An Intertemporal Capital Asset Pricing Model under Incomplete Information

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1 INTERNATIONAL JOURNAL OF BUSINESS, 4(), 009 ISSN: A Itrtmporl Cpitl Asst Pricig Modl udr Icomplt Iformtio Modhr Blllh d Zh Wu b* Profssor of Fic, Uivrsité d Crgy-Potois 33 boulvrd du port, Crgy, Frc Modhr.Blllh@u-crgy.fr b Profssor, School of Mthmtics Shdog Uivrsity, Ji 5000, P. R. Chi wuzh@sdu.du.c ABSTRACT I this ppr, w us th clssicl dymic progrmmig pricipl to obti th Hmilto- Jcobi-Bllm qutio to driv th quilibrium mrt qutio for ll ivstors. Our ppr drivs th xtdd quilibrium mrt qutio d th scurity mrt li of th clssicl cpitl sst pricig modl of Mrto (987) i cotiuous tim. It provids th cotiuous tim log to Mrto s (987) scurity mrt li. W driv th quilibrium mrt qutio d th cotiuous tim scurity mrt li of th itrtmporl cpitl sst pricig modl with icomplt iformtio. JEL Clssifictio: G, G Kywords: CAPM; Iformtio ucrtity; Mrt quilibrium; Vlutio qutios * This wor is supportd by th Ntiol Nturl Scic Foudtio (067), th Ntiol Bsic Rsrch Progrm of Chi (973 Progrm, No. 007CB84904), th Nturl Scic Foudtio of Shdog Provic (Z006A0), d Doctorl fud of Eductio Miistry of Chi.

2 48 Blllh d Wu I. INTRODUCTION Th cpitl sst pricig modl of Shrp-Litr-Mossi, CAPM, is rgrdd s o of th most commo dvlopmts i modr cpitl mrt thory. Th CAPM modl is still subjct to thorticl d mpiricl criticism. I fct, sic th modl ssums th m-vric critrio, it is subjct to ll th wll-ow thorticl objctios to this critrio. Mrto (973) dvlops quilibrium modl of th cpitl mrt. H shows tht portfolio bhvior for itrtmporl mximizr will b diffrt wh h fcs chgig ivstmt opportuity st istd of costt o. Mrto s itrtmporl modl is bsd o cosumr ivstor bhvior d cpturs ffcts, which would vr ppr i sttic modl. Ths ffcts cus sigifict diffrcs i spcifictio of th quilibrium rltioship mog sst yilds tht pprs i this modl d th clssicl modl. By rlxig th mi ssumptios usd i th CAPM, th modl hs b xtdd to mor grl coomis. Mrto s (973) modl stts tht th xpctd xcss rtur o y sst is giv by multi-bt" vrsio of th CAPM with th umbr of bts big qul to o plus th umbr of stt vribls. I th sm cotxt, Brd (979) shows tht Mrto s multi-bt pricig qutio c collps ito sigl bt qutio whr th xpctd xcss rtur o y scurity, is proportiol to its bt, with rspct to ggrgt cosumptio lo. Sic th cquisitio of iformtio d its dissmitio r ctrl ctivitis i fic, d i cpitl mrts, Mrto (987) dvlops modl of cpitl mrt quilibrium with icomplt iformtio, CAPMI, to provid som isights ito th bhvior of scurity prics. H lso studis th quilibrium structur of sst prics d its coctio with mpiricl omlis i ficil mrts. Mrto s (987) modl is two priod modl of cpitl mrt quilibrium i costly coomy whr ch ivstor hs iformtio bout oly subst of th vilbl scuritis. Th y bhviorl ssumptio is tht ivstor cosidrs icludig scurity S i his portfolio oly if h hs som iformtio o this scurity. Iformtio costs hv two compots: th costs of gthrig d procssig dt, d th costs of iformtio trsmissio. This problm is rltd to th litrtur o th pricipl-gt problm, to th siglig modls, to th diffrtil iformtio modls d to th thory of gric d glctd stocs. Mrto s modl, th CAPMI, is xtsio of th CAPM to cotxt of icomplt iformtio. As Mrto xplis Ev modst rcogitio of istitutiol structurs d iformtio costs c go log towrd xpliig ficil bhvior, th modl lso givs grl mthod for discoutig futur csh flows udr ucrtity. Not tht udr complt iformtio, th CAPMI modl rducs to th stdrd CAPM. Ficil modls bsd o complt iformtio might b idqut to cptur th complxity of rtiolity i ctio. Som fctors d costrits, li try ito th dlr busiss r ot costlss d my ifluc th short ru bhvior of scurity prics. Hc, most modls dvlopd i ficil coomics do ot xplicitly provid fuctiol rol for th complictd d dymic systm of dlrs, mrt mrs d trdrs. Bsids, th trtmt of iformtio d its ssocitd costs ply ctrl rol i

