Campbell Viceira

8/9/2019 Campbell Viceira

1/218

5|h@|i}U tti| **LU@|L?G

Lh|uL*L LUi uLh wL?}Aih4 W?it|Lht

aL? v @4TMi** @?_ wt VUih@

Th* 2ff


2/218

hiu@Ui

Wklv errn kdv lwv ruljlqv lq Mrkq .dpsehoo*v judgxdwh fodvv lq dvvhw sulf0lqj dw Kduydug Xqlyhuvlw| lq 4


3/218

.KDSWHU 31 SUHID.H

vhoyhv wkrxjkw ri dv kxpeoh/ frpshwhqw shrsoh/ rq d ohyho zlwk ghqwlvwv/

wkdw zrxog eh vsohqglg$% Iru pxfk ri wkh 53wk .hqwxu|/ hfrqrplvwv frq0fhqwudwhg rq lpsurylqj hfrqrplf rxwfrphv wkurxjk jryhuqphqw hfrqrplfsrolf|> Nh|qhv pd| kdyh lpdjlqhg wkh hfrqrplvw dv ruwkrgrqwlvw/ lqwhuyhq0lqj zlwk wkh sdlqixo exw hhfwlyh wrrov ri prqhwdu| dqg vfdo srolf|1 Wrgd|/ghqwlvwv vshqg pxfk ri wkhlu wlph jlylqj dgylfh rq rudo k|jlhqh> vlploduo|/hfrqrplvwv fdq wu| wr surylgh xvhixo dgylfh wr lpsuryh wkh p|uldg hfrqrplfghflvlrqv wkdw sulydwh lqglylgxdov duh dvnhg wr pdnh1 Wklv errn uhsuhvhqwvdq dwwhpsw dw qrupdwlyh hfrqrplfv ri wklv vruw1

Ri frxuvh/ rswlpdo sruwirolr ghflvlrqv ghshqg rq wkh ghwdlov ri wkh hqyl0urqphqw= wkh qdqfldo dvvhwv wkdw duh dydlodeoh/ wkhlu h{shfwhg uhwxuqv dqgulvnv/ dqg wkh suhihuhqfhv dqg flufxpvwdqfhv ri lqyhvwruv1 Wkhvh ghwdlovehfrph sduwlfxoduo| lpsruwdqw iru orqj0whup lqyhvwruv/ zkr duh wkh vxemhfwri wklv errn1 Vxfk lqyhvwruv pxvw frqfhuq wkhpvhoyhv qrw rqo| zlwk h{0shfwhg uhwxuqv dqg ulvnv wrgd|/ exw zlwk wkh zd| lq zklfk h{shfwhg uhwxuqvdqg ulvnv pd| fkdqjh ryhu wlph1 Wkh| pxvw dovr frqvlghu wkhlu lqfrphwrgd| dqg wkhlu lqfrph survshfwv iru wkh ixwxuh1 Dffruglqjo| wklv errnhpskdvl}hv wkh vwdwlvwlfdo dqdo|vlv ri dvvhw uhwxuqv dqg ri lqfrph1

Dfdghplf hfrqrplvwv duh qrw wkh rqo|/ ru hyhq wkh ohdglqj vrxufh ri qdq0fldo dgylfh iru orqj0whup lqyhvwruv1 D vl}hdeoh qdqfldo sodqqlqj lqgxvwu|kdv dulvhq wr khos shrsoh vdyh iru uhwluhphqw1 Wklv lqgxvwu| lv kljko| vrsklv0wlfdwhg lq vrph uhvshfwv +iru h{dpsoh lq wd{ sodqqlqj,/ exw whqgv wr uho| rquxohv ri wkxpe wr jxlgh wkh wudghr ehwzhhq ulvn dqg uhwxuq1 .rqvhuydwlyhlqyhvwruv/ iru h{dpsoh/ duh dgylvhg wr krog ihzhu htxlwlhv dqg pruh erqgv

wkdq djjuhvvlyh lqyhvwruv> |rxqjhu lqyhvwruv duh wrog wkdw lw lv dssursuldwhiru wkhp wr wdnh juhdwhu htxlw| ulvn wkdq roghu lqyhvwruv1 Dq lpsruwdqwsxusrvh ri wklv errn lv wr hydoxdwh vxfk uxohv ri wkxpe dqg sodfh wkhp rqd up vflhqwlf irxqgdwlrq1

Zh rzh wkdqnv wr pruh shrsoh wkdq zh fdq olvw e| qdph1 Glun Mhqwhu dqgMrvk Zklwh surylghg vnloohg dqg wluhohvv uhvhdufk dvvlvwdqfh1 Ohzlv .kdqzdv rxu frdxwkru rq wkh sdshu +.dpsehoo/ .kdq/ dqg Ylfhlud 5333, wkdwirupv wkh edfnerqh ri .kdswhu 71 Mrdr .rffr/ Iudqflvfr Jrphv/ dqg Sdv0fdo Pdhqkrxw zhuh frdxwkruv ri wkh sdshu +.dpsehoo/ .rffr/ Jrphv/ dqgPdhqkrxw 5334, wkdw irupv wkh edfnerqh ri .kdswhu :1 Wkh| dovr zurwhzlwk xv d sdshu rq wkh dffxudf| ri wkh orjolqhdu dssur{lpdwh dssurdfk wr

sruwirolr fkrlfh +.dpsehoo/ .rffr/ Jrphv/ Pdhqkrxw/ dqg Ylfhlud 5334,1Zh kdyh ehqhwhg iurp wkh surelqj txhvwlrqv dqg lqvljkwixo frpphqwv riwkhvh dqg rwkhu vwxghqwv dqg froohdjxhv/ hvshfldoo| Gdylg Edwhv/ PlfkdhoEuhqqdq/ Pdun Nulw}pdq/ Hjlo Pdwvhq/ Ere Phuwrq/ Udidho Uhsxoor/ PdunUxelqvwhlq/ Sdxo Vrghuolqg/ dqg Mhvvlfd Zdfkwhu1 Mrkq .dpsehoo*v sduw0

ll


4/218

.KDSWHU 31 SUHID.H

qhuv dw Duurzvwuhhw .dslwdo/ qrwdeo| Mrkq .dshfl/ Euxfh .odunh/ dqg Shwhu

Udwkmhqv/ kdyh frqvwdqwo| fkdoohqjhg xv wr uhodwh rxu dfdghplf zrun wr wkhfrqfhuqv ri lqvwlwxwlrqdo lqyhvwruv dqg wkh dvvhw pdqdjhuv zkr vhuyh wkhp1Zh dfnqrzohgjh Plfkdho Euhqqdq/ Hgxdugr Vfkzduw}/ dqg Urqdog Odjqdgr+4


5/218

@T|ih

W?|hL_U|L?

Rqh ri wkh prvw lpsruwdqw ghflvlrqv pdq| shrsoh idfh lv wkh fkrlfh ri dsruwirolr ri dvvhwv iru uhwluhphqw vdylqjv1 Wkhvh dvvhwv pd| eh khog dv dvxssohphqw wr ghqhg0ehqhw sxeolf ru sulydwh shqvlrq sodqv> ru wkh| pd|eh dffxpxodwhg lq d ghqhg0frqwulexwlrq shqvlrq sodq/ dv wkh pdmru vrxufhri uhwluhphqw lqfrph1 Lq hlwkhu fdvh/ d gl}}|lqj duud| ri dvvhwv lv dydlodeoh1

.rqvlghu iru h{dpsoh wkh lqfuhdvlqj vhw ri fkrlfhv rhuhg e| WLDD0.UHI/ wkh sulqflsdo shqvlrq rujdql}dwlrq iru xqlyhuvlw| hpsor|hhv lq wkhXqlwhg Vwdwhv1 Xqwlo 4


6/218

.KDSWHU 41 LQWURGX.WLRQ

phdq0yduldqfh dqdo|vlv ri Pdunrzlw} +4 wklv pdnhv sruwirolr fkrlfh wkh0

ru| wkh ruljlqdo vxemhfw ri prghuq qdqfh1 Pdunrzlw} vkrzhg krz lqyhvwruvvkrxog slfn dvvhwv li wkh| fduh rqo| derxw wkh phdq dqg yduldqfhru htxly0dohqwo| wkh phdq dqg vwdqgdug ghyldwlrqri sruwirolr uhwxuqv ryhu d vlqjohshulrg1

Wkh uhvxowv ri klv dqdo|vlv duh vkrzq lq wkh vwdqgdug phdq0vwdqgdugghyldwlrq gldjudp/ Iljxuh 4141 +D pxfk pruh fduhixo pdwkhpdwlfdo h{sod0qdwlrq fdq eh irxqg lq wkh qh{w fkdswhu1, Iru vlpsolflw| wkh jxuh frqvlghuvwkuhh dvvhwv/ vwrfnv/ erqgv/ dqg fdvk +qrw olwhudoo| fxuuhqf|/ exw d vkruw0whupprqh| pdunhw ixqg,1 Wkh yhuwlfdo d{lv vkrzv h{shfwhg uhwxuq/ dqg wkh krul0}rqwdo d{lv vkrzv ulvn dv phdvxuhg e| vwdqgdug ghyldwlrq1 Vwrfnv duh vkrzqdv rhulqj d kljk phdq uhwxuq dqg d kljk vwdqgdug ghyldwlrq/ erqgv d orzhuphdq dqg orzhu vwdqgdug ghyldwlrq1 .dvk kdv d orzhu phdq uhwxuq djdlq/exw lv ulvnohvv ryhu rqh shulrg/ vr lw lv sorwwhg rq wkh yhuwlfdo }hur0ulvn d{lv1+Lq wkh suhvhqfh ri lqdwlrq ulvn/ qrplqdo prqh| pdunhw lqyhvwphqwv duhqrw olwhudoo| ulvnohvv lq uhdo whupv/ exw wklv vkruw0whup lqdwlrq ulvn lv vpdoohqrxjk wkdw lw lv frqyhqwlrqdo wr ljqruh lw1 Zh iroorz wklv frqyhqwlrq khuhdqg uhwxuq wr wkh lvvxh lq wkh qh{w fkdswhu1,

Wkh fxuyhg olqh lq Iljxuh 414 vkrzv wkh vhw ri phdqv dqg vwdqgdugghyldwlrqv wkdw fdq eh dfklhyhg e| frpelqlqj vwrfnv dqg erqgv lq d ulvn|sruwirolr1 Zkhq fdvk lv dgghg wr d sruwirolr ri ulvn| dvvhwv/ wkh vhw riphdqv dqg vwdqgdug ghyldwlrqv wkdw fdq eh dfklhyhg lv d vwudljkw olqh rqwkh gldjudp frqqhfwlqj fdvk wr wkh ulvn| sruwirolr1 Dq lqyhvwru zkr fduhvrqo| derxw wkh phdq dqg vwdqgdug ghyldwlrq ri klv sruwirolr zloo fkrrvh d

srlqw rq wkh vwudljkw olqh looxvwudwhg lq wkh jxuh/ wkdw lv wdqjhqw wr wkhfxuyhg olqh1 Wklv vwudljkw olqh/ wkh phdq0yduldqfh hflhqw iurqwlhu/ rhuvwkh kljkhvw phdq uhwxuq iru dq| jlyhq vwdqgdug ghyldwlrq1 Wkh srlqw zkhuhwkh vwudljkw olqh wrxfkhv wkh fxuyhg olqh lv d _wdqjhqf| sruwirolr% ri ulvn|dvvhwv/ pdunhg lq wkh jxuh dv _Ehvw Pl{ ri Vwrfnv dqg Erqgv%1

Wkh vwulnlqj frqfoxvlrq ri wklv dqdo|vlv lv wkdw doo lqyhvwruv zkr fduh rqo|derxw phdq dqg vwdqgdug ghyldwlrq zloo krog wkh vdph sruwirolr ri ulvn| dv0vhwv/ wkh xqltxh ehvw pl{ ri vwrfnv dqg erqgv1 .rqvhuydwlyh lqyhvwruv zloofrpelqh wklv sruwirolr zlwk fdvk wr dfklhyh d srlqw rq wkh phdq0yduldqfh h0flhqw iurqwlhu wkdw lv orz grzq dqg wr wkh ohiw> prghudwh lqyhvwruv zloo uhgxfhwkhlu fdvk kroglqjv/ prylqj xs dqg wr wkh uljkw> djjuhvvlyh lqyhvwruv pd|

hyhq eruurz wr ohyhudjh wkhlu kroglqjv ri wkh wdqjhqf| sruwirolr/ uhdfklqj dsrlqw rq wkh vwudljkw olqh wkdw lv hyhq ulvnlhu wkdq wkh wdqjhqf| sruwirolr1Exw qrqh ri wkhvh lqyhvwruv vkrxog dowhu wkh uhodwlyh sursruwlrqv ri ulvn|dvvhwv lq wkh wdqjhqf| sruwirolr1 Wklv uhvxow lv wkh pxwxdo ixqg wkhruhp riWrelq +4


7/218


Expected

Return

Cash

Conservative

Investor

Risk

Best Mix of

Stocks and Bonds

Stocks

Moderate

Investor

Aggressive

Investor

Bonds

Iljxuh 414= Phdq0vwdqgdug ghyldwlrq gldjudp

6


8/218


Ilqdqfldo sodqqhuv kdyh wudglwlrqdoo| uhvlvwhg wkh vlpsoh lqyhvwphqw dg0

ylfh hperglhg lq Iljxuh 4141 Wklv uhvlvwdqfh pd| wr vrph h{whqw eh vhoi0vhuylqj> dv Shwhu Ehuqvwhlq srlqwv rxw lq klv 4


9/218


Iljxuh 415= Wkh Dvvhw Doorfdwlrq Sx}}oh

8


10/218


Rqh srvvleoh h{sodqdwlrq iru wklv sdwwhuq ri dgylfh lv wkdw djjuhvvlyh

lqyhvwruv duh xqdeoh wr eruurz dw wkh ulvnohvv lqwhuhvw udwh/ dqg wkxv fdqqrwuhdfk wkh xsshu uljkw sruwlrq ri wkh vwudljkw olqh lq Iljxuh 4141 Lq wklvvlwxdwlrq/ djjuhvvlyh lqyhvwruv vkrxog pryh dorqj wkh fxuyhg olqh/ lqfuhdvlqjwkhlu doorfdwlrq wr vwrfnv dqg uhgxflqj wkhlu doorfdwlrq wr erqgv1 Wkhglfxow| zlwk wklv h{sodqdwlrq lv wkdw lw rqo| dssolhv rqfh wkh frqvwudlqw rqeruurzlqj vwduwv wr elqg rq lqyhvwruv/ wkdw lv/ rqfh fdvk kroglqjv kdyh ehhquhgxfhg wr }hur> exw wkh erqg0vwrfn udwlr lq .dqqhu/ Pdqnlz/ dqg Zhlo*vWdeoh 4 ydulhv hyhq zkhq fdvk kroglqjv duh srvlwlyh1 Howrq dqg Juxehu+5333, uhvsrqg wr wklv glfxow| e| dujxlqj wkdw wkhvh fdvk kroglqjv duh dvshfldo oltxlglw| uhvhuyh ghwhuplqhg rxwvlgh wkh phdq0yduldqfh dqdo|vlv1 Lwlv txlwh sodxvleoh wkdw wkh 8( fdvk kroglqjv vxjjhvwhg e| Ilgholw| dqg PhuulooO|qfk duh d oltxlglw| uhvhuyh/ exw wkh rwkhu fdvk kroglqjv lq wkh wdeoh dsshduwrr odujh wr eh h{sodlqhg lq wklv pdqqhu1

Wklv errn dujxhv wkdw lw lv srvvleoh wr pdnh vhqvh ri erwk wkh dvvhw door0fdwlrq sx}}oh/ dqg wkh whqghqf| ri qdqfldo sodqqhuv wr uhfrpphqg ulvnlhusruwirolrv wr |rxqj lqyhvwruv1 Wkh nh| lv wr uhfrjql}h wkdw rswlpdo sruwir0olrv iru orqj0whup lqyhvwruv qhhg qrw eh wkh vdph dv iru vkruw0whup lqyhvwruv1Orqj0whup lqyhvwruv/ zkr ydoxh zhdowk qrw iru lwv rzq vdnh exw iru wkh vwdq0gdug ri olylqj wkdw lw fdq vxssruw/ pd| mxgjh ulvnv yhu| glhuhqwo| iurpvkruw0whup lqyhvwruv1 .dvk/ iru h{dpsoh/ lv ulvn| lq wkh orqj whup hyhqwkrxjk lw lv vdih lq wkh vkruw whup/ ehfdxvh fdvk kroglqjv pxvw eh uhlqyhvwhglq wkh ixwxuh dw xqnqrzq uhdo lqwhuhvw udwhv1 Lqdwlrq0lqgh{hg erqgv/ rqwkh rwkhu kdqg/ surylgh d nqrzq vwuhdp ri orqj0whup uhdo sd|phqwv hyhq

wkrxjk wkhlu fdslwdo ydoxh lv xqfhuwdlq lq wkh vkruw whup1 Wkhuh lv frq0vlghudeoh hylghqfh wkdw vwrfnv/ wrr/ fdq vxssruw d vwdeoh vwdqgdug ri olylqjpruh vxffhvvixoo| wkdq wkhlu vkruw0whup sulfh yduldelolw| zrxog lqglfdwh1 Iruwkhvh uhdvrqv d orqj0whup lqyhvwru pd| eh zloolqj wr krog pruh vwrfnv dqgerqgv/ dqg ohvv fdvk/ wkdq d vkruw0whup lqyhvwru zrxog gr> dqg d frqvhu0ydwlyh orqj0whup lqyhvwru pd| krog d sruwirolr wkdw lv grplqdwhg e| erqgvudwkhu wkdq fdvk1

