The Econometrics of Financial Markets - Campbell Et Al

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,,·,tlllions: one for IQ scores and the other f()(" academic perf()J"JIl'IIHT. Mul~iple observations are drawn frOIll ealh lIIarginal dislributioll alld variOilS \Ilonparametric tesls can he designed to check whether the prodllrt of Ihe ~Ilarginal distribulions equals Ihe joint distribution of the paired ohserv\ltions, Such an approach ubviously GlIIllot slJcceed if we hypothesi!.(' a l1Jiifllle marginal dislri1>ltIion fopulation values. In addi tion . it is well know n that cr'/.' and a,7 possess the following norm al limi ting distr ibut ions (see , for exam ple. Stua rt and Ord [~987»:

a/,

\

I' i

I I

,j'j;;(a/,-a~J

:.:.- N«J ,4a 1 ).

(2.'1.24)

lIow erer . we seek the limi tillg dislI 'ihut ioll of the ratio of the vari ancc s. Alth ol/g h itllIa y read ily be show n that the ratio is also asym ptot icall y Ilorm al with IIlrit mea n und er RW I. the vari ance of the limi ting dist ribu tion is not appar~nt siricc the two v;lriallce cstil llato rs arc dear ly /lot asym ptot icall y IInc orfe lated . Bilt sinc e the estim ator is asym ptot icall y crtic ient und er the nllll hypo tlles is RW I. we may \lse llall sma n's (1~7H) insig ht that the asym ptot ic

cr,;

II \\'t' a\.lIjume nonn ality only tor t·xpo . . itioll.lI ("onn 'lIicn rt'-th l' resl1l t~ in this s('nio ll "pply 1I111ch 1II0re gelle rally to log price I'lOces,,"s with liD illrn"II1lOtic varian(es,l~ II IV(' ddill(' tlw vari'lllce di(fc.'ITIIlT estimator as VI)('2) == tl\{'n C!"I.~:-\), (~.'I.~.j). and I'!ausman's result implies:

n;; - n,;.

TIll' Illll1 hypothesis" can then he t('stnlusillg (~ .. I.~:» alld allY cOllsistcnt ('stilll;llOr :!a'i of 20'1 (for example. :2(a~)~): COllstrud the standardized

sl;ltistic \'1)(:2)1 ~ which has a lilllitill~' standard lIoml;d distrihutioll 1111dn RW I. and reject the null hypothcsis at th(' :if;;, hoyd if it lies outside the illt(,ly;dl-I.%,I,%j, The as),lllptotic distrihutioll of the t,,'o-lwriod v;lriaIK(' ratio statistic \"71\(2) == a,~ now follows directl), fmlll (:2A,2!i) usillg a (irst-onierlil),lor applOxilllatioJl or the delta Illethod (se(' SeClion/\..J of the Appendix):I:1

In,;

'I'll< 111111 !!y(>othesis 110 can he t('stnl Il)' ("ollllllilillg the standardized statistic ~(VR(2)-I)/~ which is asynl()tolic;dly st,lIHttrd Jlormal-if it lies ()llIside the illtnval 1-1.~l(i. I.~HiJ. RWI Ilia), 1)(' rl'jnwd at thl' :)'}{, kvd of si~Jlili(allce,

AlillOugh litl' vari,III('C ratio is ,,1"1 ell pn'krrl'd to Ihe \',lIi,III("(' dint'n'll(T 1)('("'\\ls(' Ill(' ralio is scalc-fret", ohserv(' Iltat ir~(a,;)~ is IIsed 10 eSlilllale 2a I. lhell lite sl,llllbrd sigllilicalHT ll:Sl ofVD=() lor tilL' dilkrelllT will yidd lhl: saille inferellces as the correspollding test ofVR-I=O (l!' lhe ratio SiIlCl':

~VT)(2)

ff,i(a,; - a,;)

JET,}

~a,;

Tltnt'i"ure, ill Illis silllple cOlltexl th(' IWo tesl statistics arl: ('ljuivaklli. Ilow('\"l'r, 111('1'(' ;11"(' othl:l" reasolls that Illake tll(' I'ariallcc ratio Illore appealing \'.!