3 INTERNATIONAL JOURNAL OF BUSINESS, 4(), cpitl mrts. If ivstor dos ot ow bout trdig opportuity, h will ot ct to implmt pproprit strtgy to bfit from it. Howvr, th ivstor must dtrmi if pottil gis r sufficit to wrrt th costs of implmtig th strtgy. From Mrto s modl (987), it pprs tht tig ito ccout th ffct of icomplt iformtio o th quilibrium pric of sst is similr to pplyig dditiol discout rt to this sst s futur csh flows. I fct, th xpctd rtur o th sst is giv by th pproprit discout rt tht must b pplid to its futur csh flows. I tht cotxt, th ivstor s st is icomplt wh it dos ot coti full iformtio o xpctd rt of rtur d rtur vribility. I th Mrto s frmwor, o stoc is hld o which th ivstor dos ot hv complt iformtio. This hs th pottil to xpli why idividul d istitutiol ivstors do spd hug mouts of moy i rsrch d dvlopmt ctivitis, scuritis d iformtio lysis bfor dcidig to iclud sst i thir portfolios. Mrto (987) dopts most of th ssumptios of th origil CAPM d rlxs th ssumptio of qul iformtio cross ivstors. Bsids, h ssums tht ivstors hold oly scuritis of which thy r wr. This ssumptio is motivtd by th obsrvtio tht portfolios hld by ctul ivstors iclud oly smll frctio of ll vilbl trdd scuritis. I Mrto s (987) modl, th xpctd rturs icrs with systmtic ris, firm-spcific ris, d rltiv mrt vlu. Th xpctd rturs dcrs with rltiv siz of th firm s ivstor bs, rfrrd to i Mrto s modl s th dgr of ivstor rcogitio". Th modl shows tht icrs i th siz of th firm s ivstor bs will lowr ivstors xpctd rtur d, ll ls qul, will icrs th mrt vlu of th firm s shrs. Th mi distictio btw Mrto s modl d th stdrd CAPM is tht ivstors ivst oly i th scuritis bout which thy r wr". This ssumptio is rfrrd to s icomplt iformtio. Howvr, th mor grl implictio is tht scuritis mrts r sgmtd. Mrto s modl is bsd o th ssumptio tht thr r svrl fctors i dditio to icomplt iformtio tht my xpli this bhvior for idividuls d istitutios. Hc, th prsc of prudt-ivstig lws d trditios d othr rgultory costrits c rul out ivstmt i prticulr firm by som ivstors. Usig this ssumptio, Mrto shows tht th xpctd rtur dpd o othr fctors i dditio to mrt ris. Th mi ituitio bhid this rsult is tht th bsc of firm-spcific ris compot i th CAPM coms bout bcus such ris c b limitd (through divrsifictio) d is ot pricd. It pprs from Mrto s modl tht th ffct of icomplt iformtio o xpctd rturs is grtr th highr th firm s spcific ris d th highr th wight of th sst i th ivstor s portfolio. Th ffct of Mrto s o-mrt ris fctors o xpctd rturs dpd o whthr th sst is widly hld or ot. Th ituitio bhid Mrto s modl is tht ivstors cosidr oly prt of th opportuity st, d tht full divrsifictio is ot possibl d tht firm spcific ris is pricd i quilibrium. W dscrib i this ppr th Cpitl mrt structur, sst vlu d th