Oderu lqfrph lv dovr lpsruwdqw iru orqj0whup lqyhvwruv1 Rqh fdq wklqnri zrunlqj lqyhvwruv dv lpsolflwo| kroglqj dq dvvhw/ kxpdq zhdowk/ zkrvhglylghqgv htxdo oderu lqfrph1 Wklv dvvhw lv qrqwudgdeoh/ vr lqyhvwruv fdqqrwvhoo lw> exw wkh| fdq dgmxvw wkhlu qdqfldo dvvhw kroglqjv wr wdnh dffrxqw

ri wkhlu lpsolflw kroglqjv ri kxpdq zhdowk1 Iru prvw lqyhvwruv/ kxpdqzhdowk lv vxflhqwo| vwdeoh lq ydoxh wr wlow qdqfldo sruwirolrv wrzdugv juhdwhukroglqjv ri ulvn| dvvhwv1

Dw d wkhruhwlfdo ohyho/ wkhvh srlqwv kdyh ehhq xqghuvwrrg iru pdq| |hduv1Vdpxhovrq +4


11/218


uvw ghvfulehg wkh uhvwulfwlyh frqglwlrqv xqghu zklfk orqj0krul}rq lqyhvwruv

vkrxog pdnh wkh vdph ghflvlrqv dv vkruw0krul}rq lqyhvwruv1 Prgljoldql dqgVxwfk +4


12/218


wlrqv ri nqrzq h{dfw vroxwlrqv/ vr wkh| duh dffxudwh lq d qhljkerukrrg ri

wkrvh vroxwlrqv1 Wkh| rhu dqdo|wlfdo lqvljkwv lqwr lqyhvwru ehkdylru lq prg0hov wkdw idoo rxwvlgh wkh vwloo olplwhg fodvv wkdw fdq eh vroyhg h{dfwo|1 Wklverrn rhuv h{dpsohv ri doo wkuhh dssurdfkhv wr orqj0whup sruwirolr fkrlfh/exw wkh pdmru hpskdvlv lv rq dssur{lpdwh dqdo|wlfdo vroxwlrq phwkrgv dqgrq wkh ghyhorsphqw ri hpslulfdo prghov wkdw fdq eh xvhg e| sudfwlwlrqhuvdv zhoo dv dfdghplfv1

Wkh uhvw ri wkh errn lv rujdql}hg dv iroorzv1 .kdswhu 5 uvw uhylhzvwkh wudglwlrqdo phdq0yduldqfh dqdo|vlv/ vkrzlqj krz lw fdq eh irxqghg rqxwlolw| wkhru|1 Wkh fkdswhu dujxhv wkdw wkh ehqfkpdun prgho ri xwlolw|vkrxog dvvxph wkdw uhodwlyh ulvn dyhuvlrq lv lqghshqghqw ri zhdowk1 Zlwkwklv dvvxpswlrq/ wkhuh duh zhoo0nqrzq frqglwlrqv xqghu zklfk orqj0whup lq0yhvwruv vkrxog lqyhvw p|rslfdoo|/ fkrrvlqj wkh vdph sruwirolrv dv vkruw0whuplqyhvwruv1 .kdswhu 5 h{sodlqv wkhvh frqglwlrqv/ gxh ruljlqdoo| wr Phuwrq+4


13/218


rqh vkrxog lqfuhdvh wkhlu dyhudjh kroglqjv ri ulvn| dvvhwv zkrvh uhwxuqv duh

qhjdwlyho| fruuhodwhg zlwk wkh uhzdugv iru ulvnehdulqj> iru h{dpsoh/ wkh|vkrxog lqfuhdvh wkhlu dyhudjh doorfdwlrq wr vwrfnv ehfdxvh wkh vwrfn pdu0nhw dsshduv wr phdq0uhyhuw/ grlqj uhodwlyho| srruo| diwhu sulfh lqfuhdvhv dqguhodwlyho| zhoo diwhu sulfh ghfolqhv1 Wkhvh qglqjv duh dq hpslulfdo ghyho0rsphqw ri Phuwrq*v +4


14/218


qdqfldo sodqqhuv1 Wkh fkdswhu dovr vkrzv wkdw glhuhqw krxvhkrogv fdq

kdyh yhu| glhuhqw ulvn surshuwlhv ri wkhlu oderu lqfrph/ dqg wkdw wklv kdvodujh hhfwv rq rswlpdo sruwirolr doorfdwlrqv1 Vhoi0hpsor|hg krxvhkrogv/ iruh{dpsoh/ kdyh oderu lqfrph wkdw lv sduwlfxoduo| yrodwloh dqg kljko| fruuh0odwhg zlwk vwrfn uhwxuqv> wkhvh krxvhkrogv vkrxog eh fdxwlrxv derxw htxlw|lqyhvwphqwv1 Ilqdoo|/ wkh fkdswhu uhylhzv wkh h{lvwlqj hpslulfdo hylghqfhrq krz krxvhkrogv dfwxdoo| gr lqyhvw ryhu wkh olih f|foh1 Wkh hylghqfh lviudjphqwdu|/ dqg lw lv fohdu wkdw krxvhkrog lqyhvwphqw vwudwhjlhv duh h{wudru0glqdulo| glyhuvh> exw d ihz sdwwhuqv gr dsshdu frqvlvwhqw zlwk wkh qrupdwlyhuhfrpphqgdwlrqv ri wkh olih0f|foh prgho1 Iru h{dpsoh/ wkhuh lv vrph whq0ghqf| iru krxvhkrog ulvn h{srvxuh wr ghfolqh zlwk djh lq odwhu |hduv/ dqgzhdowk| krxvhkrogv wkdw rzq sulydwh exvlqhvvhv whqg wr rzq ihzhu sxeolfo|wudghg htxlwlhv wkdq rwkhu zhdowk| krxvhkrogv1

Wkurxjkrxw wkh errn zh ghulyh sruwirolr ghflvlrqv xvlqj vwdqgdug 0qdqfh prghov ri suhihuhqfhv1 Zh dvvxph wkdw lqyhvwruv ghulyh xwlolw| iurpfrqvxpswlrq/ zklfk lq wxuq lv vxssruwhg e| qdqfldo zhdowk +dqg srvvleo|oderu lqfrph,1 Wkh xwlolw| ixqfwlrq wdnhv d vwdqgdug irup lq zklfk uhodwlyhulvn dyhuvlrq lv frqvwdqw1e Rxu frqwulexwlrqv duh lq prghoolqj wkh qdq0fldo hqylurqphqw/ dqg ghyhorslqj wkh olqn ehwzhhq wkdw hqylurqphqw dqgsruwirolr fkrlfh/ udwkhu wkdq lq prghoolqj lqyhvwru suhihuhqfhv1

Vrph lqyhvwphqw sudfwlwlrqhuv duh xqfrpiruwdeoh zlwk wkh dfdghplf xvhri xwlolw| ixqfwlrqv1 Krzhyhu lqyhvwru suhihuhqfhvlq sduwlfxodu lpsdwlhqfhdqg dyhuvlrq wr ulvngr lqxhqfh rswlpdo sruwirolrv1 Vwxghqwv ri sruwirolrfkrlfh kdyh qr dowhuqdwlyh exw wr prgho suhihuhqfhv/ qr pdwwhu krz kdug wkdw

wdvn pd| eh1 Zh uhjdug xwlolw| ixqfwlrqv dv d frqyhqlhqw zd| wr fdswxuhlqwxlwlyh qrwlrqv vxfk dv wkh glvwlqfwlrq ehwzhhq frqvhuydwlyh dqg djjuhvvlyhlqyhvwruv1

Uhfhqwo| wkhuh kdv ehhq juhdw lqwhuhvw lq wkh hog ri ehkdylrudo qdqfh/zklfk srvwxodwhv wkdw vrph lqyhvwruv kdyh qrqvwdqgdug suhihuhqfhv1 Riwhqwkhvh suhihuhqfhv duh prghoohg xvlqj h{shulphqwdo hylghqfh iurp sv|fkrorj|1Wkh survshfw wkhru| ri Ndkqhpdq dqg Wyhuvn| + 4


16/218

@T|ih 2

)LTU Lh|uL*L LUi

Lq wklv fkdswhu zh uhylhz wkh wkhru| ri sruwirolr fkrlfh iru vkruw0whup lq0yhvwruv/ dqg h{sodlq wkrvh vshfldo fdvhv lq zklfk orqj0whup lqyhvwruv vkrxogpdnh wkh vdph fkrlfhv dv vkruw0whup lqyhvwruv1 Lq wkhvh vshfldo fdvhv wkhlqyhvwphqw krul}rq lv luuhohydqw> sruwirolr fkrlfh lv vdlg wr eh p|rslf/ eh0fdxvh lqyhvwruv ljqruh zkdw zloo kdsshq eh|rqg wkh lpphgldwh qh{w shulrg1Wkurxjkrxw wkh fkdswhu zh dvvxph wkdw lqyhvwruv kdyh qdqfldo zhdowk exwqr oderu lqfrph1

Vhfwlrq 514 ghvfulehv rswlpdo sruwirolr fkrlfh iru vkruw0whup lqyhvwruv1Zh ehjlq lq vhfwlrq 51414 zlwk wkh fodvvlf phdq0yduldqfh dqdo|vlv/ dvvxplqjwkdw lqyhvwruv fduh gluhfwo| derxw wkh phdq dqg yduldqfh ri sruwirolr uhwxuqvryhu rqh shulrg1 Wkhq lq vhfwlrq 51415 zh ghulyh vlplodu uhvxowv dvvxplqj

wkdw lqyhvwruv kdyh d xwlolw| ixqfwlrq ghqhg ryhu zhdowk dw wkh hqg ri rqhshulrg1 Zh glvfxvv dowhuqdwlyh dvvxpswlrqv wkdw fdq eh pdgh derxw xwlo0lw|/ dujxlqj wkdw wkhuh duh jrrg uhdvrqv wr suhihu vfdoh0lqghshqghqw xwlolw|ixqfwlrqv lq zklfk uhodwlyh ulvn dyhuvlrq grhv qrw ghshqg rq zhdowk1 Wkhvlpsohvw vfdoh0lqghshqghqw xwlolw| ixqfwlrq lv srzhu xwlolw|/ dqg zh vkrz krzwr ghulyh sruwirolr uhvxowv dqdorjrxv wr wkrvh ri wkh phdq0yduldqfh dqdo|vlv/dvvxplqj srzhu xwlolw| dqg orjqrupdoo| glvwulexwhg uhwxuqv1

Lq vhfwlrq 515 zh ghulyh frqglwlrqv xqghu zklfk wkh vdph sruwirolr fkrlfhlv rswlpdo iru orqj0whup lqyhvwruv1 Zh uvw dvvxph lq vhfwlrq 51514 wkdwlqyhvwruv kdyh srzhu xwlolw| ghqhg ryhu zhdowk pdq| shulrgv dkhdg/ dqgvkrz wkdw li lqyhvwruv fdq uhedodqfh wkhlu sruwirolrv hdfk shulrg/ wkh| vkrxog

lqyhvw p|rslfdoo| li dvvhw uhwxuqv duh lqghshqghqw dqg lghqwlfdoo| glvwulexwhg+LLG, ryhu wlph/ ru li xwlolw| wdnhv wkh orj irup1 Orj xwlolw| lv wkh vshfldofdvh ri srzhu xwlolw| lq zklfk erwk wkh frhflhqw ri uhodwlyh ulvn dyhuvlrqdqg wkh hodvwlflw| ri lqwhuwhpsrudo vxevwlwxwlrq lq frqvxpswlrq htxdo rqh1

45


17/218

.KDSWHU 51 P\RSL. SRUWIROLR .KRL.H

Wkh frqglwlrqv iru p|rslf sruwirolr fkrlfh duh txlwh vlpsoh/ dqg fdq

eh ghulyhg zlwkrxw wkh xvh ri dgydqfhg pdwkhpdwlfv1 Qrqhwkhohvv wkhvhfrqglwlrqv duh zlgho| plvxqghuvwrrg/ dqg rqh riwhq vhhv vshflrxv dujxphqwvwkdw wkhuh vkrxog eh krul}rq hhfwv hyhq zkhq wkh frqglwlrqv krog1 Lqvhfwlrq 51515 zh wu| wr h{srvh wkh idoodflhv lq wkhvh dujxphqwv1

Lq vhfwlrq 51516 zh frqvlghu lqyhvwruv zkr kdyh srzhu xwlolw| ghqhg ryhufrqvxpswlrq/ dqg vkrz wkdw sruwirolr fkrlfh zloo eh p|rslf xqghu wkh vdphfrqglwlrqv dv ehiruh1 Ilqdoo|/ lq vhfwlrq 51517 zh lqwurgxfh d jhqhudol}dwlrqri srzhu xwlolw|/ Hsvwhlq0]lq xwlolw|/ zklfk doorzv xv wr glvwlqjxlvk ehwzhhqwkh frhflhqw ri ulvn dyhuvlrq dqg wkh hodvwlflw| ri lqwhuwhpsrudo vxevwlwxwlrqlq frqvxpswlrq1 Srzhu xwlolw| olqnv wkhvh frqfhswv wljkwo| wrjhwkhu/ pdnlqjrqh wkh uhflsurfdo ri wkh rwkhu/ exw wkh| duh glhuhqw frqfhswv wkdw sod| txlwhglhuhqw urohv lq wkh dqdo|vlv1 Zh vkrz wkdw sruwirolr fkrlfh lv p|rslf liuhodwlyh ulvn dyhuvlrq htxdov rqh/ uhjdugohvv ri wkh ydoxh ri wkh hodvwlflw| rilqwhuwhpsrudo vxevwlwxwlrq lq frqvxpswlrq1 Hsvwhlq0]lq xwlolw| zloo eh xvhgh{whqvlyho| lq wkh uhvw ri wkh errn1

Wkurxjkrxw wkh fkdswhu wkhuh lv d vwurqj hpskdvlv rq wkh glhuhqfhehwzhhq vlpsoh uhwxuqv dqg orj uhwxuqv/ dqg rq wkh dgmxvwphqwv wkdw duhqhhghg wr wudqvodwh iurp rqh w|sh ri uhwxuq wr wkh rwkhu1 Hohphqwdu|wuhdwphqwv ri sruwirolr fkrlfh riwhq jorvv ryhu wklv glhuhqfh/ exw lw lv fhqwudowr wkh wkhru| ri sruwirolr fkrlfh iru orqj0whup lqyhvwruv1

2 5Lh||ih4 TLh|uL*L ULUi

2 i@?@h@?Ui @?@*)tt

.krrvlqj wkh zhljkw rq d vlqjoh ulvn| dvvhw

.rqvlghu wkh iroorzlqj fodvvlf sruwirolr fkrlfh sureohp1 Wzr dvvhwv duhdydlodeoh wr dq lqyhvwru dw wlph w1 Rqh lv ulvnohvv/ zlwk vlpsoh uhwxuq Usc|niurp wlph w wr wlph w 4/ dqg rqh lv ulvn|1 Wkh ulvn| dvvhw kdv vlpsohuhwxuq U|n iurp wlph w wr wlph w 4/ zlwk frqglwlrqdo phdq H|U|n dqgfrqglwlrqdo yduldqfh 2| 1 Qrwh wkh wlplqj frqyhqwlrq wkdw uhwxuqv duh jlyhqwlph vxevfulswv iru wkh gdwh dw zklfk wkh| duh uhdol}hg> wkh ulvniuhh lqwhuhvwudwh lv uhdol}hg dw w 4 exw lv nqrzq rqh shulrg lq dgydqfh dw wlph w1

Wkh frqglwlrqdo phdq dqg frqglwlrqdo yduldqfh duh wkh phdq dqg yduldqfhfrqglwlrqdo rq wkh lqyhvwru*v lqirupdwlrq dw wlph w/ wkxv wkh| duh jlyhq wvxevfulswv1

Wkh lqyhvwru sxwv d vkduh | ri klv sruwirolr lqwr wkh ulvn| dvvhw1 Wkhq

46


18/218


wkh sruwirolr uhwxuq lv

URc|n @ |U|n +4 |,Usc|n @ Usc|n |+U|n Usc|n,= +514,

Wkh phdq sruwirolr uhwxuq lv H|URc|n @ Usc|n |+H|U|nUsc|n,/ zklohwkh yduldqfh ri wkh sruwirolr uhwxuq lv 2R| @

2|

2| 1

Wkh lqyhvwru suhihuv d kljk phdq dqg d orz yduldqfh ri sruwirolr uhwxuqv1Zh dvvxph wkdw wkh lqyhvwru wudghv r phdq dqg yduldqfh lq d olqhdu idvklrq1Wkdw lv/ wkh lqyhvwru pd{lpl}hv d olqhdu frpelqdwlrq ri phdq dqg yduldqfh/zlwk d srvlwlyh zhljkw rq phdq dqg d qhjdwlyh zhljkw rq yduldqfh=

pd{k|

+H|URc|n n5

2R|,1 +515,

Vxevwlwxwlqj lq wkh phdq dqg yduldqfh ri sruwirolr uhwxuqv/ dqg vxewudfw0lqj Usc|n +zklfk grhv qrw fkdqjh wkh pd{lpl}dwlrq sureohp,/ wklv fdq ehuhzulwwhq dv

pd{k|

|+H|U|n Usc|n, n5

2|2| 1 +516,

Wkh vroxwlrq wr wklv pd{lpl}dwlrq sureohp lv

| @H|U|n Usc|n

n2|= +517,

Wkh sruwirolr vkduh lq wkh ulvn| dvvhw vkrxog htxdo wkh h{shfwhg h{fhvv uh0wxuq/ ru ulvn suhplxp/ glylghg e| frqglwlrqdo yduldqfh wlphv wkh frhflhqw

n wkdw uhsuhvhqwv dyhuvlrq wr yduldqfh1 Zh zloo vhh vlplodu h{suhvvlrqviuhtxhqwo| lq wklv errn1

