Grhv Rqh Vrurv Pdnh d GlhuhqfhB
D Wkhru| ri Fxuuhqf| Fulvhv zlwk Odujh dqg
Vpdoo Wudghuv
Jldqfduor Fruvhwwl
\doh Xqlyhuvlw|
Vwhskhq Pruulv
\doh Xqlyhuvlw|
K|xq Vrqj Vklq
R{irug Xqlyhuvlw|
Iluvw gudiw= Pdufk 4<<<
Wklv uhylvlrq= Qryhpehu 4<<<
Suholplqdu|
Devwudfw
Gr odujh lqyhvwruv lqfuhdvh wkh yxoqhudelolw| ri d frxqwu| wr vshfxodwlyh
dwwdfnv lq wkh iruhljq h{fkdqjh pdunhwvB Wr dgguhvv wklv lvvxh/ zh exlog d
prgho ri fxuuhqf| fulvhv zkhuh d vlqjoh odujh lqyhvwru dqg d frqwlqxxp ri
vpdoo lqyhvwruv vlpxowdqhrxvo| ghflgh zkhwkhu wr dwwdfn d fxuuhqf| edvhg
rq wkhlu sulydwh lqirupdwlrq derxw ixqgdphqwdov1 Lq wkh xqltxh htxloleulxp
ri wklv wudglqj jdph/ wkh suhvhqfh ri wkh odujh lqyhvwru grhv/ lqghhg/ pdnh
doo rwkhu lqyhvwruv pruh djjuhvvlyh lq wkhlu vhoolqj1 Uhodwlyh wr wkh fdvh
lq zklfk wkhuh lv qr odujh wudghu/ vpdoo lqyhvwruv dwwdfn wkh fxuuhqf| dw
kljkhu ydoxhv ri wkh ixqgdphqwdov1 \hw/ wkh glhuhqfh fdq eh vpdoo/ ru qxoo/
ghshqglqj rq wkh uhodwlyh suhflvlrq ri sulydwh lqirupdwlrq ri wkh vpdoo dqg
odujh lqyhvwruv1 D odujh exw uhodwlyho| xqlqiruphg vshfxodwru pdnhv olwwoh
glhuhqfh iru wkh ghvljq ri wkh rswlpdo vwudwhjlhv e| wkh uhvw ri wkh pdunhw1
MHO I64/ G;5
Zh wkdqn Ulfkdug Fodulgd/ Kìoëqh Uh|/ dqg vhplqdu sduwlflsdqwv dw Froxpeld Xqlyhuvlw|/
Kduydug Xqlyhuvlw| dqg Edqn ri Lwdo|1 Fruvhwwl= Ghsw1 ri Hfrqrplfv/ \doh Xqlyhuvlw|/ 5;
Klookrxvh Dy1/ Qhz Kdyhq FW 39853/ XVD/ fruvhwwlC|doh1hgx1 Pruulv= Frzohv Irxqgdwlrq/
\doh Xqlyhuvlw|/ 63 Klookrxvh Dy1/ Qhz Kdyhq FW 39853/ XVD/ vwhskhq1pruulvC|doh1hgx1 Vklq=
Qx!hog Froohjh/ R{irug R[4 4QI/ XN1
41 Lqwurgxfwlrq
D frpprqo| hqfrxqwhuhg ylhz dprqj pdunhw frpphqwdwruv lv wkdw odujh lqyhvwruv
pd| h{huflvh d glvsursruwlrqdwh lq xhqfh rq wkh olnholkrrg dqg lqflghqfh ri 0
qdqfldo fulvhv1 Odujh lqyhvwruv duh riwhq dffxvhg ri pdqlsxodwlqj pdunhwv/ gulylqj
sulfhv dzd| iurp ixqgdphqwdo ydoxhv/ lqfuhdvlqj v|vwhplf pdunhw ulvn/ dqg/ xo0
wlpdwho|/ wkuhdwhqlqj wkh vwdelolw| ri wkh lqwhuqdwlrqdo qdqfldo v|vwhp1 Wklv lv
qrw wr vd| wkdw pdunhwv frxog qrw eh ghvwdelol}hg dqg fulvhv zrxog qrw rffxu/
zhuh doo lqyhvwruv vpdoo1 Udwkhu/ wkh pdlq frqfhuq lv wkdw/ ehfdxvh ri wkhlu vl}h/
ylvlelolw| dqg uhsxwdwlrq/ lqyhvwruv fdsdeoh ri wdnlqj h{wuhpho| odujh srvlwlrqv duh
dovr fdsdeoh ri h{dfhuedwlqj plvsulflqj dqg wkhuhiruh vxevwdqwldoo| dhfw pdunhw
yxoqhudelolw| wr fulvhv1
Dowkrxjk vxfk ylhzv duh uduho| dffrpsdqlhg e| d suhflvh dffrxqw ri wkh dfwxdo
phfkdqlvpv lqyroyhg/ rqh lpsruwdqw wkhph lv wkdw wkh suhvhqfh ri odujh lqyhvwruv
lqmhfwv d ghjuhh ri iudjlolw| lqwr wkh pdunhw/ lq wkh vhqvh wkdw rwkhu wudghuv ehfrph
pruh zdu|/ dqg ehfrph pruh surqh wr edlolqj rxw dw wkh uvw klqw ri wurxeoh1 Wkh
nh| txhvwlrq lv krz/ dqg wr zkdw h{whqw wkh ehkdylru ri vpdoohu wudghuv fkdqjh lq
wkh suhvhqfh ri odujh sod|huv1 Wklv lv wkh lvvxh zh dgguhvv lq rxu sdshu1
Wkh uhfhqw olwhudwxuh rq fxuuhqf| froodsvhv dqg edqnlqj fulvhv kdv vwuhvvhg
wkh pdfurhfrqrplf lpsolfdwlrqv ri wkh frruglqdwlrq sureohp wkdw dulvhv lq wkh
suhvhqfh ri vwudwhjlf frpsohphqwdulwlhv lq sruwirolr vwudwhjlhv +dv w|slhg e| Re0
vwihog ^4<<9`,1 Dw wkh rqvhw ri d fulvlv/ d frruglqdwlrq sureohp dulvhv zkhq wkh
vxffhvv iurp d vshfxodwlyh srvlwlrq ghshqgv rq wkh srvlwlrqv vlpxowdqhrxvo| wdnhq
e| wkh rwkhu djhqwv lq wkh hfrqrp|1 Hyhq li wkh sd|r iurp d vxffhvvixo dwwdfn
grplqdwhv wkh sd|r iurp qrw dwwdfnlqj/ wkhuh duh htxloleuld lq zklfk udwlrqdo
djhqwv uhiudlq iurp dwwdfnlqj1 Zkloh/ wr rxu nqrzohgjh/ yluwxdoo| qr prgho ri
fxuuhqf| dqg qdqfldo froodsvh doorzv iru glhuhqfhv lq lqyhvwruv* vl}h/ lw lv riwhq
wkh fdvh wkdw lqirupdo glvfxvvlrq ri wkh uhvxowv iurp wklv olwhudwxuh uhihuv wr odujh
djhqwv dv ehlqj slyrwdo lq ghwhuplqlqj wkh rqvhw ri d fulvlv1 Lq rwkhu zrugv/
odujh lqyhvwruv duh frqvlghuhg pruh hhfwlyh wkdq vpdoo rqhv lq irfxvlqj pdunhw
sduwlflsdqwv* h{shfwdwlrqv rq d sduwlfxodu wudglqj vwudwhj|1 D srsxodu ylhz lv wkdw
odujh lqyhvwruv dfklhyh wklv e| vsuhdglqj uxpruv dqg pdqlsxodwlqj lqirupdwlrq/ ru
e| vljqdolqj wkhlu srvlwlrqv wr rwkhu djhqwv1 Ylhzv hpskdvl}lqj vljqdoolqj/ krz0
hyhu/ qhjohfw wkh idfw wkdw wkh phuh suhvhqfh ri odujh lqyhvwruv dhfw wkh vwuxfwxuh
ri wkh frruglqdwlrq sureohp idfhg e| doo djhqwv lq wkh pdunhwv1 Wklv idfw/ shu vh/
pd| uhvxow lq kljkhu yxoqhudelolw| wr vshfxodwlyh uxqv1
Wklv sdshu dqdo|}hv wkh frruglqdwlrq sureohp idfhg e| vpdoo dqg odujh lq0
5
yhvwruv/ glhulqj lq wkh suhflvlrq ri wkhlu lqirupdwlrq/ vlpxowdqhrxvo| ghflglqj
zkhwkhu wr dwwdfn d fxuuhqf|1 Wkh jhqhudo txhvwlrqv zh duh lqwhuhvwhg lq duh dv
iroorzv1 Krz fdq wkh suhvhqfh ri odujh lqyhvwruv vroyh ru olplw wkh lpsruwdqfh ri
wkh frruglqdwlrq sureohpB Krz odujh pxvw wkhvh lqyhvwruv ehB Frqyhuvho|/ grhv
wkh h{lvwhqfh ri vpdoo lqyhvwruv frqvwudlq wkh vwudwhj| sod|hg e| odujh lqyhvwruvB
Ilqdoo|/ gr lqirupdwlrq glhuhqfhv pdwwhu lq ghwhuplqlqj wkh uroh ri odujh sod|huv
lq qdqfldo pdunhwB Wr dgguhvv wkhvh txhvwlrqv/ zh frqvlghu wkh fdvh ri d vlq0
joh odujh lqyhvwru frh{lvwlqj zlwk d frqwlqxxp ri vpdoo lqyhvwruv/ dqg ghulyh wkh
xqltxh htxloleulxp lq d wudglqj jdph lq zklfk hdfk sod|hu fkrrvhv khu ehvw dfwlrq
edvhg rq khu sulydwh lqirupdwlrq derxw wkh vwdwh ri ixqgdphqwdov1 Wkh suhflvlrq
ri djhqwv* lqirupdwlrq lv qrw glfwdwhg e| wkhlu vl}h> zh doorz iru dowhuqdwlyh vfh0
qdulrv lq zklfk wkh sulydwh lqirupdwlrq ri wkh odujh lqyhvwruv lv hlwkhu pruh ru ohvv
suhflvh ri wkh lqirupdwlrq ri vpdoo lqyhvwruv1 Wklv lv wkh vlpsohvw iudphzrun wr
vlqjoh rxw wkh vshflf hhfw ri vl}h dqg lqirupdwlrq suhflvlrq rq wkh frruglqdwlrq
sureohp dw wkh rqvhw ri d fulvlv1
Rxu pdlq uhvxowv duh vxppdul}hg ehorz1
Wkh suhvhqfh ri wkh odujh lqyhvwru grhv/ lqghhg/ pdnh doo rwkhu lqyhvwruv pruh
djjuhvvlyh lq wkhlu vhoolqj1 Vpdoo lqyhvwruv dgrsw wkh vwudwhj| ri edlolqj rxw
iru vwurqjhu ixqgdphqwdov1
Krzhyhu/ zlwklq wklv eurdg jhqhudo qglqj/ zh pxvw glvwlqjxlvk ehwzhhq
wzr fdvhv 0 wkh fdvh lq zklfk d w|slfdo vpdoo lqyhvwru lv ehwwhu lqiruphg wkdq
wkh odujh lqyhvwru/ dqg wkh fdvh lq zklfk wkh uhyhuvh lv wuxh1