4 50 Blllh d Wu coomic modl. W us th clssicl dymic progrmmig pricipl to obti th Hmilto-Jcobi-Bllm qutio. This llows to driv th quilibrium mrt qutio for ll ivstors. Two css r studid: th costt ivstmt opportuity st d th grl cs. Our lysis drivs th xtdd quilibrium mrt qutio d th scurity mrt li of th clssicl cpitl sst pricig modl of Mrto (987) i cotiuous tim. W provid th cotiuous tim log to Mrto s (987) scurity mrt li. Th ssumptio of costt ivstmt opportuity st rprsts sufficit coditio for ivstors to bhv s if thy wr sigl-priod mximizrs. It is lso sufficit for th quilibrium rtur rltioship spcifid by Mrto s (987) simpl cpitl sst pricig modl to obti. W show tht grliztio of Mrto s (973) multi-bt sst pricig modl obtis i this coomy i th prsc of shdow costs of icomplt iformtio. W driv th corrspodig quilibrium mrt qutio d th cotiuous tim scurity mrt li of th itrtmporl cpitl sst pricig modl with icomplt iformtio. Our lysis provids two ctrl rsults. Th first rsult idicts tht i th prsc of rislss sst d iformtio costs rgrdig th risy ssts i th coomy, (i) thr xists uiqu pir of fficit portfolio ow s mutul fuds, d (ii) th rtur distributio o th risy fud is log-orml. Th scod rsult is Thr Fuds thorm. It shows tht ll idividuls i our coomy withi iformtio ucrtity, rgrdlss of thir prfrcs, my tti thir optiml portfolio positios by ivstig i t most 3 fuds. Ths fuds my b chos to b: (i) th isttously rislss sst; (ii) th sst hvig th highst corrltio with th stt vribl; d (iii) th mrt portfolio. Th structur of th ppr is s follows. Sctio II prsts our itrtmporl modl d th optiml portfolio udr icomplt iformtio. Sctio III provids som xplicit optiml solutios for th cs of CRRA utility fuctios. II. THE ECONOMIC MODEL AND THE OPTIMAL PORTFOLIO UNDER INCOMPLETE INFORMATION A. Cpitl Mrt Structur, Asst Vlu d th Ecoomic Modl Th modl is bsd o th stdrd ssumptios of prfct mrt d cotiuous trdig. Prics of ssts follow Ito procsss (cotiuous d ot diffrtibl). Udr th ssumptios of cotiuous trdig d Mrov structur, th first two momts of th distributios r sufficit sttistics. It is ssumd tht thr r risy ssts d o isttously ris-lss sst. Th rislss sst corrspods to th borrowig d ldig rt o short govrmt bods. Th modl i this ppr is vry similr i spirit to th modls i Mrto (97, 973), Brd (979), Adlr d Dums (983) d Blllh d Blllh (003). I th itrst of brvity, commo fcts of this modl will oly b stchd. Th ufmilir rdr my rfr to thos rly dvlopmts of th modl. Ivstors r pric trs i prfctly comptitiv cpitl mrts. Thy c trd cotiuously d trdig ts plc oly t quilibrium prics. I trms of Mrto (973) trmiology, th ivstmt opportuity st my b stochstic. Th stt vribls d ot b

5 INTERNATIONAL JOURNAL OF BUSINESS, 4(), rstrictd i umbr. As usul, to b cosistt with grl quilibrium, prics must b rcogizd to b dogously dtrmid usig supply d dmd. All rdom shocs my ffct both th supplis d dmds for ssts. All ths shocs to th coomy r cpturd s lmts of th stt vctor. This vctor dscribs th stt of th world. For xmpl sst prics d dividds c dpd o tim d th stt vribls. W cosidr coomy i which thr r K ivstors i th mrts. Ech ivstor c ivst his wlth i risy ssts, (stocs). Th prics of ths ssts stisfy th followig dymics dp i P i[bi i]dt P i idbi, P i(0) P i, i,,..., () Thr is rislss sst, ( bod) whos pric stisfis th followig dymics: dp 0 rp () 0 dt, Pi (0) p0 whr bi rprsts th isttous xpctd rt of rtur for diffrt stocs, i is th iformtio cost of sst i. Th trm i is th isttous voltility d is th itrst rt. Thy r ll ssumd to b boudd. Th trms B, B,, B r o-dimsiol mutully dpdt Browi motios. Thy rprst th xtrl sourcs of ucrtity i th mrts with corrltio cofficits i, j, for i, j,,. At ch momt, th th ivstor, vrious ssts. W dot by W proportio of his wlth i i th stoc. Th trm c,,, K c ivst his moy i th his wlth d by x i, i,,, th is th cosumptio rt, so th wlth of th th ivstor stisfis th followig ccumultio qutio: dw W [ xi (bi i r) r]dt i c dt W xi idbi i W (0) W0 (3) As i Brd (979), it is possibl tht fluctutios i som of th lmts of th stt vctor do ot ffct y idividul s xpctd utility of liftim cosumptio, giv th idividul s wlth. A distictio c b md btw stt vribls tht ffct t lst o idividul s xpctd utility, giv his wlth. I this cs, w dfi th stt vctor, s tht cotis thos stt vribls tht do ffct t lst o idividul s xpctd utility giv his wlth. Ech idividul s xpctd utility of liftim cosumptio my b writt s fuctio of his wlth, th vctor of stt vribls, d tim. Th stt vribls r rfrrd to s th stt vctor or s th vctor