D xvhixo frqfhsw lq sruwirolr dqdo|vlv lv wkh Vkdush udwlr V|/ ghqhg dvwkh udwlr ri phdq h{fhvv uhwxuq wr vwdqgdug ghyldwlrq=

V| @H|U|n Usc|n

|= +518,

Wkh sruwirolr vroxwlrq fdq eh uhzulwwhq dv

| @V|

n|= +519,

Wkh phdq h{fhvv uhwxuq rq wkh sruwirolr lv V2| @n dqg wkh yduldqfh ri wkh

sruwirolr lv V2| @n2/ vr wkh udwlr ri phdq wr yduldqfh lv 4@n1 Wkh vwdqgdug

ghyldwlrq ri wkh sruwirolr lv V|@n/ dqg vr wkh Vkdush udwlr ri wkh sruwirolr lvV|1 Lq wklv vlpsoh prgho/ doo sruwirolrv kdyh wkh vdph Vkdush udwlr ehfdxvhwkh| doo frqwdlq wkh vdph ulvn| dvvhw lq juhdwhu ru vpdoohu dprxqw1

47


19/218


Phdq0yduldqfh dqdo|vlv zlwk pdq| ulvn| dvvhwv

Wkhvh uhvxowv h{whqg vwudljkwiruzdugo| wr wkh fdvh zkhuh wkhuh duh pdq|ulvn| dvvhwv1 Zh ghqh wkh sruwirolr uhwxuq lq wkh vdph pdqqhu dv ehiruh/h{fhsw wkdw zh xvh orzhufdvh erogidfh ohwwhuv wr ghqrwh yhfwruv dqg xsshufdvherogidfh ohwwhuv wr ghqrwh pdwulfhv1 Wkxv U|n lv qrz d yhfwru ri ulvn|uhwxuqv zlwk Q hohphqwv1 Lw kdv d phdq yhfwru H|U|n dqg d yduldqfh0fryduldqfh pdwul{ |1 Dovr/ | lv qrz d yhfwru ri doorfdwlrqv wr wkh ulvn|dvvhwv1 Wkh pd{lpl}dwlrq sureohp +516, qrz ehfrphv

pd{|

|+H|U|n Usc|n, n

5|||1 +51:,

Khuh lv d yhfwru ri rqhv/ dqg +H|U|n

Usc|n, lv wkh yhfwru ri h{fhvvuhwxuqv rq wkh Q ulvn| dvvhwv ryhu wkh ulvnohvv lqwhuhvw udwh1 Wkh yduldqfhri wkh sruwirolr uhwxuq lv |||1

Wkh vroxwlrq wr wklv pd{lpl}dwlrq sureohp lv

| @4

n3| +H|U|n Usc|n,= +51;,

Wklv lv d vwudljkwiruzdug jhqhudol}dwlrq ri wkh vroxwlrq zlwk d vlqjoh ulvn|dvvhw1 Wkh vlqjoh h{fhvv uhwxuq lv uhsodfhg e| d yhfwru ri h{fhvv uhwxuqv/ dqgwkh uhflsurfdo ri yduldqfh lv uhsodfhg e| 3| / wkh lqyhuvh ri wkh yduldqfh0fryduldqfh pdwul{ ri uhwxuqv1

Wkh lqyhvwru*v suhihuhqfhv hqwhu wkh vroxwlrq +51;, rqo| wkurxjk wkh vfdodu

whup 4@n1 Wkxv lqyhvwruv glhu rqo| lq wkh ryhudoo vfdoh ri wkhlu ulvn| dvvhwsrvlwlrq/ qrw lq wkh frpsrvlwlrq ri wkdw srvlwlrq1 .rqvhuydwlyh lqyhvwruvzlwk d kljk n krog pruh ri wkh ulvnohvv dvvhw dqg ohvv ri doo ulvn| dvvhwv/ exwwkh| gr qrw fkdqjh wkh uhodwlyh sursruwlrqv ri wkhlu ulvn| dvvhwv zklfk duhghwhuplqhg e| wkh yhfwru 3| +H|U|n Usc|n,1 Wklv lv wkh pxwxdo ixqgwkhruhp ri Wrelq +4


20/218


ri fryduldqfhv ehwzhhq h{fhvv uhwxuqv dqg wkh vxevfulswhg yduldeoh1 Dexvlqj

wklv qrwdwlrq voljkwo|/ zh zloo dovr zulwh 2

| iru wkh yhfwru ri h{fhvv0uhwxuqyduldqfhv1,Zlwk qr ulvnohvv dvvhw/ wkh vroxwlrq ehfrphv

| @4

n3| +H|U|n Ufc|n, 3| f|= +51 lqvwhdg wkh| krog vrph frpelqdwlrq ri wzrulvn| pxwxdo ixqgv/ zkrvh sursruwlrqv duh jlyhq e| wkh wzr whupv rq wkhuljkw0kdqg vlgh ri +51


21/218


+4@5,X+Z| J, +4@5,X+Z|J,/ zklfk lv ohvv wkdq X+Z|, ehfdxvh ri wkhfxuydwxuh ri wkh xwlolw| ixqfwlrq1 Wkxv wkh lqyhvwru wxuqv grzq wkh jdpeoh>lw rhuv rqo| ulvn zlwkrxw dq| dffrpsdq|lqj uhzdug/ dqg lv xqdwwudfwlyh wrd ulvn0dyhuvh lqyhvwru1

Wkh ghjuhh ri fxuydwxuh ri wkh xwlolw| ixqfwlrq ghwhuplqhv wkh lqwhqvlw|ri wkh lqyhvwru*v ulvn dyhuvlrq1 .xuydwxuh fdq eh phdvxuhg e| wkh vhfrqgghulydwlyh ri wkh xwlolw| ixqfwlrq zlwk uhvshfw wr zhdowk/ vfdohg e| wkh uvwghulydwlyh wr holplqdwh dq| ghshqghqfh ri wkh phdvxuh ri fxuydwxuh rq wkhduelwudu| xqlwv lq zklfk xwlolw| lv phdvxuhg1 Wkh frhflhqw ri devroxwh ulvndyhuvlrq lv wkhq ghqhg dv

D+Z, @ X+Z,

X+Z,> +5145,

dqg wkh frhflhqw ri uhodwlyh ulvn dyhuvlrq lv ghqhg dv

U+Z, @ Z D+Z, @ Z X+Z,

X+Z,= +5146,

Wkh uhflsurfdov ri wkhvh phdvxuhv duh fdoohg devroxwh dqg uhodwlyh ulvn wrohu0dqfh1

.odvvlf uhvxowv ri Sudww +4 wklv lpsolhv wkdw lqyhvwruv duh zloolqj wr sd| doprvw

4:


22/218


1+tWGW

t+

tWGWt

( )GWUt

( )t

WU

( )GWU t +

( )1+tWU

( )

( )GWU

GWU

t

t

+

+

21

21

Iljxuh 514= .rqfdyh xwlolw| ri zhdowk

4;


23/218


wkh vdph uhodwlyh frvwv wr dyrlg jlyhq uhodwlyh ulvnv dv wkh| glg zkhq wkh|

zhuh pxfk srruhu/ zklfk lv rqo| srvvleoh li uhodwlyh ulvn dyhuvlrq lv doprvwlqghshqghqw ri zhdowk1

Wkh irup ri wkh xwlolw| ixqfwlrq

Wudfwdeoh prghov ri sruwirolr fkrlfh uhtxluh dvvxpswlrqv derxw wkh irupri wkh xwlolw| ixqfwlrq/ dqg srvvleo| glvwulexwlrqdo dvvxpswlrqv derxw dvvhwuhwxuqv1 Wkuhh dowhuqdwlyh vhwv ri dvvxpswlrqv surgxfh vlpsoh uhvxowv wkdwduh frqvlvwhqw zlwk wkrvh ri phdq0yduldqfh dqdo|vlv=

41 Lqyhvwruv kdyh txdgudwlf xwlolw| ghqhg ryhu zhdowk1 Lq wklv fdvhX+Z|n, @ dZ|n eZ2|n1 Xqghu wklv dvvxpswlrq pd{lpl}lqj

h{shfwhg xwlolw|/ dv lq +5143,/ lv htxlydohqw wr pd{lpl}lqj d olqhdufrpelqdwlrq ri phdq dqg yduldqfh/ dv lq +515,1 Qr glvwulexwlrqdodvvxpswlrqv duh qhhghg rq dvvhw uhwxuqv1 Txdgudwlf xwlolw| lpsolhvwkdw devroxwh ulvn dyhuvlrq dqg uhodwlyh ulvn dyhuvlrq duh lqfuhdvlqj lqzhdowk1

51 Lqyhvwruv kdyh h{srqhqwldo xwlolw|/ X+Z|n, @ h{s+Z|n,/ dqgdvvhw uhwxuqv duh qrupdoo| glvwulexwhg1 H{srqhqwldo xwlolw| lpsolhvwkdw devroxwh ulvn dyhuvlrq lv d frqvwdqw / zkloh uhodwlyh ulvn dyhuvlrqlqfuhdvhv lq zhdowk1

61 Lqyhvwruv kdyh srzhu xwlolw|/ X+Z|n, @ +Z3|n 4,@+4 ,/ dqg

dvvhw uhwxuqv duh orjqrupdoo| glvwulexwhg1 Srzhu xwlolw| lpsolhv wkdwdevroxwh ulvn dyhuvlrq lv ghfolqlqj lq zhdowk/ zkloh uhodwlyh ulvn dyhuvlrqlv d frqvwdqw 1 Wkh olplw dv dssurdfkhv rqh lv orj xwlolw|= X+Z|n, @orj+Z|n,1

Zh kdyh douhdg| dujxhg wkdw devroxwh ulvn dyhuvlrq vkrxog ghfolqh/ rudw wkh yhu| ohdvw vkrxog qrw lqfuhdvh/ zlwk zhdowk1 Wklv uxohv rxw wkhdvvxpswlrq ri txdgudwlf xwlolw|/ dqg idyruv srzhu xwlolw| ryhu h{srqhqwldoxwlolw|1 Wkh srzhu0xwlolw| surshuw| ri frqvwdqw uhodwlyh ulvn dyhuvlrq lvlqkhuhqwo| dwwudfwlyh/ dqg lv uhtxluhg wr h{sodlq wkh vwdelolw| ri qdqfldoyduldeohv lq wkh idfh ri vhfxodu hfrqrplf jurzwk1

Wkh fkrlfh ehwzhhq h{srqhqwldo dqg srzhu xwlolw| dovr lpsolhv glvwulex0

wlrqdo dvvxpswlrqv rq uhwxuqv1 H{srqhqwldo xwlolw| surgxfhv vlpsoh uhvxowvli dvvhw uhwxuqv duh qrupdoo| glvwulexwhg/ zkloh srzhu xwlolw| surgxfhv vlpsohuhvxowv li dvvhw uhwxuqv duh orjqrupdo +wkdw lv/ li wkhlu orjv duh qrupdo,1

Wkh dvvxpswlrq ri qrupdo uhwxuqv lv dsshdolqj iru vrph sxusrvhv/ exw lwlv lqdssursuldwh iru wkh vwxg| ri orqj0whup sruwirolr fkrlfh ehfdxvh lw fdqqrw

4


24/218


krog dw pruh wkdq rqh wlph krul}rq1 Li uhwxuqv duh qrupdoo| glvwulexwhg dw

d prqwko| iuhtxhqf|/ wkhq wzr0prqwk uhwxuqv duh qrw qrupdo ehfdxvh wkh|duh wkh surgxfw ri wzr vxffhvvlyh qrupdo uhwxuqv dqg vxpv ri qrupdov/ qrwsurgxfwv ri qrupdov/ duh wkhpvhoyhv qrupdo1 Wkh dvvxpswlrq ri orjqrupdouhwxuqv/ rq wkh rwkhu kdqg/ fdq krog dw hyhu| wlph krul}rq vlqfh surgxfwvri orjqrupdo udqgrp yduldeohv duh wkhpvhoyhv orjqrupdo1 Dovr/ orjqrupdoudqgrp yduldeohv fdq qhyhu eh qhjdwlyh vr wkh dvvxpswlrq ri orjqrupdolw|lv frqvlvwhqw zlwk wkh olplwhg oldelolw| ihdwxuh ri prvw qdqfldo dvvhwv1

Wkh dvvxpswlrq ri orjqrupdo uhwxuqv uxqv lqwr dqrwkhu glfxow|/ krz0hyhu1 Lw grhv qrw fduu| ryhu vwudljkwiruzdugo| iurp lqglylgxdo dvvhwv wr sruw0irolrv1 D sruwirolr lv d olqhdu frpelqdwlrq ri lqglylgxdo dvvhwv> li hdfk dvvhwuhwxuq lv orjqrupdo/ wkh sruwirolr uhwxuq lv d zhljkwhg dyhudjh ri orjqrupdovzklfk lv qrw lwvhoi orjqrupdo1 Wklv glfxow| fdq eh dyrlghg e| frqvlghulqjvkruw wlph lqwhuydov1 Dv wkh wlph lqwhuydo vkulqnv/ wkh qrq0orjqrupdolw| riwkh sruwirolr uhwxuq glplqlvkhv/ dqg lw glvdsshduv dowrjhwkhu lq wkh olplw rifrqwlqxrxv wlph1 Lq wklv dqg wkh qh{w ihz fkdswhuv zh xvh d glvfuhwh0wlphdssur{lpdwlrq wr wkh uhodwlrq ehwzhhq wkh orj uhwxuq rq d sruwirolr dqg wkhorj uhwxuqv rq lqglylgxdo dvvhwv1 Wkh dssur{lpdwlrq ehfrphv pruh dffxudwhdv wkh wlph lqwhuydo vkulqnv1 Lq .kdswhu 8 zh ghyhors h{solflw prghov vhwlq frqwlqxrxv wlph1

2 *L}?Lh4@* 4L_i* | TLih |*|)

Zh qrz ghyhors sruwirolr fkrlfh uhvxowv xqghu wkh dvvxpswlrq wkdw lqyhvwruv

kdyh srzhu xwlolw| dqg wkdw dvvhw uhwxuqv duh orjqrupdo1 Zh uhshdwhgo|dsso| d nh| uhvxow derxw wkh h{shfwdwlrq ri d orjqrupdo udqgrp yduldeoh [=

orjH|[|n @ H| orj [|n 4

5Ydu| orj [|n @ H|{|n

4

52%|= +5147,

Khuh dqg wkurxjkrxw wkh errn/ wkh qrwdwlrq orj uhihuv wr wkh qdwxudo orjd0ulwkp/ dqg orzhu0fdvh ohwwhuv duh xvhg wr ghqrwh wkh orjv ri wkh fruuhvsrqglqjxsshu0fdvh ohwwhuv1 Htxdwlrq +5147, fdq eh xqghuvwrrg lqwxlwlyho| e| uhihu0hqfh wr Iljxuh 5141 Wkh orj lv d frqfdyh ixqfwlrq olnh wkh xwlolw| ixqfwlrqlooxvwudwhg lq Iljxuh 5141 Wkxv wkh phdq ri wkh orj ri d udqgrp yduldeoh [lv vpdoohu wkdq wkh orj ri wkh phdq/ dqg wkh glhuhqfh lv lqfuhdvlqj lq wkh

yduldelolw| ri[1 Wkh htxdwlrq txdqwlhv wklv glhuhqfh iru wkh vshfldo fdvhlq zklfk orj [ lv qrupdoo| glvwulexwhg1Zh ehjlq e| dvvxplqj wkdw wkh uhwxuq rq dq lqyhvwru*v sruwirolr lv orj0

qrupdo/ dqg khqfh wkdw qh{w0shulrg zhdowk lv orjqrupdo1 Zh wkhq jr rq wrdvn zkhwkhu wklv lv frqvlvwhqw zlwk orjqrupdolw| ri wkh uhwxuqv rq xqghuo|0

53


25/218


lqj dvvhwv1 Xqghu wkh dvvxpswlrq ri srzhu xwlolw|/ htxdwlrq +5143, fdq eh

zulwwhq dv pd{H|Z3|n @+4 ,= +5148,

Pd{lpl}lqj wklv h{shfwdwlrq lv htxlydohqw wr pd{lpl}lqj wkh orj ri wkh h{0shfwdwlrq/ dqg wkh vfdoh idfwru 4@+4 , fdq eh rplwwhg vlqfh lw grhv qrwdhfw wkh vroxwlrq1 Li qh{w0shulrg zhdowk lv orjqrupdo/ zh fdq dsso| htxd0wlrq +5147, wr uhzulwh dv

pd{orjH|Z3|n @ +4 ,H|z|n

4

5+4 ,22|= +5149,

Wkh exgjhw frqvwudlqw +5144, fdq eh uhzulwwhq lq orj irup dv

z|n @ uRc|n z|> +514:,

zkhuh uRc|n @ orj+4 URc|n, lv wkh orj uhwxuq rq wkh sruwirolr/ wkh qdwxudoorjdulwkp ri wkh jurvv vlpsoh uhwxuq/ dovr nqrzq dv wkh frqwlqxrxvo| frp0srxqghg sruwirolr uhwxuq1 Glylglqj +5149, e| +4 , dqg xvlqj +514:,/ zhuhvwdwh wkh sureohp dv

pd{H|uRc|n 4

5+4 ,2R|> +514;,

zkhuh 2R| lv wkh frqglwlrqdo yduldqfh ri wkh orj sruwirolr uhwxuq1Wr xqghuvwdqg wklv htxdwlrq/ lw lv khosixo wr qrwh wkdw

H|uRc|n 2R|@5 @ orj H|+4 URc|n, +514


26/218


shqdol}lqj wkh yduldqfh ri orj uhwxuqv> zkhq ? 4/ wkh lqyhvwru dfwxdoo|

vhhnv d ulvnlhu sruwirolr ehfdxvh d kljkhu yduldqfh/ zlwk wkh vdph phdq orjuhwxuq/ fruuhvsrqgv wr d kljkhu phdq vlpsoh uhwxuq1 Wkh fdvh @ 4 lvwkh erxqgdu| zkhuh wkhvh wzr rssrvlqj frqvlghudwlrqv h{dfwo| fdqfho rqhdqrwkhu rxw1 Wklv fdvh sod|v dq lpsruwdqw uroh wkurxjkrxw wkh errn1