Zkhq d w|slfdo vpdoo lqyhvwru lv ehwwhu lqiruphg wkdq wkh odujh lq0
yhvwru/ wkh ghjuhh wr zklfk wkh| dgrsw dq djjuhvvlyh vhoolqj vwudwhj| lv
prghudwh1 Dowkrxjk wkh| gr dgrsw d pruh djjuhvvlyh vhoolqj vwudwhj|
uhodwlyh wr wkh fdvh lq zklfk wkhuh lv qr odujh lqyhvwru/ wkh glhuhqfh
fdq eh vpdoo1
Zkhq wkh odujh lqyhvwru lv ehwwhu lqiruphg wkdq d w|slfdo vpdoo lqyhvwru/
wkh ghjuhh ri djjuhvvlyhqhvv lq wkh vhoolqj vwudwhj| ri wkh vpdoo wudghuv
lv wkdw pxfk kljkhu1 Khuh/ wkh hhfw fdq eh txlwh odujh1
Wkh prgho dovr suhglfwv wkdw/ iru d jlyhq uhdol}dwlrq ri ixqgdphqwdov/ wkhuh
zloo eh d vwurqj/ exw qrw shuihfw fruuhodwlrq ri sruwirolr srvlwlrqv dfurvv djhqwv lq
wkh hfrqrp|1 D kljk fruuhodwlrq ri sruwirolr vwudwhjlhv lq wkh gdwd lv vrphwlphv
frqvlghuhg dv hylghqfh ri khuglqj ri vpdoo lqyhvwruv1 Lq rxu prgho/ krzhyhu/ kljk
6
fruuhodwlrq lv gulyhq e| sulydwh lqirupdwlrq/ dqg lv qrw dvvrfldwhg zlwk pdunhw
pdqlsxodwlrq dqg vwudwhjlf xvh ri lqirupdwlrq e| wkh odujh wudghu1 Wkh srlqw ri
wklv sdshu lv wkdw odujh lqyhvwruv fdq lq xhqfh wkh pdunhw surfhvv/ dqg lqfuhdvh
yxoqhudelolw| wr fulvhv/ zlwkrxw dq| gholehudwh dwwhpsw wr frruglqdwh pdunhw sdu0
wlflsdqwv* ehkdylru1 Zh vkrxog dovr qrwh khuh wkdw lq rxu dqdo|vlv wkh odujh lqyhvwru
grhv qrw qhfhvvdulo| wdnhv wkh h{0srvw zlqqlqj vlgh ri wkh pdunhw1
Lq oljkw ri rxu uhvxowv/ zh qg wkdw wkh lqirupdo dujxphqwv srlqwlqj wkh qjhu
dw d zhoo0lqiruphg odujh wudghu iru lqmhfwlqj lqfuhdvhg iudjlolw| lqwr wkh pdunhw
lv ylqglfdwhg e| rxu irupdo dqdo|vlv1 E| surylglqj d prgho ri fulvhv zlwk vpdoo
dqg odujh wudghuv glhulqj lq wkh suhflvlrq ri wkhlu lqirupdwlrq/ rxu dqdo|wlfdo
iudphzrun uhsuhvhqwv d uvw vwhs wrzdugv rxu xqghuvwdqglqj ri wkh uroh ri odujh
lqyhvwruv lq d frqghqfh fulvhv1
Wkh sdshu lv rujdql}hg dv iroorzv1 Vhfwlrq 5 od|v rxw wkh edvlf iudphzrun/
surylglqj d eulhi ryhuylhz ri wkh olwhudwxuh rq fxuuhqf| dqg qdqfldo fulvhv edvhg
rq lqfrpsohwh lqirupdwlrq diwhu Pruulv dqg Vklq +4<<;,1 Vhfwlrq 6 fkdudfwhul}hv
wkh xqltxh htxloleulxp lq rxu prgho1 Vhfwlrq 7 h{soruhv wkh olplwlqj surshuwlhv
ri wkh htxloleulxp dqg ghulyhv rxu pdlq sursrvlwlrqv1 Vhfwlrq 8 vxppdul}hv rxu
pdlq uhvxowv1 Vhfwlrq 9 frqfoxghv1
51 Wrzdugv d wkhru| ri frqghqfh fulvhv
5141 Wkh vhwxs
Frqvlghu dq hfrqrp| zlwk wzr w|shv ri wudghuv1 Wkhuh lv d vlqjoh odujh wudghu
dqg d frqwlqxxp ri vpdoo wudghuv1 Wkh glvwlqjxlvklqj ihdwxuh ri d odujh wudghu
lv wkdw kh kdv dffhvv wr d vx!flhqwo| odujh olqh ri fuhglw lq wkh grphvwlf fxuuhqf|
wr wdnh d vkruw srvlwlrq xs wr wkh olplw ri b 1 Lq frqwudvw/ wkh vhw ri doo vpdoo
wudghuv wdnhq wrjhwkhu kdyh d frpelqhg wudglqj olplw ri b1
Zh hqylvdjh wkh vkruw vhoolqj dv frqvlvwlqj ri eruurzlqj wkh grphvwlf fxuuhqf|
dqg vhoolqj lw iru grooduv1 Wkhuh lv d frvw wr hqjdjlqj lq wkh vkruw vhoolqj/ ghqrwhg
e| | : f1 Lq wkh vwdqgdug fdvh/ | fdq eh ylhzhg odujho| dv frqvlvwlqj ri wkh lqwhuhvw
udwh glhuhqwldo ehwzhhq wkh grphvwlf fxuuhqf| dqg grooduv/ soxv wudqvdfwlrq frvwv1
Wklv frvw | lv qrupdol}hg/ vr wkdw wkh sd|r wr d vxffhvvixo dwwdfn rq wkh fxuuhqf|
lv jlyhq e| / dqg wkh sd|r iurp uhiudlqlqj iurp dwwdfn lv jlyhq e| f1 Wkxv/ wkh
qhw sd|r wr d vxffhvvixo dwwdfn rq wkh fxuuhqf| lv |/ zkloh wkh sd|r wr dq
xqvxffhvvixo dwwdfn lv jlyhq e| |1
Zkhwkhu wkh fxuuhqw h{fkdqjh udwh sdulw| lv yldeoh ghshqgv rq erwk hfrqrplf
7
ixqgdphqwdov dqg wkh ghjuhh ri vshfxodwlyh suhvvxuh lq wkh iruhljq h{fkdqjh pdu0
nhw1 Hfrqrplf ixqgdphqwdov duh lqgh{hg e| d vwrfkdvwlf yduldeoh w / wkdw lv dv0
vxphg wr eh xqlirupo| glvwulexwhg rq wkh uhdo olqh14 Vshfxodwlyh suhvvxuh lv phd0
vxuhg e| wkh pdvv ri lqyhvwruv dwwdfnlqj wkh fxuuhqf| lq wkh iruhljq h{fkdqjh
pdunhw/ wr eh hqgrjhqrxvo| ghulyhg1 Zh dvvxph wkdw wkh fxuuhqw h{fkdqjh sdulw|
lv ghydoxhg li wkh pdvv ri lqyhvwruv dwwdfnlqj wkh fxuuhqf| lv dw ohdvw dv odujh dv
wkh lqgh{ ri ixqgdphqwdov w1
Lq zkdw iroorzv/ lw pd| eh khosixo wr nhhs lq plqg wkh h{dpsoh ri dq hfrqrp|
hqgrzhg zlwk d odujh vwrfn ri lqwhuqdwlrqdo uhvhuyhv/ zkhuh wkh fhqwudo edqn lv
zloolqj wr ghihqg wkh h{fkdqjh udwh dv orqj dv uhvhuyhv gr qrw idoo ehorz d suh0
ghwhuplqhg fulwlfdo ohyho1 Wkh fhqwudo edqn suhghwhuplqhv wklv ohyho edvhg rq lwv
dvvhvvphqw ri wkh hfrqrplf ixqgdphqwdov ri wkh frxqwu|1 Wkh fulwlfdo ohyho lv orz
zkhq ixqgdphqwdov duh vwurqj +w lv kljk,= wkh fhqwudo edqn lv zloolqj wr xvh d
odujh dprxqw ri +qrq0eruurzhg dqg eruurzhg, uhvhuyhv lq ghihqglqj wkh h{fkdqjh
udwh1 Frqyhuvho|/ wkh fulwlfdo ohyho lv kljk zkhq ixqgdphqwdov duh zhdn +w lv orz,1
Hyhq d plog vshfxodwlyh dwwdfn fdq frqylqfh wkh fhqwudo edqn wr dedqgrq wkh shj1
5151 Frruglqdwlrq sureohp zkhq wkh ixqgdphqwdo lv frpprq nqrzo0
hgjh
Frqvlghu wkh fdvh wkdw lv xvxdoo| glvfxvvhg lq wkh olwhudwxuh rq fxuuhqf| fulvhv1
Doo lqyhvwruv duh vpdoo/ revhuyh d vlqjoh sxeolf vljqdo derxw w/ vr wkdw lwv ydoxh
lq wkh shulrg lv frpprq nqrzohgjh/ dqg lqghshqghqwo| ghflgh zkhwkhu ru qrw
wr dwwdfn wkh fxuuhqf|1 Dv qrwhg e| Revwihog ^4<<9`/ wkh rxwfrph ghshqgv rq
d wulsduwlwh sduwlwlrq ri srvvleoh uhdol}dwlrqv ri w1 Wkhuh lv d xqltxh rxwfrph lq
zklfk doo djhqwv dwwdfn li w f> qr lqyhvwru zloo dwwdfn li w > exw wkhuh zloo
eh qr xqltxh rxwfrph iru ydoxhv ri w lq wkh lqwhuydo ^fc `1 Lq idfw/ lq wkh odwwhu
lqwhuydo/ hdfk lqglylgxdo lqyhvwru zloo qg lw rswlpdo wr dwwdfn wkh fxuuhqf| rqo|
surylghg d sursruwlrq ri lqyhvwruv dw ohdvw dv odujh dv w dovr dwwdfn wkh fxuuhqf|1
Vlqfh wklv lv wuxh iru doo djhqwv/ rqfh h{shfwdwlrqv frruglqdwh rq dq dwwdfn htxl0
oleulxp/ wkh fxuuhqf| zloo lqghhg froodsvh1 Frqyhuvho|/ surylghg wkh sursruwlrq
ri lqyhvwruv dwwdfnlqj lv ohvv wkdq w/ wkh lqglylgxdo rswlpdo uhvsrqvh lv wr uhiudlq
iurp vshfxodwlqj djdlqvw wkh fxuuhqf|1 Lq hlwkhu rxwfrph/ wkh ixqgdphqwdov lq rxu vshflf h{dpsoh= wkh vwrfn ri uhvhuyhv dw wkh fhqwudo edqn/ wkh qdqfldo
4 Khqfh/
wkh sulru lv lpsurshu lq wkdw wkh lqwhjudo ryhu wkh ghqvlw| lv lqqlwh1 Zh dgrsw
wklv dvvxpswlrq iru dojheudlf wudfwdelolw| iru wkh dqdo|vlv ri wkh jhqhudo fdvh/ exw wkh uhvxowv iru
wkh olplwlqj fdvhv +uhsruwhg lq rxu wkhruhpv, duh ydolg iru dq| sulru zlwk d frqwlqxrxv ghqvlw|1
