in IBM stock. Exactly onc year from now, you will have to liquidate your elllire IHM holdings, and you are concerned that it may be an inauspicious time to sell your position. eLM Financial Products Corporation has come to you with the following proposal: For a fee to be negotiated, th!!y will agree to buy your entire mM holdings exactly one year from now, but at a price per share equal to the maximum of the daily closing prices o~r the olle-year period. What fee should you expect in your negotiations with eLM? Specifically: 9.4.1 Estimate the current (time 0) fair market price H(O) of the option to sell OIl the maximum using Monte Carlo simulation. For simplicity, assume that IBM's stock price P(t) follows a geometric Brownian motion (9.2.2) so that J>(l~)

log - - ~ J>(lI)

N (ll(1~

2 - 'I), a (/2 -

til),

(9.5.1)

and lise daily returns of IBM stock over the most recent five-year period to estimate Ihe parameters 11 and a~ to calihrate your simulations. Assume

394

9. /)nivlltilll' ",icillg Model!.

lhalthere arc 2:)3 tradillg days in a ycar ancllhal market ptin's volatility when markets arc dosed, i.e., weekcnds. holidays.

\1;I\'C 110

9.4.2 Provide a !)!i% cOlllidclln: illlerval for ('(0) and alll'Slilll;Itl' oflhe \lumber ofsilllulatiolls Ileeded 10 yield a price estimate Ihal is wilhin $.0[, of the true price. 9.4.3 How does Ihis pricc co III pare wilh the price givcn hy Ill!' (;oldlllallSosin·Gatto forl\lula? Can you explain the discrepancy? Which price would you lise to decide whether to accept or reject el.M's propos;I!?

\ \ \

\ \

\

1 1

Fixcd-

[:-.: T I (I SC

s tu d ) , b

lL \I 'T

Iocool

10\

c Secu rities

E It

alld t ht o a n ' 1'11 n d s Ih;1l hav\- ~ lII.'X! WI.' \I ll 'l l 0 \1 1 11)' ~I'( ' nu ca ' ,1 \l { ' ll pnJV dlil'd ., ...·III·ili ,I H io i~i"n ill a h a n ~,1 IsseI. H li x c d - il ' d their o \lt t l t t 'n b e ,' p p l i c d ln J ll J ( w n I I lS e v e l o p e d s e to ' a ' f ix e d n s ,' c n lr il i{ p a r 'l l d s c v ( 'r .t li acadL'u 's , F ir i \,('i\SO r Jic s t u w li o ll a l s t r l l s l, Ill( IlS tt l c t u r e a f r o ll i IIU' l'i{ dy of ti lt ' !1Ia ' f ix e d f ix e d - in nd the tlily m rk('\~ ·i ll c o m ,l I 'k e ls ir own Ii)!' TrL come e si/,l.' is , s e c u .- il ';\sury S They tllc; ie s hilS le l' ll li ll o lo g ) ', L 'c u l' il have il K o ll le J~\lr('d h y q u k il s I S .i ,l k iT c x o w n tr a ll li ti e sCCllrit trcllld a d it io n c w is t' t h e sO ics h a v n o c ls y hir){< s , Se( lH lo \Y e a s p e H lS la ll d il lg o r ' rq?;an 'o ll ( l, I ln C l· n c ia l ph lk-ss o f lI lt 'l ll ti s lu d ) ,i a il ll y , \{ w h " ,t il li ll g f i x n '( t' ' s il I l f in a n raded. so Ihd c l' l ·i n c o t:\111\ c e ll w o T r p r it T lllc se r a il '' ' w S vary r y h e c a l l i r d , li x e d c u r il ie it h o u t ( 'x ! ,, 'c o s ll u s w ly e th e } ' h'l\'il\ (' ook s" .. 11 a.' F"holl.; ""ti F"hll lli (I\I\I;l) Ill' F:llltll.zi (t\I\lIi) for !'lInh er ti .. lails un 1h(" mark,'(:o, for lJS TI('~I'lIry St'fw ili('.\.

W.l. nalic Concl'pts IO.l.l lJiS({)Ullt/JOlldl We first define and illustrate basic hond market concepts for discount bonds. The yidd to maturity on a bond is that discoullt rate which equates the present value of the bond's payments to iL~ price. Thus if Pn , is the time t price of a discollnt hond that makes a sin~1c paylllellt 0(" $1 0\\ timc 1+ n, and Y", is the hond's yield to maturity, we have 1'", = (I

+ 1'",)",

(10.1.1)

so the yield can be found from the price as

( I+ V) 1,,(

::::

l'-U) "f •

( 10.1.2)

It is COlllmon in the empirical finance literature to work with log or continuously compounded variables. This has the usual adv:lIltage that it transforms the nonlinear equation (10.1.2) into a linear olle. Using lowercase letters (i)!' logs the relationship betweell log yield OIlHllog price is (10.1.3)

The In7ll slmdure of ill tere.1 I mle5 is the set of yields to maturity, at a given tilllC', Oil honds of different matnrities. The .'lirld JIJread S", == Y", - YI " orin tog terllls .I", = y,,1 - YII, is the difference between the yield on an'n-period bOlld and the yield on a one-period bond, a measure of the shape of the tcrlll structure. The yield curve is a plot of the term structure, that is, a plot of Y"I or y", against 11 on some particular date t. The solid line in Figure 10.1.1 shows the log zero-cou[>on yield curve for US Treasury securities at the end of January 1987. 1 This particular yield curvc rises at first, then falls at longer maturities so that it has a hUlllp shape. This is not unusual, altholl~h the yield curve is most cOllllllonly upw;u!l-sloping over the whole . I rall~(, of maturities. Sometimes the yield curve is illver/ed, sloping down oVfr tlte whole range of malurities.

l1o/tlillg-l'rriod Rrturns The /w/tlillg1Jrriod return on a bond is the return over some holding peri~ less than the bond's maturity. In order to economize on notation,. we specialize at once to the case where the holding period is a single period:~ We ~Tllb rurvc is nOI hased on 'l"ott·d slrip pri(t·~, which M~ r~adily aV;lilable only for rnet yea .. ,. hili is estilllaled frolll Ihe prices of cOIII'0J1-I)('arin~ Treas"ry lx,"ds. FiKure 10.1.\ is clll11 1-;" III .. /ir,sl sllhsCl'ipl n'/iTs itl 111,' II II 11 t1){'r orp(,l'io 0, Macaulay's duralion is kss lhan 1I\;llurity and it declines wilh the coupon rate. For a given cuupon rate, duratiun declines with the bond yield because a higher yield redllCl's the weighl Oil more dislanl paYlllenls in the average (10.1.10). The duration finllluia simplifies wi,en a coupon bond is selling al par or has an ill finite nl.lllirity. A par bond has price 1'",/ = I and yield F,,,, = C, so dlll'alion becumes /),,,, = (I - (I -I- j'",()-")/(i - (I -I- 1',,,,)-1). A consol bond with infinite maturilY has yield Y'CVI = C/I',ool so duration hecomes /),,,,, = (I -I- l"OOI)/l"OOI' NUlllerical examples thal illllstrate these properties are given in 'EIhlc 10.1. The tahle shows Mac,ltllay's dur'ltioll (and Inodiiied dur,lIion, defilled in (10.1.12) helow, in parenlheses) IiII' bonds with yields and coupon ~I~t;lfalll;ty abo sllgg('sl~ that 011 lilt' ('0 11 pOll hOlld

Ollt' could usc.' yit'lds Oil di~("OIIlIt honds r.uil,,!" than th(~ }'idcl to rairul.ttc- the prt"St"JI ,";,dlu' of (,~lfh COUPOII l'aYIl1(,1I1. IloWl'\'('r Ihi.'\

:'lppro;\("h I('qllil('~ IIl;I'

OI}('I11('i\slIre.1 cOlllpl(,l(" Jt'IIH'OtlIHHlIt'JIII.\III1( 1111(',

·HI·.

/0.

"'amlill/.~:' 1/111/ IIItH/illrt!

Table 10.1.

FiXl'll'/lInJlIIl'

SI'I IIritin

dum/i'lll j"r .\f!fftn! !JIII/d,.

Malurily (yt'ars) .J

".-

-. -

otY"

rau-

(:OUI'01l

O(,~,

Yit'ld

rIll

,I",

IO'X.

',.(H)1l

:!.OOO (:!.()I)O)

1.000

(1.000)

(:d)OO)

11l.1l0()

:!.OOO

:>.I)O{)

( 1.%1)

(·I.H7H)

(~).7:,I')

IO.O()O

:!.()OO

".()OO

( 1.~IO")

(·1.762) .. -"-

:IO.()OO

(10.000) C!O.OOO)

1.000

I.OO()

\H.OOO

(~J':':!4 )

:10.000 (:!'I.:!liH) :~().()()()

(:!H571)

-'--~-~'-"-'----.

( :IIl1pllll

OfX.

:.'~J

ll.tlHH

I. ~ 1:1:!

1':',,0

(1l.!IHH)

(I.\J:I:!)

(·I.:.r,O)

HAI7 21.1 ,,0 (HAI7) (:! l.l :,0)

l.tl~H

7.!lH!1 I',.H·II :!o."oo (7.7 t l:,) ( I r •. ·I:,·I) (:.!().()IJlI)

)

(I.HHI)

·I.-IW, (·1.:171\)

(O.tl·to)

1.!I:!·1 (I.!';:!:!)

-1.'1 J.I (-I.:W·\ )

O.~IHH (O.~Hi·1

I o 'X,

O.!IHH

- - -_.._.-

--

..

(:0111'011

ralt'

Yield

-- _.---- --".

r,JI(I I'!

rail'

Vit'll!

--.---

:xc

-----

(O.~)7Ii)

(O.~l":!)

:10

10 --~.-

..

7AH~)

(7.1:~:!)

IO.!I:,7

10.:,00

( 10.·1:\1,) (10.0011)

_-_.-----.--._----------

I f )t}~1 O'y"

O.~)77

I.H7', ( I.H;:,)

(O.~I77)

:,(Ycl

lOry"

,1.2,,0 ('1.2:'0)

7.1,2:,

O.~I77

I.HliH

1.1:,li

7.107

(O.!I',:I)

(I.H:!:I)

('LO,d)

(li.~I:I:1)

O,~J7(i

I,Hli:!

,1.0;'4

6.',4~

«(),~):~O)

(1.77:\)

(:I,HIiI)

(1i.2:-11)

_..

IH.'I~H

0.62:,) ( IH.!l:~H) :!0500 1·1.02" ( l:tliH:~) (:!O.OOO) ~I,!I:~H 10."O() (!)AIi;,) (IO.()OO)

------------------.--

I.. hlt- It'IUIII'' ~l;tLlIIl.ly's dill ;11;011 alld, lit pall'ntlu:s(.·s. l1\odili"tl dur,uinn fur honds \dth st'lt·'"h.'cI yi(·ld . . ,lIul lUallltiti"", Dllration, \'it'1d, ;llItllllalurity an' Mau'd ill ;tlHllIallillics Inll Ihe uudt" lying t·~,h \lbt1tHl'o i\"'~\lIH(' 111;11 hcmortalll implicalions fur the relalioll between the yield spread alld lulure inl('I'esl.r;III'S. It Illeans that lhe yield spread is (lip to a const,lIlt) the ol'tilllall()rn:ast(')' of the challge ill the IOllg-bond yield over the life of the short hUlld, alld the optilllal f(H'CcaSler of changes ill shOrl rates over the lil'c of the long hondo \tecilling that we h,l\'(' dropped all nlllstallt terlllS, the rdaliolls alt'

(n~I)S"1 =

i':lly,,-I,I+t-Yllrl.

(10.2.14)

and S",

:=

E,

L(I - i/Il)6,YI.t+i "-1

[

]

'

(10.2.15)

I~t

Efjllalioll (10.2.14) can be obtained hy sllhstitutill!-: the ddillilion of r".'1- I, (lO.!'!,,», illto (10.2.9) alld rearrangillg. It shows thaI whell the yield spread is hi!-:h, the 10llg rate is expected to risl'. This is hecalls.. a high yield ~jlr('ad gives Ihe IOllg hOlld a yield advalltage whirh IIIlIst he ollset hy an anticipatccl Gtpital loss. Surh a capital loss call only COIllC ahout through all ill('f'ease ill lhe long-bolld yield. Equatioll (10.2.1:1) lilllow~ dir(,ctly 110111 (10.2.1:\) with constant expected excess returns. 1\ shows Ih;ll whell the yicld spread is pro{"('ss

i:-. clisfll.'ist'(\ 11I111u.'r ill show tllal yield spH'ads tend lO forecast declines ill long·bond yields.

i

1"

~J1d rim

10.

FiXI'tl-/II('(IIIII' .vm,.itir,~

the regression

J:', =

J1."

+ y" .';", + fill'

(IO,\!,IH) tH

The expectations hypothesis implies that y" == I for all /I. Table 10.3 reporlS estimated y" coemcienL~ with standard errors, correcting for heteroskedasticity and overlap in the equation errors ill the manner discussed in the Appendix. The estimated coef[jcienL~ have a U shape: For small n they arc smaller th;11I one bllt significantly positivt'; lip to a year or so they decline with 11, becoming insignificantly dilferent fl'Om zero; beyond one year the coellicienlS increase and at ten years the coefficient is even significantly greater than olle. Thus Table 10.3 shows that yield spreads have forecasting power ror short-rate 1ll0vemenlS over a horiwn of two or three months, anel again over horizolls of several ye'II's. Around one yt';lr, however, yield-spread variation sn~l1Is almost lin related to suhseqllent JIIovemenlS in short rates. The regression equation (I O.:!.I H) cOlllaills Ihe same infill'lnalioll as a regression of (1/11) times the excess II-period return Oil :III II-period hond onto the yield spread s",. The relation hetween excess [etunls and yields implies that the excess-relllrn regl'essiolJ would h;lve a coefficient of (I - y,,), Table 10.3 implies that yield spreads forecast excess returns olltlO horizons of several years, but the forecasting power diminishes towards ten years, There arc several econometric difficulties with the direct approachjust descrihed. First, one los('s II p('riods or dala at the (~IJ(I of the salllpk period, This can be quite serious: For example, the len-year regression in 'Iitble 10.3 ends in 1981, whereas the three-mollth rq;ression ends in 1991. This makes a substantial difference to the reslIllS, as discussed hy Camphell alld Shiller (1991). Second, the error term {IO' is a movillg average of order (n - 1), so standard errors must he conected in lhe mallller descrih(~d in the App\:ndix, This can lead to IInite-sample problems when (II - I) is not small relative to the sample size. Thinl, the regressor is serially correlated ai'll correlated with lags of the dependellt variahle, and lhis too call ('allse fil/ite-sample prohlems (see M,lIlkiw and Shapiro [1~8fiJ. Rkhanlsoll alld St6ck [1990], ;lI\d Stambaugh 11!)H{ij). Although these econollletric prohklllS are imj)onallt, they do lIot seem to laccount lor the U-shaped lJ O. 1IIIIIili\('II'. IIII' 111011e;lrillg a IIlIil of ris!... ,\111'1'1 lal i\'('h', ronll"'illg \';Isin').. (1!177) alld Ollll'lS,

11'('

Illiglll (';IIclllal.(I+ I J). X, is a linear fUllclioll of expected consumption growth, aud E/+I is proportional to the innovation in consumption growth. The terlll-structure model of this senioJl then implies that expected consumption growth is an AR( 1) process. so that realized consumption growth is an ARMA( 1,1 i process. The ('odlirielll ~ governs the covariance hetween consumption innov1)/11 +1>l x l,+X"

xv+,

(I - 1>~)IL~

~I.t+"

+ 1/1'2 X~I + '>:'21/2 1 ~2.H-I.

