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+5153,

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FKDSWHU 51 P\RSLF SRUWIROLR FKRLFH Dssur{lpdwlrq ri wkh sruwirolr uhwxuq Wr surfhhg ixuwkhu/ zh qhhg wr uhodwh wkh orj sruwirolr uhwxuq wr wkh orj uhwxuqv rq xqghuo|lqj dvvhwv1 Frqvlghu uvw wkh vlpsoh fdvh zkhuh wkhuh duh wzr dvvhwv/ rqh ulvn| dqg rqh ulvnohvv1 Wkhq iurp +514, wkh vlpsoh uhwxuq rq wkh sruwirolr lv d olqhdu frpelqdwlrq ri wkh vlpsoh uhwxuqv rq wkh ulvn| dqg ulvnohvv dvvhwv1 Wkh orj uhwxuq rq wkh sruwirolr lv wkh orj ri wklv olqhdu frpelqdwlrq/ zklfk lv qrw wkh vdph dv d olqhdu frpelqdwlrq ri orjv1 Ryhu vkruw wlph lqwhuydov/ krzhyhu/ zh fdq xvh d Wd|oru dssur{lpdwlrq ri wkh qrqolqhdu ixqfwlrq uhodwlqj orj lqglylgxdo0dvvhw uhwxuqv wr orj sruwirolr uhwxuqv1 Ixoo ghwdlov duh jlyhq lq wkh Dsshqgl{> wkh uhvxowlqj h{suhvvlrq lv 4 +5154, uRc|n  usc|n @ | +u|n  usc|n , . | +4  | ,2| = 5 Wkh glhuhqfh ehwzhhq wkh orj sruwirolr uhwxuq dqg d olqhdu frpelqdwlrq ri orj lqglylgxdo0dvvhw uhwxuqv lv jlyhq e| | +4  | ,2| @51 Wkh glhuhqfh glvdsshduv li wkh sruwirolr zhljkw lq wkh ulvn| dvvhw lv }hur +iru wkhq wkh orj sruwirolr uhwxuq lv mxvw wkh orj ulvnohvv uhwxuq,/ ru li wkh zhljkw lq wkh ulvn| dvvhw lv rqh +iru wkhq wkh orj sruwirolr uhwxuq lv mxvw wkh orj ulvn| uhwxuq,1 Zkhq 3 ? | ? 4/ wkh sruwirolr lv d zhljkwhg dyhudjh ri wkh lqglylgxdo dvvhwv dqg wkh whup | +4  | ,2| @5 lv srvlwlyh1 Wr xqghuvwdqg wklv/ uhfdoo wkdw wkh orj ri dq dyhudjh lv juhdwhu wkdq dq dyhudjh ri orjv dv looxvwudwhg lq Iljxuh 5141 Wkh dssur{lpdwlrq lq +5154, fdq eh mxvwlhg uljrurxvo| e| frqvlghulqj vkruwhu dqg vkruwhu wlph lqwhuydov1 Dv wkh wlph lqwhuydo vkulqnv/ wkh kljkhu0 rughu whupv wkdw duh qhjohfwhg lq +5154, ehfrph qhjoljleoh uhodwlyh wr wkrvh wkdw duh lqfoxghg1 Lq wkh olplw ri frqwlqxrxv wlph zlwk frqwlqxrxv sdwkv +glxvlrqv, iru dvvhw sulfhv/ +5154, lv h{dfw dqg fdq eh ghulyhg xvlqj Lwr*v Ohppd1 Zh glvfxvv wkh frqwlqxrxv0wlph dssurdfk lq pruh ghwdlo lq Fkdswhu 81 Rqh lpsruwdqw surshuw| ri wkh dssur{lpdwh sruwirolr uhwxuq lv wkdw lw uxohv rxw wkh srvvlelolw| ri edqnuxswf|/ hyhq zkhq wkh lqyhvwru krogv d vkruw srvlwlrq +| ? 3, ru d ohyhudjhg srvlwlrq lq wkh ulvn| dvvhw qdqfhg e| eru0 urzlqj +| A 4,1 Wkh orj sruwirolr uhwxuq lv dozd|v qlwh/ qr pdwwhu zkdw wkh uhwxuqv rq wkh xqghuo|lqj dvvhwv/ dqg wkxv lw lv qhyhu srvvleoh wr h{0 kdxvw zhdowk frpsohwho|1 Frqwlqxrxv0wlph glxvlrq prghov dovr kdyh wklv surshuw|> lq vxfk prghov sruwirolrv duh uhedodqfhg ryhu vxfk vkruw lqwhu0 ydov wkdw orvvhv fdq dozd|v eh vwhpphg e| uhedodqflqj ehiruh wkh| ohdg wr edqnuxswf|1 Lq prvw dssolfdwlrqv lw lv uhdvrqdeoh wr h{foxgh wkh srvvlelo0 lw| ri edqnuxswf|/ exw wklv dssurdfk pd| qrw eh vxlwdeoh zkhq lw lpsolhv rswlpdo sruwirolrv zlwk h{wuhpho| kljk ohyhudjh1 4<

FKDSWHU 51 P\RSLF SRUWIROLR FKRLFH Wkh dssur{lpdwlrq +5154, jhqhudol}hv vwudljkwiruzdugo| wr wkh fdvh zkhuh wkhuh duh pdq| ulvn| dvvhwv/ mrlqwo| orjqrupdoo| glvwulexwhg zlwk frqglwlrqdo yduldqfh0fryduldqfh pdwul{ ri orj uhwxuqv | 1 Zh zulwh  2| iru wkh yhfwru frqwdlqlqj wkh gldjrqdo hohphqwv ri | / wkh yduldqfhv ri dvvhw uhwxuqv1 +Uh0 fdoo rxu qrwdwlrqdo frqyhqwlrqv wkdw erogidfh orzhufdvh ohwwhuv ghqrwh yhf0 wruv/ erogidfh orzhufdvh vljpd ghqrwhv d yhfwru ri fryduldqfhv/ dqg zlwk d voljkw dexvh ri qrwdwlrq  2| ghqrwhv d yhfwru ri yduldqfhv1, Wkh dssur{lpd0 wlrq wr wkh sruwirolr uhwxuq ehfrphv 4 4 uRc|n  usc|n @ | +u|n  usc|n , . |  2|  | | | = 5 5

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+5157,

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FKDSWHU 51 P\RSLF SRUWIROLR FKRLFH wkh rswlpdo sruwirolr zhljkw lv wkh ulvn suhplxp glylghg e| ulvn dyhuvlrq wlphv yduldqfh1 Lq d prgho zlwk pdq| ulvn| dvvhwv/ wkh vroxwlrq iru wkh yhfwru ri rswlpdo sruwirolr zhljkwv lv | @

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+5163,

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+5164,

vlqfh erwk | dqg |n duh ghwhuplqlvwlf dqg khqfh duh nqrzq dw wlph w1 Wkh phdq 50shulrg orj uhwxuq/ dgmxvwhg e| dgglqj rqh0kdoi wkh yduldqfh ri wkh 50shulrg uhwxuq/ lv 4 H| +uRc|c|n2 , . Ydu| +uRc|c|n2 , @ 5us . +| . |n ,+Hu  us . 2 @5,= +5165, 5 Wkh remhfwlyh ri d 50shulrg lqyhvwru zlwk srzhu xwlolw| fdq eh zulwwhq dv  4 pd{ H| +uRc|c|n2 , . Ydu| +uRc|c|n2 ,  Ydu| +uRc|c|n2 ,= 5 5

+5166,

Wkxv wkh lqyhvwru zlwk  A 3 dozd|v suhihu d orzhu yduldqfh ri orj uhwxuqv iru d jlyhq yduldqfh0dgmxvwhg phdq1 Exw iurp +5165,/ wkh yduldqfh0dgmxvwhg phdq ghshqgv rqo| rq wkh vxp +| .|n ,1 Wkh lqyhvwru fdq { wklv vxp dqg dgmxvw wkh lqglylgxdo vkduhv/ | dqg |n / wr plqlpl}h wkh yduldqfh1 Vlqfh yduldqfh ghshqgv rq wkh vxp ri vtxduhv +2| . 2|n ,/ wklv lv dffrpsolvkhg e| vhwwlqj | @ |n / d frqvwdqw sruwirolr uxoh1 57

FKDSWHU 51 P\RSLF SRUWIROLR FKRLFH Rqfh zh nqrz wkdw wkh sruwirolr uxoh lv frqvwdqw/ zh dovr nqrz wkdw lw pxvw eh wkh vdph dv wkh rswlpdo uxoh iru d vkruw0whup lqyhvwru1 Wkh uhdvrq lv wkdw lq wkh odvw shulrg ehiruh wkh krul}rq/ wkh orqj0whup lqyhvwru kdv ehfrph d vkruw0whup lqyhvwru dqg zloo fkrrvh wkh rswlpdo vkruw0whup sruwirolr1 Wklv dujxphqw iru p|rslf sruwirolr fkrlfh vwudljkwiruzdugo| h{whqgv wr dq| orqj krul}rq N1 Lw fdq eh uhodwhg wr d sx}}oh srvhg e| Pdun Nulw}pdq +5333, lq klv errn Sx}}ohv ri Ilqdqfh1 Lq d fkdswhu hqwlwohg _Kdoi Vwrfnv Doo wkh Wlph ru Doo Vwrfnv Kdoi wkh WlphB%/ Nulw}pdq srlqwv rxw wkdw wkhvh wzr vwudwhjlhv kdyh wkh vdph h{shfwhg vlpsoh uhwxuq/ exw wkh odwwhu vwudwhj| lv ulvnlhu> wkxv d ulvn0dyhuvh lqyhvwru vkrxog dozd|v suhihu wkh iruphu1 Wklv lv suhflvho| wkh hhfw wkdw xqghuolhv wkh dujxphqw iru d frqvwdqw sruwirolr uxoh1 Wkh vhfrqg Vdpxhovrq0Phuwrq frqglwlrq iru p|rslf sruwirolr fkrlfh lv wkdw wkh lqyhvwru kdv orj xwlolw|1 Lq wklv fdvh sruwirolr fkrlfh zloo eh p|rslf hyhq li dvvhw uhwxuqv duh qrw LLG1 Wkh dujxphqw khuh lv sduwlfxoduo| vlpsoh1 Uhfdoo wkdw wkh orj xwlolw| lqyhvwru fkrrvhv d sruwirolr wkdw pd{lpl}hv wkh h{shfwhg orj uhwxuq1 Htxdwlrq +515;, vkrzv wkdw wkh N0shulrg orj uhwxuq lv mxvw wkh vxp ri 40shulrg orj uhwxuqv1 Vlqfh wkh sruwirolr fdq eh fkrvhq iuhho| hdfk shulrg/ wkh vxp lv pd{lpl}hg e| pd{lpl}lqj hdfk ri lwv hohphqwv vhsdudwho|/ wkdw lv/ e| fkrrvlqj hdfk shulrg wkh sruwirolr wkdw lv rswlpdo iru d 40shulrg orj xwlolw| lqyhvwru1