8
uhsrzhu ri vshfxodwruv dqg wkh sd|r iurp dwwdfnlqj wkh fxuuhqf| duh wkh
vdph1 \hw/ iru jlyhq ixqgdphqwdov/ wkh h{fkdqjh udwh sdulw| hlwkhu froodsvhv/ ru lv
pdlqwdlqhg/ ghshqglqj rq zkhwkhu djhqwv frruglqdwh rq wkh uvw ru rq wkh vhfrqg
htxloleulxp15
Hvvhqwldo wr wklv dujxphqw lv wkh dvvxpswlrq ri frpprq nqrzohgjh uhjduglqj
wkh ehkdylru ri djhqwv lq htxloleulxp= hlwkhu d pdvv dv odujh dv w lv nqrzq wr
dwwdfn/ ru lw lv qrw1 Wkhq/ hdfk htxloleulxp lv frqvlvwhqw zlwk lqglylgxdo udwlrqdo0
lw|1 Fxuuhqf| fulvhv pdqlihvw wkhpvhoyhv dv d vxgghq vzlwfk lq wkh frruglqdwlrq
ri djhqwv dfurvv srvvleoh htxloleuld1
Qrwh wkdw/ iurp wkh srlqw ri ylhz ri lqglylgxdo lqyhvwruv/ wkh sd|r iurp d
frruglqdwhg dwwdfn fohduo| grplqdwhv wkh sd|r lq wkh htxloleulxp zlwk qr dwwdfn
+ | : f,1 Wkh frruglqdwlrq sureohp wkxv uhihuv wr wkh frqglwlrqv wkdw suhyhqw
djhqwv iurp dwwdfnlqj iru dq| w 1 \hw/ lw pd| zhoo eh wkdw/ iurp d vrfldo
srlqw ri ylhz/ hfrqrplf zhoiduh ghwhulrudwhv dv d frqvhtxhqfh ri d fulvlv1 Wkh
olwhudwxuh surylghv h{dpsohv ri erwk d idoo ru dq lqfuhdvh lq vrfldo zhoiduh diwhu d
ghydoxdwlrq lq glvfxvvlqj wkh prghov lq wklv sdshu zh duh djqrvwlf derxw wkh
vrfldo glphqvlrq ri wkh cfrruglqdwlrq sureohp* lq wkh fxuuhqf| pdunhw1
Qrwh/ dovr/ wkdw d vlqjoh odujh lqyhvwru ru d ixqg pdqdjhu froohfwlqj lqgl0
ylgxdo qdqfldo uhvrxufhv zrxog wulyldoo| surylgh d vroxwlrq wr wkh frruglqdwlrq
sureohp ghqhg deryh1 Diwhu fhqwudol}lqj ghflvlrq pdnlqj/ dq dwwdfn zloo rffxu
iru dq| w 1 D glhuhqw dqg pruh lqwhuhvwlqj dssurdfk wr wkh sureohp frqvlvwv
ri uhirupxodwlqj wkh sureohp e| wdnlqj d ihz vwhsv dzd| iurp frpprq nqrzohgjh
ri ixqgdphqwdov dqg djhqwv* ehkdylru lq htxloleulxp1
5161 Ehqfkpdun prgho L= vpdoo lqyhvwruv zlwk lqfrpsohwh lqirupdwlrq
Iroorzlqj dq hduolhu iudphzrun wkdw wzr ri xv kdv xvhg ehiruh +Pruulv dqg Vklq
^4<<;`,/ wkh wudghuv gr qrw nqrz w zlwk fhuwdlqw|/ exw rqo| revhuyh d sulydwh vljqdo
derxw lw1 Lq sduwlfxodu/ djhqw revhuyhv d vljqdo % ' w n j0 / zkhuh wkh 0 duh
glvwulexwhg dffruglqj wr vprrwk v|pphwulf ghqvlw| s E ^zulwh 8 E iru wkh f1g1i1`1
Rxu dvvxpswlrqv wkdw w lv glvwulexwhg xqlirupo| dqg 0 kdv v|pphwulf glvwulexwlrq
lpso| wkdw/ iru d vshfxodwru revhuylqj % / w3j% lv glvwulexwhg dffruglqj wr ghqvlw|
s1
Zkdw lv frpprq nqrzohgjh lq wklv prgho lv wkh glvwulexwlrq ri erwk wkh ixq0
5 Dq
hduo| glvfxvvlrq ri wkh uroh ri wkh frruglqdwlrq sureohp lq prghov ri fxuuhqf| fulvhv
gudzlqj rq wkh prgho e| Nuxjpdq ^4<:<` dqg Iorrg dqg Jduehu ^4<;7` fdq eh irxqg lq Revwihog
^4<;9`/ ss1 :80:9/ dqg Julool ^4<;9` ss1 48604881
9
gdphqwdov dqg wkh sulydwh vljqdo/ zkhuhdv wkh huuru % w lv lqghshqghqw dfurvv
djhqwv1 Xsrq uhfhlylqj klv rzq vljqdo/ wkh uhsuhvhqwdwlyh djhqw fdq jxhvv wkh
ydoxh ri w/ dqg wkh glvwulexwlrq ri vljqdov uhdfklqj wkh rwkhu djhqwv lq wkh hfrq0
rp|/ dv zhoo dv ri wkhlu hvwlpdwh ri w1 Kh fdqqrw/ krzhyhu/ frxqw rq wkh rwkhu
djhqwv wr nqrz zkdw kh nqrzv1 Wklv lv ehfdxvh wkh rwkhu djhqwv zloo kdyh wr uho|
h{foxvlyho| rq wkhlu rzq vljqdov wr irup wkhlu rzq eholhiv derxw djhqw hvwlpdwh
ri wkh ixqgdphqwdo w1 Wklv ghsduwxuh iurp wkh dvvxpswlrq ri frpprq nqrzohgjh
ri wkh ixqgdphqwdo lv nh| wr zkdw iroorzv1
Wkh xqltxh htxloleulxp fdq eh fkdudfwhul}hg e| d fulwlfdo ydoxh wW ehorz zklfk
wkh fxuuhqf| zloo froodsvh/ dqg d fulwlfdo ydoxh ri wkh lqglylgxdo vljqdo %W ehorz
zklfk lqglylgxdov zloo dozd|v dwwdfn1 Zh fdq ghulyh wkhvh fulwlfdo ydoxhv e| qrwlqj
uvw wkdw li wkh wuxh vwdwh zhuh w dqg lqyhvwruv dwwdfnhg rqo| li wkh| revhuyhg d
W
W
vljqdo ehorz
%W 3w % / wkh djjuhjdwh sursruwlrq ri dwwdfnhuv zrxog eh RoJKE% % m
w ' 8 j 1 Wkxv wkh uvw htxloleulxp frqglwlrq d fulwlfdo pdvv frqglwlrq
lv
8
%W wW
j
'
wW1
+514,
Jlyhq %W / dq| uhdol}dwlrq ri wkh ixqgdphqwdo w wW lv dvvrfldwhg zlwk d vxffhvvixo
vshfxodwlyh dwwdfn rq wkh fxuuhqf|1
Vhfrqg/ jlyhq wW/ frqvlghu wkh h{shfwhg sd|r iurp dwwdfnlqj iru dq lqyhvwru
W
uhfhlylqj d vljqdo
% 1 Vlqfh wkh suredelolw| ri d vxffhvvixo dwwdfn lv RoJKEw w m
w W 3 %
% ' 8 j / wkh uhsuhvhqwdwlyh
lqyhvwru zloo dwwdfn li dqg rqo| li klv h{shfwhg
w W 3 %
jurvv sd|r 8
zloo eh dw ohdvw dv kljk dv wudqvdfwlrq frvw |16 Dv wkh
j
h{shfwhg sd|r wr dwwdfnlqj iru d pdujlqdo lqyhvwru uhfhlylqj d vljqdo %W pxvw eh
f/ wkh rswlpdo fxwr frqglwlrq lv
8
wW %W
j
'
|
+515,
Wkh v|vwhp ri htxdwlrq lqfoxglqj wkh wzr frqglwlrqv deryh lv hdvlo| vroyhg=
+515, jlyhv w W ' %W n j8 3 E|> vxevwlwxwlqj lqwr +514, jlyhv wW ' 8 E8 3 E| '
3 E| ' |1 Wkxv %W ' | j8 3 E|/ dqg wW ' |1 Wkh fxuuhqf|
8 E8
6 Khxulvwlfdoo|/
hdfk lqyhvwru zloo wu| wr dyrlg vwudwhjlhv wkdw duh vxffhvvixo rqo| iru d vpdoo
udqjh ri eholhiv derxw wkh ehkdylru iroorzhg e| wkh rwkhu djhqwv lq wkh hfrqrp|1 Kh zloo fkrrvh
wr dwwdfn rqo| zkhq/ lq klv hvwlpdwh/ wkh ixqgdphqwdo yduldeoh lv orz hqrxjk wr pdnh klp
frqghqw wkdw d vx!flhqwo| odujh qxpehu ri rwkhu djhqwv lq wkh hfrqrp| iroorzlqj d vlplodu
vwudwhj| zloo dwwdfn wkh fxuuhqf|1
:
zloo froodsvh iru dq| uhdol}dwlrq ri wkh ixqgdphqwdo w vpdoohu wkdq |/ zkloh
hdfk lqglylgxdo lqyhvwru zloo dwwdfn wkh fxuuhqf| iru dq| uhdol}dwlrq ri klv vljqdo
ehorz | j8 3 E|17
Wkhuh lv d xqltxh htxloleulxp18 Iru w lq wkh lqwhuydo dfc o/ wkh frruglqdwlrq
sureohp l1h1 wkh sureohp ri vhohfwlqj dq htxloleulxp wkdw lv prvw dwwudfwlyh wr
lqyhvwruv lv vroyhg lq wkh iroorzlqj vhqvh1 Wkhuh zloo eh dq dwwdfn iru w ehwzhhq
f dqg |/ exw wkhuh zloo eh qr dwwdfn iru dq| w deryh |1 Lq rwkhu zrugv/
dqg wklv lv dq dsshdolqj ihdwxuh ri wkh prgho/ dwwdfnv zloo eh pruh olnho| lq dq
hfrqrp| zlwk uhodwlyho| zhdn ixqgdphqwdov zlwklq wkh udqjh ri ixqgdphqwdov
iru zklfk pxowlsoh htxloleuld zrxog hphujh lq rxu uvw vshflfdwlrq dvvxplqj
frpprq nqrzohgjh derxw w1
Wkh vwuxfwxuh ri lqirupdwlrq lv nh| wr wklv uhvxow1 Pruulv dqg Vklq ^4<<<` vkrz
wkdw/ li erwk sxeolf dqg sulydwh vljqdov duh lqwurgxfhg lq wkh hfrqrp|/ wkhuh lv
vrph wkuhvkrog lq wkh uhodwlyh suhflvlrq ri wkh wzr diwhu zklfk wkh fulwlfdo ydoxhv wW
dqg %W duh qr orqjhu xqltxh1 Wkh odujhu wkh uroh ri wkh sxeolf vljqdo/ wkh pruh olnho|