( 11.1(27) (11.1.'2H)

~AhhulIKh hond relllfllS are "," I' /JIll> alld H2u ohey

Thus Ihe s!lort

(11. U~?)

J I,

Irllll-,\Innlllll' .II{)(/,'I,I'

Th(' dilf('f('lIc(' ('/(II.lIioll lill" III" is Ih(' saJlle as in Ihe singlc-hlclol" squarerool Jllod('l, (II,I,~:\), hUI 11t(' dilf('I'('II('(' equation I(lr H!" includes only a \('rlll ill Ihe own v;lri;II\(,(, of x~ I)('caus(' X~ is IIIlCOIT('\atecI wilh 111 anel clc)('s 1101 alf('cl 11)(' v;II'i;IIH,(, of

III,

Thc dilkrcllcc ('qllatioll ((II'

A" isjllslthe Stlill

of IWo lenllS, each of which has Ih(' bmiliar I(ll'lll from lite sillglc-bclor sqllare-fOOI II10de!, The exp,'('I('d (')\('"SS log hOlld r('llim ill Ihe Iwo-I;telor IIwdcl is gil'ell

Ill'

Ed ".. ", II

-

-- (:0",11'"", I,

,\'1'

1II1t"

I - V'lI',1 1'",1+1 I/~

II\." I I)(XIt-III)

-- /i~" (I -, tjl~) ( ,\-~ I

--

II ~) -

/II"

,\'1/

fJ

(1

~ (11.1.:\,1 )

This is III We 1I0W discuss how llic models call he adapted deal with this fact. To study nominal bonds we need 10 inlroduce slime lIew nolatioll. We write the nomill·.\1 price index at til\lt~ I ",IS ClJ. ;\Ill! tht" gros~ ralc of inll,lIioll frolll I 10 1+ 1 'IS n,; I == {2,; 1/{2,. We have ;Ilre;uly ddined 1'", to he the rcal pritT of all n-period n'al bond whidl P;IYS Ollt' goods unil OIL lillie I + II; we now ddille I>!I to he Ihe nominal price of ;111 n-pcriod nominal bond whirh pays $1 at Ii III(' I + II. Frolllthcsl' IkfinitiollS il follows Ihallhe IlOlllinal price o/" an II-pcriod real hOlld is I'", (~, and the real price of all to

II-period nominal bond is I~'; (li. We do nol adopl allY spccialllotatioll /CU" Ihese last two (on(cpts. If we now apply the gelleral asset pricing condilioll.

10

lile real relul'll on all II-p('riod nominal bond. WI' filld Ilwl

:0

1,$1

= E,

[I'S'~I~'I\-I

Mit I

]

.

(11.2.1)

Multiplying through by Q,. we have

( 11.'.!.2) where M:+I == M'+I/ n l + 1 can he thought of as a nOll/illal s\ochaslic dis(Ollllt t;lctor that pri(es nominal returns. The clllpiricalli\eralUre onllolllillal honds uses Ihis result in aile of two ways. Thc firsl approa(h is to take lile primiliV!" assllllll'liollS Ihal W(' made abollt M,+ 1 ill Sec lion ILl ami 10 apply Ihem ill'le'IlI \0 M~+ I' The real lerlll-SlrllrLlIre JIlodels of the lasl scclion aJ"(~ l!Jen n'inll"rpreled as numinal tnll\-stnIC[IlrC JIlodels. Brown and Dyhvig ( I ~IK()). tor exalllple. do this when ~'SOIIH' gOVCTIIIIH"It,"i, lIot;lhly 1110.\(' "I (:.III.tcl.l. b"wl, .11111111" \ lh.. h,l\"(' 1.'~IIt'd hOlld., \\1111\(' lIominall'ayofls arc linked to a nomillal prill' index. III I!J9ti IIi(' US TIC'astlry is considerillg-

isstling- ,'\imiiar securities. These judex-Ii liked hOllds approximate I('al hOTlch hilt an' nU('ly e{\llivalelll t() rca I bonds. Un,wI\ ~lIld Sc:il,l(.'lc:r (l~I~H) gi\t';' 1\l("l(( (ti~nlssi(Jll (,I Ilu,' impl·rt(·niot\s in the UK il1dl'xin~ ~ysh·lU. anet ~'pply tht' (:I)X, l1\g('I~ml1, anti R()~, (I'JH:M) 1I\{)dl'\ III UK illdt"X-link"t\ hllml •. S'T "Iso II"". .IIHI \:;11111'1 ... 11 (I~I\I:» ",HI \:""'l'bell "lid Slrilln (I~I%)"

('x~lClly

I I, 'J'nlll-S/rUr/url! ,'vlodels 11i('}, appl}' IiiI' (:0:\, Ingnsoll, alld Ross (I !IH!'ia) sCjllare-root lIlodel directly liS III Hllillal hOlld prices, Thl' sqllare-root model restricts interest rales 10 he POSilil'l', and in Ihis n'slH'n il is JIlore approprial(' for nominal inlen'sl r.II('s Ihall f(H' 1'('.11 illll'rl'si rail'S, The secolld ''IlplOarh is 10 ;ISSllnlt' Ihat Ihe two COIIIJlOI1('nts of Ihe nOlninal slodl;I~lil' disl'ounl bl'lor, 1\11/ I and II n H I, art' independent of each otht'r. 'IiI SC(' hllw Ihis .ISSlllllptioll f;u'ilitates empirical work, take lo~s ()f tht' llol1linal slodl'lstic discollnt fade))' t() get 10 dara 011

(11.2,3) \Vh(,11 tilt' COIIIIHlJlt'lliS ,It,) I ,llld IT/II ;tr(' illdq)(,lldent, we ran price nominal honds hy IIsing Ihl' illsights of Cox, lngt'rsoll, alld Ross (I!lH!'ia) and Dybvig (1 !IH!I), R('call fronl St'nion 1I,I,:~ Ihdr r('snlt that the log hond price in a l1l()del with two illd('p('Il(\cllt ('OmpOll('llts of the stochastic discount factor is the Slim ()f Ihl' log hOlld prices illlplicd hy each compont'nt. We can, for example, ''1lpl}' I he l.ollgslaff alld Schwartz (1 !)!)2) JIIodd to nOJllin.l1 bonds hy assllmillg Ihal lilt) I is desnilll'd hy a sqllare-root single-factor model, -1/11/

I = .l'Jr

+ XIII/:! 11~1.t) I, alld

Ihat

IT,) 1

,

IS known at I and ('qllal to a state

\'al'iahk,\'~I.v\'I'lhl'lIgl,t

111;)1 = "I11I1I+rr"11 .'(Jr+x:;:!fl~l.lt,+x:!1> and Ihe l.ollgslan~S('hl\';11'I1 IIllldl'l (ksl"rihl's lIoJl1illal bonds. Mort' )!;(,IIt'J'al/)" 11)(' asslIllIplioll I hal M, f I alld 1/ n 11,1 art' ill(kpcnde~tl iJl1pli\'s thaI prin's Ill' 1I011lill Jo: I

[/'~ 1/

1.,) 1 /\1'11

_(h (6 \ l ]

1':1["" 1,')I(!;"EI)I[_I_]!ltlll-~] (h (h I "

I I

r:-I.t

==

11.2. Filling 'lmn-Struclure Models to tlU'Data

=

Q, EI [1',,-1.1+1 M I+ 1 E I+ 1

=

J>"I Qs

[(LJ]

E, [_I_J . (b+"

where the last equality uses both the independence of real variables from the price level (which enables us to replace the expectation of a p r o d u f t ' i by the product of expectations). and the fact that POI == E/[P._ 1•1+ 1 MI+I~' Equation (11.2.5) is the desired result that the nominal price of a bond which pays $1 at time t + n is the nominal price of a bond which pays one unit of goods at time 1+ n, times the expected real value ofS1 at time 1+ n. Dividing (11.2.5) by 0. we can see that the same relationship holds between the real prices of nominal bonds and the real prices of real bonds. Further, (11.2.5) implies that the expected real return on a nominal bond equals the expected real return on a real bond:

EI [1~_~I+l ~] J "I

Qs+1

(11.2.6) Gibbons and Ramaswamy (1993) usc these results to test the implications of real term-structure models for econometric forecasts of real returns on nominal bonds. Although it is extremely convenient to assume that inflation is independent of the real stochastic discount factor, this assumption may be unrealistic. Ibrr and Campbell (1995), Campbell and Ammer (1993), and Pennacchi (1991), using respectively UK data on indexed and nominal bonds, rational-expectations methodology applied to US data, and survey data, all find that innovations to expected inflation are negatively correlated in the short nm with innovations to expected future real interest rates. More directly, Campbell and Shiller (1996) fwd that inflation innovations are correlated with stock returns and real consumption growth, proxies for the stochastic discount factor suggested by the traditional CAPM of Chapter 5 and the consumption CAPM of Chapter 8. 11.2.2 Em/)ineal Evidmre on Affine-Yield Models

All the models we have discllssed so far need additional error terms if they are to fit the data. To see why. consider a model in which the real stochastic' discount factor is driven by a single state variable. In such a model, returns on all real bonds are perfectly correlated because the model has only a single shock. Similarly, ret urns on all nominal bonds are perfectly correlated in any'

I

44()

I I. 'Irnll-SlruriIll1' !IIm/t'll

model where a sin!!;lc slat(' variahle drives the nOlllinal stochastic discount IiI(' \I 1/'. In reality there are 110 d(,terlllinistic Iincar relationships alllOn!!; returns Oil dillcrent honds, so these implications are bound to he n:i('Ctcd hy the data, Adding extra state variables increases the rank of the variallcccovariance matrix of bon olld relt rns can do so only as proxies 1(11' the underlying statc variables of the lllo~c1; if there are fewer state variables than forecasting variables. this pUL~ testable restrictions on I()recasting equations for bond returns. 'A general anine-yield model with K state variables takes the f(mll

I

I

-/)",

=

A"

+ Ill" XII + ... + Ill\" XI\"

(11.'2.7)

whe 'e Xk" k == I ... K, arc the state l"ariahJes, and A" alld lJ.". /1 = I ... K, arc :onstants. The model also illlpl.ies that expected exn'ss returlls Oil long hOIl Is over thc short interest ratc (an he wrillclI as

1

E,l r".H I -

,vI,)

A;, + n;" XII

+ ... + Il~" XI\"

(ll.'2.H)

whe e A~ and B;". k = I ... K, are constants. The model puts n \lSSsect ollal restricliolls 011 these constants which arc related to the tillie-series prm\css driving the state variahles. hut we ignore this aspect of the lIIodd

hen"

11.2. Filling '/i'flll-S/rurlll;l' M(I(It-/.1 /(/111/'

/)1//1/

Now ~IIPpO~C that we do not ohscrn' thc 11'111' I'XI'I'SS rcturns on lOll!!; hOllds, hili illsl!';ul ohsl'rvl' a lIoi~y IIl1'aSllrc 1'".1+1

=

(II.:!.!/)

1'".1.1 - .\'11 !·II".I.I,

\Vh I' rc 'I ",1+ I is a II ClTor tcrill. Wc aSSlllliC Ikll // ",II I i~ h, conlainillg.l inslnlllll'lliS "/" j = I ... .1:

01'1

hogolla I loa VI'I'lo.'

(I U.IO)

Till: \'cclor h, lIli!!;ht contain la~~l'(1 variahlc~, for I'xal1lpk, if the rcturtl error '/.../+1 is serially 1Il1corrclated. We further aSSIIIIle Ihat iiII' ea('h statc vari;lhlc XI .. " = I .. , K, the cxpl'ctation or the statl' \',11 iahk conditiollal 011 Ihl' illSll'IllIlClliS is linear in the illSlnlllll'IIIS: I

E

IXk' I

h,]

(\ 1.2.11)

LOk/hjl /=1

for

~ollle

cOllstallt coe/Iiciellls (l.)' These ;tsSlllllptions imply that the expectatioll of I'",tt I conditional on the illstruiliellts, which from (I I.:!. 10) is thc sallic as Ihl' expcctatioll orthe true ex('c~s rCIUI'Il 1'",'1 1 - YI, condiliollal 011 Illc illSll'llllll'lItS, is lillcar in the illstrlllllellts:

EI"",'+I I

h,l

=

E[I'".'~I-.\'II I

h,l =

,...

II:. -I- L JJ;"E lx.,

h,l

:=

"

I

.=1

,=1

,1:.+ LJJZ . LOI,hjt.

k=1

defille C'II to he the vector [/'1.111 .. , ",V.,. 11101' asselS thcli (II,:!, I:!) call he rcwritten in veftor ()l'J1I as

11'11'1'

C,+I = A' -I- Ch, -I-

1/ 11,1'

11

(11.:.1.1:.1)

I .. ,N.

(\ 1.2.1:\)

where A' is a vt'('(or whos(' IIlh dl'lIll' II I is II;, alld C is a 11I'lIrix of('ocnicicllls whose (II, jl dClllellt is (.',,/ =

L'" 1I;"lI

k/.

( I I.:!. 14 )

k=1

Equatiolls (II.:!,I:{) alld (I I.:!. H) dl'lillc a lalclll-varialllt- IIlode! ((,.. n,p(Ttnl cX('css hOlld returns with K lalclIl variahles. En/_ I • a standard (;ARCII(I,f) l1Iodel. Tht'y lind thai a 1II0dd with y = 05 lits the short rate s(· .. ies '1llite well once GARCI-I t'frccts '\IT inclllded in Ihe 1Il001d; h()\~eve .. they do nol explore Ihe illlplications or this Ii, .. hond hond-oplion pricing.

a/

0"

Cmu-Snlilllud UI',\lri,.lilll/.\

(11/

Ihl' '/lnl/ Slmrl/lll'

So br Wl' ha\'(' (,lIIpha~ill'd tilt' tilllt'-snil's implications of aflilll'-yidd lIIodds and hav(' ignored Iheir cross-secliollal implicalions. HrowlI alld Dyhvig (19Hli) and Brown and Schadi'r (I !19'1) lake tltt' opposite approach, ignoring tI\(' IIwdcls' tillle-sl'\'ics implicatiolls and eSlimating all tht' pal'alllt'\('rs frollllhe Icrlll Sll'llCllllt' ofin\(T('sl raIl'S observcd al a poilll in lilll(", If Ihis procedll .. c is repealed 0\'('1' I!lany lilllC periods, il g('n('ralcs a scqllCIIC(" or parameter estilllates whi.h shollid ill theory hl' idl'ntit',,1 for all tillll' \ll'l'iods hilI which ill practi( (. varies 0\'('1' lilliI', Thl' procedllre is analogolls 10 rhe COlIIlIHlIl practice or calculating illlplicd volalility hy ill\'('rting the BlackScho\('s t<JrI\\l\la \lsillg tradcd optioll pric('s; tllne 100 thl' model rcq\\ires Ihal volalilil), 1)(' con~lallr ov('/' Ii 111('. hilI illlplied volatililY tellds to ,\lOVl' OVl'I' ti IIIC, Of ('()\\rSl', hOlld pricillg e!Tors I\\ight calise ('still\al = O.!lH from the estimated lirst-ordn alltocorrelatioll ofthc shorl rate, but now lhe olher pOIl'allldel's of the llIudd arc simllltalleously detCl'mined. One call of course eSli1llOlIe Ihe1ll by Ccnerali/.cd Melhod or MOJllenls. The s(luare-ruol model, like lhe homosknlaslic 1II0del, produces an av('ra!-:c forward-rale curvc IhOlI approaches ils aSYlllplO!(' vCI),slowlywhell Ih(' shoJ'l rale is highly persislelll; Ihlls Ihl' lIIodel hOI, lIlany 01' Ihe sallie elllpiriral lilllilations as lhe homoskedaslic IIlOdl'!. III sllllllllary, lhe Sill~lc-racl()r aflille-yidd llIodds WI' haw dl'sl'rihc,1 ill Ihis ch;ljllcr arc 100 restrictive 10 IiI Ih(' hl'hOlvior of nOlllinal interest rates. Th(' Ialelll-variahk struClllre of Ill X)

\

-X E,[MII .,+"

.11,+" > X).