Dssur{lpdwlrq ri wkh sruwirolr uhwxuq

Wr surfhhg ixuwkhu/ zh qhhg wr uhodwh wkh orj sruwirolr uhwxuq wr wkh orjuhwxuqv rq xqghuo|lqj dvvhwv1 .rqvlghu uvw wkh vlpsoh fdvh zkhuh wkhuh duhwzr dvvhwv/ rqh ulvn| dqg rqh ulvnohvv1 Wkhq iurp +514, wkh vlpsoh uhwxuqrq 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 orjv1Ryhu vkruw wlph lqwhuydov/ krzhyhu/ zh fdq xvh d Wd|oru dssur{lpdwlrq

ri wkh qrqolqhdu ixqfwlrq uhodwlqj orj lqglylgxdo0dvvhw uhwxuqv wr orj sruwirolruhwxuqv1 Ixoo ghwdlov duh jlyhq lq wkh Dsshqgl{> wkh uhvxowlqj h{suhvvlrq lv

uRc|n usc|n @ |+u|n usc|n, 45

|+4 |,2| = +5154,

Wkh glhuhqfh ehwzhhq wkh orj sruwirolr uhwxuq dqg d olqhdu frpelqdwlrqri orj lqglylgxdo0dvvhw uhwxuqv lv jlyhq e| |+4 |,2|@51 Wkh glhuhqfhglvdsshduv li wkh sruwirolr zhljkw lq wkh ulvn| dvvhw lv }hur +iru wkhq wkh orjsruwirolr 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,1Zkhq 3 ? | ? 4/ wkh sruwirolr lv d zhljkwhg dyhudjh ri wkh lqglylgxdo dvvhwvdqg wkh whup |+4 |,2|@5 lv srvlwlyh1 Wr xqghuvwdqg wklv/ uhfdoo wkdw wkhorj ri dq dyhudjh lv juhdwhu wkdq dq dyhudjh ri orjv dv looxvwudwhg lq Iljxuh5141

Dqrwkhu zd| wr xqghuvwdqg +5154, lv wr uhzulwh lw dv

uRc|n usc|n 2R|

5@ |

u|n usc|n

2|

5

> +5155,

xvlqj wkh idfw wkdw 2R| @ 2|

2| 1 Li rqh wdnhv h{shfwdwlrqv ri wklv htxdwlrq/

rqh qgv wkdw wkh phdq ri wkh vlpsoh h{fhvv sruwirolr uhwxuq lv olqhduo|

uhodwhg wr wkh phdq ri wkh vlpsoh h{fhvv uhwxuq rq wkh ulvn| dvvhw/ dv uhtxluhge| +514,1

Wkh dssur{lpdwlrq lq +5154, fdq eh mxvwlhg uljrurxvo| e| frqvlghulqjvkruwhu dqg vkruwhu wlph lqwhuydov1 Dv wkh wlph lqwhuydo vkulqnv/ wkh kljkhu0rughu whupv wkdw duh qhjohfwhg lq +5154, ehfrph qhjoljleoh uhodwlyh wr wkrvh

55


27/218


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*vOhppd1 Zh glvfxvv wkh frqwlqxrxv0wlph dssurdfk lq pruh ghwdlo lq .kdswhu81

Rqh lpsruwdqw surshuw| ri wkh dssur{lpdwh sruwirolr uhwxuq lv wkdw lwuxohv rxw wkh srvvlelolw| ri edqnuxswf|/ hyhq zkhq wkh lqyhvwru krogv d vkruwsrvlwlrq +| ? 3, ru d ohyhudjhg srvlwlrq lq wkh ulvn| dvvhw qdqfhg e| eru0urzlqj +| A 4,1 Wkh orj sruwirolr uhwxuq lv dozd|v qlwh/ qr pdwwhu zkdwwkh uhwxuqv rq wkh xqghuo|lqj dvvhwv/ dqg wkxv lw lv qhyhu srvvleoh wr h{0kdxvw zhdowk frpsohwho|1 .rqwlqxrxv0wlph glxvlrq prghov dovr kdyh wklvsurshuw|> lq vxfk prghov sruwirolrv duh uhedodqfhg ryhu vxfk vkruw lqwhu0ydov wkdw orvvhv fdq dozd|v eh vwhpphg e| uhedodqflqj ehiruh wkh| ohdg wredqnuxswf|1 Lq prvw dssolfdwlrqv lw lv uhdvrqdeoh wr h{foxgh wkh srvvlelo0lw| ri edqnuxswf|/ exw wklv dssurdfk pd| qrw eh vxlwdeoh zkhq lw lpsolhvrswlpdo sruwirolrv zlwk h{wuhpho| kljk ohyhudjh1

Wkh dssur{lpdwlrq +5154, jhqhudol}hv vwudljkwiruzdugo| wr wkh fdvh zkhuhwkhuh duh pdq| ulvn| dvvhwv/ mrlqwo| orjqrupdoo| glvwulexwhg zlwk frqglwlrqdoyduldqfh0fryduldqfh pdwul{ ri orj uhwxuqv |1 Zh zulwh

2| iru wkh yhfwru

frqwdlqlqj wkh gldjrqdo hohphqwv ri |/ wkh yduldqfhv ri dvvhw uhwxuqv1 +Uh0fdoo rxu qrwdwlrqdo frqyhqwlrqv wkdw erogidfh orzhufdvh ohwwhuv ghqrwh yhf0wruv/ erogidfh orzhufdvh vljpd ghqrwhv d yhfwru ri fryduldqfhv/ dqg zlwk dvoljkw dexvh ri qrwdwlrq 2| ghqrwhv d yhfwru ri yduldqfhv1, Wkh dssur{lpd0wlrq wr wkh sruwirolr uhwxuq ehfrphv

uRc|n usc|n @ |+u|n usc|n, 45|2| 45|||= +5156,

Wklv dssur{lpdwlrq krogv lq h{dfwo| wkh vdph irup li zh uhsodfh wkh ulvnohvvuhwxuq usc|n zlwk d ulvn| ehqfkpdun uhwxuq ufc|n/ h{fhsw wkdw lq wklv fdvhwkh yhfwru 2| dqg wkh pdwul{ | pxvw frqwdlq yduldqfhv dqg fryduldqfhv rih{fhvv uhwxuqv rq wkh rwkhu ulvn| dvvhwv ryhu wkh ehqfkpdun uhwxuq/ udwkhuwkdq yduldqfhv dqg fryduldqfhv ri wrwdo uhwxuqv rq wkhvh dvvhwv1 +H{fhvv dqgwrwdo uhwxuqv rqo| kdyh wkh vdph yduldqfhv dqg fryduldqfhv zkhq wkh| duhphdvxuhg uhodwlyh wr d ulvnohvv uhwxuq1,

Vroxwlrq ri wkh prgho

Lq d wzr0dvvhw prgho/ htxdwlrq +5154, lpsolhv wkdw wkh phdq h{fhvv sruw0irolr uhwxuq lv H|uRc|n usc|n @ |+H|u|n usc|n, 2|+4 |,2| / zklohwkh yduldqfh ri wkh sruwirolr uhwxuq lv 2|

2| 1 Vxevwlwxwlqj lqwr wkh remhfwlyh

56


28/218


ixqfwlrq +514;,/ wkh sureohp ehfrphv

pd{ |+H|u|n usc|n, 45

|+4 |,2| 45 +4 ,2|

2| = +5157,

Wkh vroxwlrq lv

| @H|u|n usc|n 2|@5

2|= +5158,

Wklv htxdwlrq lv wkh htxlydohqw/ lq d orjqrupdo prgho zlwk srzhu xwlolw|/ri wkh phdq0yduldqfh vroxwlrq +517,1 Wkh wrs olqh lv wkh h{shfwhg h{fhvvorj uhwxuq rq wkh ulvn| dvvhw/ zlwk wkh dgglwlrq ri rqh0kdoi wkh yduldqfh wrfrqyhuw iurp orj uhwxuqv wr vlpsoh uhwxuqv wkdw duh xowlpdwho| ri frqfhuqwr wkh lqyhvwru1 +Wkh irupxod iru wkh h{shfwdwlrq ri orjqrupdo udqgrp

yduldeohv lpsolhv wkdw H|u|n usc|n 2

|@5 @ orjH|+4 U|n,@+4 Us|,1,Wkh erwwrp olqh lv wkh frhflhqw ri uhodwlyh ulvn dyhuvlrq wlphv wkh yduldqfhri wkh ulvn| dvvhw uhwxuq1 Wkxv/ mxvw dv lq d vlpsoh phdq0yduldqfh prgho/wkh rswlpdo sruwirolr zhljkw lv wkh ulvn suhplxp glylghg e| ulvn dyhuvlrqwlphv yduldqfh1

Lq d prgho zlwk pdq| ulvn| dvvhwv/ wkh vroxwlrq iru wkh yhfwru ri rswlpdosruwirolr zhljkwv lv

| @4

3| +H|u|n usc|n 2|@5,= +5159,

Wklv vroxwlrq lv wkh htxlydohqw ri wkh pxowlsoh0dvvhw phdq0yduldqfh vroxwlrq

+51;,1 Olnh wkh phdq0yduldqfh vroxwlrq/ lw kdv wkh surshuw| wkdw wkh frh0flhqw ri uhodwlyh ulvn dyhuvlrq rqo| dhfwv wkh ryhudoo vfdoh ri wkh ulvn| dvvhwsrvlwlrq dqg qrw lwv frpsrvlwlrq1 Wkxv d yhuvlrq ri Wrelq*v pxwxdo ixqgwkhruhp krogv lq wkh orjqrupdo prgho zlwk srzhu xwlolw|1

Li wkhuh lv qr wuxo| ulvnohvv dvvhw/ dqg zh zrun lqvwhdg zlwk d ulvn|ehqfkpdun uhwxuq ufc|n/ wkhq wkh vroxwlrq ehfrphv

| @4

3| +H|u|n ufc|n 2|@5,

4 4

+3| f|,> +515:,

zkhuh f| lv wkh yhfwru ri fryduldqfhv ri h{fhvv orj uhwxuqv zlwk wkh ehqfk0pdun orj uhwxuq1 Mxvw dv lq wkh vlpsoh phdq0yduldqfh dqdo|vlv +51 exwwkh lqyhvwru glvolnhv vxfk dvvhwv ehfdxvh wkh| lqfuhdvh wkh ulvn ri wkh sruw0irolr1 Wkh wzr hhfwv fdqfho zkhq @ 4> lq wklv fdvh wkh vroxwlrq wdnhv

57


29/218


h{dfwo| wkh vdph irup zkhwkhu wkh ehqfkpdun dvvhw lv ulvnohvv ru ulvn|1 Dv

lqfuhdvhv wkh rswlpdo sruwirolr dssurdfkhv wkh plqlpxp0yduldqfh sruwirolr3| f|1

22 )LTU *L?}|ih4 TLh|uL*L ULUi

22 Lih |*|) Lu i@*|

Vr idu zh kdyh dvvxphg wkdw wkh lqyhvwru kdv d vkruw lqyhvwphqw krul}rq dqgfduhv rqo| derxw wkh glvwulexwlrq ri zhdowk dw wkh hqg ri wkh qh{w shulrg1Dowhuqdwlyho|/ zh fdq dvvxph wkdw wkh lqyhvwru fduhv derxw wkh glvwulexwlrqri zhdowk N shulrgv iurp qrz/ vr wkdw wkh xwlolw| ixqfwlrq lv X+Z|ng, udwkhuwkdq X+Z|n,1 Zh frqwlqxh wr dvvxph wkdw doo zhdowk lv uhlqyhvwhg/ vr wkh

exgjhw frqvwudlqw wdnhv wkh irup

Z|ng@ +4URgc|ng,Z| @ +4URc|n,+4URc|n2,===+4URc|ng,Z|= +515;,

Khuh wkh qrwdwlrq +4 URgc|ng, lqglfdwhv wkdw wkh sruwirolr uhwxuq lv phd0vxuhg ryhu N shulrgv/ iurp w wr w N1 Wklv N0shulrg uhwxuq lv mxvw wkhsurgxfw ri N vxffhvvlyh 40shulrg uhwxuqv1 Qrwh wkdw wklv lv d fxpxodwlyhuhwxuq> wr fdofxodwh dq dqqxdol}hg uhwxuq rqh zrxog wdnh wkh Nwk urrw li wkhedvh shulrg lv d |hdu/ wkh +N@7,wk urrw li wkh edvh shulrg lv d txduwhu/ dqglq jhqhudo wkh +N@V,wk urrw li wkhuh duh V edvh shulrgv lq d |hdu1 Wdnlqjorjv/ wkh fxpxodwlyh orj uhwxuq ryhu N shulrgv lv mxvw d vxp ri N 40shulrg

uhwxuqv= uRgc|ng@ uRc|n === uRc|ng= +515


30/218


duh frqglwlrqdoo| orjqrupdoo| glvwulexwhg1 Iru vlpsolflw| zh uhwxuq wr wkh

fdvh zkhuh wkhuh lv d vlqjoh ulvn| dvvhw1Zh qrz pdnh d kljko| uhvwulfwlyh dgglwlrqdo dvvxpswlrq/ wkdw doo dvvhwuhwxuqv duh lqghshqghqw dqg lghqwlfdoo| glvwulexwhg +LLG, ryhu wlph1 Wklvlpsolhv wkdw wkh orj ulvnohvv udwh lv d frqvwdqw us dqg wkh orj N0shulrg ulvnohvvuhwxuq lv Nus> wkh phdq orj uhwxuq rq wkh ulvn| dvvhw lv d frqvwdqw Hu dqgwkh phdq orj N0shulrg uhwxuq rq wkh ulvn| dvvhw lv NHu> wkh yduldqfh ri wkhorj uhwxuq rq wkh ulvn| dvvhw lv d frqvwdqw 2 dqg wkh ulvn| dvvhw uhwxuq lvvhuldoo| xqfruuhodwhg/ vr wkh yduldqfh ri wkh orj N0shulrg uhwxuq rq wkh ulvn|dvvhw lv mxvw N2=

Ydu|ugc|ng@ Ydu|u|n === Ydu|u|ng@ N2= +5163,

Khuh wkh uvw htxdolw| iroorzv iurp wkh devhqfh ri vhuldo fruuhodwlrq lq ulvn|uhwxuqv/ dqg wkh vhfrqg iroorzv iurp wkh frqvwdqw yduldqfh ri wkh ulvn| uh0wxuq1

Zlwk LLG uhwxuqv/ wkhq/ wkh pdsslqj iurp vlqjoh0shulrg orj uhwxuqv wr N0shulrg orj uhwxuqv lv vwudljkwiruzdug1 Doo phdqv dqg yduldqfhv iru lqglylgxdodvvhwv duh vfdohg xs e| wkh vdph idfwru N1 Li zh fdq dsso| wkh vdphdssur{lpdwlrq +5154, uhodwlqj lqglylgxdo dvvhw uhwxuqv wr wkh sruwirolr uhwxuq/wklv lpsolhv wkdw wkh suhylrxv vkruw0whup sruwirolr vroxwlrq lv vwloo rswlpdoiru d orqj0whup lqyhvwru1 Lqwxlwlyho|/ wkh rswlpdo sruwirolr zhljkw rq dulvn| dvvhw lv wkh phdq h{fhvv orj uhwxuq/ soxv rqh0kdoi wkh orj yduldqfhwr frqyhuw iurp phdq orj wr phdq vlpsoh uhwxuq/ glylghg e| ulvn dyhuvlrq

wlphv yduldqfh1 Li erwk wkh phdq dqg wkh yduldqfh duh pxowlsolhg e| N/wklv vroxwlrq grhv qrw fkdqjh1 Wkh dujxphqw fdq uhdglo| eh h{whqghg wrwkh fdvh zlwk pxowlsoh ulvn| dvvhwv1 Erwk wkh vkruw0whup lqyhvwru dqg wkhorqj0whup lqyhvwru shufhlyh wkh vdph phdq0yduldqfh gldjudp/ phuho| vfdohgxs ru grzq e| d idfwru N> wkxv wkh| fkrrvh wkh vdph srlqw rq wkh gldjudp/wkdw lv/ wkh| fkrrvh wkh vdph sruwirolr1

Wkh zhdnqhvv lq wklv dqdo|vlv lv wkh dvvxpswlrq wkdw wkh dssur{lpdwhexgjhw frqvwudlqw +5154, dssolhv wr d orqj kroglqj shulrg1 Uhfdoo wkdwwklv exgjhw frqvwudlqw krogv h{dfwo| lq frqwlqxrxv wlph/ dqg lv dq dffxudwhdssur{lpdwlrq ryhu vkruw glvfuhwh wlph lqwhuydov> exw wkh txdolw| ri wkh ds0sur{lpdwlrq ghwhulrudwhv li lw lv dssolhg ryhu orqj kroglqj shulrgv1 Iru wklvuhdvrq dq h{dfw vroxwlrq wr wkh orqj0whup sruwirolr fkrlfh sureohp zlwkrxwuhedodqflqj grhv lqyroyh vrph krul}rq hhfwv hyhq li ulvn| uhwxuqv duh LLG1Lq sudfwlfh/ krzhyhu/ wkhvh krul}rq hhfwv duh vpdoo1

@hMiht E2fff T*L|t TLh|uL*L @**LU@|L? @}@?t| Lh3L? uLh @ TLh|uL*L ULUi ThLM*i4|L| hiM@*@?U?} Aiti T*L|t @TTi@h Lh3L?|@* ? |i _ U@ti( |@| tc |i Lh3L?