6 5 Blllh d Wu of stt vribls. W itroduc o rlvt stt vribl S, whos dymic is ds Sbdt S db, S(0) s0, (4) whr B is o-dimsiol Browi motio which is dpdt o B i corrltio cofficits 0, i, for i,,. I this stdrd litrtur, ch ivstor is ssumd to mximiz th xpctd vlu t ch istt of tim dditiv d stt-idpdt vo Nwm-Morgstr utility fuctio for liftim cosumptio. A qusi-cocv utility d bqust fuctios of cosumptio d trmil wlth r usd. At ch istt, th idividul chooss optiml rt of cosumptio d optiml portfolio of risy ssts. Th ivstor chooss his portfolio d cosumptio rt to mximiz th followig xpctd utility fuctio with T E [ 0 U (c,s, t)dt h)(w (T),S(T)] (5) W dot by J (W,S, t) th mximum xpctd utility of liftim cosumptio tht is obtibl with wlth d opportuitis S t tim t. From th clssicl dymic progrmmig pricipl, w c obti th Hmilto- Jcobi-Bllm qutio. J J mx{u (c,s,t) W [ x (b r) r] t i i i (x,c ) i W i J J J c bs (W ) x x S i j i j i, j W (W ) ij J J S W xi is0, j} 0, t [0,T] S W S i J (W,S,T) h(w,s) (6) From first-ordr coditios, w hv th qutios for th optiml c c (W,S,t) d xi xi (W,S, t ), i,,. I fct, first ordr coditios for itrior mximum my b sttd s: Uc (c,s,t) J w (W,S,t) J J J (bi i r) W x j i ji, j Si0,i 0 W (W ) j WS (7)

7 INTERNATIONAL JOURNAL OF BUSINESS, 4(), Th first qutio is th usul itrtmporl vlop coditio to qut th mrgil utility of currt cosumptio to th mrgil utility of futur cosumptio (wlth). Th scod qutio shows th lirity i th portfolio dmds. W dot th th ivstor s dmd fuctio s di xi W, so w hv di xi W A vi, j(b j j r) H js0, jvi, j, j j i,,, (8) whr th vi, j r th lmts of th ivrs mtrix of th isttous vric-covric mtrix of rtur, ( i, j), i, j i, ji j, d J w U A c J c U cc W W J c H WS S J c W W (9) Equtio (8) c b usd to show tht ll ivstors optiml portfolios c b rprstd s combitio of som portfolios or mutul fuds. Coditio (8) provids th idividul s optiml risy sst portfolio i th prsc of iformtio ucrtity. Th coditios provid th idividul s optiml risy sst portfolio. Thy stt tht th idirct mrgil utility of othr uit of cosumptio must qul th idirct mrgil utility of wlth for optiml policy. Usig ths xprssios for A d H, th dmd fuctio i qutio (8) c b s s hvig two compots. Th first trm A j v i, j(b j j r) is th stdrd dmd fuctio for risy sst by sigl priod m-vric mximizr. Th trm A idicts th rciprocl of th ivstor s bsolut ris vrsio. Th scod trm H j js0, jvi, j idicts th ivstor dmd for sst s vhicl to hdg gist "ufvorbl" shifts i th ivstmt opportuity st. Som furthr rsults could b gid by rstrictig th clss of utility fuctios. W c lso dd som simplifyig ssumptios to rstrict th structur of th opportuity st. I th followig lysis, w driv th quilibrium mrt qutio for ll ivstors. W cosidr two css: th costt ivstmt opportuity st d th grl cs First cs: th cs of costt ivstmt opportuity st I this situtio, th distributio of prics is logorml for ll ssts. Th ssumptio of