(I 1.,.16)

III general equation (11.3.16) lIIust be evaluated using numerical rriethoels. bllt it simplifies dramatically in one special case. Suppose that M~.I+II aile! S,+" art' jointly lognormal conditional on time I information, with ~on­ elitionalexpectations of their logs /lm and /l" and conditional variances~nd covariance of their logs am"" a". and am" All these moments may depFnd on I and II, hut we suppress this for notational simplicity. Then we have E,[M".,tll S'tll

I

exp ( 11", x

(

S'+II > X)

+ /1. +

allllll

+

a"2 + 2a,.,)

/l.+a ,,-x +a, ) . ll

a.

I I

•

(11.3.17)

alld E,IMnl+ n I S,+"

~

XJ

==

exp(ll",+a~II/)(Jl'+:~"-X).

(11.3.18)

when' (.) is the cUlllulative elistriblltioll function of a standard normal ralldoJll variahk. anel x == log(X). I I EqlJations (11.3.17) and (113.1 Il) hold for any lognormal random variahles M and S and do nol derencI 011 any otllt'r rroperties ofth('se varia hIes.

",,'
'\f
- XI

r;:yyaH

,

Ifl


("f - X-/1", r;:y-

0,,/2)

(11.:1.21 )

.

" (1"

To get Ihe sialldard oplioll pricillg IClI'IlIu\a of Hlack and Sci" lieS ( I !17:1),

we lIe(~d two fllnhel' asslIlI'IlIions. Firsl, assl\lII"lhallhe cOlldilion;" \,;lri;I\\('(' of Ihe ullderlying securilY price /I periwis ahead, a,,, is prop"rtioll,,1 III II: au = lIa~ for sOllie COllslall1 a~. Secolld, assullle Ihal Ihe lerll1 slruclure

=

is lIal so Ihat I'lff 1'-111 Ie)!' sOllie conslalll inleresl rail' I'. Wilh tllese ,l ""I"A

.,,\('

,,>

(\,-x+(/'+a~/2)1I) r::

ylla

(,,-x+(/,-rr 1/ 2)1I) .J/i rr

.

( /1.:1.22)

For fixed-illcollle deri\'alives, however, Ihl' extra asslIillpliollS 1I('('deli 10 I{el the UhlCk-SdlO\cS IC)l'Inula (11.:1.22) arc nol reasonahle. SIIPPOs(' lltal Ihe asset 011 which Ihe call oplioll is wrillcil is a I.ero-coupoll I>olld II'hicl, cUlTelltly has 1/+ r p('riods 10 malurily. IfillI' oplioll has ext'l'cis(' prin' X alld l~jl)('riOtiS 10 expir.1I iOll, III,' Opl ion's 11OI),o1LIl "X piral ion willII(' ~101 x( I'r.' I 1 / I:!Of f(}I1I'~(". for allY gi\('11 " we rail ~11\\';,y., cldillt' n:! ::::::; n,./ II alld I ::::: - {,,,,/,, '" Ih.1I II (' l\1~IC·k·SdlOlt's lonl1111;aOlpplit·s lor Iha, II. Tht, ;l~.'lIl11lJ(i()lI~ gi\'('u ;11(' IU'nll'fllor,lit' HI.,d.. ~ S; I!O!c',1Iii 101'1II111a tu apply 10 all II wilh 111(' 'i..III)(, , ;1IIc1 n:!. I

II.J. 1'l'il'illJ; Fbml-III((I/IU' Dmlllllil". Sl'l'lIl'ilil',l

41i~

.\,0). TII{' r{'\evant hOlld prin' at expiLII inll is Ihl' T -p,'1 iot! bont! prin' sillt'(, Ihl' IllI

M,IX (/';', 1+ "

-

-

X, 0) 1

XI I':',ff ",0) J

(I1.:t2!i)

where {.''' is the cdl optioll pricc that would prevail ifthe stochastic discount brlor weI"!' M". III othn words optiolls c;ln he priced using the stochaslic lertll-slnll'tlll"!' IIIOd..!, using Ihe delermillistic model only to a(Ullst Ihe exercise pritT alld 1111" filial solulion ((II' the oplion price, This approach was firsl IIsed hy 110 ;llId Lee (I (IH(i); however as nyhvig (19WI) POilllS OUI, 110 ;uull.(T (hoOSI' ;1' t1wif II-IIIOdd Ilrl' Sillgk-f;\l"lOf 1r00\Ioskedaslic 1II0del wilh t/> = I, \"lrich Ira, 1I1111}('IOIIS unappealillg properties, Black, lklmall, alld Toy (1~1!IO), Ilealh, .larrow, and Morton (1992), and 111111 and \\'hile (I!l!/Oa) IIS(, similar ;lpplOaciws wilh differenl choice.~ fill' Ihe a-lIIodel.

I 1.4 Conclusion In Ihis chapler liT ha\'(' thorollghly I'xplorcd a Iraclahle class of illttTestrale models, Ihl' so-called affine-yield models. In Ihes(' models log hond yidds arc lincar in slate variables, which simplifies Ihe .nlalysis of Ihe term slructure ofilllcresl raIl'S and offix(,d-illcoll\e derivalive securilies. We have als"o S('('II Ihat affilw-yil'ld IlIo(ld, have SOUl(' limilaliollS, particlliarly in desnihing thl' dynamics or the short-term nominal interesl rate. Th('l"(' is 'Iccordillgll' I-(reat int('l'esl ill del'dopilll-( more f1exihle models Ihal allo\\' for slich pheIlOI\\('II;1 as 1I11111ipic 1'('l-(illll'S, nonlinear nwan-n'l'ersioll, and serially corrd;tled illl('l'esl-rale l'olatililY, and thaI fully exploil IIII' informatioll ill Ihe yidd 1'1Ir\'('. A~ Ihe lel'lll-slrll(llire lill'rallllT \IIoves forward, il will he imporlanl 10 intel-(r.\I(' il with Ihe I'esl or the asset pricilll-( lileralllrc. WI' have secn thaI lenll-sll'll('\IIJ'(' lIIodds can 1)(' dewed as lilllc-series models filr the sl')chastic discollllt bctol', The research on stock retllrns discllsscd ill Chaplel H also sl'('k.~ (0 chara('\('ri/e Ihe hehavior of Ihe slochastic disco\lnl f;lclor. Ry COlllhining lilt' inlimll;!tioll in til(' prices of stocks and lixed-illcolI\e securities it sholllcll)(' possihle 10 I-(aill a hel\er IInderstanding of Ihe economic forces that d('tennil\(' Ihe prices of financial assets,

Problems-Chapter J 1 11.1 AsslllIll' Ihallhe hOllloskl'(bstic IOl-(nol'lnal I>ond pricing llIodel giV('1I h)' eqllatioJ\s (I I. I.:\) alld (11.1.:») holds wilh rp < I. 11.1.1 SIiPPOSI' VOlllillhc C\ll'rl'lIllerlll structure of ill (t'res( rates Il~illl-( a randolll w;rlk lIIodd ;\lIgllll'lIted by I\t-terrllillis(ir drift ('\'IllS, clju;lIioll ( 11.:~.4). Ileri\'(' all ex pn'ssioll rdat illl-( the drift terllls 10 Ihe stale \'ariahle .\', and the paranlt'll'Is pf lilt' trill' I>olld pricing Iliodd.

11.1.2 (:OIIlP;III' IllO. III the two cxamplcs abovc it is easy to conlirm that this is thc casc provided that f., is symmetrically distributcd, i.c., its third momcnt is zen>. For thc nonlincar moving averagc (12.1.4), for examplc. we have Elx, X,_I] == E[ Cf.,+af.;_1 )(15,-1 +af.;_2) J == aE[f!.:-ll = 0 whcn EI f.~_1 J=O. Now considcr thc bchavior of highcr momcnts of thc form

Models that arc nonlincar in the mcan allow these higher 1ll0mCIllS to he nonlcro whcn i, j. k, ... >0. Models that arc nonlincar in variance bllt obey thc martingalc propcrty havc E(x, I X,-I •.. . J=O, so their highcr momcills arc lCro whcn i. j. k • ... >0. These models Gill only havc nonzero higher momcnts if at Icast onc timc lag indcx i. j. h• ... is 1.Cro. In thc nonlinearmoving-avcragc cxamplc, (\ 2.1.4). the third momcnt with i= j= I, E[ (f,+af ;_I)(f,-I +af;_t)2 J 2 aEk/_ 11+2a E[f~2J Elf;_11

i' O.

In} tc nrst-o~dcr ~CH cxample. (12.1.5); the same third momellt [[x, x;_11

== ·(f.,Jafi_l)fi_Iaf.~21 =

o.

fillt for this model the fOllrth IIlOlllent with

i= ,j=k=l, E(x; x;_.1 = Elf; at f.:_ 1f;_21 i' O. Wc discuss ARCH and other models of changing variance in Sectioll 12.2; for ;the remainder of this scction we concentratc on nonlinear models or thc Iconditional mean. In Section 12.1.1 we explore scveral alternative ways to ~arametrize nonlinear models, and in Section 12.1.2 wc usc thcse parame ric models to motivate and explain somc COllllTlonly IIsed tests for nonlin arity in univariatc time series, including thc test of Brock, Dechert, and Sch inkman (1987).

I

J2. J. I S011lr I'llram~tri(: /I1otidl

impossible to provide an exhaustive aCCOllill of all nonlinear specilicati ns, cven when we restrict ollr alleillioll to the slIbset of parametric 1II01.els. Priestlcy (1988), Ter;isvirta, 'lj0stheilll, and Granger (\ 9!14). and Tong (1990) provide excellcnt coveragc of mailY of thc lIIost poplllar 11011Iinellr time-series lIIodels, including IllOre-speciali/.ed lIIodels with sOllie very intrigUing names, c .g., .v/frxritillg Ihrr.~/lOltllllltort'gTr.\.\i(m (SETAl{) , flm/J/ill/tirdrJJmcll'1li I'x/Jonential a utoTrwe.uio 11 (EXPAR), and Jttltr-dr/Jenlil'1lt "'f}(lrll (SDM). To provide a sense of the hreadth of this area, we discllss fOllr examples in

.17\

12. I. NtmJil/i'l/r S/rurllHl' in (/lIil>lIIill/1' Timl' Sl'ril'.1

this sectioll: polYIIOIlliallllodels, pic(Twisc-lillear lII(xlds, Illodels, alld deterministic rhaotir lllOllels.

Mark()v-llwitchill~

['o/)'Ill/millt Mor/pLf Om' way to reprcselll the fUllction g(.) is cxp.\l\d it ill .1 'bylor scrics arollnd (/_I=(/_~='" =0, which yields a .!

LIl~XI-I+

LLI';,xl-,xl._,

1;;:;1

1=1 j:.-:/ no

+L ;=1

ro

ro

,=,

k~)

LL

(~k X,_,

XI_,

X'-k + ....

( 1~.1.7)

It is also possible to ohtailllllixed autore~ressive/ll1()villfi-avna~e representatiolls, the nonlinear equivalent of ARM/\ iIlodds. Thl' hili\lear lIloclel, for example. IIses la~g('d vailles of XI. I.. g~ed v;t1ll(,S of f " and (Toss·products or illl' 111'0:

""" La,E /_;+ Lfi,xl "'-

1=1

1=1

-t-

"'v

"-

,=I

,= I

L Lv" XI_,f /·

l•

(1~.l.H)

} 2, NOI/!iI/t'l/rilil'v il/ Fil/tll/ritll

})1I111

This lIlodel call capt 111'1' lIolllilll'arities parsillloniously (with a fillite, short lal-( 1{'lIl-(th) II'hl'lI PUI'(' lIolllille;II' lIlovilll-(-;\Vcral-(' or Ilonlinl'ar alltor('~rl's­ siVl' lIlodds f;lillo 110 so, (;rangl'l' ant! Andersen (197H) anti Suhha K;'o ;mel Cabr (1!IH'I) ex pIon' bilillear lIIodels ill delail. l'il't'l7l1i.wl.i 111'11,. II/oddl Allothn popllLlr \\';\\' to lit tlotllitlt' Ie. lien' the illtenqll lot'lcdly standard normal under the liD Ilull hypothesis; it is applied ;lIld n;plail\t'd by I !sid I (1~IH9) alld Scheinklll'1I1 alld LeBaroll (1!IH!l), who provide t'xplicitexpressiolls lor a".r(/i). Iisieh (I !lH!I) and Iisieh (I !l!ll) report Monte Calio lesults on the si/,e an(1 power of the BDS statistic ill linite sampks. 'A'hill' thele are sOllie pathologicaillolllillear Itwdds fOI which C,,(k)= (;1 (h)" as in liD dala, the BDS statistic apJ>('ars to ha\'(' good power againsl Ihl' IIIOSI cOllllnonly \lsed nonlinear IllOdels. It is importallt to understand th.1l it has power against models that are IHllllinear ill \',\rian('(~ hut not in mean, as well "s models that arc nonlinear in meall, Titus" liDS Icjeuion docs not necessarily imply that a time-series has a tillle-v.Il'ring conditional mean; it could simply he evidellu' 1(.1' a time-var),illg conditiollal variance. IIsieh (1901), 1'01' example, strongly n:jnb the hypot hl'sis that (0111111011 stock returns arc liD using the liDS test. I Ie tltcn estimates models or the tiIlH:-v;lr)'illg conditional variallcc of returlls and g(,ts 11 11K It weaker e\'idencc ag;lillsl the hypothesis that tlte r('siduals 1'10111 such IIlOdcls ;1)(' III>.