HI il'll}, 11.111'1111.111 (I ~)7H) c.'xploib III(' LH t 11,,11 ,IllY .I:'I~ IIIIHllllf.dly dlidc.'11i

I..':.lilll.lIor

01

Ii,., 1II11:-.t I't):-.~c.·s., the.' p"oP('II~' Ih.11 it i~ 'I'~ IIIpIOlit';dly unfOI n'lo"t:d with Iill' 11111"1 ('lit"(' (i" -- (i,. where: (itt i~ au)' olher (·~lilll.ltor 0111. II lIot, dlt:1I tllC.'''''' t"xi!oots a liue.n ("Ollibill.tlillll (II (i,. ,lIltl (ill -fir IhOit is iliOn.' c.'Hi("it'lll IIi.lllli, (CJlllr.uli< lillg Illl' ''-,.'IIIH«,'d l'Ilirielu"y

.1 p.II'IlIl('Ic.'1 (I,

~.I\"

I

0111,

"111(' I ("'1111 1t)lIo\\'~

dirt'nly. thcll. sinn';

,'\',III,i,,1 '"

,,\';11

=>

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) ;,,1ortall! diffcren ce from standard econom clric models whit'h we shall relurn 10 shortly. NOle Ihat subscrip ls arc used to dcnote Imll,\(/rli '!Il lillle, whereas time argullle nts I. dcnotc calenda r or r1urh tilile. a COnV('lllion we shall follow through ollt Sectioll :1.33. The hearl of Ihe ordered prohil lllodel is the assllillpl ioll that ohserve d pricc changes Y. arc related to the cOlllillUOUS v;uiabks ill Ihe 1()lIowing llIanner : if )" E ;11 .II

1';

1';

,\~

y.

J",

if

• •E

)"

/I~ (~.:\.15)

if }"k c ;\",>

wherc the sets AI furm a /Hlrliliull of the statc spale S' of 1'; . i.e., S' = U;'~ I Aj and Ai n Aj = 11 for i 'I j, and the ~/s arc the diMTl'le valul's that compris e the state space S of J'., The motivati on for the ordered probil specific alioll is to ullcover thc mappin g lJetween S' and Sand relale il 10 a sel of ('Con 0 lIlic variable s. In Hausma n, I.o, and MacKill lay (I~1~12), Ihl' ,\,\ arc defilled as: 0,

-k, +k.

variable is uisncw awl is 1I -./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 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 ~ (/'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 '.

Chr 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' -

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; . .111'1i~ (:YI + P-j IJ) (B"t,-llh- 1 I

-1

(B"E,-IU')-IB"E·-\(Y.;'i·{yu))

x

t'}:"

Iihil,,}:·-I B·)-I,

We \Villus(' Ihl'se I'ariallcc n'sulls li)r infercnccs cOllccrnillg Ihc lilnor risk premia ill S('nioll (i.:1. fl, 2,,' 1,(1t lot I''''I/o/im .'i/llllllli IIg Ihl' MfilIl-l'a/'illllC/' 1-I1I1IIit·,.

\,\'hclI 1~\('lOr pOI'I«.lios span Ihe Illean-I'ariall(,(' fronlier, tIlt" inlerccpt 11'1'111 of thl' I'xact pricing rl'ialioll All is /('ro wilhonllhl' !ll'cd lill' a riskfree ass('\.

6.2. i:'sti11l(l/ion and 7fSlill/( Thus this case retains the simplicity of the first case with the riskfree asset. In the context of the APT, spanning occurs when two well-diversified portfolios arc Oil the minimum-variance boundary. Chamberlain (1983a) provides discussion of this case. The unconstrained model will be a K-factor model expressed in real retll\'lls. Define R, as an (Nx I) vector of real returns for N assets (or portfolios of assets). Then for real returns we have a K-factor linear mode!',: R,

=

I

(6.2.4&)

a+BRJ\r+f.,

I

o

(6.2.47 )

E

(6.2.4~)

1

(6.2.49) (6.2.50)

n is the (N x K) matrix offactor sensitivities, RJ\, is the (Kx I) vector offactor portfolio real returns, and a and f., are (N x \) vectors of asset return interr cepts and disturhances, respectively. 0 is a (K x N) matrix of zeroes. The restrictions on (6.2.46) imposed by the included factor portfolios spannin~ the mean-variance frontier are: a

=

0

and

Bt

=

t.

