fulfillment of regulatory requirements on var and optimal portfolio

IXOILOOPHQW RI UHJXODWRU\
UHTXLUHPHQWV RQ YDU DQG RSWLPDO
SRUWIROLR SROLFLHV
|
Holvd Oxfldqr
Xqlyhuvlw| ri Wxulq/ Lwdo|
Devwudfw
Wklv sdshu kdv d wzrirog dlp= uvw/ zh surylgh g|qdplf YdU phdvxuhv/
zklfk/ dv rssrvlwh wr wkh xvxdo/ vwdwlf rqhv/ wdnh lqwr dffrxqw fkdqjhv lq wkh
sruwirolr pl{1 Wkhvh fkdqjhv rffxu zkhqhyhu wkh YdU krul}rq/ dv lpsrvhg
e| wkh Uhjxodwru| Dxwkrulw|/ lv juhdwhu wkdq wkh lqwhuqdo xqzlqglqj shulrg1
Vhfrqg/ zh vwxg| wkh hhfwlyhqhvv ri uhjxodwru| uhtxluhphqwv edvhg rq
vwdwlf phdvxuhv/ vxfk dv wkh ELV rqhv/ zkhqhyhu wkh sruwirolr pl{ fkdqjhv
ryhu wlph1
Zh shuirup rxu vwxg| lq dq kljko| vw|ol}hg iudphzrun/ glvwlqjxlvklqj
wkh fdvh lq zklfk qr iulfwlrqv lq wkh irup ri elg0dvn vsuhdgv h{lvw iurp
wkh
rssrvlwh/
pruh
uhdolvwlf/
rqh1
Lq wkh devhqfh ri vsuhdgv/ vwdwlf phdvxuhv whqg wr xqghuhvwlpdwh +ryhu0
hvwlpdwh, ulvn zkhq ulvn dyhuvlrq lv orz +kljk,1 Zlwk elg dvn vsuhdgv/ wkh
kljkhu lv ulvn dyhuvlrq/ wkh pruh vlplodu wkh vlwxdwlrq lv wr wkh iulfwlrqohvv
fdvh dqg wkh pruh vwdwlf dqg g|qdplf phdvxuhv glhu1 Fdslwdo dghtxdf|
uhtxluhphqwv duh wkh pruh olnho| wr eh vdwlvhg wkh kljkhu lv ulvn dyhuvlrq=
wkh| wkhuhiruh surylgh d jrrg glvfulplqdwlqj hhfw/ lq vslwh ri wkh idfw wkdw
wkh| duh edvhg rq vwdwlf phdvxuhv1
Wkh Dxwkru wkdqnv Ere Mduurz/ Ulffdugr Fhvdul/ Urehuw Ndvw/ Pduwl Vxeudkpdq|dq/
Vwdyurv ]hqlrv/ sduwlflsdqwv dw wkh [Y Hxur 0 [[LY Lqirupv Frqihuhqfh +Edufhorqd/ Mxo| 4<<:,/
wkh [[L Phhwlqj ri wkh Hxur Zrunlqj Jurxs rq Ilqdqfldo Prghoolqj +Yhqlfh/ Rfwrehu 4<<:,/ wkh
[[Y HID Phhwlqj +Sdulv/ Dxjxvw 4<<;, dqg vhplqdu sduwlflsdqwv dw wkh Xqlyhuvlw| ri Dul}rqd
dw Wxfvrq1 Jldqoxfd Ixvdl surylghg frpsxwdwlrqdo dvvlvwdqfh1
| Glsduwlphqwr gl Vwdwlvwlfd h Pdwhpdwlfd Dssolfdwd/ S1 Dueduhoor ;/ L043454 Wrulqr/ who1 .
6<3449:39567 +9568,/ id{ . 6<3449:3956;/ oxfldqrChfrq1xqlwr1lw/ kwws=22pdwk1hfrq1xqlwr1lw
Ydoxh dw Ulvn +YdU,/ orrvho| ghqhg dv wkh pd{lpxp orvv zklfk d qdqfldo
lqwhuphgldu| fdq vxhu/ zlwklq d jlyhq krul}rq dqg dw d jlyhq ohyho ri frqghqfh/
lv rqh ri wkh ehvw nqrzq dqg xs wr gdwh wrrov lq Ilqdqfh1
Prvw ri wkh YdU lpsruwdqfh vwhpv iurp wkh idfw wkdw uhjxodwruv duh fxuuhqwo|
lqfolqhg wr fkrrvh lw dv d ehqfkpdun iru qdqfldo ulvn ghqlwlrq dqg iru wkh hqvxlqj
uhjxodwru| uhtxluhphqwv1 Wklv lv wkh fkrlfh grqh lq wkh XV e| wkh Jurxs ri Wkluw|
^43` dqg lq Hxursh e| wkh Edvoh Frpplwwhh ri wkh ELV ^5`1
Dovr/ upv dqg wkuliwv zklfk duh qrw reoljhg wr uhvshfw wkh uhjxodwru| uhtxluh0
phqwv whqg wr fkrrvh lw dv wkhlu phdvxuh ri ulvn/ gxh wr lwv vlpsolflw| dqg wr wkh
zlgh dydlodelolw| ri vriwzduh lq rughu wr lpsohphqw lw1
Doo wkh YdU lpsohphqwdwlrqv/ dv idu dv zh nqrz/ uhvw rq wkh vlpsoli|lqj dv0
vxpswlrq wkdw wkh sruwirolr pl{ zloo qrw eh fkdqjhg zlwklq wkh YdU krul}rq4 1 Zh
fodlp wkdw 0 dw ohdvw iru vrph sruwirolrv 0 wklv lv dq ryhuvlpsolhg dssurdfk/ vlqfh
wkh YdU krul}rq lv {hg e| wkh Uhjxodwru| Dxwkrulw| dqg grhv qrw qhfhvvdulo|
frlqflgh zlwk wkh sruwirolr xqzlqglqj shulrg1 Wklv odfn ri frlqflghqfh fdq/ lq
rxu rslqlrq/ dhfw wkh vljqlfdqfh ri doo wkh xvxdo ru vwdwlf YdU phdvxuhv1
Iluvw/ zh surylgh g|qdplf YdU phdvxuhv/ zklfk/ rssrvlwh wr wkh xvxdo ru