vxfk d vfhqdulr1 Lq oljkw ri wklv uhvxow/ wkh vlpsoh h{dpsoh ri pxowlsoh htxloleuld
sursrvhg e| Revwihog ^4<<9`/ dqg uhsuhvhqwhg lq wkh vdph vslulw e| phdqv ri rxu
uvw prgho/ fdq eh frqvlghuhg dq h{wuhph vshflfdwlrq zlwkrxw sulydwh vljqdov1
E| wkh vdph wrnhq/ wkh xqltxhqhvv ri htxloleuld lq Pruulv dqg Vklq ^4<<;` lv dq
h{wuhph vshflfdwlrq zlwk qr sxeolf vljqdo1 Lq Iljxuh 4/ zh looxvwudwh wkh pdlq
glhuhqfhv ehwzhhq wkh wzr prgho/ e| sduwlwlrqlqj wkh srvvleoh uhdol}dwlrq ri wkh
ixqgdphqwdov ghshqglqj rq zkhwkhu d vshfxodwlyh dwwdfn dqg2ru d froodsvh ri wkh
fxuuhqf| rffxu dw hdfk sduwlfxodu ydoxh ri w1
5171 Ehqfkpdun prgho LL= d vlqjoh odujh lqyhvwru
Lw lv lqvwuxfwlyh wr frpsduh zkdw zrxog kdsshq li zh uhsodfhg wkh frqwlqxxp ri
sulydwh lqyhvwruv zlwk rqh odujh lqyhvwru1 Vxssrvh wklv odujh vshfxodwru revhuyhv d
vljqdo + ' w n # / zkhuh : f dqg # lv glvwulexwhg dffruglqj wr vprrwk v|pphwulf
ghqvlw| } E ^zulwh C E iru wkh f1g1i1`1 Fohduo|/ lq dqdorj| wr wkh rswlpdo fxwr
frqglwlrq +515,/ wkh odujh vshfxodwru zloo
li dqg rqo| li kh revhuyhv d vljqdo
3dwwdfn
+
+ zkhuh wkh suredelolw| ri vxffhvv/ C / lv dw ohdvw |1 Wkxv kh dwwdfnv li dqg
w ? 4@5/ I 4 +w, lv d qhjdwlyh qxpehu/ vr wkdw { A 1 Dv $ 3/ l1h1 ohwwlqj wkh
sulydwh vljqdo ehfrph duelwudulo| suhflvh/ wkh rswlpdo fxwr srlqw zloo whqg wr wkh ixqgdphqwdo
wkuhvkrog/ { $ 1
8 Wkh surri ri xqltxhqhvv ri wklv htxloleulxp lv glvfxvvhg ehorz vhh dovr Pruulv dqg Vklq
+4<<;,1
7 Iru
;
rqo| li + + W ' C3 E|1 Wkhuh zloo eh/ krzhyhu/ qr htxloleulxp frqglwlrq
htxlydohqw wr wkh fulwlfdo pdvv frqglwlrq +514,1
61 Vshfxodwlyh dwwdfnv zlwk vpdoo dqg odujh wudghuv
Zh qrz frqvlghu wkh jhqhudo fdvh/ zkhuh wkh vlqjoh odujh wudghu fdq pxvwhu d wudg0
lqj fdsdflw| ri b/ zkloh wkh frqwlqxxp ri vpdoo wudghuv dv d jurxs fdq wudgh xs wr
wkh fdsdflw| ri b1 Hdfk vshfxodwru pxvw ghflgh zkhwkhu wr dwwdfn ru qrw1 Dv
ehiruh/ sd|r wr dwwdfnlqj lv | li wkh pdvv ri vshfxodwru dwwdfnlqj +dffrxqwlqj
iru erwk odujh dqg vpdoo wudghuv, lv dw ohdvw w/ | rwkhuzlvh1 Wkh sd|r wr qrw
dwwdfnlqj lv f1 Dvvxph w lv glvwulexwhg xqlirupo| rq wkh uhdo olqh1 Wkh odujh
vshfxodwru revhuyhv d vljqdo + ' w n # / zkhuh : f dqg # lv glvwulexwhg dffrug0
lqj wr vprrwk v|pphwulf ghqvlw| } E ^zulwh C E iru wkh f1g1i1`1 Dv ehiruh/ wkh
frqwlqxxp ri vshfxodwruv revhuyh vljqdov % ' w n j0 / zkhuh wkh 0 duh glvwulexwhg
qrupdoo| dffruglqj wr vprrwk v|pphwulf ghqvlw| s E ^zulwh 8 E iru wkh f1g1i1`1
Qrwh wkdw b ' jlyhv h{dfwo| wkh +vpdoo lqyhvwru, ehqfkpdun prgho L ghvfulehg
deryh> vhwwlqj b ' f jlyhv h{dfwo| wkh +odujh lqyhvwru, ehqfkpdun prgho LL1
6141 D Fkdudfwhul}dwlrq ri wkh xqltxh htxloleulxp
Zh zloo suryh wkdw wkhuh lv d xqltxh htxloleulxp lq wklv jhqhudo fdvh1 Wklv htxl0
oleulxp lv lq wuljjhu vwudwhjlhv/ zlwk wkh odujh lqyhvwru dwwdfnlqj rqo| li klv vljqdo
lv ehorz vrph fxwr srlqw +W > dqg hdfk vpdoo lqyhvwru dwwdfnlqj rqo| li klv vljqdo
lv ohvv wkdq vrph rwkhu fxwr srlqw %W 1
Ehfdxvh wkhuh duh d frqwlqxxp ri vpdoo lqyhvwruv/ wkhuh zloo eh qr djjuhjdwh
xqfhuwdlqw|/ frqglwlrqdo rq wkh wuxh w/ derxw wkh sursruwlrq ri vpdoo lqyhvwruv
dwwdfnlqj wkh fxuuhqf|1 Krzhyhu/ hyhq frqglwlrqdo rq wkh wuxh w/ wkhuh zloo eh
xqfhuwdlqw| derxw zkhwkhu wkh odujh lqyhvwru lv dwwdfnlqj1 Wkhuhiruh/ glhuhqwo|
iurp wkh ehqfkpdun prgho L/ zh qrz qhhg wr wklqn ri pruh wkdq rqh fulwlfdo
pdvv frqglwlrq/ iru jlyhq fxwr srlqwv %W dqg +W 1 Wr vwduw zlwk/ zh fdq zrun rxw
dw zklfk ydoxh ri wkh ixqgdphqwdo w dq dwwdfn e| vpdoo
%W 3w lqyhvwruv zloo eh vxffhvvixo
hyhq li wkh odujh lqyhvwru grhv qrw dwwdfn1 Vlqfh 8 j lv wkh sursruwlrq ri vpdoo
lqyhvwruv revhuylqj d vljqdo % %W dqg wkhuhiruh
%W 3w dwwdfnlqj dw w/ dq dwwdfn e|
vpdoo wudghuv lv vxffhvvixo rqo| li E b 8 j
w1 Wklv zloo eh wuxh rqo| li
w w/ zkhuh w lv ghqhg e|=
E
b
8
%W w
j
<
'
w
+614,
Qrwh wkdw w zloo eh frpsulvhg ehwzhhq f dqg b1 Iru dq| uhdol}dwlrq ri wkh
ixqgdphqwdo ehorz w/ vpdoo lqyhvwruv zlwk fxwr srlqw %W zloo odxqfk d vshfxodwlyh
dwwdfn wkdw lv odujh hqrxjk wr eulqj grzq wkh fxuuhqf|/ zkhwkhu ru qrw wkh odujh
lqyhvwru dovr dwwdfnv1
Vlploduo|/ zh fdq zrun rxw dw zklfk ydoxh ri wkh ixqgdphqwdo w dq dwwdfn zlwk
eh vxffhvvixo rqo| li dovr wkh odujh lqyhvwru mrlqv wkh vpdoo lqyhvwruv lq wkh dwwdfn1
E| d v|pphwulf dujxphqw/ wklv zloo eh wuxh rqo| li w w/ zkhuh w lv ghqhg e|=
b n E b 8
%W w
j
'
w
+615,
zkhuh w lv frpsulvhg ehwzhhq w dqg 1 Iru dq| uhdol}dwlrq ri wkh ixqgdphqwdo
ehwzhhq w dqg w/ vpdoo lqyhvwruv dorqh fdqqrw fdxvh d ghydoxdwlrq1 Dq dwwdfn
zloo eh vxffhvvixo rqo| zkhq wkh odujh lqyhvwru dovr wdnhv d vkruw srvlwlrq lq wkh
grphvwlf fxuuhqf|1 Wkh wkuhvkrog 7w lv wkh vxp ri wkh odujh vshfxodwru vl}h +b,
soxv wkh vkduh ri vpdoo lqyhvwruv revhuylqj % %W +dqg wkhuhiruh dwwdfnlqj, dw
w ' 7w1
Zh qrz surylgh frqglwlrqv hqvxulqj wkdw wkh vwudwhjlhv ri wkh vpdoo dqg odujh
lqyhvwruv +zlwk fxwrv %W dqg + W , duh rswlpdo
jlyhq w dqg w1 D odujh lqyhvwru
revhuylqj vljqdo + dvvljqv suredelolw| C w3 + wr wkh srvvlelolw| wkdw w w1 Vlqfh
klv sd|r wr dwwdfnlqj lv C w3 + |/ klv rswlpdo vwudwhj| lv wr iroorz d wuljjhu
vwudwhj| zlwk fxwr +W ghqhg e|=
w +W
C
'
|
+616,
Qrwh wkdw wkh rswlpdo fxwr srlqw iru wkh odujh lqyhvwru lv qrw lqghshqghqw ri
klv vl}h b/ exw b lv uhohydqw iru ghwhuplqlqj + W rqo| wkurxjk lwv hhfw rq wkh
wkuhvkrog w1 Wkh odujh lqyhvwru lv qrw frqfhuqhg zlwk uhjlrqv ri wkh ixqgdphqwdov
ehorz w/ zkhuh d fxuuhqf| froodsvhv lqghshqghqwo| ri klv ehkdylru1 Kh lv krzhyhu
frqfhuqhg zlwk wkh ehkdylru ri vpdoo wudghuv zkhq wkh ixqgdphqwdo lv vwurqj
hqrxjk wr glvfrxudjh vrph iurp mrlqlqj dq dwwdfn1
D vpdoo lqyhvwru revhuylqj vljqdo % eholhyhv w 3j % lv glvwulexwhg dffruglqj
wr
ghqvlw| s E1 Kh nqrzv wkdw li w w/ dq dwwdfn zloo ghqlwho| rffxu > li w 5 wc w /
dq dwwdfn e| d odujh lqyhvwru lv d qhfhvvdu| frqglwlrq iru wkh {hg h{fkdqjh udwh
wr froodsvh/
exw d odujh lqyhvwru zloo dwwdfn rqo| li + + W / l1h1/ zlwk suredelolw|
W
C + 3w > dqg li w : w/ dq dwwdfn zloo qrw rffxu1 Wkxv klv rswlpdo vwudwhj| lv wr