59


31/218


P|rslf sruwirolr fkrlfh zlwk uhedodqflqj

Wkh dvvxpswlrq wkdw d orqj0krul}rq lqyhvwru fdqqrw uhedodqfh klv sruw0irolr lv vxshufldoo| dsshdolqj ehfdxvh lw pdnhv wkh orqj0krul}rq sureohpirupdoo| dqdorjrxv wr wkh vkruw0krul}rq sureohp1 Xqiruwxqdwho| lw fuhdwhvd whfkqlfdo glfxow| ehfdxvh lw lqydolgdwhv wkh xvh ri wkh orjolqhdu exgjhwfrqvwudlqw +5154,1 Pruh vhulrxvo|/ wklv dvvxpswlrq grhv qrw ghvfuleh uhdolw|1Lqyhvwruv zlwk orqj krul}rqv duh iuhh wr wudgh dvvhwv dw dq| wlph/ dqg qdqfldolqwhuphgldulhv h{lvw wr uhedodqfh sruwirolrv rq ehkdoi ri lqyhvwruv zkr qgwklv wdvn frvwo| wr h{hfxwh1 +Wklv lv wkh sxusrvh ri vr0fdoohg olihvw|oh pxwxdoixqgv1, Wkhuh lv qr lqkhuhqw frqqhfwlrq ehwzhhq wkh lqyhvwphqw krul}rqdqg wkh iuhtxhqf| zlwk zklfk sruwirolrv fdq eh uhedodqfhg1 Dffruglqjo|zh qrz dvvxph wkdw wkh orqj0whup lqyhvwru fdq uhedodqfh klv sruwirolr hyhu|

shulrg> dqg zh frqwlqxh wr xvh wklv dvvxpswlrq wkurxjkrxw wkh uhvw ri wkherrn1

.odvvlf uhvxowv ri Vdpxhovrq +4


32/218


zulwh

uR2cc|n2 5us @ +uRc|n us, +uRc|n2 us,@ |+u|n us, 4

5|+4 |,2

|n+u|n2 us, 45

|n+4 |n,2> +5164,

zkhuh zh duh doorzlqj | dqg |n wr eh glhuhqw ehfdxvh wkh lqyhvwru fdqiuhho| uhedodqfh klv sruwirolr hdfk shulrg1 Wkh frqglwlrqdo yduldqfh ri wkh50shulrg orj uhwxuq lv

Ydu|+uRc|c|n2, @ +2|

2|n,

2 +5165,

vlqfh erwk | dqg |n duh ghwhuplqlvwlf dqg khqfh duh nqrzq dw wlph w1Wkh phdq 50shulrg orj uhwxuq/ dgmxvwhg e| dgglqj rqh0kdoi wkh yduldqfh riwkh 50shulrg uhwxuq/ lv

H|+uRc|c|n2, 4

5Ydu|+uRc|c|n2, @ 5us +| |n,+Hu us 2@5,= +5166,

Wkh remhfwlyh ri d 50shulrg lqyhvwru zlwk srzhu xwlolw| fdq eh zulwwhq dv

pd{H|+uRc|c|n2, 4

5Ydu|+uRc|c|n2,

5Ydu|+uRc|c|n2,= +5167,

Wkxv wkh lqyhvwru zlwk A 3 dozd|v suhihu d orzhu yduldqfh ri orj uhwxuqv

iru d jlyhq yduldqfh0dgmxvwhg phdq1 Exw iurp +5166,/ wkh yduldqfh0dgmxvwhgphdq ghshqgv rqo| rq wkh vxp +||n,1 Wkh lqyhvwru fdq { wklv vxp dqgdgmxvw wkh lqglylgxdo vkduhv/ | dqg |n/ wr plqlpl}h wkh yduldqfh1 Vlqfhyduldqfh ghshqgv rq wkh vxp ri vtxduhv +2|

2|n,/ wklv lv dffrpsolvkhg e|

vhwwlqj | @ |n/ d frqvwdqw sruwirolr uxoh1Rqfh 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 Wkhuhdvrq lv wkdw lq wkh odvw shulrg ehiruh wkh krul}rq/ wkh orqj0whup lqyhvwrukdv ehfrph d vkruw0whup lqyhvwru dqg zloo fkrrvh wkh rswlpdo vkruw0whupsruwirolr1

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 VwrfnvDoo wkh Wlph ru Doo Vwrfnv Kdoi wkh WlphB%/ Nulw}pdq srlqwv rxw wkdw wkhvhwzr 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

5;


33/218


lv suhflvho| wkh hhfw wkdw xqghuolhv wkh dujxphqw iru d frqvwdqw sruwirolr

uxoh1Wkh vhfrqg Vdpxhovrq0Phuwrq frqglwlrq iru p|rslf sruwirolr fkrlfh lvwkdw wkh lqyhvwru kdv orj xwlolw|1 Lq wklv fdvh sruwirolr fkrlfh zloo eh p|rslfhyhq li dvvhw uhwxuqv duh qrw LLG1 Wkh dujxphqw khuh lv sduwlfxoduo| vlpsoh1Uhfdoo wkdw wkh orj xwlolw| lqyhvwru fkrrvhv d sruwirolr wkdw pd{lpl}hv wkhh{shfwhg orj uhwxuq1 Htxdwlrq +515 wkh| pxvw dozd|v eh phdvxuhg ryhu d frpprq wlph lqwhuydo1

D uhodwhg idoodf| lv wr dujxh wkdw wkhuh lv d vlqjoh ehvw orqj0whup sruwirolriru doo lqyhvwruv/ uhjdugohvv ri wkhlu suhihuhqfhv1 Wkh sursrvhg sruwirolrlv wkh _jurzwk0rswlpdo% sruwirolr wkdw pd{lpl}hv wkh h{shfwhg orj uhwxuq1Dv wkh lqyhvwphqw krul}rq lqfuhdvhv/ wklv sruwirolr rxwshuirupv dq| rwkhusruwirolr zlwk kljkhu dqg kljkhu suredelolw|1 Lq wkh olplw/ wkh suredelolw|wkdw lw rxwshuirupv jrhv wr rqh1 Wr xqghuvwdqg wklv surshuw| ri wkh jurzwk0rswlpdo sruwirolr/ qrwh wkdw wkh glhuhqfh ehwzhhq wkh fxpxodwlyh jurzwk0rswlpdo orj uhwxuq dqg wkh orj uhwxuq rq dq| rwkhu sruwirolr lv d qrupdoo|

5


34/218


glvwulexwhg udqgrp yduldeoh1 Zulwh wkh phdq ri wklv glhuhqfh dv +N, dqg

wkh vwdqgdug ghyldwlrq dv +N,1 Wkh phdq +N, lv srvlwlyh ehfdxvh wkhjurzwk0rswlpdo orj uhwxuq kdv wkh kljkhvw phdq ri dq| dydlodeoh sruwirolrvwudwhj|1 Wkh suredelolw| wkdw wkh glhuhqfh lv srvlwlyh lv ++N,@+N,,/zkhuh +, ghqrwhv wkh fxpxodwlyh glvwulexwlrq ixqfwlrq ri wkh vwdqgdugqrupdo glvwulexwlrq1 Dv wkh lqyhvwphqw krul}rq N lqfuhdvhv/ wkh udwlr+N,@+N, jurzv zlwk

sN/ vr ++N,@+N,, jrhv wr rqh1

Wkh idoodf| lv wkh fodlp wkdw wklv surshuw| ri wkh jurzwk0rswlpdo sruwirolrpdnhv lw wkh ehvw sruwirolr iru doo orqj0whup lqyhvwruv1 Lw lv/ ri frxuvh/ wkhehvw sruwirolr iru dq lqyhvwru zlwk orj xwlolw|> exw lqyhvwruv zlwk kljkhu ulvndyhuvlrq vkrxog krog pruh frqvhuydwlyh sruwirolrv1 Hyhq wkrxjk wkh jurzwk0rswlpdo sruwirolr zloo doprvw dozd|v rxwshuirup vxfk frqvhuydwlyh sruwirolrvryhu orqj krul}rqv/ wkh orvv zkhq lw grhv xqghushuirup lv odujhu dw orqjkrul}rqv/ dqg wklv srvvlelolw| lv khdylo| zhljkwhg e| frqvhuydwlyh lqyhvwruv1Wklv srlqw zdv irufhixoo| pdgh e| Vdpxhovrq +4


35/218


ixwxuh yhuvxv wkh glvwdqw ixwxuh1 Wklv lv rxu prgh ri dqdo|vlv iru prvw ri

wkh uhvw ri wkh errn1Zh uvw dvvxph wkdw lqyhvwruv kdyh wlph0vhsdudeoh srzhu xwlolw|/ ghqhgryhu frqvxpswlrq=

pd{H|

"['f

X+.|n, @ H|

"['f

.3|n 4

4 = +5168,

Khuh lv wkh wlph glvfrxqw idfwru1 Zkhq lv odujh/ lqyhvwruv sodfh uhodwlyho|kljk zhljkw rq wkh glvwdqw ixwxuh1 Dv vkulqnv/ wkh| sodfh pruh dqg pruhzhljkw rq wkh qhdu ixwxuh> lq wkh olplw dv dssurdfkhv }hur/ wkh| ehkdyh olnhvlqjoh0shulrg lqyhvwruv1 Lqyhvwruv idfh wkh lqwhuwhpsrudo exgjhw frqvwudlqwwkdw 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| .|,= +5169,Wklv remhfwlyh ixqfwlrq dqg exgjhw frqvwudlqw lpso| wkh iroorzlqj uvw0

rughu frqglwlrq ru Hxohu htxdwlrq iru rswlpdo frqvxpswlrq fkrlfh=

X+.|, @ H|^X+.|n,+4 Uc|n,`> +516:,

zkhuh +4 Uc|n, ghqrwhv dq| dydlodeoh uhwxuq/ iru h{dpsoh wkh ulvnohvvuhwxuq +4 Usc|n,/ wkh ulvn| uhwxuq +4 U|n,/ ru wkh sruwirolr uhwxuq+4 URc|n,1 Htxdwlrq +516:, vd|v wkdw dw wkh rswlpxp/ wkh pdujlqdo frvwri vdylqj dq h{wud groodu iru rqh shulrg pxvw htxdo wkh pdujlqdo ehqhw1Wkh pdujlqdo frvw lv wkh pdujlqdo xwlolw| ri d groodu ri frqvxpswlrq/ X+.|,1Wkh pdujlqdo ehqhw lv wkh h{shfwdwlrq ri wkh sd|r li wkh groodu lv lqyhvwhglq dq dydlodeoh dvvhw iru rqh shulrg/ +4 Uc|n,/ wlphv wkh pdujlqdo xwlolw|ri dq h{wud groodu ri frqvxpswlrq qh{w shulrg/ X+.|n,/ glvfrxqwhg edfn wrwkh suhvhqw dw udwh 1

Zh fdq glylgh +516:, wkurxjk e| X+.|, dqg xvh wkh srzhu xwlolw| frq0glwlrq wkdw X+.|, @ .

3| wr uhzulwh dv

4 @ H|

%

.|n

.|

3+4 Uc|n,

&= +516;,

Lq wkh fdvh zkhuh wkh uhwxuq lv ulvnohvv/ Uc|n @ Usc|n/ lw lv nqrzq dw wlphw dqg wkxv fdq eh eurxjkw rxwvlgh wkh h{shfwdwlrqv rshudwru> zh kdyh

4

+4 Usc|,@ H|

%

.|n

.|

3&= +516


36/218


Wkh whup +.|n@.|,3 wkdw dsshduv lq doo wkhvh h{suhvvlrqv lv nqrzq dv

wkh _vwrfkdvwlf glvfrxqw idfwru% ru VGI ehfdxvh lw fdq eh xvhg wr glvfrxqwh{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 dqVGI1 Wkh hog ri dvvhw sulflqj dvnv krz wkh surshuwlhv ri wkh VGI h{sodlqwkh surshuwlhv ri dvvhw uhwxuqv/ dqg krz wkh VGI lv ghwhuplqhg e| wkh jhq0hudo htxloleulxp ri wkh hfrqrp|> lq wklv errn zh wdnh d sduwldo0htxloleulxpshuvshfwlyh dqg vhhn wr h{sodlq wkh surshuwlhv ri rswlpdo frqvxpswlrq dqgsruwirolr fkrlfh jlyhq dvvhw uhwxuqv1 .dpsehoo +5333, dqg .rfkudqh +5333,vxuyh| wkh VGI phwkrgrorj|1

D orjqrupdo frqvxpswlrq0edvhg prgho

Wkh qdwxudo zd| wr surfhhg/ jlyhq rxu hduolhu glvfxvvlrq/ lv wr dvvxphwkdw dvvhw uhwxuqv dqg frqvxpswlrq duh mrlqwo| orjqrupdo/ dqg wr zrun zlwkorjolqhdu yhuvlrqv ri wkhvh htxdwlrqv1 Kdqvhq dqg Vlqjohwrq +4


37/218


wklug whup lqyroyhv wkh yduldqfhv dqg fryduldqfhv ri wkh sruwirolr uhwxuq dqg

frqvxpswlrq jurzwk1 Wklv dowhuqdwlyh uhsuhvhqwdwlrq lv pruh frqyhqlhqwiru vrph sxusrvhv/ dqg zloo eh xvhg lq rxu dqdo|vlv ri Hsvwhlq0]lq xwlolw| lqwkh qh{w vhfwlrq1

Wkh Hxohu htxdwlrq iru srzhu xwlolw| fdq dovr eh xvhg wr ghvfuleh wkhulvn suhplxp rq d vlqjoh ulvn| dvvhw ryhu wkh ulvnohvv lqwhuhvw udwh1 Wkh orjirup ri wkh jhqhudo Hxohu htxdwlrq +516;,/ ohvv wlphv +5173,/ lv

H|u|n usc|n 2|

5@ .ry|+u|ncf|n,= +5174,

Wklv vd|v wkdw lq htxloleulxp/ wkh h{shfwhg h{fhvv uhwxuq rq wkh ulvn| dvvhwpxvw htxdo ulvn dyhuvlrq wlphv wkh fryduldqfh ri wkh dvvhw uhwxuq zlwkfrqvxpswlrq jurzwk1 D vlplodu htxdwlrq ghvfulehv hdfk ulvn| dvvhw*v ulvnsuhplxp 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 wkhfrqvxpswlrq fdslwdo dvvhw sulflqj prgho ru ..DSP> khuh zh wdnh wkh ulvn|dvvhw uhwxuq dv jlyhq dqg vhhn d frqvxpswlrq uxoh dqg sruwirolr vwudwhj|wkdw zloo pdnh +5174, krog1

D frqvwdqw frqvxpswlrq0zhdowk udwlr

D glfxow| zlwk wkh orjqrupdo frqvxpswlrq0edvhg prgho lv wkdw wkhlqwhuwhpsrudo exgjhw frqvwudlqw +5169, lv qrw jhqhudoo| orjolqhdu/ ehfdxvhfrqvxpswlrq lv vxewudfwhg iurp zhdowk ehiruh pxowlso|lqj e| wkh sruwirolr

uhwxuq1 Wkh frpelqdwlrq ri vxewudfwlrq dqg pxowlsolfdwlrq fuhdwhv dq lq0wudfwdeoh qrqolqhdulw|1Lq odwhu fkdswhuv ri wklv errn/ iroorzlqj .dpsehoo +4 +5176,

66


38/218


zkhuh wkh vhfrqg htxdolw| vxevwlwxwhv lq iurp +5154,1

Wkh frqvwdqw frqvxpswlrq0zhdowk udwlr +5175, dovr lpsolhv wkdw wkh jurzwkudwh ri frqvxpswlrq htxdov wkh jurzwk udwh ri zhdowk/ vr wkh whupv lq frq0vxpswlrq lq +5173, dqg +5174, fdq eh uhzulwwhq lq whupv ri zhdowk1 Wkhirupxod iru d vlqjoh ulvn| dvvhw*v h{shfwhg h{fhvv uhwxuq/ +5174,/ ehfrphv

H|u|n usc|n 2|@5 @ .ry|+u|ncz|n, @ |2| > +5177,

zkhuh wkh vhfrqg htxdolw| iroorzv iurp +5176,1 Vroylqj wklv htxdwlrq iru |/zh rqfh djdlq rewdlq wkh p|rslf vroxwlrq +5158,=

| @H|u|n usc|n 2|@5

2|=

Lw lv vwudljkwiruzdug wr vkrz wkdw wkh p|rslf vroxwlrq iru pxowlsoh ulvn|dvvhwv/ +515:,/ lv htxdoo| ydolg iru d orqj0whup lqyhvwru zlwk d frqvwdqwfrqvxpswlrq0zhdowk udwlr1