8 54 Blllh d Wu costt ivstmt opportuity st is sufficit coditio for ivstors to bhv s if thy wr sigl priod mximizrs. It is lso sufficit coditio for th quilibrium rtur rltioship spcifid by th CAPMI of Mrto (987) to obti. W ssum tht ll risy ssts r idpdt of prfrc, i.. th stt vribl, t this cs, 0, j 0, j,,. So qutio (8) idictig th dmd for th i th sst by th th ivstor rducs to di xi W A vi, j(b j j r), j i,,, (0) This dmd corrspods lso to th sm dmd tht o-priod ris-vrs m-vric ivstor would hv. I th prsc of homogous xpcttios, bout th opportuity st, th rtio of th dmds for risy ssts will b idpdt of prfrcs, d th sm for ll ivstors. Furthr, similr to th Thorm i Mrto, w hv th followig thorm: Thorm : Cosidr coomy with risy ssts whos rturs r log-ormlly distributd. I th prsc of rislss sst d iformtio costs rgrdig th risy ssts, w hv th followig rsults: (i) thr xists uiqu pir of fficit portfolios ow s mutul fuds: th first o cotis oly th rislss sst d th scod compriss oly risy ssts. Ths portfolios r idpdt of prfrcs, wlth distributio, or tim horizo. All ivstors will b idiffrt btw choosig portfolios from mog th origil ( ) ssts or from ths two fuds i th prsc of icomplt iformtio; (ii) th rtur distributio o th risy fud is log-orml; d (iii) th wight of th risy fud s ssts ivstd i th th sst is giv by th followig xprssio: j v, j(b j j r) i j vi, j(b j j r) (,,,). This thorm rprsts cotiuous-tim vrsio of th sprtio thorm i Mrowitz-Tobi d Mrto (987). Th holdigs i th risy portfolio idict th optiml combitio of risy ssts. Th w lt th ggrgt dmd fuctios D i d i, d D A A, x i i, whr M is th (quilibrium) M vlu of ll ssts i.. th mrt vlu, so xim Di A vi, j(b j j r), j i,,, d w hv

9 INTERNATIONAL JOURNAL OF BUSINESS, 4(), M b i i r x ji ji, j, i,,, () A j W dfi bm i xi (bi r) s th xpctd rtur rt o th mrt portfolio, m i xii s th iformtio cost rt o th mrt d i,m i x ji ji, j s th covric of th rtur o th i th stoc with th rtur o th mrt portfolio, M j x j j, M s th vric of th mrt portfolio rspctivly. Th w hv M bi i r i, M, i,,, () A Usig th coditio tht th mrt portfolio is fficit i quilibrium, w c show tht th quilibrium rturs will stisfy Mrto s (987) simpl modl of cpitl mrt quilibrium with icomplt iformtio. Multiplyig () by xi d summig givs: d M bm M r M A (3) bi i r i(bm M r) (4) i,m whr i is th covric of th rtur o th i th sst with th rtur o th M mrt portfolio. This is th xtdd quilibrium mrt qutio d th scurity mrt li of th clssicl cpitl sst pricig modl of Mrto (987) i cotiuous tim. I fct, this is th cotiuous tim log to Mrto s (987) scurity mrt li. Th ssumptio of costt ivstmt opportuity st rprsts sufficit coditio for ivstors to bhv s if thy wr sigl-priod mximizrs. It is lso sufficit for th quilibrium rtur rltioship spcifid by Mrto s (987) simpl cpitl sst pricig modl to obti. Scod cs: Th grl cs Ufortutly, th ssumptio of costt ivstmt opportuity st is ot cosistt with th fcts. I prctic, thr is t lst o lmt of th opportuity st which is dirctly obsrvbl. This is th cs for th itrst rt. Th ffct of chgig itrst rt is oft cosidrd s sigl istrumtl vribl rprstig shifts i th opportuity st. This is th grl cs. W ssum tht thr xists sst (by covtio, th th o) whos xpctd rtur shows th mximum corrltios with th stt vribl