12.2 Models of Changing Volatility III this sn:tiol\ W(' cOl\sider alterllati\,t' ways to IlIO), which is uhlained hy selling 1.=0, 1'= I, al1(1 b=() to gel I I

rs. .

log(a,) =

lti

+ Ii !og(a,_1 ) + a

"I", l>i"l\. (;'''''1\'''' .• ,,r(' log(g(J/,+I/a,(O)) -log(a,~(O»/2 - log(J2rr) - II; t I /'2a,~ (0) -log(a/ Ihe po\\,tT thr('(' halvl's. The volatilily proCI'SS is highly pcrSiSll'll1 ill ;rllll ... Illockis eSlilll;II('d, ;rllhollgh ,he degn'(' ol"pI'rsis'('IIIT is st'llsilin' Itl SI)('( ili,',lIioll ill IIII' p,,,I-Wodd War \I period, Additillllllll'.Ox/df/III1/III\' \ 'f/rillh"', lip 10 Ihis pOill1 \1'(' kll'(' 1110111'1("(1 volalilily IIsillg ollly Illl' pasl hislorl' of n'lIlrns tlll'lllScll'cs, II is slr;Jighll()rward 10 add olher cxplanalory \"Iriahles: For l'xalllpk, (llll' LIB II'riw all allgllH'llled CARel I (1,1) Illodd as ( I:.!:.!,IKI whe\"(' X, is ;111\" ,'ari"hl(' kl\m"l\ al tillle /. Provided thaI X,::,,:O alld )' :::0, Ihis Illodd still ("ol"lr"ills ,'o!alilill' 10 1)(' positiVI'. Altel"llalivdy, OIlC rail add ex pl'lI 1;1 101"\' vari;rhl('.~ 10 Ill(' H:i\R( :llllIodel Wilholll allY sigll reslriniolls, (;loSII'Il,J;lg'III1lalhall, alld RUllkle (I !)!):I) aclcl a shorl-Ierlllllolllinal intl'l"esl rail' 10 various (;.-\\{( :'1 I\lodds alld show Ihal il has a sigllilical\1 posiliVl' clll'!"1 1>11 slmk Ill~nk(', "oLililitl', (:/IIl/litilllllll.'VlIlIl/ll/lllldity

The CARel I llltl(I..t~ 1\'1' hOI\"(' (ollsid(')cd imply ,hal Ihc dislrii>ution or rclurus, ("(llldilioll;rl Oil til(' 1';'" hislOIY or rellll'llS, is lIonlla!. Eqllivalelltly, Ihl' sl'lIuLrnli,,'d r('siduals 01 IheS(' 1I10dds, f ' l .c0I=I/H ,In,(O), should Ill' IlOnlla\. l '1I",rlll\l;lleh', ill I'r;\clin' Iherl' is ('XCI'SS kllrtosis ill Ih(' slall(brdi/nl \"('sidllals 01 (::\\{( :11 Illllllds, ;1I1)('il kss Ih.uI ill thl' raw relllrllS (SCI', lor ('x;\lIlph-,1\011("J,I('\'II~)H71 .11111 Nelsoll 11\'\)( I),

!489

12.2. ModeLl oj Changing Volatility

Onc way to handlc this problcm is to continue to work with the conditiollal normal likelihood function defincd by (12.2.16) and (12.2.17); but to iuterpret the estimator as a quasi-maximum likelihood estimator (White [ 19H2]). Standard errors for parametcr estimates can then be calculated using a robust covariance matrix estimator as discllssed by Bollerslelanrl Wooldridge (1992). Alternatively, one can explicitly 1lI0dclthe fat-tailed distribution of the shocks driving a GAReH process. Bollerslev (1987), for example, suggests a Stll(\t-nH distribution with k dcgrees of frcedom: K(E/+I(O»=f

(

k;

1)

f

(k)-I 2

(k-2)-1/2

( l+f~+~2 (O»)-(A+I)/2 '

. (12.2.19) where fO is the gamma function. The t distribUlion converges to the normal distribution as k increases, but has excess kurtosis; indeed its fourth moment is infinite when k :s 4. In a similar spirit Nelson (1991) uses a Gcneralized Error Distribution, while Engle and Gonzalez-Rivera (1991) estimate the error density non parametrically. GARCH models can also be estimated by Generalized Method of Momellts (GMM). This is appealing when the conditional volatility can be wrillen as a fairly simple function of obsel\led past variables (past squared retllrtlS and additional variables sllch as interest rates). Then the model implies that squarcd returns, less thc appropriate function of th~ observed variables, are orthogonal to the obsel\led variables. GMM estimation has the usual attraction that one need not specify a density for shocks to returns.

a?

Slor/wslie-Volatility Models Another respollse to the lIunllorlllality of relUrns conditional upon past returns is to assume that therc is a random variable conditional upon which returns are normal, but that this variable-which we may call stochastic volati/ity-is not directly observed. This kind of assumption is often made in continuous-time theorcticalmodels, where asset prices follow diffusions with volatility parameters that also follow diffusions. Melino and Turnbull (1990) and Wiggins () 987) argue that discrete-time stochastic-volatility models are natural approximations to such processes. If we parametrize the discretetime process for stochastic volatililY, we then have a filtering problem: to process the ohsel\led data to estimate the parameters driving stochastic volatility and to estimate the level of volatility at each point in time. A simple example of a stochastic-volatility model is the following:

where €/~N((). a,2), ~/~N(O. a/), and we assume thal f, and ~, are seri~lly uncorre\ated and independent of each other. Here a, measures the dif-

I

4~0

/2. NOlllinearities in Final/rial /)(/t(l

I

ference between the conditional lo~ standard deviation of returns and its mean; it follows a zcro-Illean AR( I) process. We can rewrite this system hy squaring the return equation and taking logs to get 10g(IJ~)

::=

a, + log(E;),

a, = cpa,-l + ~,.

(12.2.21 )

This is in linear state-space form except that the first equation of ( 12.2.21 ) has an error with a log X2 distribution instead of a normal distrihution. To appreciate the importance of the nonnormality. one need only consider thl' fact that when f., is very dose to zero (an "inlier")' log(E~) is a very large negative outlier. The system can be estimated in a variety of ways. Melino and Turnhull (1990) and Wiggins (1987) use GMM estimators. While this is straigillforward. it is not emdent. Harvey. Ruiz. and Shephard (1994) suggest a (Iuasimaximum-likelihood estimator which ignores the non normality of 10g(E;> and proceeds as if both equations in (12.2.21) had normal error tenlls. More recentlY,Jacquier, Polson, and Rossi (1994) have suggested a Bayesian approach and Shephard and Kim (1994) have proposed a simulation-hased exact maximum-likelihood estimator. 12.2.2 Multivariate ModeLl

So far we have considered only the volatility of a single asset return. More generally, we may have a vector of asset retuJ'llS whose conditional covariance matrix evolves through time. Suppose we have N asseL~ with return innovations l)i.,+I, i= I . " N. We stack these innovations into a' ctor '71+1 =[ 1)1.1+1 ... I)N.l+tl' and dc/Inc aii,,= Var,(T/i.,+d and a'j.'= V,(l)i.I+I.l)j,I+I); hence l:,=[a,j.,] is the conditional covariance matrix of a I the returns. It is often convenient to stack the nonreliun fJim.l-h and the retllrns 1/i/ alld '111"; and we might nlodd till' cOllditional variance or the idiosyncratic shock III \'l'tlll'll as ;\lIothl'l' IInivariate CARCI-I( 1,1) process. The covariance matrix implied by a \1Iodl'l of this sort is guaranteed 10 he positive definite, alld the 1I111lli>('r of parameters ill thl' lIlodel grows at rate N rather than N'l , whit'h lIIakes Ihe lIIodd applicable to Illuch larger 1\lImbers orasscts. BraulI,

494

12. NUllii1lfaritil'.1 ill Fi,ul/u'i(// /)(/1(/

Nelson, and Sunicr (1995) take this approach, using EGARCII fUllctional forms for the individual componenL~ of the model.

12.2.] Links between First

01/(1

Second MUlllenls

We have reviewed some extremely sophisticated models of tillie-varying second moments in time series whose first moments are assumed to be cunstant and zero. But the essellce of finance theory is that it relatcs the first alld second moments of asset returns. Accordingly we now discuss models in which conditional mean returns may change with the conditional variances and covariances. The GARClI-M Modd Engle, Lilien, and Robins (I 9S7) suggest adding a time-varying intercept to the basic univariate model (12.2.2). Writing r,+1 for a continuously nllllpounded asset return which is the tillle series of interest (since we no longer work with a mean-zero innovation), we have

( 12.2,:~()

'rI

I where 1+ t is an lID random variable as befure, and a; can follow any (;AR( :11 I process. This GARCi/-in-mean or GARCH-M model makes the conditional €

; mcan of thc return linear in the conditional variance. It can be straiglllforI wardly estimated by maximum likelihood, although it is not known whethn thc model satisfies the regularity conditions for asymptotic normality of the maximum likelihood estimator. ' The GARCH-M model can also bc specified so that the condition'llmean is linear in the conditional standard deviation rather than the conditional variance. It has been generali7xd to a multivariate selling hy Bollerslev, Englc, and Wooldridge (1988) and others, but the number of paralllet('\'s increases rapidly with the numher of relums and the model is typically applied to only a few assets.

I

The Instrummtai V(/ri(/bles ANm)(lch As an alternative to the GARCII-M model, Campbell (1987) ali(I Ilarvcy (1989,1991) have suggested that one call estimate the parameters linking ; first and second moments by GMM. These authors stan with a lIIodel fi)I' 'the "market" return that makes the expected markct retunt linear in its OWII variance, conditional Oil sOllie vector HI cOlltaining I. inslnlllH'lIts or forccasting variablcs:

E[ r,.,I+ dH, J

(I~.:DI)

12.2. Motif/I t!/ OWlIgi IIIi Voill/iiil.v CaJllphell alld l/arvey assullle that cOIlr 1;1" (x) ()f /tI( x):

J 2.3. Non/Jllwmetllc Estimation

. t .. lI.

lI.

...

A

~

£I.

l}.lI.

lIfIrl.

. ..

£I.

~ lI.

C.

01 lloll.

t:. lloA

A lI. A

~

"

'"~I

1(L)--~~--~~"~~--~~~-w~~--~--~~~~.--~------~"w 2 " T ~"

x

• Figure 12.4. Simulation of Y,

Sin(X,) + O.5f,

Unclcr certain regularity conditions on the shape of the kernel K and the magnitudes and behavior of the weights as the sample size grows, it may be shown that mh(x) converges to m(x) asymptotically in several ways (see Hardie [1990] for further details). This convergence property holds for a wide class of kernels, but for the remainder of this chapter and in our empirical examples we shall usc the most popular choin' of kernel, Ihe Gaussian kernel: I .' K,,(x) = --e-:;:E. (12.3.10)

h.,ffii

Au Illustration oj Kernel Regression To illustrate the power of kernel regression in capturing nonlinear relations, we apply this smoothing technique to an artificial dataset constructed by MOille Carlo simulatioll. Denote by {X,} a sequence of 500 observations which lake on values between 0 and 27T at evenly spaced increments, and leI {r, }Ill' rdalcclto {X,} through the following nonlinear relation:

( 12.3.11) where {f,} is a sequellce of liD pseudorandom standard normal variates. Using the simulated data IX" Y,} (see Figure 12.4), we shall allemptto estim,lIe the conditional expectation E[ Y, I X,) = Sin(X,), using kernel

502

12. Nonlinl!(lrilies ill HI/'II/rilll I )ata

regression. To do this. we apply the Nadaraya-Watson estimator (12.:t!l) 'th a Gaussian kernel to the data. and vary the bandwidth parametn It h tween 0.10, and 0.50, where is the sample standard deviation of (X,!. y varying II in units of standard deviation. we arc implicitly nOrlnalil.illf.( the e~planatory variable XI by its own standard deviation. as (12.3.10) SIlf.(f.(cstS. I For each value of II. we plot the kernel estimator as a function of X" and t~ese plots are given in Figures 12.5a to 12.5c. Observe that for a bandwidth 0.10•• the kernel estimator is too choppy-the bandwidth is too small t provide sufficient local averaging to recover Sin(XI). While the kel'1lel e timator does pick up the cyclical nature of the data. it is also picking up dndom variations due to noise. which may be eliminated by increasing the bjllldwidth and consequently widening the range of local averaf.(inf.(. Figure 12.5b shows the kernel estimator for a larger bandwidth 01'0.30 A. W lich is much smoothcr and a closer fit to the true conditional expectation. As the bandwidth is increased. the local averaging is per/imned over M ccessivcly wider ranges. and the variability of the kernel estimator (as a \ function of x) is reduced. Figure 12.5c ploL~ the kernel estimator with a b\mdwidth ofO.50'x. which is too smooth since some of the genuine variation or the sine function has been c1imiuated along with the noise. In thc limit. tIle kernel estimator approaches the sample average of I YI ). and all the variability of YI as a function of XI is lost.

~

a.

0t

12.3.2 Optimal Bandwidth Seleclioll

It is apparent from the example in Section 12.3.1 that choosing the proper bandwidth is critical in any application of kernel regression. There arc several methods for selecting an optimal bandwidth; the most cOlllmon of these is the method of crvu-validalioll. popular because of its robustness and asymptotic optimality (see Hardie [1990. Chapter 5] for further details). III this approach. the bandwidth is chosen to minimize a weighted-average squared error of the kernel estimator. In particular. for a sample of T observations IX" Y,l:~r. let mh.j(X,l =

TI "" L cvu( Xj) Y

I

(12.:tI2l

It- ]

which is simply the kernel estilllator based 011 the datasct with observatiolJ j deleted. roaluated at the jth observation Xj' Theil the cross-validatiolJ function CV(h) is defined as (''V(h) =

.,. I"" T

.

L[ YI - mh.I(XI )] 2 o(XI ).

02.:1.1:\ )

1=1

wllC:re a(X,) is a nonnef.('llivl' weif.(bt function that is required to redlll'l' bOundary clTcCIs (sec IHrdll' I 1\I~lO. p. Hi2] for furth('r (\isntssion). Thl'

,

o.

~'.::,~I 1

I·

'''''\'1

I

~. L----~----~-·---~,.i"""'"-

'l.'f

(a)" = 0.1,;,

L-_ _ _

~

___

~

_____

. \. ::::,\. i ~.;---

_

.. '

:, OS:£1iii~'~.I 0; h('IICC WI' would lI('ed at !cast 100,000 OhS('\'V;ltiolls to ('nSIII'(' an avcrag(' ofjllst o\le dat;1 poillt ]leI' lleighhorhood! This ('w\t'"/tlillll'/lIiol/ulil.Y call 0111)' he soh-ed hy placillg \('stri( tinllS till tl((' killds .. r 1I(1l1lill('arili('s that are allowilhk, For ('"alllple, SIlPP"'" ,I lilll'f1r ('oillhillatioll of the .'l,t's is relaled I" )'1 IHHlparalll('tril'all\', This has tl\(' adv,lIIlage of captllrill~ illlportallt 11011lilll'arili('s ",hilt- prm'idillg SlIliicil'lll slrllclure to ]ll'I'IIIil eSlilllalioll wilh n'asoll\\' I (s("t' footnote 16):

r

r

(12.3.30) and we shall adopt this convention for notational simplicity. For example, a European call option on date- T aggregate consumption Cr with strike price X has a payofTfunction Y( Dr) = max [Dr - X, OJ and hence its date-I price G( is si III ply (12.3.31 ) EVt'1l the most complex path-independent derivative security can be priced and hedl!;ed accordinl!; to (12.3.30). For example, consider a security with the highly nonlinear payoff function:

Y(C)

a-b

------:--~

1 + exp[ -(l(e - a)1

a > 0, Ct

+ b,

b < 0

(12.3.32)

I c + -log( -a/ b).

(12.3.33)

fJ

This payofT function is a smoothed version of the payoff to an option portfolio commonly known as bullish verli((l/ spT1'ad, in which a call option with a I,ow strike is purchased and a call option with a high strike price is written (see Figure 12.5 and Cox and Rubinstein [1985, Chapter 1] for further details).

ExtmrtinK SPDJ from Derivativl'J Prias There is an even closer relation between option prices and SPDs than ( 12.~.30) suggests. which Ross (1976), BarlZ and Miller (1978), and Breeden alld Litl.enberger (I978) first discovered. In particular, they show that the second derivative of the call-pricing function C( with respect to the strike

12. NOlllineanlies in Financilll /)lIla

510

tI

----.--------.--.-r-----

()

~--------~----------

b 1-----'

Figure 12.6.

fl

--.----.----------------------

() ~--------~----------

b

------------------ - - --- -- - - ---

liul/ish Vrrlim/ S/m·(/(ll'ayoJ! I'iml"lioll tlwl SlIwolh,rI Ih,j'JII

price X must equal the SPD:

()~

'\ I I

G,

iJX~

(l:.!.:I.:H)

r.