Lq d prgho zlwk srzhu xwlolw|/ wkhq/ wkh dvvxpswlrq wkdw wkh frqvxpswlrq0zhdowk udwlr lv frqvwdqw lpphgldwho| ohdgv wr wkh frqfoxvlrq wkdw sruwirolrfkrlfh lv p|rslf1 Xqghu zkdw frqglwlrqv fdq wklv dvvxpswlrq eh mxvwlhgBWkh dqvzhu wr wklv txhvwlrq lv e| qrz idploldu1 Iluvw/ li uhwxuqv duh LLG/wkhuh duh qr fkdqjhv ryhu wlph lq lqyhvwphqw rssruwxqlwlhv wkdw pljkw lq0gxfh fkdqjhv lq frqvxpswlrq uhodwlyh wr zhdowk1 Wkh vfdoh0lqghshqghqfh riwkh srzhu xwlolw| ixqfwlrq lpsolhv wkdw frqvxpswlrq lv d frqvwdqw iudfwlrqri zhdowk lq wklv fdvh1 Vhfrqg/ li ulvn dyhuvlrq @ 4/ frqvxpswlrq lv djdlq

d frqvwdqw iudfwlrq ri zhdowk1 Wkh lqwxlwlrq iru wklv uhvxow lv wkdw fkdqjlqjlqyhvwphqw rssruwxqlwlhv kdyh rssrvlqj hhfwv rq frqvxpswlrq uhodwlyh wrzhdowk1 Dq lpsuryhg lqyhvwphqw rssruwxqlw|/ vd| d kljkhu ulvnohvv lqwhuhvwudwh/ udlvhv wkh dprxqw wkdw fdq eh frqvxphg hdfk shulrg zlwkrxw ghsohw0lqj zhdowk1 Wklv lqfrph hhfw whqgv wr lqfuhdvh frqvxpswlrq uhodwlyh wrzhdowk1 Rq wkh rwkhu kdqg/ dq lpsuryhg lqyhvwphqw rssruwxqlw| fuhdwhvdq lqfhqwlyh wr srvwsrqh frqvxpswlrq wr wkh ixwxuh1 Wklv vxevwlwxwlrq hi0

ihfw whqgv wr ghfuhdvh frqvxpswlrq uhodwlyh wr zhdowk1 Lq wkh srzhu0xwlolw|prgho/ wkh orj0xwlolw| fdvh @ 4 lv wkh fdvh zkhuh lqfrph dqg vxevwlwxwlrqhhfwv h{dfwo| fdqfho rxw/ vr wkdw wkh frqvxpswlrq0zhdowk udwlr lv dozd|vfrqvwdqw uhjdugohvv ri dq| xfwxdwlrqv lq lqyhvwphqw rssruwxqlwlhv1

Wr suryh wkhvh vwdwhphqwv/ rqh fdq xvh wkh exgjhw frqvwudlqw +5176, wrqg wkh frqglwlrqdo phdq dqg yduldqfh ri zhdowk/ dqg vxevwlwxwh lqwr +5173,1Wkh uhvxowlqj htxdwlrq fdq eh vroyhg iru d frqvwdqw e li dvvhw uhwxuqv duh LLGru li @ 41 Lqwxlwlyho|/ +5173, vd|v wkdw h{shfwhg frqvxpswlrq jurzwkvkrxog pryh 4@ iru rqh zlwk wkh h{shfwhg uhwxuq1 Wklv lv frqvlvwhqw zlwk

67


39/218


d frqvwdqw frqvxpswlrq0zhdowk udwlr li wkh h{shfwhg uhwxuq lv frqvwdqw/ ru li

h{shfwhg frqvxpswlrq jurzwk dgmxvwv rqh iru rqh zlwk wkh h{shfwhg uhwxuqvr wkdw wkh ghvluhg fkdqjhv lq frqvxpswlrq jurzwk fdq eh qdqfhg mxvwe| wkh yduldwlrq lq wkh h{shfwhg uhwxuq lwvhoi/ zlwkrxw dq| qhhg iru vdylqjvdgmxvwphqwv1

22e ,Tt|i?~? |*|)

Ghvslwh wkh pdq| dwwudfwlyh ihdwxuhv ri wkh srzhu0xwlolw| prgho/ lw grhv kdyhrqh kljko| uhvwulfwlyh ihdwxuh1 Srzhu xwlolw| lpsolhv wkdw wkh frqvxphu*vhodvwlflw| ri lqwhuwhpsrudo vxevwlwxwlrq/ #/ lv wkh uhflsurfdo ri wkh frhflhqwri uhodwlyh ulvn dyhuvlrq/ 1 \hw lw lv qrw fohdu wkdw wkhvh wzr frqfhswv vkrxogeh olqnhg vr wljkwo|1 Ulvn dyhuvlrq ghvfulehv wkh frqvxphu*v uhoxfwdqfh wr

vxevwlwxwh frqvxpswlrq dfurvv vwdwhv ri wkh zruog dqg lv phdqlqjixo hyhqlq dq dwhpsrudo vhwwlqj/ zkhuhdv wkh hodvwlflw| ri lqwhuwhpsrudo vxevwlwxwlrqghvfulehv wkh frqvxphu*v zloolqjqhvv wr vxevwlwxwh frqvxpswlrq ryhu wlphdqg lv phdqlqjixo hyhq lq d ghwhuplqlvwlf vhwwlqj1 Hsvwhlq dqg ]lq +4


40/218


1Elasticity of Intertemporal Substitution ()

Constant consumption-wealth ratio

Myopic portfolio choice

CoefficientofRelativeRiskAversion()

Log utility

1

Power utility

1

=

Iljxuh 515= Hsvwhlq0]lq xwlolw|

69


41/218


+wkdw lv/ li wkh lqyhvwru qdqfhv frqvxpswlrq hqwluho| iurp qdqfldo zhdowk

dqg grhv qrw uhfhlyh oderu lqfrph,/ wkhq wkhuh lv dq Hxohu htxdwlrq ri wkhirup

4 @ H|

57+.|n.|

3

,w 4

+4 URc|n,

3w+4 Uc|n,

68 > +5179,zkhuh dv ehiruh +4 Uc|n, lv wkh jurvv uhwxuq rq dq| dydlodeoh dvvhw/ lq0foxglqj wkh ulvnohvv dvvhw dqg wkh sruwirolr lwvhoi1

Htxdwlrq +5179, vlpsolhv vrphzkdw li zh vhw +4 Uc|n, @ +4 URc|n,1Li wkh sruwirolr uhwxuq dqg frqvxpswlrq duh mrlqwo| orjqrupdo/ zh wkhq qgwkdw h{shfwhg frqvxpswlrq jurzwk htxdov

H|^f|n` @ # orj #H|uRc|n 5#

Ydu|^f|n #uRc|n`= +517:,

H{shfwhg frqvxpswlrq jurzwk lv ghwhuplqhg e| wlph suhihuhqfh/ wkh h{0shfwhg sruwirolr uhwxuq/ dqg wkh hhfwv ri xqfhuwdlqw| vxppdul}hg lq wkhyduldqfh whup1 Qrwh wkdw wkh hodvwlflw| ri lqwhuwhpsrudo vxevwlwxwlrq #/ dqgqrw wkh frhflhqw ri uhodwlyh ulvn dyhuvlrq / ghwhuplqhv wkh uhvsrqvh ri h{0shfwhg frqvxpswlrq jurzwk wr yduldwlrqv lq wkh h{shfwhg uhwxuq1 Udqgrp0qhvv lq ixwxuh frqvxpswlrq jurzwk/ uhodwlyh wr sruwirolr uhwxuqv/ lqfuhdvhvsuhfdxwlrqdu| vdylqjv dqg orzhuv fxuuhqw frqvxpswlrq li A 3 +d frqgl0wlrq vdwlvhg e| srzhu xwlolw| iru zklfk @ 4,/ exw uhgxfhv suhfdxwlrqdu|vdylqjv dqg lqfuhdvhv fxuuhqw frqvxpswlrq li ? 31

Zkhq wkhuh lv d vlqjoh ulvn| dvvhw/ wkh suhplxp rq wkh ulvn| dvvhw ryhuwkh vdih dvvhw lv

H|u|nusc|n 2|

5@

.ry|+u|ncf|n,

# +4,.ry|+u|ncuRc|n,= +517;,

Wkh h{shfwhg h{fhvv uhwxuq rq wkh ulvn| dvvhw lv d zhljkwhg dyhudjh ri wkhulvn| dvvhw*v fryduldqfh zlwk frqvxpswlrq jurzwk +glylghg e| wkh hodvwlflw|ri lqwhuwhpsrudo vxevwlwxwlrq #, dqg dvvhw l*v fryduldqfh zlwk wkh sruwirolruhwxuq1 Wkh zhljkwv duh dqg 4 uhvshfwlyho|1

D vlplodu htxdwlrq krogv iru hdfk ulvn| dvvhw lq d prgho zlwk pxowlsohulvn| dvvhwv dqg qr vkruw0whup ulvnohvv dvvhw1 Lq yhfwru qrwdwlrq/ zh kdyh

H|+u|n ufc|n, 2|

5@

#S| +4 ,R| f|> +517


42/218


fryduldqfhv zlwk frqvxpswlrq jurzwk/ R| lv wkh yhfwru ri h{fhvv0uhwxuq

fryduldqfhv zlwk wkh wrwdo uhwxuq rq wkh sruwirolr/ dqg f| lv wkh yhfwru rih{fhvv0uhwxuq fryduldqfhv zlwk wkh wrwdo uhwxuq rq wkh ehqfkpdun dvvhw1 Lqdq dvvhw sulflqj frqwh{w/ wklv htxdwlrq h{sodlqv dq| dvvhw*v ulvn suhplxp e|uhihuhqfh wr erwk lwv frqvxpswlrq fryduldqfh +wkh frqvxpswlrq .DSP, dqglwv fryduldqfh zlwk wkh lqyhvwru*v ryhudoo sruwirolr +wkh wudglwlrqdo .DSP,1Lq wkh srzhu xwlolw| fdvh/ @ 4 dqg zh kdyh d sxuh frqvxpswlrq .DSP1

Wkh idploldu frqglwlrqv iru p|rslf sruwirolr fkrlfh iroorz lpphgldwho|iurp +517;,1 Li dvvhw uhwxuqv duh LLG/ wkhq frqvxpswlrq lv d frqvwdqw iudfwlrqri zhdowk dqg fryduldqfh zlwk frqvxpswlrq jurzwk htxdov fryduldqfh zlwksruwirolr uhwxuq1 Lq wklv fdvh wkh uljkw0kdqg vlgh ri +517;, fdq eh uhzulwwhqdv + @# 4 ,.ry|+u|ncuRc|n, @ .ry|+u|ncuRc|n,/ zklfk lpsolhv wkhp|rslf sruwirolr uxoh1

Dowhuqdwlyho|/ li uhodwlyh ulvn dyhuvlrq @ 4/ wkhq @ 3 dqg wkh uljkw0kdqg vlgh ri +517;, lv mxvw .ry|+u|ncuRc|n,/ zklfk djdlq lpsolhv wkh p|rslfsruwirolr uxoh1 Wklv ghulydwlrq pdnhv lw fohdu wkdw zkdw lv uhtxluhg irup|rslf sruwirolr fkrlfh lv xqlw uhodwlyh ulvn dyhuvlrq/ qrw d xqlw hodvwlflw|ri lqwhuwhpsrudo vxevwlwxwlrq1 Orj xwlolw| lv wkh vshfldo fdvh zkhuh erwkuhodwlyh ulvn dyhuvlrq dqg wkh hodvwlflw| ri lqwhuwhpsrudo vxevwlwxwlrq +HLV,htxdo rqh1

Wkh fdvh ri d xqlw HLV uhtxluhv fduhixo kdqgolqj lq wklv prgho1 Dv #dssurdfkhv rqh/ dssurdfkhv srvlwlyh ru qhjdwlyh lqqlw|1 Htxdwlrq +517:,fdq rqo| eh vdwlvhg li Ydu|^f|n uRc|n` @ 3/ zklfk lpsolhv d frqvwdqwfrqvxpswlrq0zhdowk udwlr1 Wr dqdo|}h sruwirolr fkrlfh lq wklv fdvh/ rqh

pxvw wdnh dssursuldwh olplwv ri wkh whupv rq wkh uljkw0kdqg vlgh ri +517;,1Jlrydqqlql dqg Zhlo +4


43/218


qhqwv lq +5183, lqfuhdvh zlwkrxw olplw1 Wkh ydoxh ixqfwlrq kdv d qlwh olplw/

krzhyhu/ ehfdxvh wkh udwlr .|@Z| dssurdfkhv +4 , dv vkrzq e| Jlrydq0qlql dqg Zhlo +4


44/218

@T|ih

`L 5L*_ ) wL?}Aih4

L?_tq

Lq .kdswhu 5 zh h{soruhg lq ghwdlo wkh frqglwlrqv iru p|rslf sruwirolr fkrlfh1Zh qrz ehjlq wr ghyhors d wkhru| ri orqj0whup sruwirolr fkrlfh zkhq wkhvhfrqglwlrqv idlo1 Wr nhhs wklqjv dv vlpsoh dv srvvleoh/ zh dvvxph lqlwldoo|wkdw doo wkh udqgrp yduldeohv wkdw duh uhohydqw iru lqyhvwruv duh orjqrupdoo|glvwulexwhg zlwk frqvwdqw yduldqfhv dqg fryduldqfhv1 Wklv dvvxpswlrq lvqrw lqqrfxrxv lq wkh frqwh{w ri sruwirolr fkrlfh wkhru|/ ehfdxvh lw uhtxluhverwk wkdw wkh lqyhvwru*v sruwirolr uhwxuq kdv d frqvwdqw yduldqfh/ dqg wkdwwkh uhwxuqv rq lqglylgxdo dvvhwv kdyh frqvwdqw yduldqfhv dqg d frqvwdqwfryduldqfh zlwk wkh sruwirolr1 Wkhvh frqglwlrqv duh frqvlvwhqw zlwk rqhdqrwkhu rqo| li wkh frpsrvlwlrq ri wkh sruwirolr lv frqvwdqw/ zklfk lq wxuq lvrswlpdojlyhq frqvwdqw yduldqfhvrqo| li wkh h{shfwhg h{fhvv uhwxuqv rq doodvvhwv duh frqvwdqw1 Dvvhw uhwxuqv fdq fkdqjh ryhu wlph/ exw wkh| pxvw pryhlq sdudooho zlwk wkh ulvnohvv lqwhuhvw udwh1 Lq rwkhu zrugv/ rxu lqlwldo prghodoorzv rqo| d olplwhg irup ri wlph0yduldwlrq lq lqyhvwphqw rssruwxqlwlhv/gulyhq e| pryhphqwv lq wkh vkruw0whup lqwhuhvw udwh1 Lq wkh qh{w fkdswhuzh jhqhudol}h wkh prgho wr doorz pryhphqwv lq ulvn suhpld dv zhoo1

Wkurxjkrxw wklv dqg wkh qh{w fkdswhu zh frqvlghu lqyhvwruv zkr kdyhrqo| qdqfldo zhdowk dqg qr oderu lqfrph/ ghihuulqj d frqvlghudwlrq ri oderulqfrph wr .kdswhu 91 Zh dvvxph wkdw lqyhvwruv fduh qrw derxw zhdowk iru lwvrzq vdnh/ exw derxw wkh frqvxpswlrq vwuhdp wkdw fdq eh qdqfhg e| zhdowk1

Wr nhhs wkh dqdo|vlv vlpsoh zh dvvxph wkdw wkh lqyhvwru lv lqqlwho| olyhg>zh fdq ydu| wkh hhfwlyh lqyhvwphqw krul}rq e| ydu|lqj wkh lqyhvwru*v udwh riwlph suhihuhqfh dqg wkxv ydu|lqj wkh uhodwlyh lpsruwdqfh ri wkh qhdu ixwxuhdqg wkh glvwdqw ixwxuh1 Zh dvvxph wkdw wkh lqyhvwru kdv wkh Hsvwhlq0]lq

73


45/218

.KDSWHU 61 ZKR VKRXOG EX\ ORQJ0WHUP ERQGVB

suhihuhqfhv ghyhorshg lq vhfwlrq 51517/ zlwk frqvwdqw uhodwlyh ulvn dyhuvlrq

dqg d frqvwdqw hodvwlflw| ri lqwhuwhpsrudo vxevwlwxwlrq # wkdw qhhg qrw ehuhodwhg wr rqh dqrwkhu1Wkh nh| sureohp lq vroylqj dq lqwhuwhpsrudo prgho ri wklv vruw lv wkdw wkh

lqyhvwru*v lqwhuwhpsrudo exgjhw frqvwudlqw lv qrqolqhdu1 Lq vhfwlrq 61414 zhdssurdfk wklv sureohp e| wdnlqj d orjolqhdu dssur{lpdwlrq ri wkh exgjhwfrqvwudlqw/ dv uvw sursrvhg e| .dpsehoo +4


46/218


wL?}Aih4 Lh|uL*L LUi ? @ L_i* |

L?t|@?| V@h@?Uit @?_ +t! hi4@

TThL4@|L? Lu |i ?|ih|i4TLh@* M_}i| UL?t|h@?|

Uhfdoo wkdw wkh lqwhuwhpsrudo exgjhw frqvwudlqw/ lq d prgho zlwk frqvxps0wlrq dw hyhu| gdwh/ lv

Z|n @ +4 URc|n,+Z| .|,= +614,

D fhqwudo sureohp lq wkh wkhru| ri lqwhuwhpsrudo sruwirolr fkrlfh lv wkdwwklv exgjhw frqvwudlqw lv qrqolqhdu ehfdxvh frqvxpswlrq lv vxewudfwhg iurpzhdowk ehiruh wkh sruwirolr uhwxuq pxowlsolhv wkh uhpdlqghu1 Lq rwkhu zrugv/

rqo| uhlqyhvwhg zhdowk hduqv wkh sruwirolr uhwxuq dqg qrw doo zhdowk lv uhlq0yhvwhg1Lq wkh odvw fkdswhu zh frqfhqwudwhg rq prghov lq zklfk wkh frqvxpswlrq0

zhdowk udwlr lv frqvwdqw1 Lq wklv fdvh uhlqyhvwhg zhdowk +Z| .|, lv dfrqvwdqw iudfwlrq ri wrwdo zhdowk/ dqg +614, fdq eh uhzulwwhq lq orjolqhduirup1

Zh qrz ghyhors dq dowhuqdwlyh dssurdfk wr wkh sureohp ri qrqolqhdulw|1Iroorzlqj .dpsehoo +4 +617,

zkhuh n dqg duh sdudphwhuv ri olqhdul}dwlrq1 Wkh sdudphwhu lv ghqhge| @ 4 h{s+f z,1 Zkhq wkh frqvxpswlrq0zhdowk udwlr lv frqvwdqw/wkhq fdq eh lqwhusuhwhg dv +Z .,@Z/ wkh udwlr ri uhlqyhvwhg zhdowkwr wrwdo zhdowk1 Wkh sdudphwhu n lv jlyhq e| wkh phvv| h{suhvvlrq n @orj+, +4 , orj+4 ,@1