vwdwlf rqhv/ wdnh lqwr dffrxqw wkh pl{ fkdqjhv ehiruh wkh YdU krul}rq1 Vhfrqg/ zh
vwxg| wkh hhfwlyhqhvv ri uhjxodwru| uhtxluhphqwv edvhg rq vwdwlf YdU phdvxuhv/
l1h1 wkh frqvlvwhqf| ehwzhhq wkh ixooophqw ri uhjxodwru| uhtxluhphqwv vxfk dv
wkh ELV rqhv dqg rswlpdo sruwirolr srolflhv1 Fdq d sruwirolr pdqdjhu zkr
rswlpdo| frqwurov klv sruwirolr/ vr wkdw wkh fruuhfw phdvxuh iru klp zrxog eh d
g|qdplf rqh/ frpso| zlwk fdslwdo dghtxdf| uxohv edvhg rq d vwdwlf phdvxuhB Ru/
fdq lqwhuphgldulhv zklfk duh reoljhg wr frpso| zlwk wkh uhjxodwru| uhtxluhphqwv/
rswlpdoo| doorfdwh wkhlu sruwirolrvB
Lq rughu wr shuirup rxu vwxg|/ zh frqvlghu d yhu| vlpsoh sruwirolr dqg glvwlq0
jxlvk pdunhwv zlwkrxw dqg zlwk iulfwlrqv/ qdpho| elg0dvn vsuhdgv= lw lv zhoo nqrzq
lq idfw wkdw zlwkrxw elg0dvn vsuhdgv sruwirolr dgmxvwphqwv wdnh sodfh frqwlqxrxvo|/
zlwk vsuhdgv rqo| iurp wlph wr wlph1 Wkh qr vsuhdg fdvh pxvw eh xqghuvwrrg dv
d ehqfkpdun/ gxh wr lwv dqdo|wlfdo wudfwdelolw|1 Rq wkh frqwudu|/ zh frqvlghu wkh
prgho zlwk elg0dvn vsuhdgv pxfk pruh uhsuhvhqwdwlyh ri uhdo vlwxdwlrqv/ lq zklfk
sruwirolr frqwurov duh qrw h{huwhg frqwlqxrxvo|1
Wkh sdshu lv vwuxfwxuhg dv iroorzv= vhfwlrq 4 lqwurgxfhv vwdwlf dqg g|qdplf
Dw prvw/ wkh orjlf lv uhyhuvhg= phdq0YdU iurqwlhuv duh h{shfwhg wr eh xvhg iru sruwirolr
rswlpl}dwlrq/ dv li wkh| zhuh dq dowhuqdwlyh wr phdq0yduldqfh phdvxuhv ã od Pdunrzlw}1 Krzhyhu/
h!flhqw sruwirolrv lq wkh phdq0YdU vsdfh hlwkhu frlqflgh zlwk wkh phdq0yduldqfh rqh/ ru gr qrw
uhvshfw vhfrqg rughu vwrfkdvwlf grplqdqfh +vhh ^9/ 45`,
4
5
YdU phdvxuhv1 Vhfwlrq 5 frpsduhv wkh wzr phdvxuhv dqg vwxglhv wkh ghshqghqfh
ri wkh lqhtxdolw| ehwzhhq wkhp rq wkh sdudphwhuv ri wkh prgho1 Vhfwlrq 6 dqdo|}hv
wkh hhfwlyhqhvv ri fdslwdo dghtxdf| uxohv edvhg rq vwdwlf YdU phdvxuhv1 Vhfwlrq
7 vxppdul}hv dqg frqfoxghv1
41 Vwdwlf yhuvxv g|qdplf YdU phdvxuhv
Ohw xv lpdjlqh d yhu| vlpsoh/ frqwlqxrxv0wlph vhwwlqj/ zlwk rqh ulvnohvv dqg rqh
ulvn| dvvhw/ zklfk fdq eh lghqwlhg uhvshfwlyho| zlwk prqh| dqg wkh pdunhw
sruwirolr1
Wkh iruphu/ zklfk zh ghqrwh dv / hduqv d {hg lqwhuhvw +lqwhqvlw|, o1 Lwv
g|qdplfv fdq eh ghvfulehg wkurxjk wkh glhuhqwldo htxdwlrq
_ E| ' o E|_|
zkhuh | ghqrwhv wlph/ | 5 dfc n4 dqg Ef ' f 1
+414,
Wkh odwwhu iroorzv d jhrphwulf Eurzqldq prwlrq=
_7 E| ' Eo n >7 E|_| n j7 E|_5
+415,
zkhuh > dqg j duh frqvwdqw/ 5 lv d vwdqgdug Zlhqhu surfhvv/ dqg 7 Ef ' 7f 1
g
Wkh ryhudoo zhdowk zloo eh ghqrwhg dv ` E| '
7 E| n E|/ lwv lqlwldo ydoxh dv
g
`f ' 7f n f1
4141 Vwdwlf phdvxuhv
Lq wklv frqwh{w/ wkh YdU iru wkh krul}rq A / zklfk zh ghqrwh dv T @-@ EA / fdq eh
ghqhg dv +wkh devroxwh ydoxh ri, wkh E @ff 0 shufhqwloh ri wkh glvwulexwlrq ri
orj uhwxuqv rq zhdowk=
h
E*? ` EA *`f T @-@EA '  @
f
+416,
zkhuh hf E lv wkh suredelolw| frqglwlrqdo rq ` Ef ' `f1
Li zh dvvxph qr sxufkdvhv ru vdohv ri dvvhwv ehiruh A / lw lv hdv| wr frpsxwh
dq| shufhqwloh ri orj uhwxuqv rq zhdowk/ l1h1 dq| T @-@ EA =
#
$
s T @-@EA ' oA *?  n w n  nwfw i T > j2*2 A Dj A
f
f
6
+417,
g
zkhuh D lv wkh 40@ txdqwloh ri wkh vwdqgdug qrupdo/ dqg wf '
7f*f1
Wkh T @- phdvxuh lv lpsruwdqw zkhqhyhu lw fruuhvsrqgv wr d srvlwlyh ydoxh dw
ulvn rq wkh ulvn| dvvhw rqo|1 Lq wxuq/ orj uhwxuqv rq 7 EA / ghqrwhg dv f EA / duh