-therefore. impounded in eve I]' option pricill!:: formula is the SI'D i To estimate the SPD using (12.3.34). we require a call option priciAg formula. Although many parametric pricing formulas exist (sec Ilull l}993. Chapter 17] for some poplllar examples). Ait-Sah'lli" and 1.0 (19%) clmstruct a nonparamellic pricing formula that places fewer restrictiollspritnarily smoothness and weak dependence-Gn the data-generatin!:: procbs of the underlying asset's price. While parametric formulas such as t~ose of Black and Scholes (1973) and Merton (1973) olTer !::reat advant· ges when the parametric assumptions (e.g" geometric Brownian motion) a e satisfied. nonparametric methods arc robust to violations of these ass Imptions. Since there is some empirical evidence that casts doubt Oil such a sumptions, at least for stock indexes. 17 the Ilonparametric approach may 1 I~ave some important advantages}X Giv!=n observed call option prices {Gj • Xj. r;} (wtTt:re rj == '1; - I,), the p,rices of the underlying asset {I';}. and the riskless rate of interest (r" /. we may construct the smooth nonpara,uetric call-pricing functft>u as

i

G(l',X,r.r,) = E[G I P.X.r.r,]

(12,3.35)

lIsing a multivariate kernel K. formed as a product of d=4 univariate kernels:

( 12.:t3ti)

17Secl.o and MarKinlay (1!)tlH).lor exam!'1 ()

/I

/mgnlion, and this does mimic a k.ind oflcarning hchavior (alheit a wry simplistic ol1c).19 llowever, White (I !)~2) cites a numher of practical disadvantages with hackpropag-ation (numcriCII instahilitics, on:asion'll IH>lH.:onvt'rgence, etc.), hcnce thc preferred method for cstimating the parallll'lcrs of (12.4.4) is nonlincar least-sCjuarcs, Even the singlc hidden-layer MU' (12.4.4) possesses thc 1I11i1>mnl n/~ jnoxilllntionjlrojll'Tly: It can approximate atly nonlinear function to an arhitrary degree of accuracy with a suitable nluBher of hiddell ullits (see White [1992]). Howevcr, thc univcrsal approximatioll property is shared by lIIany nonpar.llnetric estimation tcchniques, illcludinl!; the nOllparametric regression cstimator of Section 12.3, alld the techniques ill Sertiolls 12.4.2 alld I ~.'1.:~. or course, this tclls liS lIothing ahotll the pcrfonnalllT of such techttiqltl:~ ill practice, and for a given set of data it is possihk for one tedltli(llIe to dOlllinate anothcr ill accuracy and in other ways. Perhaps the most importallt advantage ofMI .f's is their ahility to approxilll.tle complex nonlillear relations through the compositioll ofa network of relatively simple functions, This specilication lends itself naturally to /mml{P1 !})of/'.lSing, allli althongh there are ollTl'ntly no financial applications that exploit this feature of MLl's, this lIIay soon change as parallel-processing softw'lre and hardware become lIIore widely available. To illustrate the MLP model, we apply it to the artificial dataset generated hy (12.:i.II). For a network wilh olle hiddell la}'"r alld live hidden unil.~, dl'tlotn\ by MLI'( I ,:J), with H(·) set to the idetltity futlction, we obtain the fol!owing l\Iodel:

- :J.I'tII Ihe tIIon' gelleral task or approximalion (sec Broollll\cad anti Lowe II ~IHH I. to. lood), and DarkclI II (IWI La lid I'ol-\gio and (;i rosi 119~10 1), III parlic"lar, I'ol-\I-\io alld (:irosi (I ~1~1O) show how RBFs (,O.OU. all aTlnual fOlllillll()II.'"ly rOll1pollllfit'd «''''preted f.\lr of 1('1111"11 J{ of I WYt" and all alJlHlal \'olatililY n of '2IJfYr,. lllldc'l lilt' gl.lc J,..·S( hole . . OI~."lInptioll of .\ ~t·\\HH·\rir

l\ro\\,ul.Ul

llH)\\tm,

til'

=

/I/'tli

•1Ild (,Iking 111(' IItllllher of day."i pCI' Y('ar to

hom the.' dblrihlltioll N(Jl/:l:):l. retllnt"i, which an' ("{Jove'fled to 1>11.

(1

'1 /:l!"):l)

1)('

+ o/'tlU .

L:):~.

'he}'

ti hlc alld offt- r a !t'ss (Llta -dep (,lId ('nt mea ns 01' mod ('1 vali dali on, AI! this SllggcSIS the l1el'd fill' all (/ Illio n [ralJ l('wo rk or spec ifica tion f(lI' Ih(' 1IJ0del hefo n' cOll frolllin!-\ IIlI' (Llta, By prop osil lg slIch a spcc ifica lioll , aloll!-\ wilh lire kind s of plll'n OIIl ('lIa ont' is seek illg 10 capi llre alld Ihe r(')('valll varia hll's to hc used ill lite sear ch, Ihe chaJl(,(, or cOlI Jing IIpOIl a spur ious ly slIcc(,ssful lIIod el is r('du ccd,

12,6 Con clus ion NOl llill( 'arili ('s ;tn' cI('a ll\' play illg a lIIor e prol llilW lll role in fina ncia l appl icalio ns, Ihal lks 10 illl'\'('aWS ill COIl lPJllill!-\ pow er and the avai lahil ilY ofla rg(' dala s('ls , Unli k(' tltc Illal nial pn's ('llle d ill ('arl iel' chap lers , SClIlle of till' id('a s ill this chap ll'l' arc less well -('sia hlisl lt'd and 11101'(' 1(,lllaliv( ', With ill a shor t tillJe Illall), 01 Ihe I('ril l Iiqlll 's we hal'( ' cove red will h(' r('li lied , and sOllJe lIJay h('co llll' ohso lete, Nl'v('J'tl ll'lcss, il is imp orta nt to dl'v clop a sells e Ihl' dire t'lio ll of rese ard I alld Ihe opel l ques tioll s 10 h(' addr esse d, ('spe ciall \' at till' carl y St;I~('S of thes e n:pl or;lt ions , Des pite Ihe /lexihililV oflh e lIoll lille ar lIlod e)s we hav( ' cOll si(lc r('d, 111('\' do have SOIllC s('rio lls lilllil;tlio Jls, TlI('y are typic ally mor e dif'li( 'J11t to eslilIJate prec isely , ilIOn ' sCIISili\'e to ollll iers, IlIlIlH~rically less stah lc, alld mOl (' pron (' 10 oV('l'liII illg alld dala -sllo opil lg hias(,s Ihal l rom para hl(' lille ar mod (,Is, COl ltl'ar y 10 pop lllar 1)('lief'. lJoll lilll' ar I1wc\els reql lir(, /lion ' ('col lolll ic Slnl ctllr c alld f/ I"'iori cOll side ralio lls, 1101 I('ss, Aile! Ihei r illie rpre lalio ll 01'1(,11 J'('()Ilin's ilion ' ('Hi"'1 alld carl' , ll()\\'('\'er, 1I0nlillL'arili('s ar(' OftC II a faCI or ('('Oll(llllil' lik, ,llId for lIIallY lilla llcia l appl icali olls Ihe sOUJ'('('S alld ll.tlllI'!'

or

525

l'r"hlelll,1

of nonlinearity can be readily idelltified Of, at the very least, characterized in sOllie fashion. In such situations, the techniques described in this chapter are powerful additions to the arlllory of the financial econometrician.

Problems-Chapter 12 12.1 Most pseudorandolllnumbergenerators implemented on digital computers are multiplicative linear (ongrumti£ll gmerators (MLCG), in which Xn '= (aX,,_1 + r) mod m, where a is some "well-chosen~ multiplier, cis an optional constant, and 1/1 is equal to or slightly smaller than the largest integer that can he represente.d ill one computer word. (For example, let a = 1664525, r = 0, and 1/1 = 232.) In contrast to MLCG numbers, consider the following two nonlinear recursions: the tent map (see Section 12.l.1) and the logistic map, respectively: \

)\" XII

=

{2X,,_1 2(1 - X,,_tl

if X,,_I < .,! if XII-I

1Xn - l (1 - X,,-I).

::: Xo

I

I

X.

E

(0, 1) (12.6.(1 )

'l

E (0, 1).

(12.6.2) "

These recursions are examples of chaotic systems, which exhibit extre~e sensitive dependence to initial conditions and unusually complex dynamic behavior. • 12.1.1 What are good properties for pseudorandom number generawrs to have, and how should you make comparisons between distinct generators in practice (not in theory)? 12.1.2 Perform various Monte Carlo simulations comparing MLCG to the tent and logistic maps to determine which is the better pseudorandom lIumber generator. Which is better and why? In deciding which criteria to USl', think about the kinds of applications for which you will be using the pseudorandom number generators. Hint: Use 1.99999999 instead of 2 in your implementation of (12.6.1), and 3.99999999 instead of 4 in your impkmentation of (12.6.2)-for extra credit: Explain why. 12.2 Estilllate a multilayer perceptron model for monthly returns on the S&P !'iOO index from 1926: 1 to 1994: 12 using five lagged returns as inputs and one hidden layer with ten units. Calculate the in-sample root-mean-squarednror (RMSE) of the one-step-ahead forecast of your model and compare it to the cOITesponding ollt-f-sample results for the test period 1986:1 to I !)!)·1: 12. Can you explain the differences in performance (if any)?

I

526

12. Ntmlinearities ill Fi1lf1ll(ial /)ata

12.3 Usc kernel regression to estimatc the relation between the 1I\0\IIhly returns of IBM and the S&P :)00 rrolll 1%:>:\ to 1994:12. How wllulcl a convclIliona! bCLa be calculated rrom the results of the kernel estil\lator? Construct at!easttwo measures that capture the incremental v K in this situatioll. Although the Newey and West (I !IH7) weighting schclllc is the most ('OIllIllOllly lls('d, there arc several altcrnative estimators in the litcratllr(" indlldillg those of Andrews (1991), Alldrews alld MOllah,1II (I!19'l) , and C.lllallt (19K7). Hallliiton (1994) pl'Ovi(\('s a llscflll overvicw. Thr j,illt'flr jl/.\lrlllllfllial Variablf.5 emf

The gl'llcr;d formulas givcn here apply in hOlllnonlinear and lincar 1J1l)(lels, hut they can hc understood lIIore silllply in liJlcar IV regressioll lIIodels. Retllm to the linl'ar model of Scction A.I, hill allow th(' enor tCI'I1l f,(Ou) to he serially corrc:lated and hetcroskcdastil'. EquatioJl (A.I.IO) I)('collles S

= lim Var[ r

I/~H' 1'(00) 1

[-':Xl

""

lilll T Ill'!:1(O,,)Il,

T-':X.J

wllerc H(lJo) is IIII' variance-nlVariallu' matrix of llIal cd h)'

,,(011)'

(1\.3.5)

This (';\11 bc ('sti(A.3.tJ)

will'l'c H,

(0 r)

is

.111

cstimator of HWo). Eqllation (A.:!.II)

IIOW

hccomcs

(A.:t7)

III the homoskedastic white lIoise case cOllsidcred carli("r, 12 == (1 "I r so wc used all estimate n"'{o'rl == where = ,[,-I 'E.::I (;cilr). Suhstituti;lg this illto (A.:U) gives (A.I.IH).

a"l/

a"

W!at'll III(' I'ITOI 11'1'111 is scriall" IIIH'olTclalcd hill hcll'roskl' 0).

The log likelihood [ of the whole data set XI •...• conditional log likelihoods:

XT

(AA.2) is just the sum

of~he \ I

(AA.3) Since I., is a conditional density. it must integrate to I:

f

I.t(Xt+l. £J)dxt+1

= I.

(AA.4)

Given certain regularity conditions. it follows that the partial derivative of L, with respect 10 0 must integrate to zero. A series of simple manipulations then shows that ()

= (A.4.5)

The partial derivative of the conditional log likelihood with respect to Ihe parameter vector. ait(xt+l. O)/ar}. is a vector with NfJ elements. It is known as the score vector. From (AA.5), it has conditional expectation zero when the data are generated by (AA.I). It also has unconditional expectation zero and th us plays a role analogous to the vector of orthogonality conditiolls f, in CMM analysis. The sample average (11 T) 2:.;"'1 3l,(XH;t. 9)/39 plays a role analogous to gT(O) in CMM analysis. The maximum likelihood estimate of the parameters isjustthe solution to Max [(fJ) == L:~I i,. The ftrst-order condition for the maximization can he written as

o

T

gT(lJ)

= rl L ili,(x,+l. O)/(){J = D.

(AA.6)

,=1

whirh also characterizes the GMM parameter estimate for a just-identified model. Thlls the ML1~ is the same as GMM based on the orthogonality conditions in (AA.5).

538 Ih.~ml)totic

Distribution

71U'1Jr)

'1~lle asymptotic distribution

or ML parameter estimates is given by the I()I-

" wing result:

I

fl(O -

00 )

~

N(o.r'(Oo)).

(1\.4.7)

wi,ere I given by:

I

I

[I ()~£.(O)J

lim -E - -.- .-, • 11 ,JOiJO

1(0) =:

a'~d is known as the

(A.'Ul)

,,~oo

jlljOn/Ulti()11 matrix. I

Gill

be estimated by tht' sample

Clbullterpart: _

I

I"

iJ~e,(iJ)

T

L

-7

=

iJOiJO' .

(A.'I.!I)

1=1

The information matrix gives \\S;I measure of the sensitivity of the vallie of tIle likelihood is to the v,llues of the paramcters in the neighborhood of tl e maximum. If slllal\ changes in the parameters produce large changes ill likelihood ncar the maximum. then the parametcrs can bc precisely esti/natcd: Since the likelihood function is lIat at the maximum. the local s~nsitivity of thc likelihood to the parametcrs is mcasured by the local curvaturc (the sccond derivative) of likelihood with respcct to thc parameters. cvaluatcd atthc maximulII.

Infimnrltioll-Matrix Equality \ An alternative estimator of thc information matrix. Tb • uses thc average ollter product or sample variance of the score vectors:

i

=:

rl ~

al,(O) iJl,d})'.

ao

L,

b

,=1

(A.'I.IIOlicdly llormal wilh aSYlllpll>li.lI~t." .I0IiJlud (~f FUUlll(ill1 r:fUlwmio, 'll, 711110.

I\"rl...('\'. (:" I \/:,Ii. "1·JfI'rti\'(, Stork Splils," //(/II'/IIt! /1'";""11 1111'1/'/1'. :',1 ( I) . .l.lIl1lal),Fehl'll"r),.IOI-lOti. - - - . 1!):.7. "Slorl... Splils ill a Bull r-Lllk'·l." 1/(/1I'1IIt! 11'";111'11 1i11"''II'. :1:,(:\). r-Lt)'.1"1"" 7'.!-7\). - - - . 1\l'iH, "E\'"I"alioll "I'Slork Di\'idends." I/(/I/"/lt! 11",/1/1'.11 IIn'I/·II'. :IIi(-I),.I"I\,. 1\II.l:U'I. \)!)-114.

1i1'/1'1i'liffl

.1'1

BaIT, D., and/. (:;""1'11('/1. 1'1'1;1, "Inflalion, R"allnlt'f('sl R'llt'S, anelllll" Bonel ~f:tr­ kl'l: ,\ SI"d\' 01 (IK NOlllinal alld hult'x-Lillk"d COVt'l'IIlIIt'lI! lIund p .. in·,." I )j"'II,,iulI 1',1/)1'1 17:\'!., I b.-\,.lnl Inslilllll' o\" Ecol1olllic Rt''''arch, 11.,,·\'anl (llli\' ....,il\'. lIartoll. A .. I~I'I:\. "(I"i\,l'Isal I\Pl'lOxilll'llioll IIOIll1(\S (il\' SlIpt'rposiliollS o\" a SiglIloi,"'1 FIIII('lion." IHJ·: 'Jilllll. 111/0. nlnll)" :{!I, !I:\O-!H:I. I~I~H.

"'\PllIoximalilln and Eslimalioll BOIIII,ls wllIks," ;\/111'11;111' /.l'IInJ;lIg. 1·1, 11:,-I:l:t

Ibll 1111. A., awl R.