10 56 Blllh d Wu S. W us th sm dfiitios s thos i th prvious cs. W lso lt H H, from (8), w hv xim A vi, j(b j r) H js0, jvi, j j j (5) Substitutig (5) ito (8), w gt A HA xi W xim (H ) js0, jvi, j, A A j i,,, (6) This qutio xtds qutio (A.) i Brd (979) to ccout for th ffcts of iformtio ucrtity. It provids th bsis for th followig lloctio thorm which rsults immditly from idividuls portfolio dmds. But, our thorm is thr fud thorm i th prsc of iformtio ucrtity. Thorm. ( Thr Fuds thorm): All idividuls i our coomy withi iformtio ucrtity, rgrdlss of thir prfrcs, my tti thir optiml portfolio positios by ivstig i t most 3 fuds. Ths fuds my b chos to b: (i) th isttously rislss sst; (ii) th sst hvig th highst corrltio with th stt vribl; d (iii) th mrt portfolio. Usig qutio (5), w c solv for th quilibrium xpctd rturs o th idividul ssts. I this cotxt, w obti th followig qutio W lt i,s is0, i bi i r M A H x ji ji, j is0,i j A s th covric of th rtur o th ith stoc with th stt vribl d M,S i xii, S s th covric of th rtur o th mrt portfolio with th stt vribl. So th quilibrium xpctd rturs o th idividul ssts c b writt s: M H bi i r i,m i, s, i,,, 7) A A It is possibl to show tht grliztio of Mrto s (973) multi-bt sst pricig modl obtis i this coomy i th prsc of shdow costs of icomplt iformtio. I fct, th modl obtis i this coomy wh bts r msurd with rspct to ggrgt wlth d th rturs of ssts tht hdg gist chgs i th vrious stt vribls, w do th followig: ggrgt idividuls portfolio dmds d substitut i quilibrium xpctd xcss rturs for th mrt portfolio bm M r d for ssts prfctly corrltd with th stt vribl b r.

11 INTERNATIONAL JOURNAL OF BUSINESS, 4(), Assumig tht ths ssts xist d multiplyig (7) by xi d summig givs, d lso, w hv So, M H bm M r M M,S (8) A A M H b r,m,s (9) A A M M,S bm M r bi i r ( i,m i,s) (0),M,S b r This qutio rvls tht i quilibrium, ivstors r compstd i trms of xpctd rturs, for brig mrt ris (or systmtic ris). Thy r compstd lso for brig th ris of ufvorbl shifts i th ivstmt opportuity st. This qutio is turl grliztio of th rsults i th stdrd scurity mrt li d th rsults i Mrto s (987) CAPMI. W lso c writ qutio (0) s whr bm M r bi i r i,ms () b r M M,S i,ms ( i,m i,s),m,s This is th corrspodig quilibrium mrt qutio d th cotiuous tim scurity mrt li of th itrtmporl cpitl sst pricig modl with icomplt iformtio. Th trm i, MS corrspods to th mtrix of multipl-rgrssio bts for ll ssts o th mrt d o th ssts which r prfctly corrltd with th stt vribls. It is importt to ot tht this fudmtl vlutio qutio my b drivd for y sst by usig Ito s Lmm to fid its xpctd isttous rtur from th sst pric fuctio, d th, by qutig this drift rt to th quilibrium drift rts implid by th multi-bt modl of (). W c provid som xplicit solutios for th cs of CRRA utility fuctios. III. EXPLICIT OPTIMAL SOLUTION FOR CRRA UTILITY FUNCTION I this sctio, w cosidr ivstor who oly ivsts i o ctgory of risy ssts, (stocs) for which th pric P stisfis, dp P [b ]dt P db P (0) P ()