75


47/218


Lw lv khosixo wr ghulyh d orqj0whup yhuvlrq ri wklv exgjhw frqvwudlqw1 Wr

gr wklv zh fdq xvh wkh wulyldo htxdolw|

z|n @ f|n +f| z|, +f|n z|n,= +618,

Htxdwlqj wkh ohiw0kdqg vlghv ri +617, dqg +618,/ zh rewdlq d glhuhqfh htxd0wlrq lq wkh orj frqvxpswlrq0zhdowk udwlr1 Zh fdq vroyh iruzdug/ dvvxplqjwkdw olp3


48/218


Wkhvh irupxodv duh dqdorjrxv wr h{suhvvlrqv ghyhorshg e| .dpsehoo dqg

Vkloohu +4


49/218


idyrudeoh lqyhvwphqw rssruwxqlwlhv> wklv lv wkh qhjdwlyh vxevwlwxwlrq hhfw ri

wkh sruwirolr uhwxuq rq frqvxpswlrq1 Wkh lqfrph hhfw grplqdwhv li# ? 4dqg wkh vxevwlwxwlrq hhfw grplqdwhv li # A 41 Li # @ 4/ wkh wzr hhfwvfdqfho dqg wkh frqvxpswlrq0zhdowk udwlr lv frqvwdqw dv suhylrxvo| qrwhg1

Htxdwlrq +6143, doorzv xv wr uhzulwh +61;, dv

f|n H|f|n @ uRc|n H|uRc|n +4 #,+H|n H|,"['

uRc|nn= +6144,

Wkh lqqrydwlrq lq frqvxpswlrq lv wkh vxusulvh frpsrqhqw ri wkh sruwirolruhwxuq/ zklfk kdv d rqh0iru0rqh hhfw ehfdxvh ri wkh vfdoh0lqghshqghqfh riwkh xwlolw| ixqfwlrq/ soxv +4 #, wlphv wkh uhylvlrq lq h{shfwdwlrqv ri ix0wxuh uhwxuqv1 Li # ? 4/ d srvlwlyh vxusulvh derxw ixwxuh uhwxuqv lqfuhdvhvfrqvxpswlrq wrgd| wkurxjk wkh grplqdqw lqfrph hhfw> li # A 4/ d srvl0wlyh vxusulvh derxw ixwxuh uhwxuqv uhgxfhv frqvxpswlrq wrgd| wkurxjk wkhgrplqdqw vxevwlwxwlrq hhfw1

Hduolhu zh dvvxphg wkdw frqvxpswlrq dqg dvvhw uhwxuqv duh mrlqwo| orj0qrupdo dqg krprvnhgdvwlf1 Zh fdq qrz vhh zkdw lv uhtxluhg wr mxvwli|wklv dvvxpswlrq1 Dffruglqj wr +6144,/ zh qhhg wkdw orj sruwirolr uhwxuqvdqg uhylvlrqv lq h{shfwdwlrqv ri ixwxuh orj sruwirolr uhwxuqv duh qrupdo dqgkrprvnhgdvwlf/ dv wkh| zloo eh iru h{dpsoh lq d olqhdu wlph0vhulhv prgho iruorj uhwxuqv dqg rwkhu vwdwh yduldeohv1

Uhfdoo wkdw xqghu Hsvwhlq0]lq xwlolw| zlwk d vlqjoh ulvn| dvvhw/ wkh suh0plxp rq wkh ulvn| dvvhw ryhu wkh vdih dvvhw lv jlyhq e| +517;,=

H|u|n usc|n 2|

5@

.ry|+u|ncf|n,

# +4 ,.ry|+u|ncuRc|n,=

Dowkrxjk zh duh dvvxplqj wkdw frqglwlrqdo yduldqfhv dqg fryduldqfhv duhfrqvwdqw ryhu wlph/ zh uhwdlq wkhlu wlph vxevfulswv wr pdnh lw fohdu wkdw wkh|duh fdofxodwhg frqglwlrqdo rq wlph w lqirupdwlrq1 Htxdwlrq +6144, lpsolhvwkdw wkh fryduldqfh zlwk frqvxpswlrq wkdw dsshduv lq wklv h{suhvvlrq fdqeh uhsodfhg e| wkh fryduldqfh zlwk wkh sruwirolr uhwxuq/ soxv +4 #, wlphvwkh fryduldqfh zlwk uhylvhg h{shfwdwlrqv derxw ixwxuh uhwxuqv1 Xvlqj wkhuhodwlrq ehwzhhq wkh sdudphwhuv / / dqg #/ zh qg wkdw

H|u|n usc|n 2|5

@ .ry|+u|ncuRc|n, +6145,

+ 4,.ry|+u|n> +H|n H|,"['

uRc|nn,=

78


50/218


.dpsehoo +4


51/218


zklfk ehfrphv qhjoljleoh dv lqfuhdvhv> d kljko| frqvhuydwlyh lqyhvwru grhv

qrw ex| wkh ulvn| dvvhw iru lwv ulvn suhplxp1 Wkh zhljkw rq wkh vhfrqgdwwulexwh lv 4 +4@,/ zklfk lv }hur iru d p|rslf lqyhvwru zlwk @ 4 exwdssurdfkhv 4 dv lqfuhdvhv1 D kljko| frqvhuydwlyh lqyhvwru krogv wkh ulvn|dvvhw rqo| li lw frydulhv zlwk ghfolqhv lq lqwhuhvw udwhv/ frpshqvdwlqj wkhsruwirolr iru wkh uhgxfwlrq lq lqfrph wkdw rffxuv zkhq lqwhuhvw udwhv idoo1

D yduldqw ri +6146, zdv ghulyhg e| Uhvwr| +4


52/218


dq h{solflw vroxwlrq frpsdudeoh wr +6146,/ uhodwlqj sruwirolr ghpdqgv wr fr0

yduldqfhv ri dvvhwv zlwk h{rjhqrxv yduldeohv1 Htxdwlrq +6146, ghshqgv rqvshflf dvvxpswlrqv/ exw lw fdq eh xvhg wr rewdlq jhqhudo lqvljkwv1 Qrwhlq sduwlfxodu wkdw lqwhuwhpsrudo khgjlqj ghpdqg lq +6146, ghshqgv rq wkhglvfrxqwhg suhvhqw ydoxh ri doo ixwxuh lqwhuhvw udwhv> wkxv sruwirolr fkrlfh riorqj0whup lqyhvwruv lv dhfwhg idu pruh vwurqjo| e| shuvlvwhqw yduldwlrqv lqlqyhvwphqw rssruwxqlwlhv wkdq e| wudqvlwru| yduldwlrqv1

e `@| t |i ht!*itt @tti|q

Wkh lghqwlw| ri wkh ulvnohvv dvvhw lv d ixqgdphqwdo lvvxh lq qdqfh1 Lw lvfrqyhqwlrqdo wr wklqn ri wkh ulvnohvv dvvhw dv dq dvvhw wkdw kdv d vwdeoh fdslwdoydoxh lq wkh vkruw whup/ vxfk dv d Wuhdvxu| eloo ru prqh| pdunhw ixqg1 Vxfk

dq dvvhw kdv d nqrzq uhwxuq ryhu rqh shulrg/ dqg zloo eh khog e| dq lqqlwho|frqvhuydwlyh vkruw0whup lqyhvwru1

Iru d orqj0whup lqyhvwru/ krzhyhu/ d vwudwhj| ri uroolqj ryhu Wuhdvxu|eloov lv qrw qhfhvvdulo| vdih ehfdxvh pdwxulqj eloov pxvw eh uhlqyhvwhg dwxqnqrzq ixwxuh uhdo lqwhuhvw udwhv1 Ryhu wkluw| |hduv djr Prgljoldql dqgVxwfk +4 wkdw lv/ kh kdvixqgv zklfk kh zloo qrw qhhg iru q shulrgv dqg zklfk/ wkhuhiruh/kh lqwhqgv wr nhhs lqyhvwhg lq erqgv iru q shulrgv1 Li kh lqyhvwvlq q shulrg erqgv/ kh zloo nqrz h{dfwo| wkh rxwfrph ri klv lq0

yhvwphqwv dv phdvxuhg e| wkh whuplqdo ydoxh ri klv zhdowk1111 Li/krzhyhu/ kh vwd|v vkruw/ klv rxwfrph lv xqfhuwdlq1111 Wkxv/ li khkdv ulvn dyhuvlrq/ kh zloo suhihu wr vwd| orqj +ss1 4;6~4;7,1

Prgljoldql dqg Vxwfk*v dvvxpswlrq/ wkdw dq lqyhvwru fduhv derxw rqo|derxw zhdowk dw d vlqjoh ixwxuh gdwh/ lv vrphzkdw duwlfldo1 Krzhyhu dvlplodu lqvljkw dssolhv wr d orqj0whup lqyhvwru ri wkh vruw prghoohg lq wklverrn/ zkr fduhv derxw wkh vwuhdp ri frqvxpswlrq ru vwdqgdug ri olylqj wkdwfdq eh vxssruwhg e| zhdowk1 Wklv srlqw zdv uvw pdgh e| Vwljolw} +4


53/218


Qrz frqvlghu wkh sulflqj ri dq lqdwlrq0lqgh{hg shushwxlw| ru frqvro/

wkdw sd|v rqh xqlw ri frqvxpswlrq hdfk shulrg iruhyhu1 .dpsehoo/ Or dqgPdfNlqod| +4


54/218


D Bi?ih@*3?} |i tL*|L?

Wkh deryh dqdo|vlv jhqhudol}hv vwudljkwiruzdugo| wr wkh fdvh zkhuh wkhuh duhpxowlsoh ulvn| dvvhwv/ zlwk ru zlwkrxw d vkruw0whup ulvnohvv dvvhw1 Dv lq wkhsuhylrxv fkdswhu/ zh zulwh wkh yduldqfh0fryduldqfh pdwul{ ri h{fhvv ulvn|dvvhw uhwxuqv dv | dqg wkh yhfwru ri h{fhvv0uhwxuq yduldqfhv/ wkh pdlq gldj0rqdo ri|/ dv

2| 1 Zh dovr ghqh d yhfwru | wkdw frqwdlqv wkh fryduldqfhv

ri hdfk ulvn| dvvhw uhwxuq zlwk ghfolqhv lq h{shfwhg ixwxuh lqwhuhvw udwhv=

| .ry|+u|n> +H|n H|,"['

usc|nn,= +614 +6154,

zkhuh f| lv wkh yhfwru ri fryduldqfhv ri hdfk ulvn| dvvhw*v h{fhvv uhwxuq ryhuwkh ehqfkpdun zlwk wkh ehqfkpdun uhwxuq lwvhoi1 Lq sudfwlfh/ dv zh kdyhqrwhg hduolhu/ wkh dgmxvwphqw iru ehqfkpdun fryduldqfhv whqgv wr eh vpdoozkhq d vkruw0whup dvvhw lv xvhg dv wkh ehqfkpdun1

Lw lv whpswlqj wr uhodwh wkhvh vroxwlrqv wr wkh lqdwlrq0lqgh{hg frqvrolqwurgxfhg lq wkh suhylrxv vhfwlrq1 Li wkh orjolqhdul}dwlrq sdudphwhu lv{hg dqg htxdo wr S/ wkhq wkh yhfwru | lv htxdo wr d yhfwru S| frqwdlq0lqj fryduldqfhv ri hdfk ulvn| dvvhw uhwxuq zlwk wkh lqdwlrq0lqgh{hg frqvro

uhwxuq1 Lq wklv fdvh zh fdq jlyh d uhjuhvvlrq lqwhusuhwdwlrq wr wkh khgjlqjwhup lq +6153,1 3| S| lv wkh yhfwru ri srsxodwlrq uhjuhvvlrq frhflhqwviurp d pxowlsoh uhjuhvvlrq ri dq lqdwlrq0lqgh{hg frqvro uhwxuq rqwr wkh vhwri ulvn| dvvhw uhwxuqv1 Li dq lqdwlrq0lqgh{hg frqvro lv lq wkh vhw ri dydlo0deoh ulvn| dvvhwv/ wkhq wklv yhfwru zloo sodfh }hur zhljkw rq doo ulvn| dvvhwv

83


55/218


h{fhsw wkh lqdwlrq0lqgh{hg frqvro lwvhoi1 Li dq lqdwlrq0lqgh{hg frqvro lv

qrw dydlodeoh/ wkhq wkh rswlpdo khgjlqj sruwirolr zloo eh wkdw frpelqdwlrqri wkh ulvn| dvvhwv wkdw ehvw dssur{lpdwhv wkh uhwxuq rq dq lqdwlrq0lqgh{hgfrqvro lq d uhjuhvvlrq vhqvh1

Wkh sureohp zlwk wklv glvfxvvlrq lv wkdw wkh orjolqhdul}dwlrq sdudphwhu ghshqgv rq suhihuhqfhv/ dqg zloo qrw h{dfwo| htxdo S xqohvv wkh lqyhvwrulv lqqlwho| ulvn0dyhuvh dqg lqqlwho| xqzloolqj wr vxevwlwxwh frqvxpswlrqlqwhuwhpsrudoo| +4@ @ # @ 3,1 Iru doo rwkhu suhihuhqfhv/ 9@ S dqgwkxv | 9@ S|/ dowkrxjk wkh glhuhqfhv duh vpdoo hpslulfdoo|1 Lq wkh fdvhzkhuh # @ 4/ zh nqrz wkdw @ / wkh wlph glvfrxqw idfwru1 Li # 9@ 4/zh pxvw vroyh iru qxphulfdoo|1 Zh gr wklv xvlqj d uhfxuvlyh surfhgxuh1Zh wdnh dq lqlwldo ydoxh ri / vroyh +6154, iru wkh rswlpdo sruwirolr/ vroyhiru wkh fruuhvsrqglqj rswlpdo frqvxpswlrq0zhdowk udwlr xvlqj +6143,/ xvhwklv frqvxpswlrq0zhdowk udwlr wr fdofxodwh d qhz ydoxh iru / dqg uhshdwxqwlo frqyhujhqfh1 Lq sudfwlfh wkhvh fdofxodwlrqv duh h{wuhpho| udslg dqgvwudljkwiruzdug1

Zh fdq dovr doorz iru eruurzlqj dqg vkruw0vdohv frqvwudlqwv1 Xqfrq0vwudlqhg sruwirolr doorfdwlrqv duh riwhq kljko| ohyhudjhg> exw wklv shuplwvwkh srvvlelolw| ri edqnuxswf| lq d glvfuhwh0wlph prgho/ dqg pdq| lqyhvwruvduh frqvwudlqhg lq wkhlu xvh ri ohyhudjh1 Ehfdxvh wkh xqfrqvwudlqhg rswlpdosruwirolr srolf| lv frqvwdqw ryhu wlph/ zh fdq lpsrvh frqvwudlqwv xvlqj uh0vxowv lq Whsod +4


56/218


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 wkdqrqh1 Li dq lqdwlrq0lqgh{hg frqvro lv qrw dydlodeoh/ d frqvhuydwlyh lqyhvwruzloo krog dvvhwv wkdw duh jrrg sur{lhv iru lw1

Lq rughu wr jr ixuwkhu/ zh qhhg wr xvh dq h{solflw prgho ri wkh whupvwuxfwxuh ri lqwhuhvw udwhv wr ghulyh txdqwlwdwlyh suhglfwlrqv iru lqyhvwruv*kroglqjv ri Wuhdvxu| eloov/ orqj0whup erqgv/ dqg htxlwlhv1 Iroorzlqj .dps0ehoo dqg Ylfhlud +5334,/ zh qrz suhvhqw vxfk d prgho1 Wkh prgho lv vhw lqglvfuhwh wlph> lw kdv wzr idfwruv/ rqh uhdo dqg rqh qrplqdo/ zlwk orjqrupdoglvwulexwlrqv dqg frqvwdqw yduldqfhv1 Wklv lv wkh vlpsohvw whup vwuxfwxuhprgho wkdw doorzv xv wr glvwlqjxlvk ehwzhhq uhdo dqg qrplqdo erqgv1 Wkhuhdo sduw ri wkh prgho lv d glvfuhwh0wlph yhuvlrq ri wkh zhoo0nqrzq Ydvlfhn+4


57/218


wkh orj uhwxuq/ ru htxlydohqwo| wkh orj |lhog/ rq d rqh0shulrg lqgh{hg erqg=

uc|n @ orjH|^P|n` @ H|^p|n`

2 Ydu|^p|n` vlqfh P|n lv orjqrupdo1Vxevwlwxwlqj lq iurp +6155,~+6157,/ zh kdyh

uc|n @ {| 45

26%

2%

26

> +6158,

zkhuh 2% Ydu|+%%c|n, dqg 26 Ydu|+%6c|n,1 Wkh vkruw0whup uhdolqwhuhvw udwh htxdov wkh vwdwh yduldeoh {| dgmxvwhg e| d frqvwdqw/ vr lw iroorzvdq DU+4, surfhvv zlwk shuvlvwhqfh !1

Orqjhu0whup lqdwlrq0lqgh{hg erqgv fdq eh sulfhg uhfxuvlyho|1 D wzr0shulrg erqg wrgd| zloo ehfrph d rqh0shulrg erqg wrpruurz/ zkhq lwv sulfhzloo eh ghvfulehg e| +6158,1 Wkxv rqh fdq vroyh iru lwv orj sulfh/ ru htxlyd0ohqwo| 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 }hur0frxsrqerqg/ |?|/ wlphv erqg pdwxulw| q/ zklfk htxdov plqxv wkh orj sulfh ri wkherqg/ 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{1Pruh 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 wkhwhup vwuxfwxuh krogv lq wklv prgho1 Wkxv wkh orj erqg |lhog lv d frqvwdqwsoxv 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 ixwxuhuhdo lqwhuhvw udwhv1 Vlqfh wkh uhdo lqwhuhvw udwh iroorzv dq DU+4, surfhvv zlwkshuvlvwhqfh frhflhqw !%/ wkh h{shfwhg uhdo lqwhuhvw udwh n shulrgv dkhdg lvd frqvwdqw soxv !&% wlphv wkh uhdo lqwhuhvw udwh wrgd|1 Vxpplqj xs ryhu qshulrgv jlyhv +615:,1