222

6@**@Uit Lu *L?}|ih4 TLh|uL*L ULUi

Dowkrxjk wkh frqglwlrqv iru p|rslf sruwirolr fkrlfh duh vlpsoh/ dqg wkhlu orjlf fdq eh xqghuvwrrg zlwkrxw uhvruw wr dgydqfhg pdwkhpdwlfv/ wkhuh kdv ehhq pxfk frqixvlrq derxw wkhvh lvvxhv ryhu wkh |hduv1 Rqh vrxufh ri frqixvlrq lv wkh frpprq whqghqf| wr phdvxuh ulvn lq xqlwv ri vwdqgdug ghyldwlrq udwkhu wkdq yduldqfh/ dqg wr zrun zlwk phdq0vwdqgdug ghyldwlrq gldjudpv +olnh Iljxuh 414, udwkhu wkdq phdq0yduldqfh gldjudpv1 Zlwk LLG uhwxuqv wkh yduldqfh ri d fxpxodwlyh ulvn| uhwxuq lv sursruwlrqdo wr wkh lqyhvwphqw krul}rq N/ exw wkh vwdqgdug ghyldwlrq lv sursruwlrqdo wr wkh vtxduh urrw ri N1 +Li uhwxuqv duh dqqxdol}hg/ wkh yduldqfh lv frqvwdqw exw wkh vwdqgdug ghyldwlrq vkulqnv lq sursruwlrq wr wkh vtxduh urrw ri N1, Wkxv wkh Vkdush udwlr ri dq| ulvn| lqyhvwphqw/ lwv phdq h{fhvv uhwxuq glylghg e| lwv vwdqgdug ghyldwlrq/ jurzv zlwk wkh vtxduh urrw ri N1 Lw lv whpswlqj wr fdofxodwh orqj0krul}rq Vkdush udwlrv/ revhuyh wkdw wkh| duh odujh/ dqg frqfoxgh wkdw ohqjwkhqlqj wkh lqyhvwphqw krul}rq vrphkrz uhgxfhv ulvn lq d pdqqhu dqdorjrxv wr wkh hhfw ri glyhuvli|lqj d sruwirolr 58

FKDSWHU 51 P\RSLF SRUWIROLR FKRLFH dfurvv xqfruuhodwhg ulvn| dvvhwv1 Exw wklv _wlph glyhuvlfdwlrq% dujxphqw lv d idoodf|1 Vkdush udwlrv fdqqrw eh frpsduhg dfurvv glhuhqw lqyhvwphqw krul}rqv> wkh| pxvw dozd|v eh phdvxuhg ryhu d frpprq wlph lqwhuydo1 D uhodwhg idoodf| lv wr dujxh wkdw wkhuh lv d vlqjoh ehvw orqj0whup sruwirolr iru doo lqyhvwruv/ uhjdugohvv ri wkhlu suhihuhqfhv1 Wkh sursrvhg sruwirolr lv wkh _jurzwk0rswlpdo% sruwirolr wkdw pd{lpl}hv wkh h{shfwhg orj uhwxuq1 Dv wkh lqyhvwphqw krul}rq lqfuhdvhv/ wklv sruwirolr rxwshuirupv dq| rwkhu sruwirolr zlwk kljkhu dqg kljkhu suredelolw|1 Lq wkh olplw/ wkh suredelolw| wkdw lw rxwshuirupv jrhv wr rqh1 Wr xqghuvwdqg wklv surshuw| ri wkh jurzwk0 rswlpdo sruwirolr/ qrwh wkdw wkh glhuhqfh ehwzhhq wkh fxpxodwlyh jurzwk0 rswlpdo orj uhwxuq dqg wkh orj uhwxuq rq dq| rwkhu sruwirolr lv d qrupdoo| glvwulexwhg udqgrp yduldeoh1 Zulwh wkh phdq ri wklv glhuhqfh dv +N, dqg wkh vwdqgdug ghyldwlrq dv +N,1 Wkh phdq +N, lv srvlwlyh ehfdxvh wkh jurzwk0rswlpdo orj uhwxuq kdv wkh kljkhvw phdq ri dq| dydlodeoh sruwirolr vwudwhj|1 Wkh suredelolw| wkdw wkh glhuhqfh lv srvlwlyh lv ++N,@+N,,/ zkhuh +, ghqrwhv wkh fxpxodwlyh glvwulexwlrq ixqfwlrq ri wkh vwdqgdug qrupdo glvwulexwlrq1 Dv s wkh lqyhvwphqw krul}rq N lqfuhdvhv/ wkh udwlr +N,@+N, jurzv zlwk N/ vr ++N,@+N,, jrhv wr rqh1 Wkh idoodf| lv wkh fodlp wkdw wklv surshuw| ri wkh jurzwk0rswlpdo sruwirolr pdnhv lw wkh ehvw sruwirolr iru doo orqj0whup lqyhvwruv1 Lw lv/ ri frxuvh/ wkh ehvw sruwirolr iru dq lqyhvwru zlwk orj xwlolw|> exw lqyhvwruv zlwk kljkhu ulvn dyhuvlrq vkrxog krog pruh frqvhuydwlyh sruwirolrv1 Hyhq wkrxjk wkh jurzwk0 rswlpdo sruwirolr zloo doprvw dozd|v rxwshuirup vxfk frqvhuydwlyh sruwirolrv ryhu orqj krul}rqv/ wkh orvv zkhq lw grhv xqghushuirup lv odujhu dw orqj krul}rqv/ dqg wklv srvvlelolw| lv khdylo| zhljkwhg e| frqvhuydwlyh lqyhvwruv1 Wklv srlqw zdv irufhixoo| pdgh e| Vdpxhovrq +4<:<, lq dq duwlfoh hqwlwohg _Zk| Zh Vkrxog Qrw Pdnh Phdq Orj ri Zhdowk Elj Wkrxjk \hduv wr Dfw Duh Orqj%1 Wkh odvw sdudjudsk ri wkh duwlfoh uhdgv dv iroorzv= _Qr qhhg wr vd| pruh1 L*yh pdgh p| srlqw1 Dqg/ vdyh iru wkh odvw zrug/ kdyh grqh vr lq survh ri exw rqh v|oodeoh1% +s1639,1 D pruh uhfhqw srsxodu glvfxvvlrq ri wkh idoodf|/ zklfk lv hdvlhu wr uhdg ehfdxvh lw doorzv lwvhoi d zlghu vhohfwlrq ri zrugv/ lv lq Nulw}pdq +5333/ Fkdswhu 6,1

22

Lih |*|) Lu UL?t4T|L?

Wkh dvvxpswlrq wkdw lqyhvwruv fduh rqo| derxw zhdowk dw d vlqjoh krul}rq lv dqdo|wlfdoo| frqyhqlhqw exw hpslulfdoo| wurxeohvrph1 Prvw lqyhvwruv/ zkhwkhu wkh| duh lqglylgxdov vdylqj iru uhwluhphqw ru lqvwlwxwlrqv vxfk dv xqlyhuvlwlhv wkdw olyh r hqgrzphqw lqfrph/ duh frqfhuqhg qrw zlwk wkh ohyho ri zhdowk iru lwv rzq vdnh/ exw zlwk wkh vwdqgdug ri olylqj wkdw wkhlu 59

FKDSWHU 51 P\RSLF SRUWIROLR FKRLFH zhdowk fdq vxssruw1 Lq rwkhu zrugv wkh| frqvxph rxw ri zhdowk dqg ghulyh xwlolw| iurp frqvxpswlrq udwkhu wkdq zhdowk1 Lq wklv vhfwlrq zh dvvxph wkdw xwlolw| lv ghqhg ryhu d vwuhdp ri frq0 vxpswlrq1 Rqfh zh pdnh wklv dvvxpswlrq/ wkh krul}rq sod|v d pxfk vpdoohu uroh lq wkh dqdo|vlv> lw vwloo ghqhv d whuplqdo gdwh/ exw frqglwlrqv dw hdfk lqwhuphgldwh gdwh duh dovr lpsruwdqw1 Li wkh whuplqdo gdwh lv glvwdqw/ lqwhu0 phgldwh frqglwlrqv grplqdwh wkh vroxwlrq/ zklfk zloo qrw ghshqg vhqvlwlyho| rq wkh h{dfw fkrlfh ri whuplqdo gdwh1 Lq idfw/ zh fdq ohw wkh whuplqdo gdwh jr wr lqqlw| dqg zrun zlwk dq dwwudfwlyho| vlpsoh lqqlwh0krul}rq prgho1 Zh fdq ydu| wkh hhfwlyh lqyhvwphqw krul}rq e| ydu|lqj wkh wlph glvfrxqw idfwru wkdw ghwhuplqhv wkh uhodwlyh zhljkwv lqyhvwruv sodfh rq wkh qhdu0whup ixwxuh yhuvxv wkh glvwdqw ixwxuh1 Wklv lv rxu prgh ri dqdo|vlv iru prvw ri wkh uhvw ri wkh errn1 Zh uvw dvvxph wkdw lqyhvwruv kdyh wlph0vhsdudeoh srzhu xwlolw|/ ghqhg ryhu frqvxpswlrq= pd{ H|

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+5167,

Khuh  lv wkh wlph glvfrxqw idfwru1 Zkhq  lv odujh/ lqyhvwruv sodfh uhodwlyho| kljk zhljkw rq wkh glvwdqw ixwxuh1 Dv  vkulqnv/ wkh| sodfh pruh dqg pruh zhljkw rq wkh qhdu ixwxuh> lq wkh olplw dv  dssurdfkhv }hur/ wkh| ehkdyh olnh vlqjoh0shulrg lqyhvwruv1 Lqyhvwruv idfh wkh lqwhuwhpsrudo exgjhw frqvwudlqw wkdw zhdowk qh{w shulrg htxdov wkh sruwirolr uhwxuq wlphv uhlqyhvwhg zhdowk/ wkdw lv/ zhdowk wrgd| ohvv zkdw lv vxewudfwhg iru frqvxpswlrq= Z|n @ +4 . URc|n ,+Z|  F| ,=

+5168,

Wklv remhfwlyh ixqfwlrq dqg exgjhw frqvwudlqw lpso| wkh iroorzlqj uvw0 rughu frqglwlrq ru Hxohu htxdwlrq iru rswlpdo frqvxpswlrq fkrlfh= X  +F| , @ H| ^X  +F|n ,+4 . Uc|n ,`>