43
iroorz d wuljjhu vwudwhj| zlwk fxwr %W ghqhg e|=
]w
w
'3"
w %W
s
_w n
j
j
]w
'
w w
w %W
+W w
s
C
j
j
_w ' |
+617,
Wkh vhfrqg whup rq wkh ohiw kdqg vlgh dffrxqwv iru wkh idfw wkdw/ iru ixqgdphqwdov
ehwzhhq wc dqg w/ wkh h{shfwhg sd|r iurp d vshfxodwlyh srvlwlrq ghshqgv rq
suredelolw| wkdw wkh odujh lqyhvwru zloo uhfhlyh d vljqdo ehorz + W jlyhq w1 Qrwh
wkdw/ dv 7w lv erxqghg iurp ehorz e| w/ wkh rswlpdo fxwr srlqw iru wkh odujh
+ W 3w
lqyhvwru
lpsolhv
wkdw
wklv
suredelolw|
lv
dovr
erxqghg
iurp
ehorz/
C
W + 3w
C ' |1
6151 D Surri ri xqltxhqhvv
Zh zloo
vkrz
wkdw wkhuh lv d xqltxh htxloleulxp/ fkdudfwhul}hg e| wkh txdgux0
W qrz
W
soh % c + c wc w ghqhg deryh1 Wkh surri ri xqltxhqhvv frphv lq wkuhh vwhsv1
Iluvw/ zh uh0zulwh wkh htxdwlrqv zlwk d fkdqjh ri yduldeoh1 Vhfrqg/ vkrz wkdw
wkhuh lv d xqltxh vroxwlrq wr wkh htxdwlrqv +dqg wkxv wkdw wkhuh lv d xqltxh wuljjhu
vwudwhj| htxloleulxp,1 Odvwo|/ zh vkrz wkdw wkh ehvw uhvsrqvh g|qdplfv lq wuljjhu
vwudwhjlhv ohdgv wr wkh xqltxh wuljjhu htxloleulxp/ dqg wkdw lwhudwhg ghohwlrq ri
grplqdwhg vwudwhjlhv iroorzv ehvw uhvsrqvh g|qdplfv1
Lq wkh uvw vwhs/ zh uh0zulwh wkh irxu htxdwlrqv xvlqj d fkdqjh ri yduldeohv1
Zh ghqh B dqg B dv iroorzv
B
w %W
w %W
dqg B j
j
Htxdwlrq +614, dqg +615,
E
b
E
b n E b
8
B
E '
8
B
%W n jB
'
%W n jB
+618,
+619,
Iru dq| jlyhq %W / zulwh B E%W dqg B E%W iru wkh xqltxh ydoxhv ri B dqg B vroylqj
htxdwlrqv +618, dqg +619, uhvshfwlyho|1 Revhuyh wkdw erwk B dqg B duh ghfuhdvlqj
44
lq wkh fxwr ydoxh %W iru wkh vpdoo lqyhvwruv9
_B
_%W
_B
dqg W
_%
'
'
b
b
E
E
s EB n j
s B
n
j
f
f
Uh0duudqjlqj +616, jlyhv=
+ W ' w C3 E|
W
3 E|
' % n jB C
+61:,
Vxevwlwxwlqj lqwr +617, jlyhv=
]w
w
'3"
w %W
_w n
s
j
j
Zulwlqj 5
'
3
w %W
/
j
3%W
]j
'3"
'
w w
C3
|
E
j
_w ' |
wklv ehfrphv
3%W
]j
w
5
ww
w %W
C
s
j
]w
w
s E5 _5 n
5
' w3j%W
w %W j5
s E5 C
C3
|
E _5 ' |
Zlwk rxu fkdqjh ri yduldeohv/ wkhq/ htxdwlrq +617, fdq eh qdoo| uhzulwwhq dv
B E%W c B E%W
'
|
+61;,
zkhuh
Bc B
]B
'
5
'3"
s E5 _5 n
]B
s E5 C
'
5 B
j B5
C3
|
E _5
+61<,
Zh duh qrz uhdg|
vkrz
W wr
wkdw wkh irxu htxdwlrqv +618,/ +619,/ +61:,
dqg
W
+61;, lq yduldeohv % c + c Bc B kdyh d xqltxh vroxwlrq1 Revhuyh wkdw Bc B lv
9 Wkdw
lv/ rwkhu wklqjv htxdo/ wkh fxuuhqf| froodsvhv iru vwurqjhu ydoxhv ri wkh ixqgdphqwdov
zkhq vpdoo wudghuv ehfrph pruh djjuhvvlyh1
45
vwulfwo| lqfuhdvlqj lq erwk dujxphqwv +iru B B ,1 Dovr Bc B | lv qhjdwlyh
iru vx!flhqwo| orz ydoxhv ri Bc B dqg Bc B | lv srvlwlyh iru vx!flhqwo| kljk
ydoxhv ri Bc B 1 Wkxv/ wkhuh lv d xqltxh ydoxh ri %W vroylqj B E%W c B E%W ' |1
Wkh rwkhu yduldeohv duh wkhq xqltxho| ghwhuplqhg e| wkh rwkhu wkuhh htxdwlrqv lq
wkh v|vwhp1
Wr wklv srlqw/ zh kdyh frqvwuxfwhg dq htxloleulxp lq wuljjhu vwudwhjlhv/ dqg
vkrzq wkdw lw lv wkh xqltxh wuljjhu vwudwhj| htxloleulxp1 Zh pxvw qrz zuds xs
wkh dujxphqw e| vkrzlqj wkdw wkhuh lv qr rwkhu htxloleulxp lqyroylqj vrph pruh
frpsolfdwhg vwudwhjlhv1 Wkh dujxphqw khuh kdv dsshduhg lq vlplodu irupv lq vrph
suhylrxv zrun e| wzr ri xv +iru lqvwdqfh/ lq Pruulv dqg Vklq ^4<<;/ 4<<<`,/ exw wkhlu
ruljlq olhv lq wkh sulru olwhudwxuh rq jdphv zlwk lqfrpsohwh lqirupdwlrq +Fduovvrq
dqg ydq Gdpph ^4<<6`,1 Lq rughu qrw wr gh hfw dwwhqwlrq iurp wkh pdlq lqqrydwlrq
ri wkh fxuuhqw sdshu/ zh zloo phuho| vnhwfk wklv qdo vwhs1 Iluvw/ lw lv hdv| wr yhuli|
wkdw li wkh odujh vshfxodwru dqg frqwlqxxp vshfxodwruv iroorz d wuljjhu vwudwhj|/
wkhq hdfk vshfxodwru*v ehvw uhvsrqvh zloo eh d wuljjhu vwudwhj|1 Lw lv dovr hdv| wr
yhuli| prqrwrqlflw|= wkh pruh vshfxodwruv dwwdfn/ wkh pruh riwhq wkhuh zloo eh d
ehvw uhvsrqvh wr dwwdfn1 Qrz ohw E%& c +& eh d vhtxhqfh ri wuljjhuv vxfk wkdw +4, wkh
wuljjhu vwudwhjlhv sdudphwhul}hg e| E%&n c +&n duh d ehvw uhvsrqvh wr wkh wuljjhu
vwudwhjlhv sdudphwhul}hg e| E%& c +& > dqg +5, E%f c +f ' E4c 41 Fohduo|/ E%& c +& lv d ghfuhdvlqj vhtxhqfh dqg wkh olplw lv d wuljjhu vwudwhj| htxloleulxp1 Wkxv
vwduwlqj zlwk doo vshfxodwruv dozd|v dwwdfnlqj/ zh frqyhujh 0 xqghu ehvw uhvsrqvh
g|qdplfv 0 wr wkh xqltxh wuljjhu vwudwhj| htxloleulxp1 Vlploduo|/ vwduwlqj zlwk doo
vshfxodwruv qhyhu dwwdfnlqj/ zh frqyhujh 0 xqghu ehvw uhvsrqvh g|qdplfv 0 wr wkh
xqltxh wuljjhu vwudwhj| htxloleulxp1 Ilqdoo|/ lw fdq eh vkrzq wkdw wkh dfwlrqv ri
dwwdfnlqj wkh fxuuhqf| duh vwudwhjlf frpsohphqwv/ lq wkh vhqvh wkdw wkh h{shfwhg
sd|r iurp dwwdfnlqj wkh fxuuhqf| lv lqfuhdvlqj lq wkh ghjuhh wr zklfk rwkhu
wudghuv duh dwwdfnlqj lw1 Iurp wklv/ lw fdq eh vkrzq wkdw wkh lwhudwhg ghohwlrq
ri vwulfwo| grplqdwhg vwudwhjlhv plplfv wklv ehvw uhvsrqvh g|qdplfv1 Wkxv/ qrw
rqo| lv wkh htxloleulxp lghqwlhg deryh wkh xqltxh Ed|hvldq Qdvk htxloleulxp/
lw vdwlvhv wkh pxfk vwurqjhu surshuw| wkdw lw lv wkh rqo| vwudwhj| suroh zklfk
vxuylyhv wkh lwhudwhg ghohwlrq ri vwulfwo| grplqdwhg vwudwhjlhv1
71 Surshuwlhv ri wkh htxloleulxp
Dowkrxjk zh gr qrw kdyh d forvhg irup vroxwlrq
wr rxu prgho/ zh fdq h{dplqh
olplwlqj surshuwlhv1 Wr wklv hqg/ zh zulwh %W Ejc c + W Ejc c w Ejc c w Ejc iru wkh xqltxh vroxwlrq dv d ixqfwlrq ri j dqg c dqg dqdo|}h wkh ehkdylru ri wkh
46
v|vwhp dv j $ f dqg $ f1 Qrwh wkdw/ zkhq wdnlqj wkhvh olplwv/ zh kdyh wr
doorz iru wkh srvvlelolw| wkdw rqh sdudphwhu jrhv wr }hur idvwhu wkdq wkh rwkhu1
Khuhdiwhu/ zh zloo irfxv rq wzr srvvleoh fdvhv1 Wkh uvw lv wkh fdvh lq zklfk/ lq
wkh olplw/ wkh lqirupdwlrq ri wkh odujh lqyhvwru lv duelwudulo| pruh suhflvh wkdq wkh
lqirupdwlrq ri wkh uhsuhvhqwdwlyh vpdoo lqyhvwru= dv j $ f dqg $ f/ zh dovr
kdyh j $ 41 Wkh vhfrqg fdvh uhyhuwv wkh udqnlqj lq wkh suhflvlrq ri lqirupdwlrq1
Lq wklv fdvh/ j $ f dqg $ f/ exw j $ f1 Lw wxuqv rxw wkdw wklv glvwlqfwlrq
surylghv dq lpsruwdqw lqvljkw rq wkh hfrqrplfv ri wkh prgho1
Dv d suholplqdu| vwhs wrzdugv dqdo|}lqj wkh olplwlqj surshuwlhv ri wkh
| prgho/
3
3
zh vkrz wkdw/ e| htxdwlrq +61;,/ B ydulhv ehwzhhq 8 E| dqg 8
3| 1 Wr
j
vhh wklv/ revhuyh wkdw/ dv B 5 f iru B 5 / wkh ydoxh ri wkh ixqfwlrq
C j B 5 C3 E| lq htxdwlrq +61<, zloo eh ehwzhhq | +fruuhvsrqglqj wr
wkh rswlpdo
fxwr iru wkh odujh lqyhvwru, dqg 1 Hydoxdwlqj +61;, dw CE ' / zh