Wkh rqh0shulrg orj uhwxuq rq dq q0shulrg lqgh{hg }hur0frxsrq erqg lvmxvw wkh fkdqjh lq lwv sulfh/ +s?3c|ns?c|,1 .rpelqlqj wklv h{suhvvlrq zlwk+6159, dqg +615:,/ wkh h{fhvv uhwxuq ryhu wkh rqh0shulrg orj lqwhuhvw udwh lv

u?c|n uc|n @ 45

E2?32% 6%E?32% E?3%%c|n> +615;,

vr wkh q0shulrg erqg lv vhqvlwlyh rqo| wr uhdo0lqwhuhvw0udwh vkrfnv %%c|n/zlwk d vhqvlwlylw| E?31 Ehfdxvh wkh uhdo sduw ri rxu prgho kdv rqo| d

86


58/218


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 lv2?| @ E

2?3

2%/ dqg wkh ulvn suhplxp lv

H| ^u?c|n uc|n` 2?|

5@ 6%E?32%= +615 wkxv wkh whup suhplxp lv frqvwdqwdv srvwxodwhg e| wkh h{shfwdwlrqv k|srwkhvlv ri wkh whup vwuxfwxuh1 VlqfhE?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|/ wkhpdujlqdo xwlolw| ri frqvxpswlrq iru d uhsuhvhqwdwlyh lqyhvwru lv kljk/ wkdw lv/zkhq zhdowk lv prvw ghvludeoh1 Lq htxloleulxp/ wkhvh erqgv kdyh d qhjdwlyhwhup suhplxp dqg wkh uhdo |lhog fxuyh lv rq dyhudjh grzqzdug0vorslqj1Zlwk d qhjdwlyh 6%/ rq wkh rwkhu kdqg/ orqj0whup lqgh{hg erqgv sd| rzkhq wkh pdujlqdo xwlolw| ri frqvxpswlrq iru d uhsuhvhqwdwlyh lqyhvwru lvorz/ dqg vr lq htxloleulxp wkh| kdyh d srvlwlyh whup suhplxp1 Lq wklv fdvhwkh uhdo |lhog fxuyh lv rq dyhudjh xszdug0vorslqj1

Htxdwlrq +615


59/218


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 riuhwxuqv dqg wkh ulvn suhplxp lq sursruwlrq wr rqh dqrwkhu1 Lqwhuhvw0udwhyduldqfh fdq rqo| dhfw wkh vroxwlrq lqgluhfwo|/ wkurxjk wkh orjolqhdul}d0wlrq sdudphwhu 1 Hpslulfdoo| zh qg wkdw fkdqjhv yhu| olwwoh zkhqwkh sdudphwhuv ri wkh prgho fkdqjh/ dqg vr wkh lqgluhfw hhfw wkurxjk lvtxdqwlwdwlyho| 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 dv0vhwv duh hqrxjk wr frpsohwh wkh pdunhw zlwk uhvshfw wr uhdo0lqwhuhvw0udwhulvn/ dqg wkxv wkh lqyhvwru fdq xvh dq| wzr lqdwlrq0lqgh{hg erqgv wr frq0vwuxfw d sruwirolr zlwk wkh rswlpdo ohyho ri lqwhuhvw0udwh vhqvlwlylw|1 Li rqo|vkruw0pdwxulw| erqgv zlwk d orz vhqvlwlylw| duh dydlodeoh/ wkh lqyhvwru fdqfrpshqvdwh e| kroglqj pruh ri wkhp ru hyhq ohyhudjlqj klv srvlwlrq> li rqo|orqj0pdwxulw| erqgv zlwk d kljk vhqvlwlylw| duh dydlodeoh/ wkh lqyhvwru fdqfrpshqvdwh e| kroglqj ihzhu ri wkhp1

Wklug/ wkh lqwhuwhpsrudo khgjlqj ghpdqg/ wkh vhfrqg whup lq +6163,/ lvlqfuhdvlqj lq wkh shuvlvwhqfh ri lqwhuhvw0udwh vkrfnv !%1 Wklv lv ehfdxvh/ dvvkrzq lq wkh jhqhudo vroxwlrq +6146,/ khgjlqj ghpdqg lv ghwhuplqhg e| wkhfryduldqfh ri wkh ulvn| dvvhw uhwxuq zlwk wkh glvfrxqwhg ydoxh ri doo ixwxuhlqwhuhvw udwhv1 D pdmru wkhph ri wkh hpslulfdo zrun lq wklv errn lv wkdwshuvlvwhqw vkrfnv wr lqyhvwphqw rssruwxqlwlhv duh pxfk pruh lpsruwdqw irusruwirolr fkrlfh wkdq duh wudqvlwru| vkrfnv wr lqyhvwphqw rssruwxqlwlhv1

Irxuwk/ dv ulvn dyhuvlrq lqfuhdvhv dqg wkh hodvwlflw| ri lqwhuwhpsrudo vxe0vwlwxwlrq ghfolqhv/ wkh olplw ri +6163, lv d sruwirolr wkdw lv htxlydohqw wrdq lqdwlrq0lqgh{hg frqvro1 Lq wklv prgho dq lqdwlrq0lqgh{hg frqvro kdvlqwhuhvw0udwh vhqvlwlylw| S@+4 S!%,/ dqg @ S zkhq 4@ @ # @ 31Wklv lv d vshfldo fdvh ri wkh jhqhudo srlqw pdgh lq vhfwlrq 614171

Htxdwlrq +6163, lv dsshdolqjo| vlpsoh/ exw lw odfnv uhdolvp ehfdxvh lwgrhv qrw doorz lqyhvwruv wr krog qrplqdo erqgv ru htxlwlhv1 Lq rughu wr jrixuwkhu/ 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 lqd0wlrq1 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{shfwhgorj lqdwlrq }| soxv dq lqdwlrq vkrfn/ dqg h{shfwhg lqdwlrq iroorzv dq

88


60/218


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 dqgy5c|n/ fdq eh fruuhodwhg zlwk hdfk rwkhu dqg zlwk wkh uhdo vkrfnv wr wkhprgho=

y5c|n @ 5%%%c|n 56%6c|n %5c|nc +6166,

yZc|n @ Z%%%c|n Z6%6c|n Z5%5c|n %Zc|nc +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 orjVGI/ wkh orj lqdwlrq udwh/ dqg wkhlu frqglwlrqdo phdqv1 Wkhvh vkrfnv duhfurvv0vhfwlrqdoo| xqfruuhodwhg/ zlwk yduldqfhv 26/

2Z/

2%/ dqg

251 Lw lv

lpsruwdqw wr qrwh wkdw }|n/ wkh h{shfwhg lqdwlrq udwh/ lv dhfwhg e| erwkd sxuh h{shfwhg0lqdwlrq vkrfn %5c|n dqg wkh vkrfnv wr wkh h{shfwhg dqgxqh{shfwhg orj VGI %%c|n dqg %6c|n1 Wkdw lv/ lqqrydwlrqv wr h{shfwhglqdwlrq fdq eh fruuhodwhg zlwk lqqrydwlrqv lq wkh orj VGI/ dqg khqfh zlwklqqrydwlrqv lq wkh vkruw0whup uhdo lqwhuhvw udwh1 Wkhvh fruuhodwlrqv phdq wkdwqrplqdo lqwhuhvw udwhv qhhg qrw pryh rqh0iru0rqh zlwk h{shfwhg lqdwlrqwkdw lv/ wkh Ilvkhu k|srwkhvlv qhhg qrw krogdqg qrplqdo erqg sulfhv fdqlqfoxgh 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 uhdovhfwru1 Exw wklv lv phuho| d pdwwhu ri qrwdwlrqdo frqyhqlhqfh1 Rxu prgho lvd uhgxfhg irup udwkhu wkdq d vwuxfwxudo prgho/ vr lw fdswxuhv fruuhodwlrqvdprqj vkrfnv wr uhdo dqg qrplqdo lqwhuhvw udwhv exw grhv qrw kdyh dq|wklqjwr vd| derxw wkh wuxh xqghuo|lqj vrxufhv ri wkhvh vkrfnv1

Wkh sulflqj ri qrplqdo erqgv iroorzv wkh vdph vwhsv dv wkh sulflqj rilqgh{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=

sm?c| @ D

m? E

mc?{| E

m2c?}|= +6168,

Wkh Dsshqgl{ jlyhv h{suhvvlrqv iru wkh frhflhqwv Dm?/ Emc? dqg E

m2c?1

Vlqfh qrplqdo erqg sulfhv duh gulyhq e| vkrfnv wr erwk uhdo lqwhuhvwudwhv dqg lqdwlrq/ wkh| kdyh d wzr0idfwru vwuxfwxuh udwkhu wkdq wkh vlqjoh0idfwru vwuxfwxuh ri lqgh{hg erqg sulfhv1 Lqdwlrq dhfwv wkh h{fhvv uhwxuq

89


61/218


rq dq q0shulrg qrplqdo erqg ryhu wkh rqh0shulrg qrplqdo lqwhuhvw udwh/ vr

ulvn suhpld lq wkh qrplqdo whup vwuxfwxuh lqfoxgh frpshqvdwlrq iru lqdwlrqulvn1 Olnh doo rwkhu ulvn suhpld lq wkh prgho/ krzhyhu/ wkh ulvn suhpld rqqrplqdo erqgv duh frqvwdqw ryhu wlph> wkxv wkh h{shfwdwlrqv k|srwkhvlv riwkh 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 ixoohydoxdwlrq ri erqg ghpdqg uhtxluhv wkdw zh lqfoxgh htxlwlhv lq rxu prghodv dq dwwudfwlyh dowhuqdwlyh orqj0whup lqyhvwphqw1 Zh gr wklv dv vlpso| dvsrvvleoh/ 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,

.dpsehoo +4


62/218


Nzrq +4


63/218


Wdeoh 614= Whup Vwuxfwxuh Prgho Hvwlpdwlrq

4


64/218


lru> wkxv wkh prgho fdq wudgh r ehwwhu w hovhzkhuh iru zruvh w ri phdq

vkruw0whup lqwhuhvw udwhv1, Dffruglqjo| zh uhtxluh wkdw wkh prgho h{dfwo|w wkh vdpsoh phdqv ri qrplqdo lqwhuhvw udwhv dqg lqdwlrq1 Wklv uhvwulfwlrqkdugo| uhgxfhv wkh olnholkrrg dw doo lq 4


65/218


rq yrodwlolw| erxqgv iru wkh vwrfkdvwlf glvfrxqw idfwru +Kdqvhq dqg Mdjdq0

qdwkdq 4


66/218


Wdeoh 615= Vdpsoh dqg Lpsolhg Prphqwv ri wkh Whup Vwuxfwxuh

Prphqw 4


67/218


lqgh{hg erqgv> wkh glhuhqfh lq yrodwlolw| lqfuhdvhv zlwk pdwxulw|/ vr wkdw 430

|hdu qrplqdo erqgv kdyh d vwdqgdug ghyldwlrq wkuhh wlphv juhdwhu wkdq 430|hdu lqgh{hg erqgv1 Wklv glhuhqfh lq yrodwlolw| pdnhv wkh Vkdush udwlr irulqgh{hg erqgv lq urz ; frqvlghudeo| kljkhu wkdq wkh Vkdush udwlr iru qrplqdoerqgv lq urz 61 Vlqfh lqgh{hg erqg uhwxuqv duh jhqhudwhg e| d vlqjoh0idfwruprgho/ wkh Vkdush udwlr iru lqgh{hg erqgv lv lqghshqghqw ri pdwxulw| dw31661 Wkh Vkdush udwlr iru qrplqdo erqgv ghfolqhv zlwk pdwxulw|> vkruw0whup qrplqdo erqgv kdyh d udwlr forvh wr wkdw iru lqgh{hg erqgv/ exw wkhVkdush udwlr iru 430|hdu qrplqdo erqgv lv rqo| 31481 Wkhvh qxpehuv lpso|wkdw lq rxu sruwirolr dqdo|vlv/ lqyhvwruv zlwk orz ulvn dyhuvlrq zloo kdyh dvwurqj p|rslf ghpdqg iru lqgh{hg erqgv1

Wdeoh 615 fdq dovr eh xvhg wr hydoxdwh wkh hpslulfdo w ri wkh prgho1 Dfrpsdulvrq ri wkh prgho*v lpsolhg prphqwv zlwk wkh vdpsoh prphqwv iruqrplqdo erqgv vkrzv wkdw wkh prgho wv wkh yrodwlolw| ri h{fhvv qrplqdoerqg uhwxuqv h{wuhpho| zhoo1 Wkh prgho vrphzkdw ryhuvwdwhv wkh dyhudjhh{fhvv qrplqdo erqg uhwxuq dqg wkh qrplqdo Vkdush udwlr/ exw wklv fdq ehdwwulexwhg lq sduw wr wkh xszdug guliw lq lqwhuhvw udwhv ryhu wkh 4


68/218


kljkhu0iuhtxhqf| yduldwlrq lq lqwhuhvw udwhv lv dwwulexwhg wr wkh uhdo lqwhuhvw

udwh/ sduwlfxoduo| diwhu wkh hqg ri wkh 4wklv lv djdlq d pdqlihvwdwlrq ri wkh htxlw| suhplxp sx}}oh1

Wkh uljkw kdqg vlghv ri Wdeohv 4 dqg 5 uhshdw wkhvh hvwlpdwhv iru wkhYrofnhu0Juhhqvsdq shulrg 4


69/218


Iljxuh 614= Ilwwhg uhdo dqg qrplqdo |lhogv dqg lqdwlrq

98


70/218


Wdeoh 616= Rswlpdo Doorfdwlrqv wr Htxlwlhv dqg Orqj0Whup Erqgv

Xqfrqvwudlqhg .rqvwudlqhg Xqfrqvwudlqhg .rqvwudlqhg

U1U1D1 Htxlw| Lqgh{hg Htxlw| Lqgh{hg Htxlw| Qrplqdo Htxlw| Qrplq

+D, Vdpsoh Shulrg= 4


71/218


Lq d zruog zlwk ixoo lqgh{dwlrq/ wkh xqfrqvwudlqhg ghpdqg iru erwk orqj0

whup lqgh{hg erqgv dqg htxlwlhv lv srvlwlyh dqg riwhq deryh 433 / lpso|lqjwkdw wkh lqyhvwru rswlpdoo| eruurzv wr qdqfh sxufkdvhv ri htxlwlhv dqglqgh{hg erqgv1 Wkh sruwirolr vkduh ri lqgh{hg erqgv h{fhhgv wkdw ri htxlwlhv/ghvslwh wkh kljkhu Vkdush udwlr ri htxlwlhv/ ehfdxvh lqgh{hg erqgv duh pxfkohvv ulvn| wkdq htxlwlhv1e Dv wkh frhflhqw ri uhodwlyh ulvn dyhuvlrq lqfuhdvhv/wkh ghpdqgv iru erwk orqj0whup lqgh{hg erqgv dqg htxlwlhv idoo/ exw wkhvkduh ri htxlwlhv idoov idvwhu1 Lq wkh olplw wkh lqqlwho| ulvn0dyhuvh lqyhvwrukrogv d sruwirolr htxlydohqw wr dq lqgh{hg shushwxlw| dv zh kdyh douhdg|glvfxvvhg1 Zkhq wkhuh duh eruurzlqj dqg vkruw0vdoh frqvwudlqwv/ lqyhvwruvzlwk orz ulvn dyhuvlrq lqyhvw ixoo| lq htxlwlhv dv d zd| wr pd{lpl}h wkhlu ulvndqg h{shfwhg uhwxuq zlwkrxw xvlqj ohyhudjh/ zkloh pruh ulvn0dyhuvh lqyhvwruvkrog erwk lqgh{hg erqgv dqg htxlwlhv1 .dvk sod|v rqo| d plqru uroh dqgrqo| lq wkh sruwirolrv ri wkh prvw ulvn0dyhuvh lqyhvwruv/ zkr duh doprvw ixoo|lqyhvwhg lq lqgh{hg erqgv1

Wkhvh qglqjv duh uhodwhg wr wkh _dvvhw doorfdwlrq sx}}oh% ri .dqqhu/Pdqnlz/ dqg Zhlo +4


72/218


xqfrqvwudlqhg lqyhvwruv zlwk orz ulvn dyhuvlrq dfwxdoo| krog vkruw srvlwlrqv

lq qrplqdo erqgv/ zkloh frqvwudlqhg lqyhvwruv krog rqo| htxlwlhv1 Dv ulvndyhuvlrq lqfuhdvhv/ lqyhvwruv pryh lqwr fdvk udwkhu wkdq orqj0whup qrplqdoerqgv1 Iljxuh 615 looxvwudwhv wklv srlqw1 Wkh wrs sdqho ri wkh jxuh sorwv frq0vwudlqhg doorfdwlrqv wr htxlwlhv/ d whq0|hdu qrplqdo erqg dqg d wkuhh0prqwkqrplqdo eloo/ zkloh wkh erwwrp sdqho sorwv frqvwudlqhg p|rslf doorfdwlrqv1Wkh krul}rqwdo d{lv phdvxuhv uhodwlyh ulvn wrohudqfh +4@, udwkhu wkdq uho0dwlyh ulvn dyhuvlrq/ ehfdxvh erwk wrwdo dqg p|rslf doorfdwlrqv du

Documents

Campbell Viceira