+5169,

zkhuh +4 . Uc|n , ghqrwhv dq| dydlodeoh uhwxuq/ iru h{dpsoh wkh ulvnohvv uhwxuq +4 . Usc|n ,/ wkh ulvn| uhwxuq +4 . U|n ,/ ru wkh sruwirolr uhwxuq +4 . URc|n ,1 Htxdwlrq +5169, vd|v wkdw dw wkh rswlpxp/ wkh pdujlqdo frvw ri vdylqj dq h{wud groodu iru rqh shulrg pxvw htxdo wkh pdujlqdo ehqhw1 Wkh pdujlqdo frvw lv wkh pdujlqdo xwlolw| ri d groodu ri frqvxpswlrq/ X  +F| ,1 Wkh pdujlqdo ehqhw lv wkh h{shfwdwlrq ri wkh sd|r li wkh groodu lv lqyhvwhg lq dq dydlodeoh dvvhw iru rqh shulrg/ +4 . Uc|n ,/ wlphv wkh pdujlqdo xwlolw| ri dq h{wud groodu ri frqvxpswlrq qh{w shulrg/ X  +F|n ,/ glvfrxqwhg edfn wr wkh suhvhqw dw udwh 1 5:

FKDSWHU 51 P\RSLF SRUWIROLR FKRLFH Zh fdq glylgh +5169, wkurxjk e| X  +F| , dqg xvh wkh srzhu xwlolw| frq0 glwlrq wkdw X  +F| , @ F|3 wr uhzulwh dv %  &  F|n 3 4 @ H|  +4 . Uc|n , = +516:, F| Lq wkh fdvh zkhuh wkh uhwxuq lv ulvnohvv/ Uc|n @ Usc|n / lw lv nqrzq dw wlph w dqg wkxv fdq eh eurxjkw rxwvlgh wkh h{shfwdwlrqv rshudwru> zh kdyh %   & F|n 3 4 = @ H|  +516;, +4 . Usc| , F| Wkh whup +F|n @F| ,3 wkdw dsshduv lq doo wkhvh h{suhvvlrqv lv nqrzq dv wkh _vwrfkdvwlf glvfrxqw idfwru% ru VGI ehfdxvh lw fdq eh xvhg wr glvfrxqw h{shfwhg sd|rv rq dq| dvvhwv wr qg wkhlu sulfhv1 Wkh frqvxpswlrq ri dq| lqyhvwru zkr lv deoh wr iuhho| wudgh lq wkh qdqfldo pdunhwv fdq eh xvhg dv dq VGI1 Wkh hog ri dvvhw sulflqj dvnv krz wkh surshuwlhv ri wkh VGI h{sodlq wkh surshuwlhv ri dvvhw uhwxuqv/ dqg krz wkh VGI lv ghwhuplqhg e| wkh jhq0 hudo htxloleulxp ri wkh hfrqrp|> lq wklv errn zh wdnh d sduwldo0htxloleulxp shuvshfwlyh dqg vhhn wr h{sodlq wkh surshuwlhv ri rswlpdo frqvxpswlrq dqg sruwirolr fkrlfh jlyhq dvvhw uhwxuqv1 Fdpsehoo +5333, dqg Frfkudqh +5333, vxuyh| wkh VGI phwkrgrorj|1 D orjqrupdo frqvxpswlrq0edvhg prgho Wkh qdwxudo zd| wr surfhhg/ jlyhq rxu hduolhu glvfxvvlrq/ lv wr dvvxph wkdw dvvhw uhwxuqv dqg frqvxpswlrq duh mrlqwo| orjqrupdo/ dqg wr zrun zlwk orjolqhdu yhuvlrqv ri wkhvh htxdwlrqv1 Kdqvhq dqg Vlqjohwrq +4<;6, slrqhhuhg wklv dssurdfk1 Wkh orj irup ri wkh ulvnohvv0udwh Hxohu htxdwlrq +516;, fdq eh zulwwhq dv 4 orj   H| ^f|n ` @ . usc|n .  2S| = +516<,   5 Wkh wkuhh whupv rq wkh uljkw0kdqg vlgh ri +516<, fruuhvsrqg wr wkuhh irufhv dfwlqj rq frqvxpswlrq1 Iluvw/ d sdwlhqw lqyhvwru zlwk d kljk wlph glv0 frxqw idfwru  lv lqkhuhqwo| zloolqj wr srvwsrqh frqvxpswlrq1 Vhfrqg/ d kljk lqwhuhvw udwh jlyhv dq lqyhvwru dq lqfhqwlyh wr srvwsrqh frqvxpswlrq1 Srvwsrqlqj frqvxpswlrq phdqv udlvlqj frqvxpswlrq lq wkh ixwxuh uhodwlyh wr frqvxpswlrq wrgd|/ wlowlqj wkh frqvxpswlrq sdwk xszdugv> krzhyhu gl0 plqlvklqj pdujlqdo xwlolw| ri frqvxpswlrq olplwv wkh lqyhvwru*v zloolqjqhvv wr wrohudwh dq| ghyldwlrq iurp d dw frqvxpswlrq sdwk1 Wkh lqyhvwru*v zloolqj0 qhvv wr wlow frqvxpswlrq lq uhvsrqvh wr lqfhqwlyhv lv nqrzq dv wkh hodvwlflw| 5;

FKDSWHU 51 P\RSLF SRUWIROLR FKRLFH ri lqwhuwhpsrudo vxevwlwxwlrq lq frqvxpswlrq +HLV,1 Lq d prgho zlwk srzhu xwlolw|/ wkh HLV htxdov wkh uhflsurfdo ri ulvn dyhuvlrq/ 4@1 Wkxv zlwk d kljk  wkh lqyhvwru lv h{wuhpho| xqzloolqj wr wlow frqvxpswlrq dqg wkh sodqqhg frqvxpswlrq jurzwk udwh zloo fkdqjh rqo| voljkwo| zlwk wkh wlph glvfrxqw idfwru dqg wkh ulvnohvv lqwhuhvw udwh1 Wkh wklug whup rq wkh uljkw0kdqg vlgh ri +516<, uhsuhvhqwv wkh hhfw ri xqfhuwdlqw|1 D ulvn0dyhuvh lqyhvwru zloo uhvsrqg wr xqfhuwdlqw| e| lqfuhdvlqj suhfdxwlrqdu| vdylqj/ djdlq wlowlqj wkh frqvxpswlrq sdwk xszdugv1 Lw lv dovr srvvleoh wr zulwh h{shfwhg frqvxpswlrq jurzwk lq whupv ri wkh h{shfwhg sruwirolr uhwxuq1 Wkh uvw whup rq wkh uljkw0kdqg vlgh ri +516<, lv xqfkdqjhg e| grlqj wklv/ wkh vhfrqg whup ehfrphv +4@,H| uRc|n / dqg wkh wklug whup lqyroyhv wkh yduldqfhv dqg fryduldqfhv ri wkh sruwirolr uhwxuq dqg frqvxpswlrq jurzwk1 Wklv dowhuqdwlyh uhsuhvhqwdwlrq lv pruh frqyhqlhqw iru vrph sxusrvhv/ dqg zloo eh xvhg lq rxu dqdo|vlv ri Hsvwhlq0]lq xwlolw| lq wkh qh{w vhfwlrq1 Wkh Hxohu htxdwlrq iru srzhu xwlolw| fdq dovr eh xvhg wr ghvfuleh wkh ulvn suhplxp rq d vlqjoh ulvn| dvvhw ryhu wkh ulvnohvv lqwhuhvw udwh1 Wkh orj irup ri wkh jhqhudo Hxohu htxdwlrq +516:,/ ohvv  wlphv +516<,/ lv H| u|n  usc|n .

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+5173,

Wklv vd|v wkdw lq htxloleulxp/ wkh h{shfwhg h{fhvv uhwxuq rq wkh ulvn| dvvhw pxvw htxdo ulvn dyhuvlrq  wlphv wkh fryduldqfh ri wkh dvvhw uhwxuq zlwk frqvxpswlrq jurzwk1 D vlplodu htxdwlrq ghvfulehv hdfk ulvn| dvvhw*v ulvn suhplxp lq d prgho zlwk pxowlsoh ulvn| dvvhwv1 Lq dvvhw sulflqj wkhru|/ wklv htxdwlrq lv xvhg wr h{sodlq dvvhwv* ulvn suhpld dqg lv nqrzq dv wkh frqvxpswlrq fdslwdo dvvhw sulflqj prgho ru FFDSP> khuh zh wdnh wkh ulvn| dvvhw uhwxuq dv jlyhq dqg vhhn d frqvxpswlrq uxoh dqg sruwirolr vwudwhj| wkdw zloo pdnh +5173, krog1 D frqvwdqw frqvxpswlrq0zhdowk udwlr D glfxow| zlwk wkh orjqrupdo frqvxpswlrq0edvhg prgho lv wkdw wkh lqwhuwhpsrudo exgjhw frqvwudlqw +5168, lv qrw jhqhudoo| orjolqhdu/ ehfdxvh frqvxpswlrq lv vxewudfwhg iurp zhdowk ehiruh pxowlso|lqj e| wkh sruwirolr uhwxuq1 Wkh frpelqdwlrq ri vxewudfwlrq dqg pxowlsolfdwlrq fuhdwhv dq lq0 wudfwdeoh qrqolqhdulw|1 Lq odwhu fkdswhuv ri wklv errn/ iroorzlqj Fdpsehoo +4<<6,/ zh zloo ds0 surdfk wklv sureohp e| dssur{lpdwlqj wkh exgjhw frqvwudlqw1 Iru qrz/ krzhyhu/ zh dvvxph wkdw wkh frqvxpswlrq0zhdowk udwlr lv frqvwdqw/ lq zklfk 5<

FKDSWHU 51 P\RSLF SRUWIROLR FKRLFH fdvh wkh exgjhw frqvwudlqw ehfrphv orjolqhdu1 Zh vroyh wkh prgho xqghu wklv dvvxpswlrq/ dqg wkhq vhhn wr qg frqglwlrqv wkdw mxvwli| wkh dvvxpswlrq1 D frqvwdqw frqvxpswlrq0zhdowk udwlr fdq eh zulwwhq dv F| @ e> Z|

+5174,

dqg wkh exgjhw frqvwudlqw +5168, fdq wkhq eh zulwwhq lq orj irup dv z|n @ uRc|n . orj+4  e, @ usc|n . | +u|n  usc|n , 4 . | +4  | ,2| . orj+4  e,> 5

+5175,

zkhuh wkh vhfrqg htxdolw| vxevwlwxwhv lq iurp +5154,1 Wkh frqvwdqw frqvxpswlrq0zhdowk udwlr +5174, dovr lpsolhv wkdw wkh jurzwk udwh ri frqvxpswlrq htxdov wkh jurzwk udwh ri zhdowk/ vr wkh whupv lq frq0 vxpswlrq lq +516<, dqg +5173, fdq eh uhzulwwhq lq whupv ri zhdowk1 Wkh irupxod iru d vlqjoh ulvn| dvvhw*v h{shfwhg h{fhvv uhwxuq/ +5173,/ ehfrphv H| u|n  usc|n .  2| @5 @ Fry| +u|nc z|n , @ | 2| >

+5176,

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H| u|n  usc|n .  2| @5 = 2|

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Coefficient of Relative Risk Aversion (γ)

Constant consumption-wealth ratio

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1

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1 Elasticity of Intertemporal Substitution (ψ)