12 58 Blllh d Wu d i o rislss sst, (th bod), whos pric stisfis: dp0 rp0 dt, P0 (0) P0 (3) whr b rprsts th isttous xpctd rt of rtur i th stoc, is th iformtio cost rt, is th isttous voltility d r is th itrst rt. Thy r ll ssumd to b boudd. Th trm B, is o -dimsiol Browi motio. It rprsts th xtrl sourcs of ucrtity i th mrt. At ch momt, th ivstor c ivst his moy i ths two ids of ssts. W dot by W his wlth d by x th proportio of his wlth i th stoc. Th trm ( x) is th proportio i th bod, c of th ivstor stisfis th followig qutio: is th cosumptio rt, so th wlth dw W[x(b r) r]dt cdt xw db W(0) W0 (4) W itroduc o rlvt stt vribl S, whos dymic is ds Sbdt S db, S(0) s0, (5) whr B is o dimsiol Browi motio which is dpdt o B with corrltio cofficits 0,. Th ivstor wts to choos his strtgy x d cosumptio rt c to mximiz th followig xpctd utility fuctio rt T rt W (T)S (T) J(W 0) mx E[ 0 c S dt ], x,c (6) whr r d r costts, r 0, 0. W rfr to this utility fuctio s th Costt Rltiv Ris Avrsio (CRRA) cs. W wt to obti th xplicit optiml proportio x *, cosumptio rt c * d vlu fuctio for this cs. Th dmissibl strtgy (x *,c * ) is clld optiml strtgy which ttis th mximum of J(W0 ). Th id to gt th optiml solutio coms from th tchiqu to solv clbrtd LQ (lir qudrtic) problms i optiml cotrol thory. This mthod is dvlopd i Wu d Xu (996). From (4) d (5), w first hv

13 INTERNATIONAL JOURNAL OF BUSINESS, 4(), dw W ( )[x(b r) r]dt W ( )ddt W ( )x dt W ( )x db W (0) W0 (7) d ds S ( )bdt S ( ) dt S ( ) db, S (0) s 0 (8) So d[w S ] W S [( )(x(b r) r) ( )b ( )x ( ) 0,( ) x]dt W S ( )cdt W S ( ) db W S ( )xdb W (0)S (0) W0 S 0 (9) W lt Q b ogtiv dtrmiistic smooth fuctio stisfyig Q(T)= whos dymics will b giv lttr. Applyig Itô s formul to rt W S Q from 0 to T d tig xpcttio o both sids, w hv rt E W (T)S (T)Q(T) W0 S 0 Q(0) rt T E0 { W S Q[ r ( )(x(b r) r) ( )b ( )x ( ) 0,( ) x] rt rt W S Q( )c W S Q}dt. So w c writ

14 60 Blllh d Wu III II I Q(0) S W ]}dt Q Q r) b ) ( )( ( Q ) ( )bq ( rq [Q S W ] r) b ) ( )( ( r))x )(b ( ) ( ( x ) ( Q[ S W ] tq W Qc )W ( [c S { E mx Q(0) S W ) J(W 0 0 0, rt 0, 0, rt T 0 rt (x,c) Hr T 0 rt (x,c) ]dt Q W Qc )W ( [c S mx E I QL(x)dt S W mx E II T 0 rt (x,c) whr 0, 0, r) b ) ( )( ( r))x )(b ( ) ( ( x ) ( L(x) d ]dt Q Q r) b ) ( )( ( Q ) ( )bq ( rq [Q S W mx E III 0, T 0 rt (x,c) Now w lt ) (t Q b th solutio of th followig ordiry diffrtil qutio of Broulli typ:

15 INTERNATIONAL JOURNAL OF BUSINESS, 4(), Q MQ Q Q(T), t [0, T] (30) d ( )( 0,( ) b r) M r ( )b ( ) (3) Lt Q ~ M(Tt) Q,th So ~ Q ~ Q ~ Q M(T t) (T), t [0,T] T M(Ts) Q ~ [ t ds] d w obti: M(Tt) T M(Ts) Q [ t ds] MT T M(Ts) Q(0) [ 0 ds] Th w c loo bc I d II. If w t t [0,T], (3) c ( ) Q W, t [0,T] (33) whr Q(T) is giv by (3) d is positiv. O c chc tht I ttis its mximum t poit c * d I=0. This is lso th fd bc form of wlth. Th w t: * 0,( ) b r x (34) whr th domitor is positiv. O c chc tht L ' (x * ) 0 d L " (x * ) 0. Thus th fuctio L(x) ttis its mximum t poit x * d L(x * )=0, II=0. So for CRRA cs, w c hv th xplicit optiml proportio x * from (34), th optiml cosumptio rt c * from (33) d th optiml vlu fuctio:

16 6 Blllh d Wu whr Q(0) is giv by (3). J(W 0) W0 S 0 Q(0) (35) IV. SUMMARY Iformtio costs, which r diffrt from trsctio costs, r justifid by th hug mouts of moy spt by idividul d istitutiol ivstors i lysis, vluig, d trtig iformtio. Iformtio is fudmtl for sst pricig sic ivstor s iformtio st is icomplt bcus it dos ot coti iformtio rgrdig th xpctd rtur d its vribility. Ths iformtio costs r ssimiltd by Mrto d hr s dditiol discout rts for futur csh flows. Wh iformtio costs r igord, our modl rducs to th stdrd modl. A itrtmporl modl of th cpitl mrt hs b dvlopd which is cosistt with both th xpctd utility mximiztio d th limitd libility of ssts. Th lysis i Mrto (973) shows tht th quilibrium rltioships mog xpctd rturs spcifid by th clssicl cpitl sst pricig modl will obti oly udr som spcil dditiol ssumptios. Mrto s modl is robust i th ss tht it c b xtdd i obvious wy to iclud som othr ffcts. I this cotxt, w driv itrtmporl cpitl sst pricig modl i coomic cotxt prmittig both stochstic cosumptio-goods prics d stochstic portfolio opportuitis. Th modl is grliztio of Mrto s (973) cotiuous-tim modl, drivig quivlt pricig qutios tht r simplr i form. Brd (979) cocluds his ppr by strssig th fct tht rs tht d dditiol thorticl dvlopmt iclud th rol of firms d thir optiml ivstmt d cpitl structur dcisios, th impct of iformtio costs d trsctio costs. W provid two thorms. Thorm shows tht i th prsc of rislss sst d iformtio costs rgrdig th risy ssts i th coomy, (i) thr xists uiqu pir of fficit portfolio ow s mutul fuds: th first o cotis oly th rislss sst d th scod compriss oly risy ssts. All ivstors will b idiffrt btw choosig portfolios from mog th origil ( ) ssts or from ths two fuds i th prsc of icomplt iformtio; d (ii) th rtur distributio o th risy fud is log-orml. Thorm is Thr Fuds thorm. It shows tht ll idividuls i our coomy withi iformtio ucrtity, rgrdlss of thir prfrcs, my tti thir optiml portfolio positios by ivstig i t most 3 fuds. Ths fuds my b chos to b: (i) th isttously rislss sst; (ii) th sst hvig th highst corrltio with th stt vribl; d (iii) th mrt portfolio. Our modl is grliztio of Mrto (973) d Brd (976) by ccoutig for th ffcts of iformtio costs. ENDNOTES. Mrto s modl my b sttd s follows: R p r p[r m r] p pm, whr

17 INTERNATIONAL JOURNAL OF BUSINESS, 4(), _ R p : th quilibrium xpctd rtur o scurity P; _ R m : th quilibrium xpctd ~ ~ cov(r p/ R m) rtur o th mrt portfolio; r: th rislss rt of itrst; p : th ~ vr(r m ) bt of scurity P, tht is th covric of th rtur o tht scurity with th rtur o th mrt portfolio, dividd by th vric of mrt rtur; p : th quilibrium ggrgt " shdow cost" for th scurity P. It is of th sm dimsio s th xpctd rt of rtur o this scurity P; m : th wightd vrg shdow cost of icomplt iformtio ovr ll scuritis. REFERENCES Adlr, M., d B. Dums, 983, Itrtiol Portfolio Choic d Corportio Fic: A Sythsis, Jourl of Fic. Blllh, M., d M. Blllh, 003, Itrtiol Portfolio Choic d th Effct of Iformtio Costs, Itrtiol Jourl of Fic. Blc, F., 974, Itrtiol Cpitl Mrt Equilibrium with Ivstmt Brrirs, Jourl of Ficil Ecoomics, pp Brd, D.,979, A Itrtmporl Asst Modl with Stochstic Cosumptio d Ivstmt Opportuitis, Jourl of Ficil Ecoomics 7, pp Mrto, R., 97, Optimum Cosumptio d Portfolio Ruls i A Cotiuous Tim Modl, Jourl of Ecoomic Thory, pp Mrto, R., 973, A Itrtmporl Cpitl Asst Pricig Modl, Ecoomtric, p Mrto, R., 987, A Simpl Modl of Cpitl Mrt Equilibrium with Icomplt Iformtio, Jourl of Fic, pp Soli, B., 974, A Equilibrium Modl of Itrtiol Cpitl Mrt, Jourl of Ecoomic Thory, pp Wu, Z., d W. Xu, 996, A Dirct Mthod i Optiml Portfolio d Cosumptio Choic, Applid Mthmtics, B, pp