(6.2.5\ )

To understand the intuition behind these restrictions, we can return to :he Black version of the CAPM from Chapter 5 and can construct a spanning example. The theory underlying the model differs but empirically the restrictions are the same as those on a two-factor APT model with spanning. The unconstrained Black model can he written as R,

= a+

f3 0mRol + f3rn Rrn, + f."

(6.2.52)

where ROIl' and Ro, arc the return on the market portfolio and the associated zcro-beta portfolio, respectively. The restrictions on the Black model are a = 0 and f30m +f3rn = L asshown in Chapter 5. These restrictions correspond to those in (6.2.5\). For the unconstrained model in (6.2.46) the maximum likelihood estimators are (6.2.53)

230

6. Multifactor PricillK MrHlrLl

(li.2.:,·1)

where

T

I '" I.L. = TL.....RI

and

1=1

To estimate the constrained lIIodel, we consider the uncolJStrained model in (6.2.46) with the matrix B partitioned into an (N xl) cO\lIlIIn vector b l and an (Nx(K-I» matrix BI illldthe factor portfolio vector partitioned into the first row RI! and the last (K-l) rows R K,!. With this partitioning the constraint B ~ == ~ can be written hi + BI ~ == to For the lI11colIStrailled model we have

Y

Substitllling a jmodel, I 1

\

== 0 and b l ==

L -

BI L into (G.2.56) gives the constrained

R, - ~RII = BI (R h ,! - ~RII)

+ E,.

(li.2.:,7)

Using (6.2.57) the maxillllllll likelihood estimators are

«(i.25H)

b; :to II

==

(G.2.IiO)

~rhe null hypothesis a e«uals I.em c'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' (,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('riol[(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".1.) .,r: ..•11., :\.:\Iil :.'.07;\ 2.·IH:! :1.21 ,I :>'9Ii:~ . __._--- ---- -- -.- - - -- - - - - - - - --'/'-/ = :lli4 Da)'s ...

I

1:.'.7:,:1 '1.\'1:1 li.'(1)11 ·1.'111 :1.111'1

10.0:.'1{ :>.dci (~).~.:!) tite (1 is a knowlI detplil's btlt with a rl'plared hy the integlal Iift-. I I()\1'l'I'('/', if (T is stochastir, till' situatioll ht'colIH'S 11101 e (olllpkx. For

,I;'

N~ (I~)!U). n.dl.lIld f{tllIl.l (Iqql), B('f"c'" (I~IXII), (AI" (;old('11I ... ,.~ (I!I~II), Ilt-slnll (I!I!I:I), 1101111, ..... , 1'1.,1"11. ,,,,,I ,,1,,\1'11," (I!I~I:l),111I1I "lid l\'loil('(11' I'l'plictl(' Ihl' opli()II's payolf. I "'"I'isticalll', slodl;lslic I'oblilill' illllOdu('('s a second sourc(' 1111('('1'1;lillll' ililo the rl'plicatillg portli>lio 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(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 clll'·I'I(,·r (J 'JXlI). IIlld(,II~'illg !'Ic'nll ily prirt',

I

'4'i2

II. ·1i'I'II/·SlnHIIIIl'M"dd,

nut W(' kllow frolll asset jlril-in):: 1IIl'IIry 111011 11ll' ulHkrlying s'Tllrily pri, (" S, I"IIISI salisfy

I Sf

~

==

.

,

(

== ('xp /1",+/1,+ rr""" + a"2 + 2a",,) '

1,·tlt\llf,f-I"·\f~,,1

(11.:~,I'I)

We also kilo\\, Ihal Ille pi ice of an II-pel iod lero-coupon hond, I'"f. III1ISI ';lIisIY

I

I'''f

=

EtlM".f+,,1

== eXP(ll",+

a~",).