qrupdo zlwk phdq
o n > j2*2 A
zklfk zh dvvxph wr eh srvlwlyh/ dqg yduldqfh j2 A 1 Wkhlu YdU/ T @-f@/ lv
s
T @-f@EA ' o n > j2*2 A n Dj A
dqg lv srvlwlyh li dqg rqo| li
A
D2j2
Eo n > j2 *22
+418,
Wklv frqglwlrq lv dvvxphg wr krog lq wkh vhtxho1
Vlqfh wkh deryh YdU phdvxuhv wkh ulvnlqhvv ri uhwxuqv xqghu wkh dvvxpswlrq
wkdw qr sxufkdvhv ru vdohv ri dvvhwv duh fduulhg rxw lq dfc A / lw lv d vwdwlf phdvxuh1
Lq rughu wr surylgh d g|qdplf phdvxuh/ l1h1 wr phdvxuh YdU zlwkrxw wkh qr0
dgmxvwphqw dvvxpswlrq/ zh pxvw uvw ghqh zklfk vruw ri sruwirolr dgmxvwphqwv
zh frqvlghu1 Zh gr wklv vhsdudwho| iru wkh qr elg0dvn dqg elg0dvn fdvh/ vlqfh/ dv
zh zloo dujxh/ wkh iruphu fdvh lv d ehqfkpdun vlwxdwlrq/ zkloh wkh vhfrqg lv pxfk
pruh uhdolvwlf1
4151 G|qdplf YdU phdvxuhv zlwkrxw elg0dvn vsuhdgv
Zh dqdo|}h uvw wkh lqwhuyhqwlrqv rq wkh sruwirolr zklfk frqvlvw lq pdlqwdlqlqj
dozd|v dq rswlpdo pl{1 Vlqfh iru wudfwdelolw| zh dvvxphg rqo| wzr dvvhwv/ wkh
rswlpdo srolflhv hqwdlo nhhslqj wkhlu udwlr
wE| 'g 7 EE||
dw dq rswlpdo ohyho wW 1
Wkhvh srolflhv duh irupdoo| rewdlqhg dvvxplqj wkdw wkh sruwirolr pdqdjhu
pd{lpl}hv wkh orqj0uxq h{shfwhg xwlolw| ri qdo zhdowk ` E|/ zlwk d srzhu xwlolw|
ixqfwlrq ri h{srqhqw R 1 Li zh ghqrwh dv .f h{shfwhg ydoxhv dw wlph }hur dqg
ohw wkh glvfrxqw idfwru b eh fkrvhq vr dv wr nhhs wkh remhfwlyh ixqfwlrq erxqghg5 /
wkh rswlpdo srolflhv duh wkh vwdwlrqdu| vroxwlrqv ri wkh sruwirolr pdqdjhu sureohp
5 Iru wklv frqglwlrq dqg wkh surri ri wkh rswlpdolw| ri wkh srolf| vhh ^8`1
7
tT
` E| f d1h1
*4 i TEbl Hf d` El R *Ro
t E|c7 E|c|MEfcl l <n"
+419,
iru hyhu| srvlwlyh |1
Pruh suhflvho|/ wkh frqvwdqw rswlpdo srolf| fruuhvsrqglqj wr wklv sureohp lv
wkh Phuwrq*v rqh=
>
wW '
+41:,
2
v1w1
E Rj
>
zkhuh  R lv wkh uhodwlyh ulvn dyhuvlrq frh!flhqw1 Iru wkh vdnh ri vlpsolflw|/ zh
dvvxph wW : fc l1h1 >*j2  R1
Rqfh wkh rswlpdo srolf| lv iroorzhg/ wkh g|qdplfv ri ` E| zloo eh jryhuqhg e|
wkh vwrfkdvwlf glhuhqwldo htxdwlrq
W
W
w
w
_` E| ' Eo n >  n wW ` E|_| n j  n wW ` E|_5
dqg wkh orjdulwkplf uhwxuqv rq ` E| duh qrupdo zlwk lqvwdqwdqhrxv phdq >W jW *2 dqg vwdqgdug ghyldwlrq jW/ zkhuh
W
>W 'g o n >  nw wW
W
jW 'g j  nw wW
vr wkdw
W
>W j2 ' o n > E 2R
2 E R j
Lw iroorzv wkdw wkh dssursuldwh phdvxuh ri YdU dw wkh @ ohyho ri frqghqfh/ ghqrwhg
dv T @-W@ EA / lv
#
$
W
s
j
W
W
T @-@EA ' > 2 A n DjW A
+41;,
2
2
2
2
2
2
%
>
>A  2R DsA
' oA jE R j 2 E R
&
Wklv phdvxuh lv g|qdplf/ lq wkh vhqvh wkdw lw wdnhv sruwirolr dgmxvwphqwv lqwr
frqvlghudwlrq1 Lw lv zruwkzkloh wr qrwlfh wkdw lw ghshqgv h{solflwo| rq ulvn dyhuvlrq/
vlqfh wkh rswlpdo srolf|/ zklfk ghwhuplqhv wkh guliw dqg glxvlrq ri frqwuroohg
8
zhdowk/ grhv1 Rssrvlwh wr lw/ wkh vwdwlf YdU phdvxuh/ +417, deryh/ grhv qrw
ghshqg h{solflwo| rq ulvn dyhuvlrq1 Erwk phdvxuhv duh lpsolflwo| dhfwhg e| ulvn
dyhuvlrq wkurxjk wkh fkrlfh ri wkh frqghqfh ohyho @= krzhyhu/ zkloh wkh frqghqfh
ohyho fdq dovr eh lpsrvhg e| uhjxodwruv/ wkh R sdudphwhu hqwhulqj wkh g|qdplf
YdU phdvxuh lv surshu ri wkh sruwirolr pdqdjhu1
4161 G|qdplf YdU phdvxuhv zlwk elg0dvn vsuhdgv
Dv vrrq dv dgmxvwphqw lv frvwo|/ lw lv qr orqjhu udwlrqdo wr uhylvh wkh sruwirolr
frpsrvlwlrq frqwlqxrxvo| lq wlph1 Rq wkh frqwudu|/ lq wkh suhvhqw vhw0xs/ lqwhu0