B;"T"".

((II'

Arlilici;II Nt'mal

""1-

I ~IHH. "SI,lIisli .. all.t'anrill~ N"lwork'jf,A ( IlIir)'ing Vit'\\'." in

20lh S\'/11IItIl;II/IIIIIII//I'/ull'll"rr: f.'1I1II1'lIlillgSrimrf IIntlSIIII;llin."". I!I'!.-'!.O:I,

Rt'slllll, Virgilli;). Ihrsk)', R., andJ, Ill- I.on~, HI!I:I, "Wh)' Do,'s Ihl' Srock Markel FlnCln'llt'~," C!.IIIIII/·"r '/OIlI'l/II/o/hlllwlII;n. 10H, '!.!II-:I II. lIaslI. S., 1!177, "Thl' 11I\·.'slllll'l\l I't'rli>nllanf(' of COlllllHlII Stlld.. s in Relation 10 TIIl'ir Prill' III F."nin~s Ralios: /\ 'Ii'sl oftht, Ellident M'llkt'III),!,Olh",i,," ./11110111/111 ""1111111'. :I'!.. titi:l-tiH'!.. II .. .-I.. ,'rs, S., I!IKO. "Th,' (:,,"slalll Ebslilily of Varialle,' Mod ..1 alld It, IlIIpli"ali, ,'" iiII' (1IlIi'''1 Pricing. "jllllntlli o/I-IllIlItrI', :1:1, liti 1-li7:t - - - , I!IH:{, "\'01 ria II ... ', "I' Sl'lmil\' Prin' R.. I\I!'us lIasi ..

Ili~h.

I ...' .... and (:\,,,illg

I...s.s An'r.,ioll and 110 .. E'l"ill' 1',t'llIi,"n

1'11111.· ... (!IIf/llnhjlllllllll/ °//':,.,,1111/11;1'1. 110. 7:1-!1'!..

111-11',,111. C .••11)(1 IC I Llgt'n",,". 1\/7·1, "nl'\t·llllin.lllls of l\itl-t\.sl..l'd Spr"ads ill Ihl' (1\'1'1'-111,,-( :1111111 .... ~larl."l."jllllml/l o//;illl/nrillll·:mllolll;n. I. :1:,:I-:\(i·1.

IlI-m·t·I\(·Io. It, I'}H·I. "Thl' (knIlT"nn' of S.·tllll'\Ke "allt·nts ill Er~o(li( :-'1a .. l.o\, (:lIains." Slm}III.,/;{ /'W{I·''''.' Itlld '/'11/';" AN,lim/io/l.,. 17. :\ti~I-:I7:I.

1I.... llanl. V.• 1!IH7 ... ( :l'IIss-S"l'IiOllal III'p"lId"II(,(' alld "rohl"lIIs ill IlIfi'n'IIIT II, I\I"rl..'·I-lIa,,,d A'Tolllllillg R'·"'OI,,·h." ./(1111'11,,1

(lrAnollll/i"~

lI/v/1Il1,.

~r,.

1-

·IK. Ilt-mdt. E .• II. II~"I. R. 11.111. .,,"1,/. I 1.11 IS" 1;)1 I. 1!174. "Eslilllalioll and Inrl'lt'net· ill NOlllin""r Sln'l'I"r,,1 ~llldt'I .• alld A. 1.11. 1'}'lIi, "()"lilll;11 (:1111'1',,1 "I' EXl'l'IlIi"lI (:'"Is." \"'''rking I'ap"r IYF.-III'!.:I-'lIi. ~Ia";,, IIIISI'lh Illstillll" "rT':chllol"h/\' 1."hor.lll1rl' f(,,' Fill;IIII i.11 Fllgill,·.... ing. (:0111,1>, idg'·. o\IA. Ikll,illl;". n .. I.. hog.lIl. ;11)(1 ,\. l.o. l'I~Hi. "WIll'1I Is Timt' CllllliIIllOIlS?," 'v\'orkill~ I'apl'r I.FF.-III·I:I-!Jli. !,-1;."a.-lllls,·I" IIISlillll" "fTi.'chnllloh/\' I.ahoralon· IIII' Fill",,('i;,1 Ellgill'·'·lillg. IIJ.'II I.... harp. t\1.. I '.IH:I, "Tr~""~\l1 iOIl' \);11.1 '1',"1, of Eflirit'nry ,,\" I h .. ( :hil~lg() l\o~II" (1I'Ii'>lI' F.~, h""gl'." ./"/11110/ "/ Iliek, ,\.,

I'I~IIJ,

"'"11"/ iI/I /·:,.,,/llIlIIin.

I:!. I HI-I W,.

.. ( hI \',,1111,· l>illl"ioll 1'1 i, " P'li' ,·ss,·s (If Ih .. Mal k"1 l'oll«,lio ... .//I"n"tI ,,/ Fill/HI,,'. 1'>. ti7:1--liH'I.

54!>' :MB' "'.. . .

Jll'jn"fnm

"!I.,~ .IIl''''' ..

nierwag-, C., G. Kaufman, and A. Toevs (cds.), 19H3, Innuvations in Bond PortJolio '1" .. Management: Dumtion A 11 a Iys is and Immu11iUltioll.JAI Press, Westport. cr. lIigg-ans,.J., and C. ('..annings, 19H7, "Markov Renewal Processes. Counters and Repeated Sequences in Markov Chains: Aduanm in Applied Probability, 19! :'i21-:'i4:'i.

Billing-sky, P.• 196H, Conllt'1gmrl' of Probability MflLlllTI'J, John Wiley and Sons, Ne':i York. , mack. E. 1971, "Toward a Fully Automated Stock Exchange: Financial Analys15journlll,July-August, 29--44. - - - , F., 1972, "Capital Market Equilibrium with Restricted Borrowing," Journal oj

/lusinf5S, 45,July, 444-454. - - - , 197fi, "Studies of Stock Price Volatility Changes: in Promdings oJtht 1976

Meetings oj the BusinfSJ find r)-onomie Statistics Section, Amnican Statistical Association. pp. 177-18 I. - - - , 1993, "Return and Beta," journal of Portfolio Managnnent, 20, 8-18.

Black, F., alld P. KArasinski, 1991, "nond and Option Pricing when Short Rates Are I.og-normal; Financial Analysl5joumal,July-Au/-,'ust, 52-59. Black, E, and M. Scholes, 1972, "The Valuation of Option Contracts and a Test of Mark"1 Efficiency." jounwl oj Finanrf. 27, ~99-4IH. - - - , 1973, "The Pricing of Options and Corporate Uabilities," journal of Political /:'rrl1lomy. HI. 637--654.

Black, F., E. Derman, and W. Toy, 1990, "A One-Factor Model of Interest Rates and Ils Application to Treasury Bond Options," Financial AnalySlJjoumal, January-February, 33-39. lIIack, F., M.Jensen, and M. Scholes. 1972. "The Capital Asset Pricing Model: Some Empirical Tests: in Jensen, M. (cd.), Studies in tht Theory of Capital Marluts, i'raeg-er, New York. Blanchard, 0., and M. Walson, 19H2, "Bubbles, Rational Expectations and Financial Markets: in P. Wachtel (ed.), Crises i1l the f.'eollomie and Financial StructUft: lllll,b/ps, Rlll.~ts, and Sharks, I.cxin~ton, Lexin~ton, MA. lIIattherg-, R., and N. Gonedes, 1974, "A Comparison of Stable and Student Distributions as Statistical Models te)r Stock Prices," journal of lIusiness, 47. 244-2HO. BIIIII1t', M., ,lIld I. Friend, 197:\ "A New Look at the Capital Asset Pricing Model:

jOlmul1 of Financf, 2H, 19-33. - - - , I ~)7H. Tilf Uuwgillg IInlJo of thf /rulit'itiual Inllf.l/or,John Wiley and Sons, New York.

1\11111"', M., ,1\\3-1HI. IIhIllH', M., C. MacKinlay, and B. Terker, 19H9, 'Order Imbalances and Stock Price Movt'ments on Octoi>er 19 and 20, I ~lK7." journal of Finanrf, 44, 827-84R.

>("k SplilS, Stork 1'1 ices, and Transactioll Co.~t."" jor(l7lfll (lj Filla II cia I h(}//()/IIin, 'l'l, H:I-IOI.

IIn."II'''"', M., and E. Schwartz, 1977a, "Convertihle lIonds: Vaillation "lid Optimal StratCf;ies for Call and Conversion," JUllnuzl ajhllll1l(r, 3'l, lIiY9-1715. - - - , E/77b, "The Valuation of American Put OptiollS," jOllnlll1 oj 1'/IUIIlCf, 32, 1·I!H(i2. - - - , 197!i, "Finilc DifTl'H'llce Mcthods ilnd/lIlllpl'lOlTsscS ArisillJ.: in the Pticing of COlltillJ.:Cllt Claims: A Sylllhesis," ./ountlli ()./ /'illfwrill{ alld QUlllllillllivr 1\l/l//pi.l, 1:1, 4(i1~171. IY7!I, "/\ Continuous-Tillie Appmach to Ihc PritillJ.: of BOlllb," journa/ oj lil/I/ki"~ 1/1/(1 Fil/l/l/iP. :1, 133--1 !.!•. 1I ...·nll.\l1, M., N ..ll'J.:ati!"esh, allti n. Swarnillathan, 1\)\/:1, ''In\"esllllt"llI Analysis anti Ihe Adjustlllellt of Stock I'ri1 •• 1 ;\sSI'I 1'. i, l\\g Will" 1\\1 (:()IlS1111l1'1 io" I bla." ;1 1II1')/{ fill ":IIIII(JJllil 1:1"'/1",1, ~(\. ·11'\7-:-.1~.

549

IIl'jl'll'lIC1'S

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- - - , EmO, "Permanent Income, Current Income, and Consumption," Jouitzal of Rusiness and Economic Statistics, 8, 265-278.

J., and P. Perron, 1991, "Pilfalls and Opportunities: What Macroeconomists Should Know about Unit Roots," NB1~R MllCTOtconomics Annual, fl, 141-201.

(~lIl1phl'll,

(:alllphell •.J., ami R. Shiller, I !lH7, "C()inle~rati()n and Tests of Present Value Mo<Jels: ./ounwl '11'0Ii(i((lll~fOllorny,!l5, IOfi'2-10S7. I - - - , I !lSHa, "The Divirlend·Price Ratio anrl Expectations of Future Dividend.and Discollnt Factor-s.· Un,in" of FilUmria/ S/Uf/i,s, I, 195-227. - - - , I!'HHb. "Stock Prices, Eaminv;s, and Expected Dividends," Journal of Finallu, 4:\, (i(i 1-()7(i. l!l!)\, "Yield Spreads and Interest R:lte Movements: A Bird's Eye View," UnJim of Hrollorrlir .\'Irulif.l, SH, 495-514. 1!l!Hi. "A Scorecard for Indexed Co\,ernment Debt," NRER Macrorconom;cs A 111111111, f(lnhc()lIIin~. Call1pbl'lI,.I .. S. Crossman, andJ. Wang, 1!193, "Trading Volume and Serial Correiation in Stock R(~tllrns," QI/flrlnly./ounlal of t.collomirs, 108,905-939. (:aristl'iu. E., II. Muller, and D. SieV;l1Illnd, cds., 1994, Clrangr-l'oinll'rob/'ms,lnstitute of Mathematical Statistics, I IaY'vard , ('.1\.

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nil'

/Jo",,\'

,'f

SllIlh((\l;r I'IIIII·W·'. (:10;'1'"0;'" a'ld 11.a11.

/(I'/n-rllrt's

Cox,.J., 1975, "Notes on Option Pricing I: Constant Elasticity of Variance Diffusions," unpublished lecture notes, Graduate School of Business, Stanford University. Cox, J., and S. Ross, 1976, "The Valuation of Options for Alternative Stochastic Processes," Joun/1l1 of Finllllcial f.(olloll/irs, 3, 145-166. Cox. J, and M. Rllbinstein, I ~IK:), O/Jtil>lls Mllrkds, Prentice-Hall, Englewood Cliffs, NcwJcrsey. Cox,J.,.J. Ingersoll, and S. R{)s.~, 19H1a, ·A Reexamination ofTrdditiona1 Hypotheses ahout the Terlll Structure of Interest Rates," journal of Finanu, 36, 769-799. - - - , 19H I h, "The Relation between Forward Prices and Futures Prices," journal oj Fillflllcilli /~(ollornics, 9, :Vn-346. - - - , 19H5a, "A Theory of the Term Structure of Interest Rates," Econometrica, 53, :'\HS-40H. - - - , 19H5b, "An Intertemporal Ceneral Equilibrium Model of Asset Prices," 1':C(),lOmfirira, 53, 3fi3-3H4. (:ox,.l., S. Ross, and M. Rubinstein, 1979, ·Option Pricing: A Simplified Approach." JOlin/ai/if Financi,,1 /:co'lOmics, 7,229-264. Crack, T. and O. Ledoit, 1996, "Robust Structure Without Predictability: The 'Compass Rose' Pallern of the Stock Market," journal of Finance, 51, 751-762. C."ig. S., J. Kohlase, and D. Papell, 199 I, "Chaos Theory and Microeconomics: An Application to Model Specification and Hedonic Estimation: Revinv of /-.'rm/IJII/ics and .'illlli.llic.l, 73, 20R-215. CUlIlh)" R., aJl(I D.. Modest, 1987, "Testing for Market Timing Ability: A Framework ti,r Forecast Evaluation," journal of Finmlcial },co nom ics, 19, 169-190. Clltll-r. D., J. POIerha, and L. Summers, 1991, ·Speculative Dynamics." Reuiew pf /o:('o1/omir Slur/ies, 5K, !,29-:,9-RO. J)c·lIonelt. W., and R. Thal('r, I!IW" "Do('s the Stock Market Overrean?," journlll /if FiIl//II(f, 40, 7!J3-H05.

554 - - - , 1987, "Further Evidence on InVl'slor Overreaction and Stock M,II!;.I·1 SI',I' sonalilY," jtlltnlfll '11-/1/f1l/rt'. 12, :.:.7-:,M2. [)chn~lI,

(;., 1959, 'I1trory '1 Vttlup,.I0hn Wiley and Suns, Ncw Yurko

DeLong, B., A. Shleiicr, L. SUlIllllers, and R. W"ldlll,III1l, 1\l\lOa, "PositiVI' FI'I',llMrk [m'estmenl Slrategies and Destabilil.inl( Speculalion: jounllli '11-111111111', ,I:., 379-396. - - - , 19!JOb, "Noisc Trader Risk in Fin"nci,,1 MarkeL""./lIunwl oJI'olililllll':IIIIIIIIII.\'. 9M, 703-73M. Dcmsctl, H., 196M, "Thc Cost of Transaninl(: Quar/l'rJy jounltlloj Erollolll;I.I. li2. 33-53. Derman, E., and I. Kani, 1\.191, "Riding on Ihe Smik," IUSK, 7, February, :{2-:1\1. Dhl)'mcs, 1'.,1. ~'Iicnd, B. Gultekin, and M. Gultekin, 19tH, "A Critical Rl'l'xaminalion (If thc F.mpiriral Evidence on Ihe Arbilrage Pricinl( Theory." '/II/mllt! 0( Fillfwa, 3\.1, :{2:~34(j. Dial 'Jnis, P., 19MM, GrollI' IV/IrPSe1lIIl/ioIlJ ill I'm/illbilily IlIlfl SllIliJI;(.I, IIISlillltl' or 1\1"t IJI cmatical Statistics, Hayward, CA.

Diac~>Ilis, P., and M. Shahshahani,

I \.IH1, "On Nonlinear Functions of I.inear COInbinations: SIAMjollrn1l1 on Scimlijir linn SliIli.l/iCIII Compuling, :,( I), 17,,-1\11.