(11.:~.20)

I

Usillg (11.3.1 !)) alld (11.:t20) losilllplify (11.:\.17) alld (11.:U H), ;1\111 slIhSliIlIlillg inlo (11.:~.!(i), we gel all expression for Ihe price ofa call ol'lioll whell Ihe IInderlyillg securily isjoilll\Y \ogllol'lna\ wilh Ille Illllhipniod stochaslic dis(,Ollllt I;IClor:

, (/I,+a",,-.\'+a,,)

.'i,I

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' Ill. 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 ('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 IlII I..

II/

fI/"d/'1I1 IIltil SIII'I'I. (-,,,.,.

II,""illlas. 1> .• 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

- - - , 1!193b, ·Why Lon~ Horizons? A Study of Power a~ainst Persistent Alternatives: unpublished paper, Princeton University. - - - , 1!)\):l, ·Some Lessons from the Yield Curve: jllltl7wl oJ r:Conomic Pm~cljVf'j, !J(3),129-152. - - - , 199(ia, "Understandinv; Risk and Return: jounwl oJ Political fcollorny, \04, 29H-:H5. I !1!Hib, ·Consumption and the Stock Market: Interpreting International Evidence: NBER Working Paper 5610, National Bureau of Economic Research, Cambridv;e, MA. (;;11111'1>('11, J., alld.l. AlIlmer, I!l!n, "What Moves the Stock and Bond Markets? A Variance Decomposition f'lr I.ong-Term A~set Returns: Journal oj Finana, el1, J., and A. Kyle, 1993, ·Smart Money, Noise Trading, and Stock ;Price B~havior," Review of l:'collomic Studies, 60, 1-34. ' Cllllpbell,J., and N. G. Mankiw, 19H7, "Arc Output Fluctuations Tr,lI1sitory?," Quartnlyjoumal oj Economics, 102, 857-8S0.

- - - , 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 Mo1" tions Rl'search Centre, University ofWarwi~k, UK.

Carverhill, A., and N, Webber, 19HH, "American Options: Validation of NlIm('liral Methods," Preprint HH/~, Financial Options Research Centre, l 'nivelsil), of Warwick, UK. Cecchelli, S., P. l.am, and N. Mark, 1(1!IO, "Mean Reven;ion in Equilihrilllll Ass(,1 Prices," Amenwtl Jo;r"'IOmir Un,it''''', HO, 3!IH-41 H. 1994, "TeSlin~ Volatility Restrictions on Intertemporal Mar~inal Rates of Substitution Implied by EllieI' Eqllations and A.set Returns," .Iollll/{/l of "'. /Irma, 49,123-152. Chamberl"ain, G., 198301, "Funds, Factors, and DivCfsilication in Arbitraf.:(' I'ricinf.: Models," fcolI,,"vlnrn, 51, 1305-13~3. - - - , 19831.>, "A Characterization of thl' Distrihutions that Imply MeanNal ian('e Utility Func:lions," Jour/wi o/I:'{,(lIIolllil' 'I7lfory, ~9, 19H:)-~O I. Chamberlain, G., and M. Rothschild, 1983, "Arbitrage, Factor Structure, and MeanVariance Analysis Oil Large A.set Markel.," 1:'clJllOmelrim, 51, I ~HI-1304. Chan, It, 19H8, "On the I 47-lli3.

COI\II~lrian

Investment Strateh'Y." ./oumal

0/ HlI.lilll'.l,l,

Ii I,

Chan, It, N. Chen, and D. Hsieh, 1!185, "An Exploratory Investigation of the Finn Si7.e Effect," Journal 0/ Fillflllri(JlI:'conomin, 14, 451-4 n.

C0'III, It: W. Christie,

and r. Schultz, 199:" "Market Structure anti tht' Illtraday Evolution of Bid-Ask SI>reaos for NASDAQ Securities," ./(}lIma/ of Ilm;lIr.I,I,. 68, 3:)-liO.

1I

,

,

ellan, It, G. Karolyi, F. Lon~s!afr, and A. Sanders, 1!19~, "An Empirical Comparison I of Alternative Modc:ls of the Short-Term Interest Rate," jOltnltl{ 'if hllflllCf, 47,1209-1227. I

(;11an, 1,., and]. Lakonishok, 1993O. FII~I,·.