yhqwlrq wdnhv sodfh rqo| zkhq wkh udwlr ri ulvn| wr ulvnohvv dvvhwv uhdfkhv dq xsshu
ru d orzhu eduulhu1 Wkh hhfw ri elg0dvn vsuhdgv lv wkdw ri lqwurgxflqj wrohudqfh
erxqgv/ zlwklq zklfk sruwirolr pl{hv glhuhqw iurp wkh rswlpdo rqh duh wrohudwhg1
Iru wkh vdnh ri vlpsolflw|/ lq zkdw iroorzv zh zloo frqfhqwudwh rq elg0dvn
vsuhdgv sursruwlrqdo wr wkh ydoxh ri wkh wudqvdfwlrq dffruglqj wr wkh frqvwdqw
r1 Xqghu wklv dvvxpswlrq wkh g|qdplfv ri wkh ulvnohvv dqg ulvn| dvvhw fdq eh
uhzulwwhq dv
_ E| ' o E|_| E n r_uE| n E r_L E|
+41<,
_7 E| ' Eo n >7 E|_| n j7 E|_5 n _uE| _L E|
+4143,
zkhuh u dqg L duh wkh surfhvvhv ri fxpxodwlyh sxufkdvhv dqg vdohv ri wkh ulvn|
dvvhw1
Zh fdq dvvxph6 dv lq wkh suhylrxv vhfwlrq wkdw wkh sruwirolr pdqdjhu
pd{lpl}hv wkh orqj0uxq h{shfwhg xwlolw| ri wkh oltxlgdwlrq ydoxh ri zhdowk/
7̀ E| '
E| n E r7 E|
ru wkdw kh pd{lpl}hv wkh fruuhvsrqglqj h{shfwhg jurzwk udwh +ERl 3 *?Hf ` El R ,/
vlqfh erwk sureohpv duh vdwlvhg e| wkh vdph frqwuro srolf|7 =
gr qrw wudgh zkhqhyhu , w / zkhuh , dqg zklfk duh vxfk wkdw
, wW duh rswlpdoo| fkrvhq dffruglqj wr ^8`/
Wkh fruuhvsrqglqj sruwirolr sureohp kdv ehhq vwxglhg lq ^8` zlwk wkh remhfwlyh ixqfwlrqdo
+419, rq / e| ^:` zlwk wkh remhfwlyh ri pd{lpl}lqj wkh h{shfwhg udwh ri jurzwk ri zhdowk1 D
glhuhqw remhfwlyh ixqfwlrq lv lq ^7`1 Wkh qrq0srzhu xwlolw| fdvh lv h{dplqhg lq ^6`1
7 D srolf| ri wklv w|sh/ zklfk lv ixqgdphqwdo wr rxu glvfxvvlrq ri uhjxodwru| lvvxhv/ lv rswlpdo
dovr zkhq wkh xwlolw| ixqfwlrq lv qrw d srzhu rqh= vhh ^6`1
6
Z
9
wudgh zkhq w ' , ru w ' / vr dv wr nhhs wkh udwlr w dorqj wkh fruuhvsrqglqj
eduulhu1
Lw iroorzv wkdw/ xqghu wkh rswlpdo sruwirolr srolf|
` EA ' 7f i T > j2 *2 A n j5EA n uEA *, L EA *n
n i TEoA f n
]
f
A
i T Eo EA
| dE r _L E| E n K _uE|o
+4144,
Wkh surfhvvhv u dqg L wxuq rxw wr eh wkh orfdo wlphv ri 7 dw wkh ohyhov , dqg uhvshfwlyho|1
Wkh suhvhqfh ri wkh orfdo wlphv u dqg L grhv qrw doorz xv uhvwdwh wkh g|qdplf
YdU iru orjdulwkplf uhwxuqv rq frqwuroohg zhdowk/ ghqrwhg dv T @-S@ EA / lq whupv ri
wkh prphqwv ri wkh xqghuo|lqj vwrfkdvwlf surfhvv= T @-S@ EA fdq rqo| eh rewdlqhg
e| vlpxodwlrq/ iru jlyhq ydoxhv ri wkh sdudphwhuv dqg frqvhtxhqwo| ri wkh
frqwuro eduulhuv1
Dv dq h{dpsoh/ zh {
@ ' bbIc o ' eIc > ' DIc j ' 2fIc `f ' fffc r ' DI
dqg ohw R eh htxdo wr c Dc 21 Iru hdfk vhwwlqj zh uxq dffruglqj wr dq
dojrulwkp lq ^46` 43333 vlpxodwlrqv ri wkh surfhvv w/ uhjxodwhg ehwzhhq wkh
dssursuldwh eduulhuv , dqg +iru R '  , ' fSc ' 51:</ iru R ' D
, ' SDc ' D/ li R ' 2/ , ' e.c ' fD1 Wkh lqlwldo ydoxh ri wc wf / lv vhw wr
lwv rswlpdo ohyho/ zklfk lv wW ' 7S zkhq R ' c wW '  zkhq R ' Dc wW ' f.e
zkhq R ' 2,1 Hdfk vlpxodwlrq uxqv ryhu 4 |hdu/ dqg zh uhjlvwhu lwv g|qdplf YdU
dw wkh krul}rqv A ' f(c c c Sc t / zkhuh ( vwdqgv iru gd|v/  vwdqgv
iru prqwk2v dqg
t
iru |hdu1 Iljxuh 714 uhsruwv wkh vlpxodwlrq uhvxowv= lw fdq eh
R pryhv iurp 04 wr 05/
wW dqg lwv YdUW ghfuhdvh/
W
eduulhuv , dqg duh forvhu wr w 1
qrwlfhg wkdw dv ulvn dyhuvlrq lqfuhdvhv iurp 5 wr 6/ vlqfh
YdUS ghfuhdvhv1 Wkhuh duh wzr uhdvrqv iru wklv= qrw rqo|
dv vkrzq lq wkh suhylrxv vhfwlrq/ exw dovr wkh
Wkh odwwhu idfw pdnhv lqwhuyhqwlrqv pruh iuhtxhqw/ vrphwklqj zklfk lq wxuq nhhsv
ulvn xqghu ehwwhu frqwuro1