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66

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FKDSWHU 61 ZKR VKRXOG EX\ ORQJ0WHUP ERQGVB lqdwlrq0lqgh{hg frqvro1 Li wklv dvvhw lv dydlodeoh lw zloo sod| vrph uroh lq wkh rswlpdo sruwirolr ri dq| lqyhvwru zkr kdv uhodwlyh ulvn dyhuvlrq juhdwhu wkdq rqh1 Li dq lqdwlrq0lqgh{hg frqvro lv qrw dydlodeoh/ d frqvhuydwlyh lqyhvwru zloo krog dvvhwv wkdw duh jrrg sur{lhv iru lw1 Lq rughu wr jr ixuwkhu/ zh qhhg wr xvh dq h{solflw prgho ri wkh whup vwuxfwxuh ri lqwhuhvw udwhv wr ghulyh txdqwlwdwlyh suhglfwlrqv iru lqyhvwruv* kroglqjv ri Wuhdvxu| eloov/ orqj0whup erqgv/ dqg htxlwlhv1 Iroorzlqj Fdps0 ehoo dqg Ylfhlud +5334,/ zh qrz suhvhqw vxfk d prgho1 Wkh prgho lv vhw lq glvfuhwh wlph> lw kdv wzr idfwruv/ rqh uhdo dqg rqh qrplqdo/ zlwk orjqrupdo glvwulexwlrqv dqg frqvwdqw yduldqfhv1 Wklv lv wkh vlpsohvw whup vwuxfwxuh prgho wkdw doorzv xv wr glvwlqjxlvk ehwzhhq uhdo dqg qrplqdo erqgv1 Wkh uhdo sduw ri wkh prgho lv d glvfuhwh0wlph yhuvlrq ri wkh zhoo0nqrzq Ydvlfhn +4<::, frqwlqxrxv0wlph prgho1 Ehfdxvh erqgv kdyh ghwhuplqlvwlf sd|rv/ wkh| fdq eh sulfhg e| zulwlqj grzq d wlph0vhulhv prgho iru wkh vwrfkdvwlf glvfrxqw idfwru +VGI, P|n 1 Wkh VGI ghwhuplqhv wkh sulfhv ri doo dvvhwv lq wkh hfrqrp|/ exw wkh olqn lv sduwlfxoduo| gluhfw zlwk erqgv vlqfh zh gr qrw kdyh wr prgho wkhlu sd|rv1 Lq d uhsuhvhqwdwlyh0djhqw iudphzrun wkh VGI fdq eh uhodwhg wr wkh pdujlqdo xwlolw| ri d uhsuhvhqwdwlyh lqyhvwru/ exw khuh zh vlpso| xvh lw dv d ghylfh wr jhqhudwh d frpsohwh vhw ri erqg sulfhv1 D pruh ghwdlohg h{sodqdwlrq ri wklv w|sh ri prgho lv jlyhq e| Fdpsehoo/ Or/ dqg PdfNlqod| +4<<:/ Fkdswhu 44, dqg Fdpsehoo +5333,1 Zh uvw wdnh orjv dqg zrun zlwk p|n  orj+P|n ,1 Zh euhdn wklv lqwr lwv frqglwlrqdo h{shfwdwlrq dw wlph w/ {| / dqg d vkrfn uhdol}hg dw wlph w . 4/ y6c|n = p|n @ {| . y6c|n = +6155, Zh dvvxph wkdw {| iroorzv dq DU+4, surfhvv= {|n @ +4  !% , % . !% {| . %%c|n =

+6156,

Zh doorz wkh lqqrydwlrqv wr wkh orj VGI wr eh fruuhodwhg zlwk lqqrydwlrqv wr lwv frqglwlrqdo h{shfwdwlrq= y6c|n @  6% %%c|n . %6c|n =

+6157,

Wkh whup vwuxfwxuh ri uhdo lqwhuhvw udwhv Wkh hfrqrplf phdqlqj ri wkhvh dvvxpswlrqv fdq ehvw eh dssuhfldwhg e| zrunlqj rxw wkhlu lpsolfdwlrqv iru wkh whup vwuxfwxuh ri uhdo lqwhuhvw udwhv1 Wkhuh lv d gluhfw olqn ehwzhhq wkh vwrfkdvwlf glvfrxqw idfwru dqg 7;

FKDSWHU 61 ZKR VKRXOG EX\ ORQJ0WHUP ERQGVB wkh orj uhwxuq/ ru htxlydohqwo| wkh orj |lhog/ rq d rqh0shulrg lqgh{hg erqg= uc|n @  orj H| ^P|n ` @ H| ^p|n ` 2 Ydu| ^p|n ` vlqfh P|n lv orjqrupdo1 Vxevwlwxwlqj lq iurp +6155,~+6157,/ zh kdyh uc|n @ {| 

 4 2 2  6% % .  26 > 5

+6158,

zkhuh  2%  Ydu| +%%c|n , dqg 26  Ydu| +%6c|n ,1 Wkh vkruw0whup uhdo lqwhuhvw udwh htxdov wkh vwdwh yduldeoh {| dgmxvwhg e| d frqvwdqw/ vr lw iroorzv dq DU+4, surfhvv zlwk shuvlvwhqfh !1 Orqjhu0whup lqdwlrq0lqgh{hg erqgv fdq eh sulfhg uhfxuvlyho|1 D wzr0 shulrg erqg wrgd| zloo ehfrph d rqh0shulrg erqg wrpruurz/ zkhq lwv sulfh zloo eh ghvfulehg e| +6158,1 Wkxv rqh fdq vroyh iru lwv orj sulfh/ ru htxlyd0 ohqwo| lwv orj |lhog/ dqg wkhq wkh orj sulfhv dqg |lhogv ri doo orqjhu0pdwxulw| erqgv1 Wkh vroxwlrq iru wkh orj |lhog rq dq q0shulrg lqgh{hg }hur0frxsrq erqg/ |?| / wlphv erqg pdwxulw| q/ zklfk htxdov plqxv wkh orj sulfh ri wkh erqg/ s?| / lv jlyhq e| q|?| @ s?| @ D? . E? {| >

+6159,

zkhuh D? dqg E? duh ixqfwlrqv ri erqg pdwxulw| q exw qrw ri wlph w1 D uhfxuvlyh h{suhvvlrq iru wkh frhflhqw D? lv jlyhq lq wkh Dsshqgl{1 Pruh lpsruwdqw iru rxu suhvhqw glvfxvvlrq/ wkh frhflhqw E? lv jlyhq e| E? @ 4 . !% E?3 @

4  !?% = 4  !%

+615:,

Wr xqghuvwdqg wklv htxdwlrq/ qrwh wkdw wkh h{shfwdwlrqv k|srwkhvlv ri wkh whup vwuxfwxuh krogv lq wklv prgho1 Wkxv wkh orj erqg |lhog lv d frqvwdqw soxv dq dyhudjh ri h{shfwhg ixwxuh uhdo lqwhuhvw udwhv ryhu wkh qh{w q shulrgv/ dqg q wlphv wkh orj erqg |lhog lv d frqvwdqw soxv d vxp ri h{shfwhg ixwxuh uhdo lqwhuhvw udwhv1 Vlqfh wkh uhdo lqwhuhvw udwh iroorzv dq DU+4, surfhvv zlwk shuvlvwhqfh frhflhqw !% / wkh h{shfwhg uhdo lqwhuhvw udwh n shulrgv dkhdg lv d frqvwdqw soxv !&% wlphv wkh uhdo lqwhuhvw udwh wrgd|1 Vxpplqj xs ryhu q shulrgv jlyhv +615:,1 Wkh rqh0shulrg orj uhwxuq rq dq q0shulrg lqgh{hg }hur0frxsrq erqg lv mxvw wkh fkdqjh lq lwv sulfh/ +s?3c|n s?c| ,1 Frpelqlqj wklv h{suhvvlrq zlwk +6159, dqg +615:,/ wkh h{fhvv uhwxuq ryhu wkh rqh0shulrg orj lqwhuhvw udwh lv 4 2 u?c|n  uc|n @  E?3 2%   6% E?3  2%  E?3 %%c|n > 5

+615;,

vr wkh q0shulrg erqg lv vhqvlwlyh rqo| wr uhdo0lqwhuhvw0udwh vkrfnv %%c|n / zlwk d vhqvlwlylw| E?3 1 Ehfdxvh wkh uhdo sduw ri rxu prgho kdv rqo| d 7<

FKDSWHU 61 ZKR VKRXOG EX\ ORQJ0WHUP ERQGVB vlqjoh idfwru/ |lhogv dqg sulfhv ri lqdwlrq0lqgh{hg erqgv ri doo pdwxulwlhv duh gulyhq rqo| e| wkh vkruw0whup uhdo lqwhuhvw udwh dqg wkxv duh shuihfwo| fruuhodwhg zlwk rqh dqrwkhu1 Wkh yduldqfh ri wkh q0shulrg erqg uhwxuq lv 2  2 / dqg wkh ulvn suhplxp lv 2?| @ E?3 % H| ^u?c|n  uc|n ` .