rewdlq 8 B ' |1 Frqyhuvho|/ hydoxdwlqj
+61;, dw CE ' |/ 8 E7B zloo eh dw
prvw 3| | 1 Zh frqfoxgh wkdw | 8 B 3| | 1
7141 Wkh fdvh ri d odujh lqyhvwru zlwk pruh suhflvh lqirupdwlrq + j
$ 4,
Zh qrz wdnh wkh olplw ri wkh htxloleulxp h{suhvvlrqv dv j $ 4/ l1h1 zh ohw
wkh odujh wudghu lqirupdwlrq ehfrph duelwudulo|
uhodwlyh
wr wkh vpdoo
j pruh
suhflvh
3
sod|huv1 Revhuyh wkdw wkh h{suhvvlrq C B 5 C E| lq htxdwlrq +61<,
whqgv wr / iru doo 5 B dqg B 8 3 E|1 Lqwxlwlyho|/ zkhq wkh suhflvlrq ri wkh
odujh wudghu lqirupdwlrq ehfrphv duelwudulo| odujh/ iru doo sudfwlfdo sxusrvhv wkh
vpdoo wudghuv frqvlghuv wkh vljqdo ri wkh odujh lqyhvwru dv rqh wklqj zlwk wkh vwdwh
ri wkh ixqgdphqwdo1 Vr htxdwlrq +61;, ehfrphv
8 B
'
|
Iru d vpdoo wudghu/ fdofxodwlqj wkh h{shfwhg sd|r iurp dq dwwdfn rqo| ghshqgv
rq wkh glvwulexwlrq ri ixqgdphqwdov frqglwlrqdo rq klv vljqdo1 Vxevwlwxwlqj lqwr
wkh rwkhu htxloleulxp frqglwlrqv/ zh rewdlq rxu uvw sursrvlwlrq=
47
Sursrvlwlrq 7141
Li
j
$4
/ wkhq
w Ejc $ b n E b E |
w Ejc %W Ejc $
8 3 E|
j
w Ejc +W Ejc $ C3 E|
w Ejc $ wn Ebc %W Ejc c j zkhuh
wn Ebc %W c j lv wkh xqltxh ydoxh ri w
E
b
8
vroylqj
%W w
j
'
w1
Wkh htxloleulxp h{suhvvlrq iru wkh olplw ri w kdv d vlpsoh lqwxlwlyh h{sod0
qdwlrq1 Uhfdoo wkdw wkh vkduh ri vpdoo wudghuviroorzlqj
wkh wuljjhu vwudwhj| %W
W
7
dqg revhuylqj % %W frqglwlrqdo rq 7w lv 8 % j3w 1 Wkhq/ e| xvlqj wkh ghi0
lqlwlrq ri 7B/ dqg wkh idfw wkdw zlwk d odujh lqyhvwru zlwk vxshulru lqirupdwlrq
8 E7B ' |/ wkh vkduh ri wkh srsxodwlrq ri vpdoo wudghuv E b dwwdfnlqj dw w ' 7w
lv 8 E7B ' 8 E7B ' |1 Wkxv w $ b n E bE |1
Frpsdulqj wklv uhvxow zlwk wkh vroxwlrq ri wkh prgho zlwkrxw d odujh wudghu
suhvhqwhg lq vhfwlrq 5 +ehqfkpdun prgho L,/ zh vhh wkdw wkh ixqgdphqwdo wkuhvkrog
w Ejc lv vwulfwo| juhdwhu wkdq wW/ dqg lqfuhdvlqj lq wkh vl}h ri wkh odujh lqyhvwru1
Zlwk d odujh wudghu/ wkh fxuuhqf| froodsvhv dw kljkhu ydoxhv ri ixqgdphqwdov1 Vlqfh
7
w lv lqfuhdvlqj lq b/ wkh fxwr iru vpdoo lqyhvwruv/ %W / lv dovr lqfuhdvlqj lq wkh vl}h
ri wkh odujh lqyhvwruv1 Lqwxlwlyho|/ zkhq wkh lqirupdwlrq ri wkh odujh lqyhvwru lv
suhflvh/ vpdoo lqyhvwruv duh frqghqw wkdw wkh odujh wudghu*v vljqdo zloo qrw eh
idu dzd| iurp wkhlu ehvw hvwlpdwhv ri wkh ydoxh ri wkh ixqgdphqwdo1 Wkxv/ wkh|
ehfrph pruh djjuhvvlyh/ lq wkh vhqvh wkdw wkh| qg lw rswlpdo wr edlo rxw iru
kljkhu ydoxhv ri wkh ixqgdphqwdo1
Lq frqwudvw/ uhodwlyh wr wkh ehqfkpdun prgho LL/ wkh suhvhqfh ri d frqwlqxxp
ri vpdoo wudghuv pdnhv wkh odujh lqyhvwruv pruh fdxwlrxv1 Wkh fxwr srlqw + W lv
orzhu/ idoolqj iurp C3E| wr 7w C3 E|1 Wklv lv ehfdxvh wkh odujh wudghu
kdv qrz wr doorz iru wkh idfw wkdw vrph sursruwlrq ri vpdoo lqyhvwruv zloo idlo wr
mrlq klp lq wkh dwwdfn1:
: Iru w ? 4@5/ erwk { dqg | duh deryh wkh wkuhvkrog 1 Dv $ 4/ | zloo eh ehorz { /
dqg forvhu wr 1
48
Ixuwkhu lqvljkw fdq eh jdlqhg e| ohwwlqj wkh lqirupdwlrq ri erwk vpdoo dqg odujh
sod|huv ehfrph duelwudulo| suhflvh j $ f/ $ f/ zkloh nhhslqj wkh lqirupdwlrq
ri wkh odujh lqyhvwruv duelwudulo| pruh suhflvh +l1h1 j $ 4,1 Wkh pdlq uhvxowv duh
vxppdul}hg e| rxu uvw fruroodu|1
Fruroodu| 7151
Li
j $ f/ %W Ejc +W Ejc w Ejc w Ejc $
$b
$b
$b
$ i
f dqg
n E
n E
n E
4?
$4
b |
b |
b |
bc b b
j
/ wkhq
E
E
E
n E
E
| j
Zkhq wkh odujh wudghu lv zhoo lqiruphg uhodwlyho| wr vpdoo wudghuv/ lq wkh olplw
doo wuljjhu srlqwv whqg wr wkh xqltxh ydoxh b n E b E |1 Wklv lv vwulfwo|
juhdwhu wkdq wkh wuljjhu srlqw rewdlqhg zlwk vpdoo wudghuv rqo| +vhh wkh ehqfkpdun
prgho L deryh,/ dqg lwv ydoxh lv lqfuhdvlqj lq wkh vl}h ri wkh odujh wudghu1
7151 Wkh fdvh ri d uhodwlyho| xqlqiruphg odujh lqyhvwru + j
$
,
f
Zkdw li wkh lqirupdwlrq ri wkh odujh lqyhvwru lv uhodwlyho| ohvv suhflvh wkdq wkdw
ri vpdoo lqyhvwruvB Wr
dgguhvv
wklv txhvwlrq/
zh qrz ohw j $ f/ dqg revhuyh wkdw
wkh h{suhvvlrq C j B 5 C3 E| lq htxdwlrq +61<, whqgv wr |/ iru doo 5
dqg B 8 3 E 3| | 1 Vr htxdwlrq +61;, ehfrphv
8 EB n E | 8 B
8 B |
+714,
$ wkh uhsuhvhqwdwlyh vpdoo wudghu
E '
Glhuhqw iurp wkh suhylrxv fdvh/ zlwk j
f
zloo qrz frqvlghu klv rzq vljqdo +dv rssrvhg wr wkh vljqdo ri wkh odujh wudghu, dv
dssur{lpdwho| htxdo wr wkh ixqgdphqwdo1 Dw wkh vdph wlph/ kh zloo eh frqfhuqhg
zlwk wkh srru suhflvlrq ri wkh odujh lqyhvwru lqirupdwlrq uhfdoo wkdw wkh ehkdylru
ri wkh odujh lqyhvwru lv fuxfldo iru wkh vpdoo lqyhvwruv* sd|r zkhq wkh ixqgdphqwdo
lv ehwzhhq w dqg 7w1
Xvlqj +616, wkh uhsuhvhqwdwlyh vpdoo lqyhvwru fdq fdofxodwh wkh h{dfw suredelolw|
wkdw wkh odujh lqyhvwru
vljqdo zloo eh ehorz +W li wkh ixqgdphqwdo lv htxdo wr 7w1 Wklv
W suredelolw| lv C + 3w ' |1 Zkdw fdq d vpdoo lqyhvwru frqfoxgh/ krzhyhu/
li klv vljqdo % whoov klp wkdw w lv glhuhqw iurp 7wB Nh| wr xqghuvwdqglqj +714,
W
lv wkdw/ dv wkh vljqdo ri wkh odujh lqyhvwru lv txlwh qrlv|/ wkh suredelolw| C + 3w
49
zloo uhpdlq dssur{lpdwho| frqvwdqw iru ydoxhv ri w durxqg 7w1 Vshflfdoo|/ lw zloo
eh dssur{lpdwho| htxdo wr | iru ixqgdphqwdov lq wkh lqwhuydo ehwzhhq w dqg 7w1
Wkxv/ wkh vpdoo lqyhvwruv zloo wklqn wkdw wkh suredelolw| ri dq dwwdfn e| d odujh
xqlqiruphg lqyhvwru lv frqvwdqw ryhu wkh uhohydqw udqjh ri wkh ixqgdphqwdo1
Lq oljkw ri wklv frqvlghudwlrq/ wkh wzr whupv rq wkh uljkw kdqg wkh rswlpdo
fxwr frqglwlrq +714, iru wkh uhsuhvhqwdwlyh vpdoo lqyhvwru kdyh d vlpsoh lqwxlwlyh
h{sodqdwlrq1 Wkh uvw whup/ 8 EB/ lv wkh h{shfwhg jurvv sd|r iru ydoxhv ri wkh
ixqgdphqwdo ehorz w 1 Wkh vhfrqg whup lv wkh h{shfwhg jurvv sd|r iru w lq wkh
lqwhuydo ehwzhhq w dqg 7w/ doorzlqj
iru d frqvwdqw suredelolw| E | ri dqdwwdfn
e| d odujh lqyhvwru= E |E8 B 8 EB1 Qrwh wkdw ht1 +714, ehfrphv 8 B ' |
dv B $ B/ dqg 8 B ' 3| | dv B $ 41
Frpelqlqj htxdwlrqv +618, dqg +619,/ zh dovr kdyh
b ' E b 8 B
8
B
E n
j BB
+715,
zulwlqj B Ebc |c j dqg B Ebc |c j iru wkh xqltxh vroxwlrqv wr wkh sdlu ri htxdwlrqv
+714, +715,/ zh vwdwh rxu vhfrqg sursrvlwlrq1
Sursrvlwlrq 7161
Li
j
$
f/ wkhq
w Ejc $ b n E b 8 B Ebc |c j
w Ejc %W Ejc $ B Ebc |c j
j
w Ejc + W Ejc $ C3 E|
w Ejc $ E b E 8 EB Ebc |c j