Wkurxjk qrq0uhsruwhg vlpxodwlrqv zh kdyh fkhfnhg wkdw wkh suhylrxv uhvxowv
duh vwdeoh zlwk uhvshfw wr fkdqjhv lq vsuhdgv1
dlqvhuw
khuh jxuh 714o
:
51 Frpsdulqj vwdwlf dqg g|qdplf phdvxuhv
Wklv vhfwlrq frpsduhv wkh vwdwlf dqg g|qdplf phdvxuhv rewdlqhg deryh1
5141 Iulfwlrqohvv fdvh
Dv iru wkh ehqfkpdun/ qr elg0dvn fdvh/ wkh wzr phdvxuhv/ vwdwlf +T @-, dqg g|0
qdplf +T @-W ,/ fdq eh frpsduhg dqdo|wlfdoo|1
Ohw xv vwxg| wkh lqhtxdolw|
T @-@W EA T @-@ EA +514,
zkhq wkh lqlwldo sruwirolr pl{ iru wkh vwdwlf fdvh frlqflghv zlwk wkh rswlpdo rqh/
l1h1
wf ' wW 1
Vlqfh wkh vwdwlf YdU fdq eh uh0zulwwhq dv
3
I
2
>no3 j2 A 3j A
E
7f e
T -@ EA ' *? E
C
`
f
4
n f eoA F
F
D
zkloh wkh g|qdplf lv +41;,/ wkh lqhtxdolw| +514, fdq eh h{sdqghg dv
# 7f e
I
>no3 j2 A 3j A
2
I $
E>W 3 2 EjW 2 A 3jW A
e
eE>W3
f
 Ej W 2 A
2
3jW
I
A
eoA
Wklv rqh lv vdwlvhg zkhq ulvn dyhuvlrq lqfuhdvhv8 / vr wkdw wW whqgv wr }hur= wkh
vwdqgdug phdvxuh lv xszdugv eldvhg dqg ryhuhvwlpdwhv wkh g|qdplf rqh1 Lq wxuq/
wklv lv gxh wr wkh idfw wkdw zkhq ulvn dyhuvlrq lqfuhdvhv d frqwuroohg sruwirolr lv
pdgh pxfk vdihu wkdq zlwkrxw frqwurov1
Zkhq ulvn dyhuvlrq ghfuhdvhv vr dv wr gulyh wW wr 04 zh kdyh9 T @-@W EA :
T @-@ EA = wkh vwdqgdug phdvxuh lv grzqzdugv eldvhg/ ru xqghuhvwlpdwhv wkh
g|qdplf rqh1 D grzqzdugv eldv pd| dovr rffxu iru srvlwlyh udwlrv wW / l1h1 iru
erxqghg ulvn dyhuvlrq1 D vdpsoh frpsdulvrq ri YdUv iru wklv fdvh lv vkrzq lq wkh
jxuh ehorz=
ulvn dyhuvlrq
lqfuhdvhv/ vr wkdw whqgv wr }hur/ wkh o1k1v1 ri wkh lqhtxdolw| whqgv
Y
dU[
uW
wr V3 h
dvvhw wr rffxu dw
h / zklfk lv qhjdwlyh/ vlqfh zh dvvxphg d orvv rq wkh ulvn|
wkh jlyhq ohyho ri frqghqfh/ l1h1 Y dU[ ? 31 Wkh u1k1v1 whqgv wr E3 huW huW @ 3= Wkh
8 Zkhqhyhu
frqfoxvlrq iroorzv1
9 Zkhq jrhv wr 04/ dqg frqvhtxhqwo| wr 0 4 dqg 5 wr . 4/ wkh o1k1v1 whqgv wr V hY dU[ /
3
zkloh wkh u1k1v1 jrhv wr 0 E3 huW ? 31 Wkh frqfoxvlrq iroorzv1
;
lqvhuw khuh jxuh 715o
d
Iurp wkh jxuh/ zh dujxh wkdw wkh glhuhqfh ehwzhhq YdUv lqfuhdvhv zlwk
wkh krul}rq A = wkh xqghuhvwlpdwh ri ulvn rewdlqhg wkurxjk d vwdwlf phdvxuh lv
h{dfhuedwhg ryhu orqj krul}rqv/ vlqfh rswlpdo srolflhv shuvlvw ryhu lw> wkh glhuhqfh
lqfuhdvh lv wkh orzhu wkh kljkhu lv uhodwlyh ulvn dyhuvlrq1
5151 Fdvh zlwk iulfwlrqv
Zkhqhyhu frvwv frph lqwr wkh slfwxuh/ wkh frpsdulvrq ehwzhhq vwdwlf dqg g|qdplf
YdU phdvxuhv fdqqrw eh zrunhg rxw dqdo|wlfdoo|/ vlqfh T @- fdqqrw eh frpsxwhg
zlwk d forvhg irupxod1 Lq jxuh 716 krzhyhu zh frpsduh wkh ghqvlwlhv ri wkh qdo
zhdowk/ iru wkh sdudphwhu ydoxhv ri vhfwlrq 416 dqg R ' / xqghu wkh qr frvw dqg
qr dgmxvwphqw dvvxpswlrq dqg xqghu wkh frvw dqg rswlpdo srolflhv dvvxpswlrq1
Wkh uhdghu fdq hdvlo| qrwlfh wkh ryhudoo glhuhqfh ehwzhhq wkhp= lq sduwlfxodu/
wkh| hqwdlo d T @- zlwk qr vsuhdgv dqg qr dgmxvwphqw juhdwhu wkdq T @- / dw d
rqh |hdu krul}rq1
S
S
dlqvhuw
khuh jxuh 716o
Wklv uhvxow fdqqrw eh jhqhudol}hg/ vlqfh sduw ri wkh glhuhqfh ehwzhhq wkhp
lv wkh rqh ehwzhhq T @- dqg T @-W / zklfk fdq eh hlwkhu srvlwlyh ru qhjdwlyh/
ghshqglqj rq ulvn dyhuvlrq1 Rq wkh rqh vlgh/ wkh glhuhqfh lv h{dfhuedwhg e| wkh
suhvhqfh ri wudqvdfwlrq frvwv/ zklfk whqg wr orzhu uhwxuqv dqg wr lqfuhdvh T @zlwk uhvshfw wr T @-W1 Rq wkh rwkhu/ lqwhuyhqwlrqv zlwk frvwv duh ohvv iuhtxhqw