I

Djal~()nd, P., I Diba~ B.,

I

1965, "National Debt in a Neoclassical Growlh Model: Amrt111111 Eronomic lu-t,il'w, 55, I 12C..-I I :,0.

and H. Grossman, 1988, "The Theory or Rational Bubble,; in Stol!;. PriITS." /:'conomir Jmmwl, 98,71(;-7,.7.

Dickdy, D., and W. Fuller, 1979, "Distribution of the ESlimalOrs Ii,r Autoregressive Time Series with a Unit Root: ./lIllnltll oj lite Amrrimn SllIli.ll;mlll.,.III1';II/;O", 74, 427-431.

I

Dims

III,

E., 1979, "Risk Mcasurement When Shares Arc Subject 10 Infrequellt Trading." journal oj Hllaluitlll-:Collomin, 7, 197-226.

Ding, Z., C. Granger, and R. Engle, I \.193, "A Lon~ Memory Property of Stock Returns and a Ncw Model," journlll of l~mpiriCtlI Finflll(r, I, H3-1 06. Dollcy,J., 1933, "Chardctcristics and Procedurc of Com mOil Stock Split-Ups: /1(/" I tlflrt/lJusillw Ilroil'l", 31 (;-32(;. Donaldson, R., and M. Kamstra, 1990, "A New Dividcnd Forccastinl( I'rocedlll'c that . RcjeCl~ Bubhles in A~sct Pricl's: The Cas" of 1929's Stock Crash," /111';171' of Finllluilll Stlltl;,." \), 333-:{Ii:t Donoho, D., and I.Johnstone, 19H\.I, "Projection-Base,\ Approximation and A Dualil), with Kernel Methods: AlI1l1lLI 'iSlali,lif.l, 17. ',Ii-lOo. Dunil', D., 1992, Dy"tlmi( A.lSrll,,.;(jll~ nlfIJry,l'rinCl'loll University Press, I'rinn·ton,

NJ. DuBie, D., and C. Huang, [91l5, "Implemcnting Arrow-Dehreu Equilihria by Conlinuous Trading of Few LonwLivcd Securities: 1~'rollome/riw, 53, 1337-1 :F.(;. Duffie, D., and R. Kan, 1993, "A Yield-Factor Model of Inlerest R,ltes." unpuhlished paper, Stanford University.

Jir/PII'l/fI'.1 DlIhHII.}, I ~)H I. "Rallk T(·s(., lor Snial D"IwlIcklln· ... ./"1111111/0/ Timr .\"1'.1 Allilly.I;'I. l,117-llH. DlIll>llI.}. ,1\1\1 R. Roy. 1~)Wl. "SOllll' Rolll"l E,," I IZI'slIlboll S.lIn)!1e Alllol'olld,lIiollS alld ·1i.·,IS 01" Randomness." '/0111"11,,1 "l /':""IIIIIII'/li, I. :l!), '2:,7-'2.7:1. DIIIIII. K .• alld K. Sin~letol\, 1!)Hli, "Moddlill~ the 'krlll SlIlIrlllll' 0(" Intclt'st Ratl's lI11dn Habit Formalion alld 1)1I1.,bilil)' of (;ol>d'."./II"I"II"/ /ll"'I/I"lfi(//I-."Ci~ 110m in, 17. 27-!);). Dll)!ilt', II., I!)\).I, "Pricing with a Smile," II/SI\, 7,.I.lIl1lary. IH-:!o. Dllrlallf, S .. and R. Ibll, 19HV, "Mcasurillg Noise in Stock I'riCl's." IInpllhlished paper, Sianford University. D'1I"1a1l1", S., and 1'. Phillips, 19HH, "Trends Vel'SIIS Ralldom W;lIks in Time Snil's Analysis: /o:n)IJ"//Iptrim, !iii. I :1:1:1- 1:\:,7. Ihh\'ig. P.. I\)H:,. "An Explidl BIHII\(I 0\\ IlIdi"i,IIl,,1 t\ssl'b' lkviatiolls hom APT Pricing ill a Fillite Economy: ./111/1"1111/0/ /'11111111 ill/I:·/Il1l1l1l1i(.'2. Ddn·ig. 1'., and S. Rms, I \JH!i. "Yes. Thl' APT Is ·I'·slabl,· ... ./""1"1111/ II/I'/Ilill/ir. ,10. I 17:'>-11 HH, D)'hl'ig. p,,} Ingnsoll . .Ir.• alld S, Ross. l\l!)ti, "Long Forward .1I1111.1·lo·Conpoll Rail'S LUI Ncver ~"IIl." j,mmal '1 /I/(Iilln.llllil illll.II I !t-ll'I'mk",IOIsI iril)' wilh ESI illl'.,I!" III' Ihl' V;lIi.llln· III I'" 11I1l;lIillll." /-.',ol/tllI/"/,im, :,0, 'IH7-IOOH.

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11.1,hrullck . .\.. alld T. 110. I !lH7. "( )I(kr i\niv.lI. a, -Th .. COlllpOlll'nts of the Bid-Ask Spread: A Cen('rOIl Approach: Financial Mark("1S Research Celller Working Paper 94-~~, OW("1l (;r'lduatt· School of Manag('J1ll'nt, Vanderbilt University, - - - , I \)\):>h, "Dealer Verslts Auction Markets: A Paired Comparison of Execntiull Costs on NASDAQ anrl the NYSE: Financial Market~ Research Center W"rking Paper \):>-1/), Owen Cra(\uatl' School of Manag~m~nt, Vandl'rhill llni\'('rsity, llulll'r, 1'.. I !)W>. "Projection Pursuit." AllllfliJ "ISlalislin, 13(2) • .j35--~2~, IltlhcrJnall, (;" 19H2, "A Sill1pk Approach til 1'.i()lIl1mir 771f(1I),. 2H, I H:{-I!I I.

Arhitrag~

Pricing

Th~ory:

j()unlalllf

.

II Uhl'lJ1l;ln. C., S. "-mdel. ,1Ild R. StaJllhaugh, I!lH7. "Mimicking Portfolios and Exact Arhitrage Pricing," jOlll7wl iif Fi/l(llllf. 42. 1-9. 111IllI'rJnan. (; .. and S. Kandel, 19H7. "Mean-Variance Spanning: jounUlI oj hnflllU, '1~(4). H7~HHH.

J IIIII . ./.. I '1~}3, 0l,liolls. l·illllrl',I, 1I11t! 011",1' /)1'17I'lIliw Sfrun'lifl (2d ed,), Prentice-Hall. Englewood Clitls. New.lersey. 111111, ,I., and A. Whitt', 19H7, "The Pricing of Options OJl V"latilities ... ./llurlllll t1'Fil/fll/ff, 42, 2KI-~()0.

A~set~

with

SlOcha~tic

- - - , I '}\IOa. "I~ricing (Jlt,'resi-Ratt'-DerivI. "l.ong Term Storage Capacity of Resen'oirs," 'li-nnsllrliolls Ilf tI" Alllflil'fll/ Swirl), '1Th,il Hligil/mI. 1 Iii. 770-799, IIIllChillsoll . .l .. A. 1.0. alJ(l T. Poggio. 19~)4 ... A Nonparametric Approach to the Pricillg and lIedgill~ of Dl'rivativc Secllrities Via l.~arning Networks," JOlin/fit "I Fil/Illllf. 4!1. H:>I-KH!I. Illgl'lsoll.J, I!IH7, 'f1/fIll)' of I-';'/III/(il/[ /)1'(iJiOIl fIIl/killg, Rowman & LiuldicJd. Totowa, ~I· 111),\"1',011,

J.I r.. J. Sk,'holl. ,Illd R. \\'(·il.

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I-/l1ll1/rilllal/" QIIIII/lilll/it'r /\l/lIl)'.\i.I, 1:,.1;27-1;:>0.

It .... K.. 1'1:>1. "Oil StorhaSlir Dillt'n'lliial Equations," MOl/oin IIf IIIl/lim/.'iorirly, '1,1-:>1.

a,r AIII"i,."" MIIIII,...

JII\-II'('Iill.J. ,lIld M. Rubin.stein, III!):" "Rt'co\'t'rin~ I'rohahility Di,'lrihutioll,S ffl/,111 (:"'lIt'lllporary S('curity Priccs," working p.lpcr, Jlaas School of l\mill(~ss, lllliYt'rsilY llfCalii'ortlia al IkrlliOIl I'ri,. .. s alld \'.'1>1"' 11·,11'.,,· o'IS." ./"lIm,,1 tI{Nlllllldlll 1·:flllIlIlI/;n. 1\1. :li:I-:IHH. - - - . 1\IHH. "Maxim'lIll Likelihood Eslimalioll of{;'·II ....alil.,·,IIt .. , 1' ... 11 ('",·s with Pis(I"l'll'I), Salllplt'd Dala." !:rllllllll//·tr;,. "1111.",,,,, '1. 2:{1-2-t7.

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---.1'11.. 19\):1. Tit, 11/I/lI.\lrill/ O'glll/il.lllitll/ 1/1/1/ /l1'K1I/lllillll II/II" .~""I/Iilil"\ lilt/win (NIIER (;ollkr(,lll"l' R'·pon). l 'lIiH'r,ilr 0) (:hi ..ago I'rl'ss. (:hi,.a).\o. II .. - - - . ("\.. I\)\)(i. M",III'I El/llil'II'): .~/"I"

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A.. and A. C. MacKinlay. I\IHH. "SIIKk Mad;"'l l'ri ...~s Do Nol Follow Randoll\ Walks: E\·idl'n(·(' lrolll a Silllp!.. Sp"ri'icalioll Tl'sl." /In';111' II/ 1·/I1II11,.;nl SllId;r.I.I.1Hili.

- - - . Ill!!!). "TIt(' Sill' and POWl'r of Iltl' Variallcl' Ralio TI'SI ill Fillill' Salllples: ,\ I\lonl(' Carlo Ill\"('sligalion." .f"l/rnlll o//\rt/l1l1l11l'1r;r.\. 40. 20:{-2:\H. I!)90a. "All

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or Fillallcial

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I.o'jgsla.f. F.• I!)H!I. "A NOlllillear (;"II('ral Etp'ilihrilllll Mod..! o/" IiiI' ·Ii·, II. SI, III III' ,. I of hll"n'sl Ralcs ... jll//llIlIllI/ 1-1110//(;01 I·.illll"'";"'. ~:l. 1!1:,-22·1.

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1.1l1t1l1l·1. L. I~I~H, "A''''I I'l'icilll{ ill 1-:... """lIi('.\ with hi, liolls." IIIIPllhli~ll1r.1, (;01111hridl{e University Press, C,II11hridl{e, UK. \I,ldh""'\ll, A., and S. Smidt, 1991, "A lIayesian Mod,,1 01 Inlrad"y SI'l''';"list l'ricill!{: .f0Il,.",,1 'if foi'llIlIIriall:(OIl/IlIIin, 30, 9!1- 1:14. \Lt~nlls,

J,

alltl II. Nelldecker, I!)HH, Ma/r;x Oil{t'll'lIlilll CtlllIlilll, .lohll Wi"'y and SOliS, New York.

\LtI'Ill'Sla. 1'.. and R. TholllpSOII, I~IW" "I""li.tlly Alllicil'",('d EVl'lIts: /\ Model or Stock I'ril'C Reactions with all Apl'li('alion 10 (:orporal,· i\c'IlIisitions,"jlllllI/rli oj Fillflllrilll/~'roll{Jmin, I-I, ~:~7-~r)o. \1"lki('l. B., I ~1\12, "Elfil'iellt M;IIlet IIYl'oth"sis," ill N('\\'nt;lIl, 1'., M. Mill{a\(', alld .1. L,t'I',,1I (,'tis.), ;\'/1(1/'aIK'I/1'" IJil/;III/lIn' 01 ,\{"I/11' lil/,I hl/'iill", Manllill,llI, I.ondoll. \I;IIHklhl'Ol, B.. I !Hi:I, "The Variatioll or (:,., tain Sp'" 1I1"li,,· 1', i('cs," .lolIl/"d u( 11,";, 1//,11. :\(;, :1\H-41 !I. I 'Hi7. "The Varialion or c,"I"ill Sp\'(,III",iv(' I', i, "'." ./"1111,,11 01 1I11.\IIit'I.I, :\(i. :\~H-II!I.

Rt>Jm'/lI"/'.f

1'171, "\\'11l'1I C:'111 1'1 i( c· lie- ,hhi ll OIg('d ElJic il'llfl y? A I,illli l 10 Ihc' Valid il,· of lIlt' ({," It (01 II \\',rlk 'lIId ~IMI illgal c' ~lodl'ls," IIn,in ll,,/t- :I"/lI/ lIlIIin til/(l SIt/li ,li[ '.

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II .. '"111 .\1. '!;"I' I"' 1<J7~1, "Roh llSI RIS '\lIal ),sis of LOllg RIIII SC·l'i.rI ( :01'1(''''1 iOIl." IIl/l/d il/ II/Ihi ' 11111' 1 I/I/Iil llltli SI(/I i,lim l ""liII lIP. ,II< (Boo k :!). :1'/101.

~lallell'lhllll.

B., ;11111 II. 'Lldo l',

('lIn 's,"

1~)(i7,

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"C)II rill' Disl l'i""l ioll of Slo(' k I'l'icl ' llillc lI r., IW.7 -IOli :!.

\1;11111.'/"1'01, B., '"I1I. J. \';111 ,'\lc·". IClfil-16 I. . Mei,.J., 1993, "A Semiautoregression Approach to the Arhitrage Pricing Theory: journal of Finance, 48, 599-620. MelillO, A., ) 988, "The Teml Structure of Imerest Rates: Evidence and Theop.: journal oj Economic Sumeys, 2, 335-366. I Melino, A., and S. Turn hull, 1990, "Pricing Foreign Currency Options with Stochastic Volatility," journal of Econometrics, 45, 239-265. Merton, R., 1969, "Lifetime PorLfolio Selection under Uncertainty: The Continuous Time Case," Revil'UJ of Economirs and Statistics, 51, 247-257. - - - , 1972, "An Analytic DeriV'ation of the Efficient Portfolio Frontier: journal of Financial and Quanlitative Analysis, 7, 1851-1872. ; 1973a, "An Intertemporal Capital A,set Pricing Mode[: Econometrica, 41, H67-887. 1973h, "Rational Theory of Option Pricing,· Bell journal oj Economicj a,ind Mmwgement Science, 4, 141-[8:1. . , - - - , I 976a, "The Impact on Option Pricing of Specification Error in the Underlying Stock Price Distrihution," journal of FinanCf, 31, 333--350. - - - , E176h, "Option PricingwhenUnderJying Stock Returns Are Discontinuous: journal of Finan cia I Economics, 3, 125-144.

- - - , I!JHO, "Oil Estimatil lg the Expel·ted Rl,turn Oil till' Market: An Explorat ory hl\·estiga tion,· joul"llal o/l-IwlI/ri all:t"IIlIlIl IIir.l. H, :~~:l-:l(; I. - - - , I!JHI, "On Market Timing alldlll\'(' stml'nt l'erfimn ann:,I: An E, "The Persistence of Volatility and Stock Market Fluctuations: A1IIl!riwnl:conomic Review, 76,1142-1151.

- - - . I'JHH, "Mean Reversion in Stock Returns: Evidence and Implications: JourlIal of Finallciall';collo1llics, 22, 27-60. Powell, M., 19R7, "Radial Basis Functions for Multivariable Interpolation: A Review." in.J. M;lson and M. C.ox (cds.), Algorithm.lfor Approximation, Clarendon Press, Oxford. UK. Prabhal;I, N., 1995, "Condilional Methods in Event-Studies and An Equilibrium .Justification for Using Standard Event-Study Procedures," School of Managemefll, Yale University, New l'laven, CT. I'ral'tI. P.. 1972.