 2?| @  6% E?3 2% = 5

+615<,

Wkh ulvn suhplxp rq wkh orqj0whup erqg/ ru whup suhplxp/ lv ghwhuplqhg e| wkh frqglwlrqdo fryduldqfh ri wkh h{fhvv erqg uhwxuq zlwk wkh orj VGI1 Lq rxu krprvnhgdvwlf prgho wkh frqglwlrqdo fryduldqfh lv frqvwdqw wkurxjk wlph exw ghshqghqw rq wkh erqg pdwxulw|> wkxv wkh whup suhplxp lv frqvwdqw dv srvwxodwhg e| wkh h{shfwdwlrqv k|srwkhvlv ri wkh whup vwuxfwxuh1 Vlqfh E?3 A 3/ wkh whup suhplxp kdv wkh rssrvlwh vljq wr  6% 1 Zlwk d srvlwlyh  6% / orqj0whup lqgh{hg erqgv sd| r zkhq wkh VGI ru/ htxlydohqwo|/ wkh pdujlqdo xwlolw| ri frqvxpswlrq iru d uhsuhvhqwdwlyh lqyhvwru lv kljk/ wkdw lv/ zkhq zhdowk lv prvw ghvludeoh1 Lq htxloleulxp/ wkhvh erqgv kdyh d qhjdwlyh whup suhplxp dqg wkh uhdo |lhog fxuyh lv rq dyhudjh grzqzdug0vorslqj1 Zlwk d qhjdwlyh  6% / rq wkh rwkhu kdqg/ orqj0whup lqgh{hg erqgv sd| r zkhq wkh pdujlqdo xwlolw| ri frqvxpswlrq iru d uhsuhvhqwdwlyh lqyhvwru lv orz/ dqg vr lq htxloleulxp wkh| kdyh d srvlwlyh whup suhplxp1 Lq wklv fdvh wkh uhdo |lhog fxuyh lv rq dyhudjh xszdug0vorslqj1 Htxdwlrq +615<, lpsolhv wkdw wkh Vkdush udwlr iru lqgh{hg erqgv lv  6% % / zklfk lv lqghshqghqw ri erqg pdwxulw|1 Wkh lqyduldqfh ri wkh Vkdush udwlr wr erqg pdwxulw| iroorzv iurp wkh vlqjoh0idfwru vwuxfwxuh ri wkh uhdo vhfwru ri wkh prgho1 Wkh udwlr ri wkh ulvn suhplxp wr wkh yduldqfh ri wkh h{fhvv uhwxuq/ zklfk ghwhuplqhv d p|rslf lqyhvwru*v doorfdwlrq wr orqj0whup erqgv/ lv  6% @E?3 1 Wklv grhv ghshqg rq erqg pdwxulw| exw qrw rq wkh yrodwlolw| ri wkh uhdo lqwhuhvw udwh1 Wkh rswlpdo sruwirolr ri lqdwlrq0lqgh{hg erqgv Li rqo| lqdwlrq0lqgh{hg erqgv duh dydlodeoh wr wkh lqyhvwru/ wkh uhdo vhfwru ri wkh prgho lv doo zh qhhg wr ghulyh dq h{solflw vroxwlrq wr wkh sruw0 irolr sureohp1 Ehfdxvh doo orqj0whup lqdwlrq0lqgh{hg erqgv duh shuihfwo| fruuhodwhg/ zh fdq frqvlghu zlwkrxw orvv ri jhqhudolw| wkh fkrlfh ehwzhhq wzr dvvhwv= d vlqjoh0shulrg lqdwlrq0lqgh{hg erqg dqg dq q0shulrg lqdwlrq0 lqgh{hg erqg1 Xvlqj +615;, wr fdofxodwh wkh whupv lq +6146,/ zh qg wkdw wkh rswlpdo sruwirolr zhljkw rq wkh q0shulrg erqg lv        4 4  6% 4 +6163, . 4 = ? @  E?3  E?3 4  !% 83

FKDSWHU 61 ZKR VKRXOG EX\ ORQJ0WHUP ERQGVB Wklv vroxwlrq kdv vhyhudo lqwhuhvwlqj surshuwlhv1 Iluvw/ wkh yduldqfh ri uhdo lqwhuhvw udwh vkrfnv/ 2% / grhv qrw gluhfwo| dhfw wkh sruwirolr zhljkw ? ehfdxvh/ jlyhq wkh sdudphwhul}dwlrq ri rxu prgho/ lw pryhv wkh yduldqfh ri uhwxuqv dqg wkh ulvn suhplxp lq sursruwlrq wr rqh dqrwkhu1 Lqwhuhvw0udwh yduldqfh fdq rqo| dhfw wkh vroxwlrq lqgluhfwo|/ wkurxjk wkh orjolqhdul}d0 wlrq sdudphwhu 1 Hpslulfdoo| zh qg wkdw  fkdqjhv yhu| olwwoh zkhq wkh sdudphwhuv ri wkh prgho fkdqjh/ dqg vr wkh lqgluhfw hhfw wkurxjk  lv txdqwlwdwlyho| qhjoljleoh1 Vhfrqg/ wkh lqwhuhvw0udwh vhqvlwlylw| ri wkh rswlpdo sruwirolr lv jlyhq e| ? E?3 / dqg wklv grhv qrw ghshqg rq wkh erqg pdwxulw| q1 Wzr dv0 vhwv duh hqrxjk wr frpsohwh wkh pdunhw zlwk uhvshfw wr uhdo0lqwhuhvw0udwh ulvn/ dqg wkxv wkh lqyhvwru fdq xvh dq| wzr lqdwlrq0lqgh{hg erqgv wr frq0 vwuxfw d sruwirolr zlwk wkh rswlpdo ohyho ri lqwhuhvw0udwh vhqvlwlylw|1 Li rqo| vkruw0pdwxulw| erqgv zlwk d orz vhqvlwlylw| duh dydlodeoh/ wkh lqyhvwru fdq frpshqvdwh e| kroglqj pruh ri wkhp ru hyhq ohyhudjlqj klv srvlwlrq> li rqo| orqj0pdwxulw| erqgv zlwk d kljk vhqvlwlylw| duh dydlodeoh/ wkh lqyhvwru fdq frpshqvdwh e| kroglqj ihzhu ri wkhp1 Wklug/ wkh lqwhuwhpsrudo khgjlqj ghpdqg/ wkh vhfrqg whup lq +6163,/ lv lqfuhdvlqj lq wkh shuvlvwhqfh ri lqwhuhvw0udwh vkrfnv !% 1 Wklv lv ehfdxvh/ dv vkrzq lq wkh jhqhudo vroxwlrq +6146,/ khgjlqj ghpdqg lv ghwhuplqhg e| wkh fryduldqfh ri wkh ulvn| dvvhw uhwxuq zlwk wkh glvfrxqwhg ydoxh ri doo ixwxuh lqwhuhvw udwhv1 D pdmru wkhph ri wkh hpslulfdo zrun lq wklv errn lv wkdw shuvlvwhqw vkrfnv wr lqyhvwphqw rssruwxqlwlhv duh pxfk pruh lpsruwdqw iru sruwirolr fkrlfh wkdq duh wudqvlwru| vkrfnv wr lqyhvwphqw rssruwxqlwlhv1 Irxuwk/ dv ulvn dyhuvlrq lqfuhdvhv dqg wkh hodvwlflw| ri lqwhuwhpsrudo vxe0 vwlwxwlrq ghfolqhv/ wkh olplw ri +6163, lv d sruwirolr wkdw lv htxlydohqw wr dq lqdwlrq0lqgh{hg frqvro1 Lq wklv prgho dq lqdwlrq0lqgh{hg frqvro kdv lqwhuhvw0udwh vhqvlwlylw| S @+4  S !% ,/ dqg  @ S zkhq 4@ @ # @ 31 Wklv lv d vshfldo fdvh ri wkh jhqhudo srlqw pdgh lq vhfwlrq 614171 Htxdwlrq +6163, lv dsshdolqjo| vlpsoh/ exw lw odfnv uhdolvp ehfdxvh lw grhv qrw doorz lqyhvwruv wr krog qrplqdo erqgv ru htxlwlhv1 Lq rughu wr jr ixuwkhu/ zh qhhg wr dxjphqw wkh prgho wr lqfoxgh vxfk dvvhwv1 Wkh whup vwuxfwxuh ri qrplqdo lqwhuhvw udwhv Lq rughu wr sulfh qrplqdo erqgv/ zh pxvw prgho wkh surfhvv gulylqj lqd0 wlrq1 Zh dvvxph wkdw wklv surfhvv kdv wkh vdph irup wkdw zh kdyh douhdg| dvvxphg iru wkh VGI1 Wkdw lv/ uhdol}hg orj lqdwlrq |n htxdov h{shfwhg orj lqdwlrq }| soxv dq lqdwlrq vkrfn/ dqg h{shfwhg lqdwlrq iroorzv dq

84

FKDSWHU 61 ZKR VKRXOG EX\ ORQJ0WHUP ERQGVB DU+4, surfhvv=  |n @ }| . yZc|n >

+6164,

}|n @ +4  !5 , 5 . !5 }| . y5c|n =

+6165,

Zh dvvxph wkdw wkh vkrfnv wr uhdol}hg dqg h{shfwhg lqdwlrq/ yZc|n dqg y5c|n / fdq eh fruuhodwhg zlwk hdfk rwkhu dqg zlwk wkh uhdo vkrfnv wr wkh prgho= y5c|n @  5% %%c|n .  56 %6c|n . %5c|nc

+6166,

yZc|n @  Z% %%c|n .  Z6 %6c|n .  Z5 %5c|n . %Zc|nc

+6167,

Wkh prgho lv gulyhq e| irxu qrupdoo| glvwulexwhg/ zklwh qrlvh vkrfnv %6c|n / %Zc|n / %%c|n / dqg %5c|n wkdw ghwhuplqh wkh lqqrydwlrqv wr wkh orj VGI/ wkh orj lqdwlrq udwh/ dqg wkhlu frqglwlrqdo phdqv1 Wkhvh vkrfnv duh furvv0vhfwlrqdoo| xqfruuhodwhg/ zlwk yduldqfhv  26 / 2Z /  2% / dqg  25 1 Lw lv lpsruwdqw wr qrwh wkdw }|n / wkh h{shfwhg lqdwlrq udwh/ lv dhfwhg e| erwk d sxuh h{shfwhg0lqdwlrq vkrfn %5c|n dqg wkh vkrfnv wr wkh h{shfwhg dqg xqh{shfwhg orj VGI %%c|n dqg %6c|n 1 Wkdw lv/ lqqrydwlrqv wr h{shfwhg lqdwlrq fdq eh fruuhodwhg zlwk lqqrydwlrqv lq wkh orj VGI/ dqg khqfh zlwk lqqrydwlrqv lq wkh vkruw0whup uhdo lqwhuhvw udwh1 Wkhvh fruuhodwlrqv phdq wkdw qrplqdo lqwhuhvw udwhv qhhg qrw pryh rqh0iru0rqh zlwk h{shfwhg lqdwlrq wkdw lv/ wkh Ilvkhu k|srwkhvlv qhhg qrw krogdqg qrplqdo erqg sulfhv fdq lqfoxgh dq lqdwlrq ulvn suhplxp dv zhoo dv d uhdo whup suhplxp1 Zh kdyh zulwwhq wkh prgho zlwk d vhoi0frqwdlqhg uhdo vhfwru +6155,~+6157, dqg d qrplqdo vhfwru +6164,~+6167, wkdw lv dhfwhg e| vkrfnv wr wkh uhdo vhfwru1 Exw wklv lv phuho| d pdwwhu ri qrwdwlrqdo frqyhqlhqfh1 Rxu prgho lv d uhgxfhg irup udwkhu wkdq d vwuxfwxudo prgho/ vr lw fdswxuhv fruuhodwlrqv dprqj vkrfnv wr uhdo dqg qrplqdo lqwhuhvw udwhv exw grhv qrw kdyh dq|wklqj wr vd| derxw wkh wuxh xqghuo|lqj vrxufhv ri wkhvh vkrfnv1 Wkh sulflqj ri qrplqdo erqgv iroorzv wkh vdph vwhsv dv wkh sulflqj ri lqgh{hg erqgv1 Wkh orj sulfh ri dq q0shulrg qrplqdo }hur0frxsrq erqg/ sm?c| / lv d olqhdu frpelqdwlrq ri {| dqg }| zkrvh frhflhqwv duh wlph0lqyduldqw/ wkrxjk wkh| ydu| zlwk wkh pdwxulw| ri wkh erqg= m m sm?c| @ Dm? . Ec? {| . E2c? }| =