wkdq zlwkrxw= T @- dqg T @- whqg wkhuhiruh wr eh forvhu wkdq T @- dqg T @-W 1
Dq h{whqvlyh qxphulfdo frpsdulvrq lv grqh lq d frpsdqlrq sdshu/ ^<`/ iurp
zklfk zh kdyh rewdlqhg jxuh 7171
S
S
dlqvhuw
khuh jxuh 717o
Wkh pdlq frqfoxvlrq lv wkdw/ dv lq wkh qr0elg0dvn fdvh/ wkh glhuhqfh ehwzhhq
vwdwlf dqg g|qdplf YdUv fdq eh srvlwlyh dv zhoo dv qhjdwlyh= vwdqgdug/ vwdwlf
phdvxuhv pd| ryhuhvwlpdwh dv zhoo dv xqghuhvwlpdwh dfwxdo/ g|qdplf orvvhv1 Lq
wkh elg0dvn fdvh/ wkh zhgjh lv qrw prqrwrqlf zlwk uhvshfw wr wkh krul}rq A ru
wkh glvwdqfh ehwzhhq wkh lqlwldo sruwirolr pl{ dqg wkh rswlpdo rqh1 Lw fdq eh
frqvlghudeoh dovr ryhu vkruw krul}rqv/ vxfk dv rqh gd| +A ' ,/ ru zkhq wkh lqlwldo
sruwirolr pl{ lv forvh ru htxdo wr wkh rswlpdo rqh +wf ' wW ,1
<
Lq vslwh ri wklv xqghwhuplqdf|/ wkh hhfw ri ulvn dyhuvlrq fdq eh fohduo| hy0
lghqfhg= ulvn dyhuvlrq dhfwv qrw rqo| wkh rswlpdo pl{/ exw dovr wkh glvwdqfh
ehwzhhq lw dqg wkh wrohudqfh erxqgv ru wkh qxpehu ri lqwhuyhqwlrqv1 Wkh kljkhu
lv ulvn dyhuvlrq/ wkh forvhu , dqg duh wr wW/ dqg wkh pruh iuhtxhqw duh lqwhuyhq0
wlrqv1 Lq wxuq/ wkh pruh rqh lqwhuyhqhv/ wkh kljkhu lv wkh h{shfwhg eldv ehwzhhq
g|qdplf dqg vwdwlf phdvxuhv1 Lq wklv vhqvh/ wkh kljkhu lv ulvn dyhuvlrq/ wkh pruh
vlplodu wkh vlwxdwlrq lv wr wkh iulfwlrqohvv rqh dqg wkh pruh vwdwlf dqg g|qdplf
phdvxuhv glhu1
Dovr/ lq vslwh ri wkh idfw wkdw T @- fdqqrw eh dqdo|wlfdoo| frpsxwhg dqg wkh
glhuhqfh ehwzhhq T @- dqg T @- fdqqrw eh suhglfwhg/ zh fdq glvfxvv wkh ixooo0
phqw ri uhjxodwru| uhtxluhphqwv zkhq zhdowk lv frqwuroohg yld wrohudqfh erxqgv
rq wkh pl{1
S
S
61 Ixooophqw ri uhjxodwru| uhtxluhphqwv
Lq Hxursh wkh Edvoh Frpplwwhh ri wkh ELV ^5`/ zklfk kdg ehhq hvshfldoo| fuhdwhg
lq rughu wr vxshuylvh qdqfldo dfwlylwlhv/ vxjjhvwhg wkdw edqnv vkrxog vhw dvlgh
uhjxodwru| fdslwdo dprxqwlqj wr d pxowlsoh ri wkh ryhudoo ydoxh dw ulvn1 Wkh odwwhu
lv ghqhg lq whupv ri uhwxuqv/ dw d <<( ohyho ri frqghqfh1 Wkh Edvoh Frpplwwhh
doorzv edqnv wr xvh wkhlu lqwhuqdo ulvn0pdqdjhphqw prghov wr fdofxodwh YdU= d
vwurqj suhihuhqfh iru prghov edvhg rq wkh qrupdolw| dvvxpswlrq iru uhwxuqv lv
fxuuhqwo| ghprqvwudwhg e| prvw lqwhuphgldulhv1
Wkh sxusrvh ri wklv vhfwlrq lv wkdw ri vwxg|lqj zkhwkhu d sruwirolr pdqdjhu
zkr frqwurov klv sruwirolr +vr wkdw wkh dssursuldwh ulvn phdvxuh iru klp zrxog eh
d g|dplf rqh, dovr frpso| zlwk d fdslwdo uhtxluhphqw edvhg rq d vwdwlf phdvxuh1
Dw d vhfrqg vwdjh/ lq vhfwlrq 616/ zh dvn dovr wkh rssrvlwh txhvwlrq= fdq d pdqdjhu
zkr lv reoljhg wr uhvshfw wkh ELV uhtxluhphqwv iroorz dq rswlpdo sruwirolr srolf|B
Lq rughu wr dgguhvv wkhvh txhvwlrqv/ ohw xv uvw irupdol}h wkh ELV uhtxluhphqwv lq
rxu vlpsolhg vhwwlqj1
Rssrvlwh wr T @- EA / wkdw frqvlghuv udwhv ri uhwxuq/ wkh ryhudoo ydoxh dw ulvn
lv ghqhg lq whupv ri uhwxuqv1 Iru wkh bbI ohyho ri frqghqfh/ li zh ghqrwh lw dv
T @- bb EA / lw vdwlvhv
@
h E` EA f
`f T @- bbEA ' ff
Dgrswlqj d vwdwlf phdvxuh/ l1h1 xvlqj wkh k|srwkhvlv lpsolflw lq wkh Edvoh
dssurdfk wkdw wkh sruwirolr pl{ zloo qrw eh fkdqjhg ehiruh wkh krul}rq A / wkh
43
ryhudoo YdU iru rxu vlpsoh sruwirolr ri vwrfnv dqg erqgv fdq eh frpsxwhg dv
T @- bbEA ' 7f di
T E T @-fbb EA o n fEi
TEoA 
zkhuh T @-f bb EA lv dv ghqhg deryh1