Zkdw gr zh nqrz derxw B Ebc |c j dqg B Ebc |c jB Wr dgguhvv wklv txhvwlrq/
zh dgrsw wkh iroorzlqj vwudwhj|1 Iluvw/ zh frpelqh wkh sdlu ri htxdwlrqv lq +714,
dqg +715, dv wr rewdlq dq h{solflw h{suhvvlrq iru wkh ydoxh ri j dv d ixqfwlrq ri |/
b/ 7B dqg B=;
j ' jh |c bc B
'
htxdwlrq +715,/ zh rewdlq @ I 4 4 dqg uh0duudqjlqj jlyhv wkh h{suhvvlrq lq wkh wh{w1
; Iurp
b E b | 8 B
1
B 8 3 3| | 8 B
4:
4w w I 1 Vxevwlwxwlqj lqwr htxdwlrq +715,
Zh kdyh douhdg| hvwdeolvkhg
deryh wkdw B ydulhv ehwzhhq wkh wzr erxqgdulhv
|
3
3
8 E| dqg 8
3| Zh fdq wkhuhiruh dqdo|}h wkh ehkdylru ri wkh ixqfwlrq
jh |c bc B dv B ydulhv ehwzhhq wkhvh wzr erxqgdulhv1 Vhfrqg/ zh xvh wkh uhvxowv
iurp wklv dqdo|vlv wr gudz frqfoxvlrqv derxw wkh ehkdylru ri 7B dqg B dv j $ f iru
dq| qlwh 1
Ehiruh surfhhglqj
lv
ixuwkhu/ qrwh wkdw wkh qxphudwru ri wkh deryh h{suhvvlrq
}hur iru 7B ' 8 3 3| b 1 Jlyhq wkdw 7B lv erxqghg iurp deryh e| 8 3 3| | / lw lv
hdv| wr yhuli| wkdw wklv lv d uhohydqw vroxwlrq rqo| li b |/ wkdw lv wr vd|/
li wkh
odujh lqyhvwru lv qrw wrr odujh1 Wkxv/ lq vwxg|lqj wkh ixqfwlrq j
h |c bc B zh qhhg
wr frqvlghu wzr fdvhv1
Iru d odujh lqyhvwru zlwk b |/ j
h |c bc B whqgv wr lqqlwh dv B $ 8 3 E|/
lv vwulfwo| srvlwlyh iru doo B 5 8 3 E| c 8 3 3| | / dqg whqgv wr }hur dv B $
8 3 3| | 1 Wkxv zh frqfoxgh wkdw *4 B Ebc |c j ' 8 3 3| | / lpso|lqj wkdw
j
B Ebc |c j ' 41
j<f
*4
<f
Li/ lqvwhdg/ wkh odujh lqyhvwru lv vxfk wkdw b |/ wkhq j
h |c bc B whqgv wr lqqlwh
3
3
dv B $ 8 E|/ lv vwulfwo| srvlwlyh iru doo B 5 8 E| c 8 3 3| b / dqg htxdov
}hur zkhq B ' 8 3 3| b / zklfk lv ehorz wkh xsshu erxqgdu| iru 7B 1 Zh dovr
qrwh wkdw j
h |c bc B lv vwulfwo| qhjdwlyh iru doo B 5 8 3 3| b c 8 3 3| |
dqg
| 3
duh qrw uhohydqw iru rxu dqdo|vlv1
whqgv wr }hur dv B $ 8
3| / exw wkhvh ydoxhv
| 3
Wkxv/ zh frqfoxgh wkdw *4 B Ebc |c j ' 8
3b 1 Wklv lpsolhv *4 B Ebc |c j '
8 3
3| | 3| b
'
j <
f 8 3 |33bb 1
j
<f
Zh vxppdul}h wkh uhohydqw qglqjv zlwk wkh iroorzlqj fruroodu|=
Fruroodu| 7171 Li b |/ j $ f/ $ f dqg j $ f/ wkhq %W Ejc / + W Ejc dqg
w Ejc doo whqg wr b n E b 3| | / zkloh w Ejc whqgv wr b> li b |/
j $ f/ $ f dqg j $ f/ wkhq %W Ejc / +W Ejc / w Ejc dqg w Ejc doo whqg wr
|1
Zkhq b |/ wkh vkduh ri vpdoo lqyhvwruv dwwdfnlqj wkh fxuuhqf|
zkhq wkh
|
7
7
ixqgdphqwdo lv dw wkh wkuhvkrog w lv htxdo wr 8 EB ' 3| 1 Wkh vkduh ri
lqyhvwruv dwwdfnlqj wkh fxuuhqf| dw w ' 7w lv wkhuhiruh vxevwdqwldoo| orzhu wkdq wkh
fdvh glvfxvvhg lq wkh suhylrxv vhfwlrq1 Ghvslwh wkh orz suhflvlrq ri wkh odujh wudghu
lqirupdwlrq/ krzhyhu/ lwv vl}h vwloo dhfwv doo wkh uhohydqw wkuhvkrogv1 Wkh odujhu
7
b/ wkh kljkhu wkh suredelolw|
| ri d froodsvh1 Qrwh wkdw wkh glvwdqfh ehwzhhq w dqg
7
w lv htxdo wr b E b 3| / vr wkdw wkh glhuhqfh ehwzhhq w dqg w whqgv }hur
4;
dv wkh vl}h ri wkh odujh lqyhvwru b dssurdfkhv |1 Wkh vhfrqg sduw ri rxu fruroodu|
hvwdeolvkhv wkdw wklv glhuhqfh lv lghqwlfdoo| htxdo wr }hur iru dq| b ehorz |1< Prvw
fuxfldoo|/ zkhq b |/ ydu|lqj wkh vl}h ri wkh odujh lqyhvwru grhv qrw dhfw wkh
suredelolw| ri wkh froodsvh1 Doo rswlpdo fxwr srlqwv duh lqghshqghqw ri b1
Wr vkhg oljkw rq wklv lpsruwdqw ihdwxuh ri wkh htxloleulxp/ qrwh wkdw/ jlyhq
rxu dvvxpswlrq derxw wkh glvwulexwlrq ri wkh ixqgdphqwdo w/ hdfk vpdoo wudghu
kdv d xqlirup sulru rq wkh shufhqwdjh ri vpdoo wudghuv dwwdfnlqj wkh fxuuhqf|1
Ghqrwh wklv shufhqwdjh zlwk k
h1 Vlqfh d odujh xqlqiruphg lqyhvwru lv shufhlyhg wr
dwwdfn wkh fxuuhqf| zlwk suredelolw| E |/ zkloh wkh vpdoo wudghuv* lqirupdwlrq
lv duelwudulo| suhflvh/ wkh h{shfwhg jurvv sd|r iurp dwwdfnlqj wkh fxuuhqf| iru d
vpdoo lqyhvwru uhfhlylqj d vljqdo % fdq eh zulwwhq dv
5 Ew ' % ' E | RoJK Eb n E b k
h w
n
| ERoJK E b k
h w
zkhuh k
h 5 dfc o1 Dvvxplqj wkdw erwk b dqg | duh ehorz 425/ zh fdq fdofxodwh wkh
h{shfwhg jurvv sd|r iru glhuhqw ydoxh ri wkh vljqdo dv iroorzv=
5 Ew ' % '
;
A
A
A
A
?
A
A
A
A
=
3b3w
E | n |
33bb3w
3
w
E |
n|
3b 3w 3b
E |
3b
f
wf
f w b
b w b
b w
w
Frqvlghu wkh fdvh lq zklfk b lv vx!flhqwo| vpdoo uhodwlyh wr |1 Lq wklv fdvh/
wkh fxwr srlqw %W G 5 Ew ' %W ' | idoov lq wkh lqwhuydo +bc b1 Vpdoo lqyhvwruv
nqrz wkdw wkhlu rzq pdvv lv hqrxjk wr fdxvh d froodsvh ri wkh shj/ zkloh wkh odujh
lqyhvwru lv ehkdylqj qrlvlo|1 Wkh| zloo fkrrvh d %W wkdw lv lqghshqghqw ri b1 Iru
d jlyhq %W/ dq| fkdqjh lq wkh vl}h ri wkh odujh djhqw lv frpshqvdwhg e| d fkdqjh
lq wkh pdvv ri vpdoo lqyhvwruv dwwdfnlqj wkh fxuuhqf|1 Dv phqwlrqhg deryh/ wkh
wzr wkuhvkrogv zloo 7w dqg w frlqflgh1 Li Vrurv lv odujh uhodwlyh wr |/ lqvwhdg/ wkh
rswlpdo fxwr srlqw zloo idoo lq wkh lqwhuydo + bc 1 Wkhq/ glhuhqfhv lq wkh
vl}h ri wkh odujh wudghu fdq qr orqjhu eh frpshqvdwhg e| fkdqjhv lq wkh pdvv ri
vpdoo lqyhvwruv dwwdfnlqj1
Wr vxp xs/ lq wkh fdvh ri d odujh wudghu ohvv lqiruphg uhodwlyh wr vpdoo wudghuv/
wkh lpsolfdwlrqv ri klv suhvhqfh rq wkh olnholkrrg ri d fulvlv lv pxfk ohvv sur0
qrxqfhg1 Lq wkh olplw/ li b |/ wkh fxwr srlqw ri doo wudghuv whqgv wr frlqflgh
< Lq wklv fdvh/ { rqo| iru w 5 + 4 > 4 ,1 Wkh lqhtxdolw| uhyhuvhv iru vpdoohu w1 Lqvwhdg/ | iru w 4 1
5
6 5
4<
zlwk wkh fxwr srlqw lq wkh fdvh ri qr odujh wudghu/ l1h1 b ' f= wkh suhvhqfh ri d
odujh wudghu grhv qrw pdwwhu1 Li b |/ lqvwhdg/ wkh fxwr srlqw lv kljkhu/ dowkrxjk
qrw dv kljk dv lq wkh fdvh glvfxvvhg lq wkh suhylrxv vxevhfwlrq1
7161 Frqwlqxlw|
Iluvw/ revhuyh wkdw hyhu|wklqj frqyhujhv qlfho| wr wkh wzr ehqfkpdunv1 Lq rwkhu
zrugv/ uhjdugohvv ri wkh uhodwlyh suhflvlrq ri lqirupdwlrq +l1h1 uhjdugohvv ri zkhwkhu
j
$ 4 ru j $ f,/ zh kdyh %W Ejc / +W Ejc / w Ejc dqg w Ejc doo frqyhujlqj
wr | dv j $ f/ $ f dqg b $ f> dqg doo frqyhujlqj wr dv j $ f/ $ f dqg
b $ 1
Vhfrqg/ li wkh odujh vshfxodwru*v vljqdo lv lqirupdwlyh uhodwlyh wr wkh frqwlqxxp
vshfxodwruv* vljqdov +l1h1/ li j $ f/ $ f dqg j $ 4, wkhq wkh frqyhujhqfh lv
yhu| vprrwk/ dqg lq sduwlfxodu %W Ejc / + W Ejc dqg w Ejc duh doo olqhdu lq b1
Exw li wkh odujh vshfxodwru*v vljqdo lv xqlqirupdwlyh uhodwlyh wr wkh frqwlqxxp
vshfxodwruv* vljqdov +l1h1/ li j $ f/ $ f dqg j $ f, wkhq wkh h{lvwhqfh ri wkh