R" ;IlId \', N~, I~I'I:I, "~kO\,,"ill~ ;",.1 '!','slillg Ih,' 1111110\('\ "f Nl'ws "" VII/.ltilitl," /1111/1"'/ "I "'1I1/1I11',IH, 17·1 ' 1-I77H,

ElIgII'. It, }), l.ilil·lI. OIlId R, R"hillS, 1~IH7, "E'lilll;llillg Tilll"-V;II)'illg Risk PIt'lniOi ill Iht' Tl'nll SlnH'llIle'; TI\(' ARC :II-~t I>-\od"'," /'.'wI/II/1lrlr;m, ~)~" :IC) 1~I07, EII~"',

It, \'. N~, ;,,"1 ~1. /{olll,cllild, I ~I'IO, ",\''''1 Pricing with a FarlOl' ARCII (:""; .. ;. all,',' SII· ..... " ... ·; hlll'i, ;1',,1 E,t;IIl"h's I'llI' Tn'"slIl,)' 1\;lIs," ./"11""0/0/ Frol/o1l/"', 'In, ·1~>, 'l I :I-:!:IH.

Epstein, \.., allel S. Zill, I\)H'I, "SIIh.'lillllioll, Risk AVI'rsioll, alld Ihl' Tl'mpo .. al llt-!.av;o .. of COI""I1'IHioll alld ,\SS(' I 1~l'lllrllS: II Tllt·o ...·lical Fr;lIl1c"·O .. !.. ... fUIIIII""·/,i,,l, :>7, \1:17-'lIiH. -'-,1\1110, "FilSt·( IId"1 Ri,k AI'I'/Sioll "lid IIH' EI}llilY I'rl'lIlillllll'lIl/h-,"./OI1J'11l1 i

til

MOIIt'/I/I,I hlll/olllin, 'lfi. :lin-·107.

- - - , 1\)\11, "SIIIl\lillltioll, R;,k A\'(· .. ,ioll, ;111,1 Ih,' T,'IIlj>orallkhavior 01' (:OIlSIl'"I'I;nll alld "\.'st'l R"""I1" '\11 E"'I'; .. i... ,1 11l\',·sli~atioll." ./11111/10/ o( /'oliliud /.:, OllH"'.V. ~J~l, ':!(;:~-':!H(;.

Eslldla. ,\ .. alld C;. II;lI'IIIItt\'cli" I C)~II, "'I'll,' '(('nn Slnlctlll':,:,-:,7Ii. I-:Y"III, '1'., ;11,,1 (;, 11011'1'.11, l'IKli, "TIll' 1'1 iring of Fill ,,""S alld Optiolls (:01111'''''''' lhl' \'"hu- l.illl' IlIdl'x," '/011,.,,1// of /-I/lill/a, ,II. IH:I-H:,li.

1111

Fallolli, F., 1\)\lIi, 110/1" '\/(/)/11'/1" I/lo/pil olld .\·/lol~}[il'.1 (:1 re I ('11.), 1'''l'lIlin'-llall, l'ppl'r Saddll' Ri\'l' 1', :-.11. F;"lolli. I':, allel T. Faholl; (I',k). 11 1'1:., nil' /JII/ld/lltole o(/'/.\'I'I/,/",·01l/".\·,'(//I-1lil'l (·111> ,.,\.), ""ill, 11," I ItidJ.\'·, 1\ .. 1'.111101,1'.. , I 'lIi r,. "T!.l'lkkl\';"r "I Slnr!.. ~I.II 1.('1 Prin·: " NelV "ppI IIlil l{hlllLllI'ill[I':uJ/Wlllin, Ill, :I-:D.

0", 11." '/11111-

Lihh,,"S, M.. all(1 W. F,' rso II , 19H!,. "ll'still~ As,,·t I'ri('ill~ 1\lod"'s with Chan~in~ 1':XP(,(,("lions and "n Unohs('l'v"hk Mar\..el I'ol'llolio." ./111/111111 "I NlWlllill1 l'.'mll(}III;I.I. 14, 217-2:Hi. (;,I'\)OIlS,

~I., "lid K. Ramaswamy, 19!/:\, "ATest ortbe Cox, Ill~elsol\, and Ross Mudd of Ihe Tel'''' Structure: H"";,,II' tif H,lIlIIc;1l1 SIUt/i,.I, Ii, Ii 19-ti:,H.