Wkh ELV uxohv { d frqvwdqw ri sursruwlrqdolw| e| zklfk wkh YdU phdvxuh kdv
wr eh pxowlsolhg lq rughu wr rewdlq wkh uhjxodwru| fdslwdo/ ghshqglqj rq wkh sdvw
suhglfwlyh shuirupdqfh ri wkh ulvn prgho: 1 Ohw & eh wklv frqvwdqw/ zkhuh & 5
dc n4= ohwwlqj uhjxodwru| fdslwdo frlqflgh zlwk prqh|/ zh fdq wkhq irupdol}h
wkh ELV fdslwdo dghtxdf| uhtxluhphqw dv
f &T @- bb EA +614,
Lq wxuq/ lqwurgxflqj wkh ghqlwlrq
g
g E&c Ac oc T @-f '


n i TEoA 
i T E T @-fbb EA &
wkh uhtxluhphqw fdq eh uh0zulwwhq/ iru rxu prgho/ dv
w f g E&c Ac oc T @-f +615,
Lq vxfk d yhuvlrq/ lw fdoov iru wkh udwlr ri ulvn| wr ulvnohvv dvvhwv dw wlph 3 zkhq
wkh uhtxluhphqw lv lpsrvhg qrw wr h{fhhg d jlyhq wkuhvkrog g / zklfk ghshqgv
rq wkh frqvwdqw & / rq wkh YdU krul}rq A / rq wkh lqwhuhvw udwh o dqg rq wkh ulvn|
dvvhw YdU/ T @-f EA 1
Lq zkdw iroorzv zh zloo vwxg|/ iru d sruwirolr pdqdjhu zkr lv qrw reoljhg wr
uhvshfw wkh fdslwdo uhtxluhphqw/ zkhwkhu wkh odwwhu lv vdwlvhg iru | ' fc l1h1 lq lwv
ruljlqdo yhuvlrq/ +615,/ dqg zkhwkhu lw lv wuxh zlwk suredelolw| rqh diwhu wkdw gdwh/
iru | 5 Efc A o G
wE| g E&c Ac oc T @-f +616,
Lq fdvh lw grhv qrw krog zlwk suredelolw| rqh/ zh zloo dvvhvv wkh suredelolw| ri
ixooophqw/ hf EwE| g 1 Wkh vwxg| iru | : f jlyhv xv d g|qdplf ylhz ri ulvn/
Wkh ELV uxohv dovr glvwlqjxlvk ydulrxv wlhuv dqg frqvlghu dv uhohydqw YdU wkh pd{lpxp
ehwzhhq wkh suhylrxv gd| YdU dqg wkh dyhudjh YdU ryhu wkh odvw vl{w| gd|v1 Wkh uhdghu fdq
wklqn ri rxu prgho dv uhsuhvhqwdwlyh ri wlhu rqh fdslwdo/ l1h1 sursulhwdu| dvvhwv1 Vlqfh lwv uhwxuqv
duh xqfrqglwlrqdoo| qrupdov/ zlwk frqvwdqw sdudphwhuv/ wkh pd{lpxp YdU ryhu 93 gd|v ru wkh
suhylrxv gd| rqh duh qrw h{shfwhg wr glhu vlvwhpdwlfdoo|1 Iru wklv uhdvrq/ zh uxoh wkh dowhuqdwlyh
rxw frqvlghulqj wkh odvw gd| +lq frqwlqxrxv wlph/ wkh wlph 3, YdU rqo|1
:
44
rssrvlwh wr wkh vwdwlf rqh ruljlqdoo| sursrvhg e| wkh Frpplwwhh dqg hpehgghg lq
+615,1 Lw dlpv dw dvvhvvlqj wkh h!flhqf| ri wkh ELV uxohv lq pdlqwdlqlqj d vdihw|
fxvklrq ryhu wlph/ li wkh sruwirolr/ dv dvvxphg/ grhv qrw fkdqjh1
Zh frxog kdyh vwxglhg lqvwhdg zkhwkhu dq lqvwlwxwlrq zklfk lv reoljhg wr uh0
vshfw wkh uhjxodwru| uhtxluhphqwv fdq dovr iroorz rswlpdo sruwirolr srolflhv ri wkh
w|sh ghvfulehg deryh1 Fdq lw pd{lpl}h dq remhfwlyh ixqfwlrq ri wkh w|sh +419,
deryhB Wklv vwxg| pdnhv vhqvh iru | ' f rqo|/ vlqfh dw | : f wkh lqwhuphgldu|
lv reoljhg wr dgdsw wr dqrwkhu fdslwdo uhtxluhphqw +wkh rqh iru A n |,= zh zloo
glvfxvv lw eulh | dw wkh hqg ri wkh iroorzlqj vhfwlrq1
6141 Iulfwlrqohvv fdvh
Dv iru wkh suhvhqw sureohp/ zh vwduw e| frqvlghulqj wkh erughuolqh fdvh lq zklfk wkh
sruwirolr lv frqwlqxrxvo| uh0khgjhg/ vlqfh qr elg0dvn vsuhdgv h{lvw= wkh sruwirolr
pdqdjhu pdlqwdlqv w ' wW 1 Frqvhtxhqwo|/ wkh uhtxluhphqw lv vdwlvhg li dqg rqo|
li wW g / ru;
n i TEoA 
+617,
E >  i T ET @-f bbEA Vlqfh wW lv lqfuhdvlqj zlwk uhodwlyh ulvn wrohudqfh * E R/ zkloh g lv lqgh0

>
Rj 2
&
shqghqw ri ulvn wrohudqfh/ wkh uhjxodwru| uhtxluhphqw lv pruh olnho| wr eh vdwlvhg
wkh orzhu lv ulvn wrohudqfh1 Wkhuh lv wkhq d jrrg glvfulplqdwlqj hhfw ri uhjxod0
wru| uhtxluhphqwv rq qdqfldo lqwhuphgldulhv= rqo| wkh kljko| ulvn0dyhuvh rqhv duh
olnho| wr frpso| zlwk wkh uhjxodwru| uhtxluhphqw1