odujh vshfxodwru kdv qr lpsdfw xqwlo kh furvvhv d fhuwdlq vl}h wkuhvkrog +b,1
81 Wkh uroh ri d odujh lqyhvwru lq d fxuuhqf| fulvlv= d v|q0
wkhvlv ri rxu uhvxowv
Kdylqj frqgxfwhg wkh dqdo|vlv/ zh duh qrz lq d srvlwlrq wr jdwkhu wrjhwkhu wkh
ydulrxv vwudqgv lq rxu wd{rqrp| ri fdvhv1 Zh qrwh wkh iroorzlqj srlqwv1
Wkh suhvhqfh ri wkh odujh wudghu lqgxfhv wkh vpdoo wudghuv wr ehfrph pruh
djjuhvvlyh vhoohuv1 Zkhq b ' f/ wkh wuljjhu srlqw iru wkh vpdoo wudghuv/ %W/
lv jlyhq e| %W ' | lq wkh olplw1 Zkhq b : f/ wklv wuljjhu srlqw fdq eh
kljkhu/ luuhvshfwlyh ri wkh uhodwlyh lqirupdwlrqdo txdolw| ri wkh odujh wudghu
Lq sduwlfxodu/ zkhq wkh odujh wudghu lv zhoo lqiruphg uhodwlyh wr wkh vpdoo
wudghuv +jlyhq e| Fruroodu| 715,/ wkh vpdoo wudghu*v wuljjhu srlqw whqgv wr
%W ' b n E b E | Wklv lv vwulfwo| odujhu wkdq
wudghu1
+814,
|/ dqg lv lqfuhdvlqj lq wkh vl}h ri wkh odujh
Krzhyhu/ zkhq wkh odujh wudghu lv ohvv zhoo lqiruphg uhodwlyh wr wkh vpdoo
wudghuv +jlyhq e| Fruroodu| 717,/ wkh lpsdfw ri wkh odujh wudghu rq wkh wudglqj
53
vwudwhjlhv ri wkh vpdoo wudghuv lv vpdoohu1 Iluvwo|/ li b |/ wkh odujh wudghu
kdv qr lpsdfw rq wkh wudglqj vwudwhjlhv ri wkh vpdoo wudghuv +vlqfh %W ' |,1
Zkhq b : |/ wkh wuljjhu srlqw whqgv wr
%W
'
'
b n E b
|
2
| | |b
b n E b E
|
E
zklfk lv ohvv wkdq +814,1 Khqfh/ wkh vpdoo wudghuv duh ohvv djjuhvvlyh vhoohuv
wkdq li wkh| zhuh glvdgydqwdjhg uhodwlyh wr wkh odujh wudghu lq whupv ri wkhlu
lqirupdwlrq1
91 Frqfoxvlrq
Wkh wkhru| ri fxuuhqf| dqg qdqfldo fulvhv wr zklfk wklv prgho frqwulexwhv lv vwloo
lq lwv lqidqf|1 \hw/ wkhuh duh d ihz frqvlghudwlrqv wkdw duh srwhqwldoo| uhohydqw iru
srolf| dqdo|vlv dqg ghvljq1 Kdylqj vxppdul}hg rxu pdlq uhvxowv lq vhfwlrq 8/ wklv
forvlqj vhfwlrq zloo eh ghyrwhg wr d glvfxvvlrq ri irxu lpsolfdwlrqv ri rxu prgho
wkdw vkhg oljkw rq wkh g|qdplfv ri fulvlv1
Iluvw/ rxu prgho +iroorzlqj Pruulv dqg Vklq ^4<<;`, vwuhvvhv wkdw vpdoo fkdqjhv
lq lqirupdwlrq derxw ixqgdphqwdov pd| kdyh d odujh lpsdfw rq djhqwv ehkdylru/
dqg wuljjhu d fulvlv1 Lq rxu dqdo|vlv/ d froodsvh pd| eh dvvrfldwhg zlwk d vpdoo
fkdqjh lq wkh sulydwh vljqdo uhfhlyhg e| wkh odujh lqyhvwruv durxqg wkh fulwlfdo ydoxh
+ W1 Wkh phfkdqlvp wkurxjk zklfk lqirupdwlrq wudqvodwhv lqwr d fulvlv lv krzhyhu
qrw duelwudu|/ dv lv lq prghov wkdw vhh fulvhv dv gulyhq e| vxqvsrwv1 Fulvhv whqg wr
kdsshq zkhq ixqgdphqwdov duh uhodwlyho| zhdn1 Wklv lv dq lpsruwdqw glhuhqfh
uhodwlyh wr vhfrqg jhqhudwlrq prghov ri fxuuhqf| fulvlv +Revwihog ^4<<9`,/ ru prghov
gudzlqj rq wkh Gldprqg dqg G|eylj ^4<;6` edqn uxq iudphzrun1
Vhfrqg/ wkh fxuuhqw olwhudwxuh edvhg rq vxqvsrwv suhglfw wkdw doo djhqwv sxuvxh
wkh vdph dfwlrq lq htxloleulxp1 Wklv lv qrw wkh fdvh lq wkh htxloleulxp ghvfulehg
deryh1 Iru dq| uhdol}dwlrq ri wkh ixqgdphqwdov/ vrph vpdoo wudghuv zloo uhfhlyh d
vljqdo deryh wkh wuljjhu/ dqg ghflgh qrw wr dwwdfn wkh fxuuhqf|1 Prvw lpsruwdqw/
wkh odujh wudghu pd| idlo wr dwwdfn wkh fxuuhqf|/ hyhq zkhq klv dwwdfn zrxog kdyh
fdxvhg d froodsvh iru vxuh1 Wklv pxowlsolflw| ri rxwfrph lv qrw wr eh frqixvhg zlwk
pxowlsolflw| ri htxloleuld1 Lw iroorzv lqvwhdg iurp wkh vwuxfwxuh ri lqirupdwlrq1
Wklug/ e| dhfwlqj wkh h{shfwhg sd|r iurp dwwdfnlqj wkh fxuuhqf| frqglwlrqdo
rq rqh*v eholhiv/ wkh rssruwxqlw| frvwv ri vshfxodwlrq +| lq rxu qrwdwlrq, lq xhqfhv
54
wkh olnholkrrg ri d fulvlv1 Qrwdeo|/ lq prghov zlwk vxqvsrwv wkh rssruwxqlw| frvwv ri
vshfxodwlrq pd| dhfw wkh uhjlrq ri wkh vxssruw ri wkh ixqgdphqwdov fkdudfwhul}hg
e| pxowlsoh htxloleuld/ exw qrw wkh olnholkrrg ri dq dwwdfn/ wkdw lv hvvhqwldoo|
duelwudu|1 Dw wklv vwdjh lq wkh ghyhorsphqw ri wkh olwhudwxuh srolf| dqg qrupdwlyh
suhvfulswlrqv duh suhpdwxuh1 Qrqhwkhohvv/ zh eholhyh wkdw/ iurp d srvlwlyh srlqw ri
ylhz/ rxu uhvxowv frxog hqkdqfh rxu xqghuvwdqglqj ri wkh phfkdqlfv ri vshfxodwlyh
dwwdfnv/ dqg surylgh exloglqj eorfnv iru ixuwkhu srolf|0rulhqwhg dqdo|vlv1
Irxuwk/ hfrqrplvwv dqg srolf| pdnhuv kdyh orqj ghedwhg zkhwkhu vshfxodwlrq/
hvshfldoo| vshfxodwlrq e| odujh lqyhvwruv/ lv ghvwdelol}lqj1 Lq wkh suhvhqfh ri d
frruglqdwlrq sureohp/ zh kdyh vkrzq wkdw lqfoxglqj d odujh lqyhvwru lq wkh pdunhw
lqfuhdvhv wkh olnholkrrg ri d fulvlv/ dqg uhqghuv vpdoo lqyhvwruv pruh djjuhvvlyh/
exw wkh hhfw fdq eh vpdoo li wkh odujh lqyhvwruv lv uhodwlyho| xqlqiruphg1 Odujh
sod|huv zlwk lqihulru lqirupdwlrq hhfwlyho| lqfuhdvh wkh dprxqw ri qrlvh wkdw lv
uhohydqw lq wkh sruwirolr ghflvlrq e| wkh uhvw ri wkh pdunhw1 Wkxv/ lw lv wkh lqwhusod|
ehwzhhq vl}h dqg lqirupdwlrq dffxudf| wkdw pxvw eh wdnhq lqwr dffrxqw wr dgguhvv
wkh deryh lvvxh1
55
Uhihuhqfhv
^4` Fduovvrq/ K1 dqg H1 ydq Gdpph ^4<<6`1 Joredo Jdphv dqg Htxloleulxp Vh0
ohfwlrq/ Hfrqrphwulfd 94/ <;<0434;1
^5` Fruvhwwl/ J1/ S1 Shvhqwl dqg Q1 Urxelql ^4<<;`1 Zkdw Fdxvhg wkh Dvldq Il0
qdqfldo FulvlvB QEHU Zrunlqj Sdshu 9;6609;671
^6` Gldprqg/ G1 dqg S1 G|eylj ^4<;6`1 Edqn Uxqv/ Ghsrvlw Lqvxudqfh/ dqg Olt0
xlglw|/ Mrxuqdo ri Srolwlfdo Hfrqrp| <4/ 734074<1
^7` Lqwhuqdwlrqdo Prqhwdu| Ixqg ^4<<<`/ LPI0Vxssruwhg Surjudpv lq Lqgrqhvld/
Nruhd/ dqg Wkdlodqg= D Suholplqdu| Dvvhvvphqw/ Mdqxdu|
^8` Pruulv/ V1 dqg K1 V1 Vklq ^4<<;`1 Xqltxh Htxloleulxp lq d Prgho ri Vhoi0
Ixooolqj Fulvhv/ Dphulfdq Hfrqrplf Uhylhz ;;/ 8;:08<:1
^9` Pruulv/ V1 dqg K1 V1 Vklq ^4<<<`1 Sulydwh yhuvxv Sxeolf Lqirupdwlrq lq Frru0
glqdwlrq Sureohpv lq surjuhvv1
^:` Revwihog/ P1 ^4<;9`1 Udwlrqdo dqg Vhoi0ixooolqj Edodqfh0ri0Sd|phqwv Fulvhv/
Dphulfdq Hfrqrplf Uhylhz :9/ :5;41
^;` Revwihog/ P1 ^4<<9`1 Prghov ri Fxuuhqf| Fulvhv zlwk Vhoi0Ixooolqj Ihdwxuhv/
Hxurshdq Hfrqrplf Uhylhz 73/ 436:0437:1
56
Figure 1
Fundamentals and currency collapse: a comparison across models
with and without common knowledge
A continuum of investors with common knowledge of the
fundamental
fundamental
0
Attack
Collapse
1
Multiple equilibria:
(attack and collapse)
(no attack, no collapse)
No Attack
No Collapse
Benchmark model I
A continuum of investors with incomplete information
fundamental
1-t
Large attack and
collapse
No collapse
Some investors attack
© Copyright 2026 Paperzz