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Riller, J.. 1990. "Long-Run i'erf\'lliotl of the A,hill'''gl' ",iling Thl'or),," .11111/'111/11111'/11111/0', :I;l, 107:1-1 10:\' - - - , 19H,I. "A Crilkal Rl'l'xaminalion of lit" Empirical El'i11. Fanor An;ll),sis: 1".vdwlI/I'·

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Rubinstein. M .. 1\176. "The Valuatioll 01 Uncertain I n;lIa ./t1l11'll1l!1I{1.1I11'1/1It1 I·.illi/li/I/in,

I!IX!I, "\\,111'

n",·,

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':!:I-:I1. S,lol'l', S,. 1!IX:la, "TIlIIt"-Snin S'·glllt'III.llioll: "Model alld a r.lt-lllOd." 11//111/111/1 ill II Sl i"IUI", ~~t. 7-~!L

I'IX:\( •. "( III S"V,III"IlI;lIillll III Tilllt, Snit's: ill S. "-arlill, T. l\nlt'lIli\';!, ;(11,1 [" (;o'''[lIIall, I"" .. .'illlll'l·\ ill t-:mllllll/l'I)';n. 'filii' S")'il',l. 111/1/1111111;1'10;1111' SII//IIIii I, ,\, .ul"llIi,' 1'It'''. N.. \\, \'''1 \... S'OIl, I ... I!IW" "Tilt, l'It's"1I1 \'ahll' ~lo.It'1 "I' Stock Pric('s: ){eg-r('ssit>1I 'ksl.' ,lIld ~Iolllt, (:allo R""llts," 1I/1·il'/lllI/ Fml/lllllit'.l 111/11 SllIli,ll;n. (i7. :,~I'I-(iOi, - - - . 1!IX7, "( ll'lioll I'ricilll!: wht'll Ih.· Varialln' Chan!!;t's ){allc!ollll\,: Thl'or~ hliIllation. ;IIHI ~'II Applicatioll," ./0"1""0/0/ Pi 1111 II (';11/ (l"d Quall/italitl(' AU'l(' \i\, ':!:!, ·11 !I-·I:\H, and F.xchang,t.· (:Ol1\1Hl'\sion, l~I~H. j\t"rH.1'1 2()()(J: Au /':xflminal;on (~ICII""1I1 I"tjuilv '\/l/tI"'1 /1.'("'/"/1/11/'/1/1, [IS (;owrnlllt'nl I'rillling OlH ..." W.lShillglllll.

St'Cliritic."!\

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5M2

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!ihorark, (;., al\(l.J. WdlllCl', I !IHti, [o;III/,iriw//'U/(I'\.\(·.1 ",ilh tlNdiwlilli/.1 III SllIll\lin ..!"hn Wiley ,lIId SOliS, Nt'w York.

Sia'~ It, and L, Starks, 19\/1, "Inslilulions, lllIlividlials and Relurll AII\Ilflllll·Ll1illl\\."

I

Working l'apt'r, Ullivt'rsilY of Tl'xas, AIISlill.

Sicgd,.J., 1!J!J4, SII/rkJ )ilr Illp LOIIK HIIII, Norton, Nt'w YOlk. Sih'!-r, S" I !175, .'i11l/i.l/im/llI)nr'III', (:hapIllan and Iiall, l.ondon. !

Simkowil/., M" and W, 1\c"llIcs, I !IHO, "A'rIllII"'lric Slahlt' Disl/illlll('d SI', :~Oti--:\ I~. ' Silllr'

c.,

1!171, "OIUpUI and Lahol Inl'llI in ~I,\lI\,rac lllrinJ.:," lI/C1okillK' 1'''11/'11

1111

i

ErlllHllllir tlrlil,it)', :1, ti!I;,-7:!H,

I

1977, "ExoJ.:cJlcity alld Callsal OrderillJ.: in Macl'Oc('ollollli,' ~Iod"'s," in ,\'/'1/1 Mpl/wti.! ;11 IlIuillP,l,1 eyrlp 1II',lfllrtl/.· /'/CI(('I'dillgl )111111 II (/llIft'II'IIII', Ft'dt', "I RI'.crvC Balik of Millll!'apolis.

I I

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SI;II\lhallJ.:h, R., 19W1, "On thc Exclusion of Ass~ls from Tcsts of thl~ Two P:. Ilt'rlin. \':lSil I'k, ()., 1'177, "1\11 1'.ll'lilil>rilllll (:11:11;1< Ini';lIioll "I' \Ill' 'li'1'111 SI 1'11. "(li\,id .. lld I 11110\,;11 iOIl' ,lIld SIll( k PriCI' \'"I;lIilill·." 1'.i'I/IIIIII,'I";III. :17-1i1.

,.1;,

\\'III';IIII'\', S., (\1:->:->, """"'" ·!\-s" .. 1 II,,· (:OI\Sttlllplioll·(I;I,,'tI ,\",'1 I'li. :.'Hli, :1 I\. :11 :1-:11:., :117, :1 I H. :1:.'0, :1:.' I, :1:.':1-:1:.'7, :1:10, :I:t!. :17:.'. :I!I:., ·IOX. ·11,., ·II!I,I:! 1,1:.':.'.1:1,>, ·I·n, .\-1:>. IIH, 1\11. ·1'.17 (:,11"'11111, ·10\1 \ :;11 "It'ill, ·17:.' (:,'" 1"'lIi, :10·1. :1 III (:Io.IIII""II"ill, 'I:.', :.':.'\1. :.':11'. '.01 (:10 .. 11, K" 110. 11'1 Cloall. 1..,107.110 (:1 ... 11.:-"" :.':1'1. :.'111.1:.'1 (:I,", 117. I:.' I-I :.':1 (:hoi.IO:.' Choll. :11' I. 11' I

(:hrisli,', A., :17!I, ·I'l? Christi .. , W., 107, 110 Chllllg, C., !,I)·I (:lllIlIg, K., Ii Cochralll', ·IH, .1 1 ), :.:.', :!7·I, :11):.'. :1:.'7. :1:\0, :1:1:.' Coli (' II , H·I, W., HH, 111-1. Ill? (:"lIiIlS, IIi7 COllt', 110 Cunno1", ~'.! I. '.!:\7-'.!·1t c, .1IS(;1l1l illi,!.'s, :1 \(i, :1 I H, :1:.'li. :1:.'7. :\:10, +1:.', ,>07 (:oo\l('I', ·to!)

(:00(11('0',/;,> COIH'lalld, 10:1, 107, 1111' Corrado, II:!, 17:1 (:owl('s, :.'0. :1:>-:17, Ii'. Cox, D., 7, ).II) Cox,.J., :1 I. :1 I H, :1·10, :1·11, :1·1'1. :F.'•. :17!), :IHO, :IH:.', ·11·1, ·I:.'!), ·1:1'•. ·I:~I'. ·1·10, ·1·11, ·1·1:\, ·1·1·1, ·I·I!), ·\:.1 •. \".s. ·11.:1, :.OH, (:r;lI'k, 11:1 (:';Iig, ·17·1 (:'II('io, 107, 10'1 (:1111,'1'. :117

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DUllill', :)07 DurI:luf, '27t>---27H Dyhvig. 220, 221, :1:>1, ,n~, 140, 44:1, -1+1. '1.~2. ';!ili, 4!i4 Llsky, 44, 99. 10~, 107. 140 Eckho, II .. 17'1 Eckho. E., 17:) Edwards,4:{ EiclH'nhaulIl, :l2ti Eikdu)()IIl.I02 Eillswill. Ili. :12 I/I>gi. :\ I :>. :I I ~), :{20, :\:14 EslI ,'11;1, ·12·1 Eq:ll>. '12 I-',Iho"i, :1!Hi, '10", ·lIl1i 1-';01>;1,20, 2'2, ·11---4~, S4, (i", 7H, 79, 1:>0.1:>7, '20H, 211, 212, '21:>, 221, '2·10. '241. '24H, 249. 21iti-2tiR. 274. :17!I. ·11 H. '1'21. 4'22, 4'24 Fang. :IH7. :~HH 10';111.'1. ·IR. ·I~I. :)'2 • Fell9, 2HH' :1:1:1, 424 Fuller, 4!), 46, Ii!) FllrhllSh, 110

I

Cahr,472 Calai, 10:1, 107 Gallant, A., 522, 5~5 Galli, H4 Garber, 258 Garcia, ~34 Garman, :180, 4H I, 507 Gatto, :185, 391 George, 107, 1:15 Genler, 316 Gihb()ns, 193, 196, 199,206, 24S, 24ti,2YH,317,44S,446,448,455 Gilles, 275 Ciovannini,A4,320 Girosi, 512, S 16, .517, 522 Gleick,47::1 Glosten, 101---103, 106, 107, 1::15,486, 4HH,497 Godek, IIIl (;o"tzmann, ::III Goldberger, 504 Goldenherg, :'179 Co\clrnan, 3W>, :191 Goldstein, 102 Gonedes, 17, 20H, :179 GOl1lale7.-Rivera, 4H9 (;oodhart, 107, 1()\l Conlon, 256 (;oltlieh,12I---12:1 Graham, 114 Crandmont, 474 Granger, 17, .59, 60, 6.5, 91, 257, 470, 472,4A6 Cranito, 406 Grassuerger, 4 7C)-<j 7A Gray, 4')2, 4:)5 (;regOl)" 4::15

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11",11 ick, 2Ii7-270. 21,1. 2W•. 2Xli, 'I'IH. :>:\(i Ilul'!. 7. :\oI22 Ilosking, :,\1, 611 Ilsil·h. '17,1, 17:>, -17!1 IIsII,I7 IllIang. C., H, :IHO. :1!1I. 4(il. :IOH IIl1ang, R., 107, 110, IY> IIl1hl'nmn, 221, 2:1l 111111. :1,/0, :I'I!I. :17H-:IK I. 'I'>!I. 'ltil. :>10 I11IIsl. :.!I. (i2 1llIlchinsoll. :1,10. :1'12. :> I O. :.12. :> 1!I. :,~2

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I

• Subject Index

ahsolllie vallie GARCH Illodel, 'IR5 arlivatioll funnioll, :) 13 arlillc-yield 1Il0dds of the term stl"llClllre, 12H, 441, 44:. aggrq~att' consluilptioll aggrl'gatioll,305 (:OIlSIIIIIPlioll (~lpital A,sl't Pricing Model, :~\(i '\lIlni("all oplion, :149 :t1l11)litIHit--dl'pendent eX(l(JIll'lltial autoregressioll (EXPAR) m()dd~,

470 ;lIlIipnsisll'nn', nO alllillll'tir variates lIlethod, 3HH arhitrage opportunities, 339 ,1;lIt' price vector, 29" hond excess relUrllS, 4 H Merton's approach !O option pricing, :l51 arhil ragl' portfolios, :l!'i I Mhilrag(' Pricing Theory (APT), R, W., ~)2, 21 ~). Srr also Capital Asset Pricillg Modd, IIIl1ltil;lctor Ill1 ddlt'rllliniSlie volalilily, :17!1 eSlimalor for ,,', :lli\, :17,1, :17" swchaslic \"Olalilil)" :11\0 implied volatilily, 377 oplion scnsilivilies. 3:,1 horrowinll ("onslraillls. 31:, Box·Cox lransl()JJnalion, 1,10 lIox-Pinfc Q'SI-lloll).;la,' IIlilil)', :1:!li C(){'IJ'ICit.'lIt IUllnion:-o. :F',r) C1Jilllt'gl;lliulI.

(OllIHJII 1;111',

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die,

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Ii 11'111 III",

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II", 11101,1:/:/, I:>H ,narket-a(ljusted-returll lTlodel, I !)l;

nons)'llchronous trading, 177 Ilormal return, 151 p"st-evellt window, 157 sampling illlel>'al, 175 SkC:\\fIH'SS

of

rt~tll .. nSJ

172

,standardized cumulative abnormal return, 160 t('st pOWt'r, I (ill (');al'l factor pricing, 221 illtl'rpr('ting deviations. 242 llu';llI-variance effiri('1l1 set Inathematics, 243 ll(lIlrisk-haSl'd alternatives. 24H optilllal orthogonal portfolio, 24:~, 2-\:',21H I i,k-lJas('d alterllatiV(", 247 Sh;tlp(' ratio, 245. 247. 24H, 2')2 t;lllp;ency pOllfolio, 24:>, 2,17 exn'"s kurtosis 17, 4HH, :> 12. S,'" fI!.1I1 kurtosis, returns ('xcess n'ttlrllS, 12, I H2, 2f}l{, 2~1l ('X('ITis(' III in' , :H9

exmic securilies, 391 expansion of lhe slales, 357 EXPAR models, 470 expectations hypothesis (EH), 413, 41 Il, 419, See also pure expectations hypothesis, term slruclUre of inlerest r.lIes empirical evidence, 418 log expectations hypothesi. 432, 4:17 preferred habitats, 418 yield spreads, 418 expected discounted value, See discounted value expont>lltial GARCI-I model, 486. 4HH exponential spline, 412

I

face value, 396 factor analysis. 234 factor model, I:):'. Su aLso multifactor models f;lir ganlt>. Sef martingale rat tail, 16, 4HO. See also kurtosis finite-dimt>lISional distributions (FDDs), 344, 364 Fisher inromlation matrix, See information matrix fixed-income derivative securities, 4:,5 Black-Scholes formula, 462 IIt>ath:larrow-Mor!on model, 457 Ilo-Lee model, 456 hOllloskedastic single-factor model,463 option pricing. 461 term struclUre of implied volatililY, 41i3 fixl'd-income securitit>s, 395 floor function, 114 Fokker-Planck eCJuation, 359 foreign currency, 5, 3H2, 386, 390 forw,lrvard ratt>, 399, 438, 440, See qlso term structure of interest rates coupon-bearing term structurrl , 4011

Su/Jjnl/I/I/"":

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II ..-~.ec modt'!, 4[,Ii, 4(i.1, St't' 11£111 i)ricin~ fixcd-incOllle dnivati"c 1l 't:lIrities hOI~HlskecliL~lic singk-hlt'lor ~lIoclel,129,1f'2, 4f,7 la(('ill-variable \llodels, 'lolll LOII~sIOlI1:Schwanllllodcl, '1:11l s'llIal,(,-l'IIol sillgl('-I;lI'lol' lI,o,kl, 4:1:1, ,1:"1 stochaslic c.IiscounllilClor, '127 IIv(.-filctor moclel, 4:lH Vasicek mocld, 4:1-1 (('slillg for no"linl'ar sl 1'111'1 lin'

t I

Iln)('k-lkchert-Sch('inklllall II'SI. -17H Iisieh (('SI. ·17:, 'Isa), lest. 471i Theta. :If,:l thn'shold, 472 threshold aUloregression (TAR), ,172 lillie aggregation. !}1. 12!1 lillie inconsislcncy. :tH tillle-nonseparabilily ill Ihe ulilit), functioll, 327, :tm, Srp ullll hallil "\\'Inalion modds . Iran' operator, 74 Trades ancl Quotes (TAQ) dalaha,,,', 107 trai1\il\~ a It.'t\r1\in~ 'H.~twurk, !) I r,. r) I X Irailling palh. [,I!I Iransat'LioJls t'OSL"'I, 1t1!",

lransanions dala. 107, 1:1fi trallsitioll densit), fUllnioll, :I"H Tr('asury S('curities, ~!I:, STRIPS, :I!lIi Trc;\s\ll)' bills, :I!)(i Treasul)' nOles ,\tid 1I""11