+6168,

m m Wkh Dsshqgl{ jlyhv h{suhvvlrqv iru wkh frhflhqwv Dm? / Ec? dqg E2c? 1 Vlqfh qrplqdo erqg sulfhv duh gulyhq e| vkrfnv wr erwk uhdo lqwhuhvw udwhv dqg lqdwlrq/ wkh| kdyh d wzr0idfwru vwuxfwxuh udwkhu wkdq wkh vlqjoh0 idfwru vwuxfwxuh ri lqgh{hg erqg sulfhv1 Lqdwlrq dhfwv wkh h{fhvv uhwxuq

85

FKDSWHU 61 ZKR VKRXOG EX\ ORQJ0WHUP ERQGVB rq dq q0shulrg qrplqdo erqg ryhu wkh rqh0shulrg qrplqdo lqwhuhvw udwh/ vr ulvn suhpld lq wkh qrplqdo whup vwuxfwxuh lqfoxgh frpshqvdwlrq iru lqdwlrq ulvn1 Olnh doo rwkhu ulvn suhpld lq wkh prgho/ krzhyhu/ wkh ulvn suhpld rq qrplqdo erqgv duh frqvwdqw ryhu wlph> wkxv wkh h{shfwdwlrqv k|srwkhvlv ri wkh whup vwuxfwxuh krogv iru qrplqdo dv zhoo dv iru uhdo erqgv1 Sulflqj htxlwlhv Hyhq wkrxjk rxu irfxv lq wklv fkdswhu lv rq orqj0whup erqgv/ d ixoo hydoxdwlrq ri erqg ghpdqg uhtxluhv wkdw zh lqfoxgh htxlwlhv lq rxu prgho dv dq dwwudfwlyh dowhuqdwlyh orqj0whup lqyhvwphqw1 Zh gr wklv dv vlpso| dv srvvleoh/ dvvxplqj wkdw wkh xqh{shfwhg orj uhwxuq rq htxlwlhv lv dhfwhg e| vkrfnv wr erwk wkh h{shfwhg dqg xqh{shfwhg orj VGI= uec|n  H| uec|n @  e% %%c|n .  e6 %6c|n =

+6169,

Fdpsehoo +4<<<, vkrzv wkdw wklv ghfrpsrvlwlrq ri wkh xqh{shfwhg orj htxlw| uhwxuq lqwr d olqhdu frpelqdwlrq ri wkh vkrfnv wr wkh h{shfwhg dqg xqh{0 shfwhg orj VGI lv frqvlvwhqw zlwk d uhsuhvhqwdwlyh0djhqw hqgrzphqw prgho zkhuh h{shfwhg djjuhjdwh frqvxpswlrq jurzwk iroorzv dq DU+4,1 Iurp wkh ixqgdphqwdo sulflqj htxdwlrq 4 @ H| ^P|n U|n ` dqg wkh orjqrupdo vwuxf0 wxuh ri wkh prgho lw lv hdv| wr vkrz wkdw wkh ulvn suhplxp rq htxlwlhv/ ryhu d rqh0shulrg ulvnohvv uhwxuq uc|n / lv jlyhq e| H| ^uec|n  uc|n ` .

 2e| @  6%  e%  2% .  e6 26 = 5

+616:,

Olnh doo rwkhu fryduldqfhv lq wkh prgho/ wklv lv frqvwdqw ryhu wlph vr wkh htxlw| suhplxp lv d frqvwdqw1 Rqfh qrplqdo erqgv dqg htxlwlhv duh lqfoxghg lq wkh prgho/ wkh dojheudlf sruwirolr vroxwlrqv ehfrph vxflhqwo| frpsolfdwhg wkdw lw lv qr orqjhu khosixo wr zulwh wkhp rxw h{solflwo|1 Lqvwhdg/ zh hvwlpdwh wkh prgho rq klvwrulfdo gdwd dqg suhvhqw qxphulfdo vroxwlrqv1

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+718, +719,

Wkdw lv/ wkh rswlpdo sruwirolr uxoh lv olqhdu lq wkh YDU vwdwh yhfwru exw wkh rswlpdo frqvxpswlrq uxoh lv txdgudwlf1 df lv dq q0yhfwru/ D lv dq q  p pdwul{/ ef lv d vfdodu/ dqg e dqg e2 duh p0yhfwruv1 Wkh prwlydwlrq iru wklv jxhvv lv wkdw +718, lv wkh vlpsohvw sruwirolr uxoh wkdw doorzv wkh lqyhvwru wr vkliw klv sruwirolr lq uhvsrqvh wr fkdqjlqj ulvn suhpld1 Doo wkh yduldeohv lq wkh vwdwh yhfwru }| fdq srwhqwldoo| dhfw ulvn suhpld/ dqg wkxv wkh sruwirolr yhfwru | pxvw eh iuhh wr uhvsrqg wr wkhp> zh dvvxph wkdw lw grhv vr olqhduo|1 Jlyhq d olqhdu sruwirolr uxoh/ wkh h{shfwhg sruwirolr uhwxuq lpsolhg e| +5155, lv txdgudwlf lq wkh vwdwh yduldeohv1 Vwdwh yduldeohv dhfw wkh h{shfwhg sruwirolr uhwxuq erwk gluhfwo|/ e| vkliwlqj wkh h{shfwhg uhwxuqv rq h{lvwlqj dvvhw kroglqjv/ dqg lqgluhfwo|/ e| vkliwlqj wkh dvvhw doorfdwlrqv1 Vlqfh hdfk ri wkhvh hhfwv lv olqhdu/ wkhlu lqwhudfwlrq lv txdgudwlf dqg wklv pdnhv wkh h{shfwhg sruwirolr uhwxuq txdgudwlf1 Exw wkhq wkh frqvxpswlrq Hxohu htxdwlrq +716, lpsolhv wkdw wkh orj frqvxpswlrq0 zhdowk udwlr pxvw dovr eh dw ohdvw txdgudwlf lq wkh vwdwh yduldeohv1 Wkh frqvxpswlrq Hxohu htxdwlrq dovr kdv d frqglwlrqdo yduldqfh whup/ exw wklv wrr wxuqv rxw wr eh txdgudwlf lq wkh vwdwh yduldeohv jlyhq rxu krprvnhgdvwlf YDU vshflfdwlrq1 Nlp dqg Rpehuj +4<<9, ghulyhg d vlplodu olqhdu0 txdgudwlf vroxwlrq iru d qlwh0krul}rq frqwlqxrxv0wlph prgho lq zklfk wkh lqyhvwru kdv srzhu xwlolw| ghqhg ryhu whuplqdo zhdowk1 Wr yhuli| rxu jxhvv dqg vroyh iru wkh sdudphwhuv ri wkh vroxwlrq/ zh zulwh wkh frqglwlrqdo prphqwv wkdw dsshdu lq +717, dv ixqfwlrqv ri wkh YDU frhflhqwv dqg wkh xqnqrzq sdudphwhuv ri +718, dqg +719,1 Zh wkhq vroyh iru wkh sdudphwhuv wkdw vdwlvi| +717,1 Wklv jlyhv xv wkh sdudphwhuv lq +718, dv ixqfwlrqv ri wkh YDU frhflhqwv dqg wkh vwloo xqnqrzq sdudphwhuv ri +719,1 Qh{w zh vxevwlwxwh lqwr +716,/ erwk vlghv ri zklfk duh txdgudwlf lq wkh YDU vwdwh yduldeohv jlyhq rxu frqmhfwxuhg txdgudwlf irup iru wkh rswlpdo frqvxpswlrq0zhdowk udwlr1 Ilqdoo| zh vroyh wklv v|vwhp ri txdgudwlf htxdwlrqv iru wkh sdudphwhuv ri +719,1 Jlyhq wkh orjolqhdul}dwlrq sdudphwhu / wklv vroxwlrq lv dqdo|wlfdo1 Fdps0 ehoo dqg Ylfhlud +4<<<, zulwh lw rxw h{solflwo| iru wkh vshfldo fdvh lq zklfk wkhuh lv d frqvwdqw ulvnohvv lqwhuhvw udwh/ d vlqjoh ulvn| dvvhw/ dqg d vlqjoh iruhfdvwlqj yduldeoh iru wkh h{fhvv ulvn| uhwxuq zklfk lwvhoi iroorzv dq DU+4, surfhvv1 Zh h{sodlq wklv fdvh lq vhfwlrq 714161 Fdpsehoo/ Fkdq/ dqg Yl0 :6

FKDSWHU 71 LV WKH VWRFN PDUNHW VDIHU IRU ORQJ0WHUP LQYHVWRUVB fhlud +5333, h{whqg wkh dssurdfk wr wkh jhqhudo YDU fdvh/ iru zklfk lw lv pruh frqyhqlhqw wr vroyh wkh olqhdu0txdgudwlf htxdwlrqv qxphulfdoo|1 Wkh vroxwlrq suhvhqwhg khuh lv h{dfw lq frqwlqxrxv wlph zkhq dvvhw sulfhv iroorz frqwlqxrxv glxvlrq surfhvvhv/ li wkh frqvxpswlrq0zhdowk udwlr lv frq0 vwdqw1 Wkh frqvxpswlrq0zhdowk udwlr lv frqvwdqw li wkh hodvwlflw| ri lqwhuwhp0 srudo vxevwlwxwlrq # @ 4/ exw lq doo rwkhu fdvhv/ wkh vroxwlrq lv rqo| dq dssur{lpdwlrq1 Fdpsehoo dqg Nrr +4<<:, dqg Fdpsehoo/ Frffr/ Jrphv/ Pdhqkrxw/ dqg Ylfhlud +4<<<, hydoxdwh wkh dffxudf| ri wkh dssur{lpdwh vroxwlrq/ uhvshfwlyho| lq prghov zlwk h{rjhqrxv sruwirolr uhwxuqv dqg h{rjh0 qrxv uhwxuqv rq xqghuo|lqj dvvhwv/ dqg qg wkdw lw lv dffhswdeo| dffxudwh iru hodvwlflwlhv ri lqwhuwhpsrudo vxevwlwxwlrq xs wr derxw 61 Lq sduwlfxodu/ wklv lpsolhv wkdw orz hodvwlflwlhv ri lqwhuwhpsrudo vxevwlwxwlrq ri wkh vruw hvwlpdwhg e| pdfurhfrqrplvwv vkrxog eh frqvlvwhqw zlwk wkh xvh ri wkh ds0 sur{lpdwh vroxwlrq1 Wkh fdvh # @ 4 lv dovr lpsruwdqw ehfdxvh rqo| lq wklv fdvh gr zh nqrz wkdw wkh ydoxh ri  htxdov 1 Lq doo rwkhu fdvhv zh pxvw vroyh iru  dorqj zlwk wkh rwkhu sdudphwhuv ri wkh prgho1 Zh gr wklv qxphulfdoo|> zh lql0 wldol}h  @ / vroyh iru wkh rwkhu sdudphwhuv ri wkh prgho/ fdofxodwh wkh lpsolhg phdq orj frqvxpswlrq0zhdowk udwlr iurp +719,/ uhfdofxodwh / dqg uhshdw xqwlo frqyhujhqfh1 Wklv surfhvv lv h{wuhpho| udslg h{fhsw lq fdvhv zkhuh wkh lqqlwh0krul}rq rswlpl}dwlrq sureohp lv loo0ghqhg +iru h{dpsoh/ ehfdxvh dyhudjh udwhv ri uhwxuq duh wrr kljk uhodwlyh wr wkh lqyhvwru*v udwh ri wlph suhihuhqfh/ vr wkdw d qlwh0krul}rq lqyhvwru*v xwlolw| glyhujhv dv wkh lqyhvwphqw krul}rq lqfuhdvhv,1 Dq lpsruwdqw surshuw| ri wkh prgho lv wkdw jlyhq wkh orjolqhdul}dwlrq sdudphwhu / wkh rswlpdo sruwirolr uxoh grhv qrw ghshqg rq wkh lqwhuwhpsrudo hodvwlflw| ri vxevwlwxwlrq #1 # rqo| dhfwv sruwirolr fkrlfh wr wkh h{whqw wkdw lw hqwhuv lqwr wkh ghwhuplqdwlrq ri = Hpslulfdoo|/ wklv lqgluhfw hhfw wkurxjk  vhhpv wr eh plqru1