Clhh,,"S, M., S. Ross, alld J. Shallken, I!lH!/, "A Test of thl' Elliciellcy of a Givt'll l'ort!(,lio," 1,'COllolII,trim, ;,7, 1121-11:,2. (;i1ks. (:., and S. l,eRoy, I!l!ll, "Econollletric /\spcets 01' thc V,II ian('(' 1I0unds Tests: A SllrVl'Y." IVlIil'w oj Financial Studips, 4,7:,:\-7\11. Ci()\'''"llilli, A., ali(I 1'. Wcil, I!lH!l, "Risk i\version and Inll'riCllll'OIal Substitution in the Capital A~sl'tl'ricillg Modl'l: Workin~ l'ap!'J' ~H~'I, NIIER, Cambridge, ~Ii\.

(;kick . .J.,

1~IH7,

Chllll.l:

Milkil/~

(I

Nntl Scil'lllr, Vikin~ l'("n~lIill IIIL, New YOlk.

CloStCll, 1... 1!IH7, "Componenl~ of the !\id-A.sk Spr('ad "ll(ltlie Statistiral Propertil's "fTransaclion Prices: jounJllllIlh,/Ill!cp, '12, I 29:{....I:107. (;I""I,·n. 1.., and I.. Harris, I!lIlH, "Esti\11atin~ thl' Components of the lIid/ Ask SpIl'ad," jll/l/'l/al of /-llUl/lrjall,;'()/wlllirs, 21. 12:\-142. (;lost("lI. 1.., and 1'. Mil~rom, I !lH:" "lIid, A.sk and TI allsaetioll I''';('('s in a SpCli,llist Market witli !letero~enl·o\l.sly Infol'l)\l'(\ Tr;,,\l'I's: .Il11lnll,i IIf Fi,lIIl1rilif 1·.'IIlIlIlIIin, 14.71-100. (;10.'1"11, 1... R. .l,,~annathan, and D. Rllnkle. I \/\/:1. "On Ihl' Rl'lation 1I"twl'en thl' Expe('(l'd Vallie and the Volatility ol'tll,· :--JOl1lill,tI Exress Rellllll on Stocks: .I"II11UIl lifFi'lIl11rf, 41l, 177!l-IHIlI. (;Ild,·k. 1'..

l~I%,

"Why NASDAQ t.tllk,·t M,ll"IS Avoid

()dd-F.i~hth Q,Hlt,·,."'/OIirt/(/f

,,/f'il/lll1rilll hOI/lim in, 41, 'lIi;I-·17·1. (;"ld"lIhl'r~,

1>., 1991. "/\ Unilicd Mcthod

l'( ......"i(.'!\,"

,/I'lirlltll

«,I'

(If l'i·lIflllriflll~·(lJlw",in.

I'ri('in~

:!~).

Optiolls

Oil

DilillSioll 1' ... >-

:\-:\,1.

(;"ldIO,'", II .. II. Sosin, alld M. (;allo. 1!17!1. "Path Ikl'(',,,I"1I1 Optioll.s: 'Blly at thl' l.ow, S,'II at til,· Hi)!;h:'./lJun/(lf O{fillllllf'('. :1-1.1111-1127.

(;"ldlll.llI. II .. II. S",ill .. 11,,1 I.. SI I. 1',,11111. allel n. l'ala,lI11ik, I\)H\), (.'III1t'l.101 gc·mlt·l'II. I ~170, 1'1t·t1it/lIJ,i/ily I.('xillgloll. I.c·,illgloll. MA.

of Slork A/tllkfl 1'lia.I, llealil·

Crallil", 111., I!IH·I, /lo"d l'lIIl/olio /lIIlIIlIlIiwlioll. I.t'Xilll\llIril-,,,,,1. It. ;1I,d I~. '\k," 'II. I 'IK I. "()" ~Llll, "Nl's,ed Tesls of Altel'llative '('1'1111 Slrul'III"" Thl'mi,'s." 1I"1;'~" lltJ1uin and .')Ial;.,lir.\, (i;), 11 [)-I ~:t

or Fro-

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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 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

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