e

 tTiU@* U@tiG ?i ht!) @tti| @?_ @ UL?t|@?| hi@* ?|ihit| h@|i

Fdpsehoo dqg Ylfhlud +4<<<, vwxg| d vshfldo fdvh lq zklfk wkhuh lv d vkruw0 whup ulvnohvv dvvhw zlwk d frqvwdqw uhdo orj uhwxuq us 1 Ehfdxvh wklv ulvnohvv uhdo uhwxuq lv frqvwdqw/ lw lv d vdih lqyhvwphqw erwk iru vkruw0whup lqyhvwruv dqg iru orqj0whup lqyhvwruv1 Wkxv wkh lvvxh hpskdvl}hg lq Fkdswhu 6/ wkdw wkh lghqwlw| ri wkh ulvnohvv dvvhw pd| eh glhuhqw iru lqyhvwruv zlwk glhuhqw krul}rqv/ grhv qrw dulvh khuh1 Fdpsehoo dqg Ylfhlud +4<<<, dovr dvvxph

:7

FKDSWHU 71 LV WKH VWRFN PDUNHW VDIHU IRU ORQJ0WHUP LQYHVWRUVB wkdw wkhuh lv d vlqjoh ulvn| dvvhw +_vwrfnv%, zlwk orj uhwxuq u|n jlyhq e| u|n  H| u|n @ x|n >

+71:,

zkhuh x|n lv wkh lqqrydwlrq wr wkh ulvn| dvvhw uhwxuq/ qrupdoo| glvwulexwhg zlwk phdq }hur dqg yduldqfh  2 1 Wkh h{shfwhg h{fhvv orj uhwxuq rq wkh ulvn| dvvhw/ dgmxvwhg iru rqh0kdoi lwv yduldqfh lq wkh idploldu pdqqhu/ htxdov d vwdwh yduldeoh {| = 2 H| u|n  us .  @ {| = +71;, 5 Ilqdoo|/ {| iroorzv dq DU+4, surfhvv zlwk phdq  dqg shuvlvwhqfh !1 Wkh lqqrydwlrq wr {|n lv zulwwhq |n / dvvxphg wr eh qrupdoo| glvwulexwhg zlwk phdq }hur dqg yduldqfh 2# = {|n @  . !+{|  , . |n =

+71<,

Prghoolqj phdq uhyhuvlrq Wkh lqqrydwlrqv x|n dqg |n fdq eh fruuhodwhg/ zlwk fryduldqfh # 1 Lq idfw/ wklv fryduldqfh lv zkdw jhqhudwhv lqwhuwhpsrudo khgjlqj ghpdqg iru wkh ulvn| dvvhw e| orqj0whup lqyhvwruv1 Wkh vwdwh yduldeoh {| vxppdul}hv lqyhvwphqw rssruwxqlwlhv dw wlph w1 Wkxv wkh frqglwlrqdo fryduldqfh ehwzhhq wkh ulvn| dvvhw uhwxuq dqg wkh vwdwh yduldeoh phdvxuhv wkh delolw| ri wkh ulvn| dvvhw wr khgjh wlph0yduldwlrq lq lqyhvwphqw rssruwxqlwlhv1 Wklv fryduldqfh lv jlyhq e| +7143, Fry| +u|n > {|n , @ Fry| +u|n > u|n2 , @ # = Wkh prgho fdq eh vroyhg iru duelwudu|  # / exw zh irfxv dwwhqwlrq rq wkh fdvh zkhuh  # ? 31 Wklv fdvh fdswxuhv wkh qrwlrq wkdw vwrfnv duh _phdq0 uhyhuwlqj%> dq xqh{shfwhgo| kljk uhwxuq wrgd| uhgxfhv h{shfwhg uhwxuqv lq wkh ixwxuh/ dqg wkxv kljk vkruw0whup uhwxuqv whqg wr eh rvhw e| orzhu uhwxuqv ryhu wkh orqj whup1 Wklv rvhw uhgxfhv wkh frqglwlrqdo yduldqfh ri orqj0whup vwrfn uhwxuqv/ vlqfh Ydu| +u|n . u|n2 , @ 5Ydu| +u|n , . 5Fry| +u|n > u|n2 , @ 5Ydu| +u|n , . 5# ? 5Ydu| +u|n ,= 

+7144,

At t @ t*}| U@?}i uhL4 |i T@h@4i|ih3@|L? Lu @4TMi** @?_ VUih@ Ebbbc U L4||i_ |i |ih4 j 2 *2 L? |i *iu|@?_ t_i Lu EeH UULh_?}*) tL4i i^@|L?t ihi @hi t*}|*) @*|ihi_ uhL4 |i ULhhitTL?_?} i^@|L?t ? @4TMi** @?_ VUih@

:8

FKDSWHU 71 LV WKH VWRFN PDUNHW VDIHU IRU ORQJ0WHUP LQYHVWRUVB Wkdw lv/ wkh frqglwlrqdo yduldqfh ri vwrfn uhwxuqv grhv qrw jurz lq sursruwlrq zlwk wkh lqyhvwphqw krul}rq/ exw jurzv pruh vorzo|1 Li zh fdofxodwh d frqglwlrqdo yduldqfh udwlr/ Y U| +N, 

Ydu| +u|n . u|n2 . ===u|ng , > NYdu| +u|n ,

+7145,

wkh udwlr zloo eh ohvv wkdq rqh dw doo krul}rqv N1 Vwrfnv zloo dsshdu uhodwlyho| vdihu wr orqj0whup lqyhvwruv1 Wkh glvfxvvlrq deryh frqfhqwudwhv rq frqglwlrqdo yduldqfhv/ vlqfh wkhvh duh zkdw pdwwhu wr lqyhvwruv1 Wkh hpslulfdo olwhudwxuh rq phdq0uhyhuvlrq/ lqlwldwhg e| Srwhued dqg Vxpphuv +4<;;, dqg Idpd dqg Iuhqfk +4<;;e,/ w|slfdoo| hpskdvl}hv xqfrqglwlrqdo yduldqfhv lqvwhdg1 Wkhvh fdq ehkdyh txlwh glhuhqwo| iurp frqglwlrqdo yduldqfhv vlqfh Fry+u|n > {|n , @ Fry+{| . x|n > {|n , @ !2% .  # =

+7146,

Shuvlvwhqfh lq wkh surfhvv iru {|n fdq pdnh wklv xqfrqglwlrqdo fryduldqfh }hur/ hyhq li wkh frqglwlrqdo fryduldqfh lv qhjdwlyh1 Fdpsehoo +4<<4, dqg Fdpsehoo/ Or/ dqg PdfNlqod| +4<<:/ Fkdswhu :, hpskdvl}h wklv srlqw1 Pruh jhqhudoo|/ wkh xqfrqglwlrqdo yduldqfh udwlr/ Y U+N, 

Ydu+u|n . u|n2 . ===u|ng , Y U| +N, @ > NYdu+u|n , 4  U2 +N,

+7147,

zkhuh U2 +N, lv wkh h{sodqdwru| srzhu ri d uhjuhvvlrq ri wkh N0shulrg vwrfn uhwxuq rqwr wkh vwdwh yduldeoh {| 1 Wkxv wkh xqfrqglwlrqdo yduldqfh udwlr lv dozd|v juhdwhu wkdq wkh frqglwlrqdo yduldqfh udwlr> hpslulfdo uhvxowv xvlqj wkh iruphu xqghuvwdwh wkh ulvn0uhgxfwlrq wkdw lv uhohydqw iru orqj0whup lq0 yhvwruv1 Wkh glhuhqfh ehwzhhq wkh wzr yduldqfh udwlrv fdq eh vxevwdqwldo/ vlqfh dxwkruv vxfk dv Idpd dqg Iuhqfk +4<;;d, dqg Fdpsehoo dqg Vkloohu +4<;;d/e, kdyh irxqg orqj0krul}rq U2 vwdwlvwlfv dv odujh dv 73(1 Zh h{soruh wklv lvvxh hpslulfdoo| lq vhfwlrq 715151 Vroylqj wkh prgho Wkh prgho lv d vshfldo fdvh ri wkh jhqhudo YDU v|vwhp1 vroxwlrq wdnhv wkh irup | @ df . d {|c f|  z| @ ef . e {| . e2 {2| =

:9

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70

80

90

100

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0

10

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70

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Wklv lv htxdwlrq +5154, zkhq lqyhvwphqw rssruwxqlwlhv duh frqvwdqw1 Zh kdyh vkrzq lq vhfwlrq 51516 wkdw wkh Hxohu htxdwlrq iru dq lqyhvwru zlwk srzhu xwlolw| ri frqvxpswlrq dqg qr oderu lqfrph lpsolhv wkh iroorzlqj h{suhvvlrq iru wkh ulvn0suhplxp rq wkh ulvn| dvvhw +vhh htxdwlrq ^5176`,= H| u|n  us .

  4 Ydu| +u|n , @  Fry| fo|n  fo| > u|n = 5

+D58,

ru/ jlyhq rxu dvvxpswlrqv derxw wkh lqyhvwphqw rssruwxqlw| vhw/   4  .  2 @  Fry| fo|n  fo| > u|n = 5

+D59,

Zh fdq frpsxwh wkh fryduldqfh whup lq wkh uljkw0kdqg vlgh ri htxdwlrq +D59, e| qrwlqj wkdw +9175, lpsolhv fo|n  fo| @ eo +z|on  z| ,> 47

+D5:,

FKDSWHU 51 DSSHQGL[= PDWKHPDWLFDO GHULYDWLRQV vr wkdw

    Fry| fo|n  fo| > u|n @ eo Fry| z|on  z| > u|n @ eo Fry| +uRc|n > u|n , @ eo o| 2 >

zkhuh wkh vhfrqg htxdolw| rewdlqv iurp +D56,/ dqg wkh wklug htxdolw| rewdlqv iurp +D57,1 Wkhuhiruh/ 4  .  2 @ eo o| 2 > 5 iurp zklfk +9176, lq wh{w iroorzv1 Rswlpdo frqvxpswlrq uxoh Wr ghulyh wkh rswlpdo frqvxpswlrq uxoh/ qrwh wkdw wkh orj ri wkh Hxohu htxdwlrq +9174, zlwk l @ s |lhogv wkh iroorzlqj htxdwlrq iru h{shfwhg orj frqvxpswlrq jurzwk=    o  o  o  o  4   4 o o o > H uRc|n . orj  . Ydu uRc|n   f|n  f| H| f|n  f| @  5 +D5;, zkhuh  o  +4   _ ,/ dqg L kdyh vxssuhvvhg wkh vxevfulsw w iurp wkh frq0 glwlrqdo prphqwv rq wkh uljkw0kdqg0vlgh ehfdxvh o|  o lpsolhv wkdw wkh| o o o duh frqvwdqw1 Pruhryhu/ htxdwlrq +D5:, lpsolhv Ydu| ^uRc| n  +f|n  f| ,` @ o +4  eo ,2 Ydu+uRc| n ,1 Rq wkh rwkhu kdqg/ iurp htxdwlrq +D5:, dqg wkh orj exgjhw frqvwudlqw +D56, zh kdyh  o  o  o o H| f|n  f| @ e H| z|n  z|   o o o o o o o o o @ eo H uRc| n  eS ef . en . e S +4  e, z|+= D5<, Htxdol}lqj wkh uljkw0kdqg vlgh ri htxdwlrqv +D5;, dqg +D5<,/ dqg lghq0 wli|lqj frhflhqwv/ zh rewdlq wzr htxdwlrqv1 Wkh uvw rqh lpsolhv eo @ 4> zkloh wkh vhfrqg rqh lpsolhv      o  4 4 4 . orj  o eof @  o o  eo H uRc| n e S      o 4  eo no = . +4  eo ,2 Ydu uRc| n 5 48

+D63,

FKDSWHU 51 DSSHQGL[= PDWKHPDWLFDO GHULYDWLRQV Vlqfh eo @ 4/ zh kdyh eof  H^fo|  z| ` @ orj+oS ,  orj+4 . oS ,1 Zh fdq hdvlo| vxevwlwxwh wkh orj0olqhdul}dwlrq frqvwdqwv oS dqg n o rxw iurp htxdwlrq +D63, dqg rewdlq      o  o  4  4 4 2 o o o o  e H uRc|n . orj  . +4  e , Ydu uRc|n = ef @ orj 4  h{s   5 +D64,

2e2

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Wr ghulyh wkh rswlpdo sruwirolr uxoh lq wkh hpsor|phqw vwdwh zh uvw qhhg wr ghulyh orjolqhdu h{suhvvlrqv iru wkh lqwhuwhpsrudo exgjhw frqvwudlqw +916<, dqg wkh Hxohu htxdwlrq +9173, iru dq hpsor|hg lqyhvwru1 Orjolqhdu lqwhuwhpsrudo exgjhw frqvwudlqw Zh fdq uhzulwh wkh lqwhuwhpsrudo exgjhw frqvwudlqw +916<, dv    Z|en Z| F|e O| e @ 4.  URc| n> O|n O| O| O|n ru/ lq orjv/ e z|en  o|n @ orj +h{s iz|  o| j  h{s ife|  o| j,  o|n . uRc| n=

+D65,

Zh fdq qrz olqhdul}h htxdwlrq +D65, e| wdnlqj d uvw0rughu Wd|oru h{0 sdqvlrq durxqg +fe|  o| , @ H^fe|  o| ` dqg +z|e  o| , @ H^z|e  o| `1 Wklv jlyhv e z|en  o|n  ne . e +z|  o| ,  eS +fe|  o| ,  o|n . uRc| n>

+D66,

zkhuh h{s iH^z|e  o| `j > 4 . h{s iH^z|e  o| `j  h{s iH^fe|  o| `j h{s iH^fe|  o| `j > eS @ 4 . h{s iH^z|e  o| `j  h{s iH^fe|  o| `j

e @

+D67, +D68,

dqg ne @ +4  e . eS , orj+4  e . eS ,  e orj+e , . eS orj+eS ,=

+D69,

Qrwh wkdw e > eS A 3 ehfdxvh Z| . O|  F|e A 3 dorqj wkh rswlpdo sdwk1 49

FKDSWHU 51 DSSHQGL[= PDWKHPDWLFDO GHULYDWLRQV Orjolqhdu Hxohu htxdwlrq Zh fdq zulwh htxdwlrq +9173, dv      4 @  e H| h{s orj    fe|n  fe| . uc|n      . +4   e , H| h{s orj o   fo|n  fe| . uc|n   e H| ^h{s i{|n j` . +4   e , H| ^h{s i||n j` >

+D6:,

zkhuh wkh qrwdwlrqdo fruuhvsrqghqfh ehwzhhq wkh uvw dqg vhfrqg olqh lv reylrxv/ dqg  o  +4   _ ,1 Wdnlqj d vhfrqg rughu Wd|oru h{sdqvlrq ri h{s i{|n j dqg h{s i||n j durxqg {| @ H| ^{|n ` dqg | | @ H| ^||n ` zh fdq zulwh=    4 2 e 4   H| h{s i{| j 4 . +{|n  {| , . +{|n  {| , 5    4 2 e . +4   , H| h{s i| | j 4 . +||n  | | , . +||n  || , 5     4 4 e e @  h{s i{| j 4 . Ydu| +{|n , . +4   , h{s i| | j 4 . Ydu| +||n , = 5 5 Ilqdoo|/ d uvw0rughu Wd|oru h{sdqvlrq durxqg }hur jlyhv     4 4 e e 4   4 . {| . Ydu| +{|n , . +4   , 4 . | | . Ydu| +||n , > 5 5 ru

   3 @  e orj   H| fe|n  fe| . H| ^uc|n `     4 . Ydu| uc|n   fe|n  fe| 5    . +4   e , orj  o  H| fo|n  fe| . H| ^uc|n `   o   4 e Ydu| uc|n   f|n  f| = 5

+D6;,

Rswlpdo sruwirolr uxoh Zh vwduw jxhvvlqj wkh ixqfwlrqdo irup ri wkh rswlpdo srolflhv lq wkh hp0 sor|phqw vwdwh lv= fe|  o| @ eef . ee +z|  o| , > e| @ e = 4:

+D6<,

FKDSWHU 51 DSSHQGL[= PDWKHPDWLFDO GHULYDWLRQV Qrwh wkdw zh fdq dovr zulwh wkh rswlpdo frqvxpswlrq srolf| lq wkh uhwluhphqw vwdwh +9175, lq wkh vdph irup dv htxdwlrq +BB,= fo|n  o|n @ eof . eo +z|n  o|n , >

+D73,

zkhuh eo @ 41 Vxewudfwlqj wkh orj Hxohu htxdwlrq +D6;, iru u|n @ us iurp wkh orj Hxohu htxdwlrq +D6;, iru u|n @ u|n |lhogv= H| u|n  us .

  4 Ydu| +u|n , @  e Fry| fe|n  fe| > u|n 5   . +4   e , Fry| fo|n  fe| > u|n +D74, =

Exw htxdwlrqv +D6<, dqg +D73,/ wkh orj0olqhdu lqwhuwhpsrudo exgjhw frq0 vwudlqw +D66, dqg wkh wulyldo htxdolw| fr|n  fe| @ +fr|n  o|n ,  +fe|  o| , . +o|n  o| ,>

+D75,

lpso| wkdw    e  r Fry| fr|n  fe| > u|n @ Fry| er uRc| n . +4  e, +o|n  o| , > u|n @ er e| 2 . +4  er , 1 >

+D76,

iru v @ h> u1 Wkh vhfrqg olqh iroorzv iurp wkh dvvxpswlrqv rq dvvhw uhwxuqv dqg oderu lqfrph1 Vxevwlwxwlqj edfn lqwr htxdwlrq +D74, dqg xvlqj htxdwlrq +D59, zh qg   4  .  2 @  + e ee . +4   e ,, e| 2 .  e +4  ee ,  1 > 5

+D77,

iurp zklfk htxdwlrq +9178, lq wh{w rewdlqv lpphgldwho|1 Rswlpdo frqvxpswlrq uxoh Wkh orj0Hxohu htxdwlrq +D6;, iru l @ s dqg wkh wulyldo htxdolw| +D75, lpso|      e H| fe|n  o|n . +4   e , H| fo|n  o|n @ +fe|  o| , . e| > +D78, zkhuh 4

e|  e @ 

   4 e  e e e o H uRc|n . Y . orj  . +4   , orj   j> 5 4;

+D79,

FKDSWHU 51 DSSHQGL[= PDWKHPDWLFDO GHULYDWLRQV dqg Y

e

e @ ^ e +4  ee ,2 . +4   e ,+4  eo ,2 ` Ydu^uRc| n` e e . +4  e , Ydu^o|n ` e 5 e +4  ee ,+4  ee , Fry^uRc| n > o|n`=

+D7:,

Li zh vxevwlwxwh htxdwlrqv +D6<, dqg +D73, lqwr htxdwlrq +D78, zh rewdlq ef . e H| ^z|n  o|n ` @ e . eef . ee +z|  o| , >

+D7;,

zkhuh ef @  e eef .+4 e ,eof dqg e @  e ee .+4 e ,eo 1 Ixuwkhu vxevwlwxwlrq ri wkh orj exgjhw frqvwudlqw lq wkh hpsor|phqw vwdwh +D66, dqg jxhvv +D6<, lq wkh ohiw0kdqg vlgh ri htxdwlrq +D7;, |lhogv   e ef . e +e  eS ee , +z|  o| , . e n e  eS eef  j . H uRc| n @ e . eef . ee +z|e  o| , = Lghqwli|lqj frhflhqwv rq erwk vlghv ri wklv htxdwlrq zh jhw wkh iroorzlqj wzr0htxdwlrq v|vwhp= e +e  eS ee , @ ee >  e  e e e ef . e n  eS eef  j . H uRc| n @ . ef= Zh fdq vroyh wklv v|vwhp uhfxuvlyho|/ vlqfh wkh uvw htxdwlrq ghshqgv rqo| rq ee dqg wkh vhfrqg rq ee dqg eef 1 Vlpsoh dojheudlf pdqlsxodwlrq ri wkh uvw htxdwlrq jlyhv wkh iroorzlqj txdgudwlf htxdwlrq iru ee = 3 @  e eS +ee ,2 . ^4   e e . +4   e , eS ` ee  +4   e , e =

+D7<,

Wkh h{suhvvlrq iru eef lv jlyhq e|    4 e  e 4 4 e . + orj  . +4   e , orj  o ,+D83, ef @   e H uRc| n n    4 e e e e o e . Y  +4  e , j  +4   , ef  n > 5 zlwk

n @ +4   e , . eS e A 3=

4<

+D84,