Dv idu dv wlph A lv frqfhuqhg/ wW grhv qrw ghshqg rq lw/ zkloh g ehkdyhv zlwk
uhvshfw wr lw rssrvlwh wr T @-f bb EA 1 Lw kdv d plqlpxp dw
2c 2 j 2
e Eo n >
j2*22
+618,
Wkh uhjxodwru| uhtxluhphqw lv wkhq wkh ohvv +pruh, olnho| wr eh vdwlvhg wkh ixuwkhu
lv wkh krul}rq/ dv orqj dv A lv vpdoohu +juhdwhu, wkdq +618,1 Wkh iruphu fdvh lv
vkrzq lq jxuh 7181
Sohdvh qrwh wkdw/ dv srlqwhg rxw e| Pduwl Vxeudkpdq|dp/ lq rughu wr vwxg| wkh frqvlvwhqf|
ehwzhhq sruwirolr srolflhv dqg uhjxodwlrq zh fdq dovr lqfoxgh wkh frqvwudlqw N +W , wrjhwkhu
;
zlwk +41</ 4143, dqg wkhlu lqlwldo frqglwlrqv lq sureohp +419,1 Wkh sureohp/ hqulfkhg zlwk
wklv dgglwlrqdo frqvwudlqw/ kdv wkh vdph vroxwlrqv dv deryh l +617, krogv1 Lw kdv qr vroxwlrqv
rwkhuzlvh1
45
lqvhuw khuh jxuh 718o
d
Dv frqfhuqv > dqg j / wW lv lqfuhdvlqj zlwk > dqg ghfuhdvlqj zlwk j / zkloh
T @-f bb EA lv lqfuhdvlqj zlwk j dqg ghfuhdvlqj zlwk >/ vr wkdw g uhdfwv olnh w W1
Dv d frqvhtxhqfh/ zh fdqqrw dvvhvv wkh hhfw rq wkh uhtxluhphqw ixooophqw ri
fkdqjhv lq > dqg j
Wr vxp xs/ lq wkh ehqfkpdun/ qr0vsuhdg fdvh/ uhjxodwru| uhtxluhphqwv ri wkh
ELV w|sh edvhg rq d vwdwlf phdvxuh duh hdvlhu wr vdwlvi| wkh kljkhu lv ulvn
dyhuvlrq1
6151 Fdvh zlwk iulfwlrqv
Zlwk elg0dvn vsuhdgv/ zh qhhg wr glvwlqjxlvk wkuhh vxefdvhv/ dffruglqj wr wkh
uhodwlyh pdjqlwxgh ri g dqg wkh frqwuro eduulhuv1 Lq wzr fdvhv wkh uhtxluhphqw
lv hlwkhu dozd|v ru qhyhu vdwlvhg/ zkloh lq wkh wklug rqh lw fdq eh vdwlvhg zlwk
srvlwlyh suredelolw|/ ohvv wkdq rqh1
Frqvlghu uvw wkh fdvh lq zklfk wkh kljkhvw wrohudwhg udwlr grhv vdwlvi| wkh
uhtxluhphqw/ l1h1 g = wkh odwwhu lv vdwlvhg zlwk suredelolw| rqh iru hyhu|
| 5 dfc A o/ vlqfh lw lv sduw ri wkh rswlpdo srolf|/ w Li/ rq wkh frqwudu|/ wkh orzhvw wrohudwhg udwlr , grhv qrw vdwlvi| wkh uhtxluhphqw/
l1h1 g ,/ wkh odwwhu lv lqfrqvlvwhqw zlwk sruwirolr srolflhv ri wkh eduulhu w|sh iru
hyhu| |1
Wkh lqwhuphgldwh fdvh/ lq zklfk wkh uhtxluhphqw lv qhlwkhu dozd|v qru qhyhu
vdwlvhg/ lv fkdudfwhul}hg e| wkh idfw wkdw wkh yduldeoh g lv ehwzhhq wkh frqwuro
eduulhuv= , g 1 Lq wklv fdvh zh kdyh qr jxdudqwhh wkdw iroorzlqj wkh
rswlpdo sruwirolr srolf| lv frqvlvwhqw zlwk ixooolqj wkh uhtxluhphqw/ xqohvv | ' f
dqg wf ,1
Krzhyhu/ zh fdq frpsxwh/ dw ohdvw qxphulfdoo|/ wkh dfwxdo suredelolw| zlwk
zklfk wkh uhtxluhphqw lv vdwlvhg ru rewdlq dq hvwlpdwh ri lw xvlqj wkh vwdwlrqdu|
ghqvlw| ri wkh udwlr w1
Dv iru wkh iruphu/ zh uxq d qxphulfdo dqdo|vlv/ lq rughu wr dssuhfldwh wkh gh0
shqghqf| ri wkh suredelolw| ri ixooolqj wkh uhtxluhphqw rq wkh prgho sdudphwhuv1
Dv iru wkh odwwhu/ iroorzlqj ^44`/ wkh udqgrp yduldeoh w fdq eh suryhg wr kdyh
wkh vwdwlrqdu| ghqvlw|
2
2 E> j2 *2 w2E>*j 3
R Ew ' 2 2>*j 2 3 2>*j2 3
j E
,
46
+619,
vr wkdw zh fdq zulwh wkh suredelolw| lq wkh fdslwdo uhtxluhphqw dv
2
2
g 2>*j 3 ,2>*j 3
h EwE| g ' 2>*j2 3
f
,2>*j23
Dv dq h{dpsoh/ jxuh 719 uhsruwv wkh ydoxhv ri wkh dfwxdo suredelolw|/ iru & ' dqg glhuhqw gdwhv A = A ' f(c c c Sc t 1 Wkhvh ydoxhv uhvxow iurp wkh
vlpxodwlrqv ri w ghvfulehg lq vhfwlrq 41611 Iru hdfk vlpxodwlrq krzhyhu zh xvh
6 glhuhqw vwduwlqj ydoxhv iru w/ uhvshfwlyho| ehorz dqg deryh wkh rswlpdo rqh1
Wkh jxuh suhvhqwv dovr/ iru hdfk sdudphwhu vhw/ wkh fruuhvsrqglqj ydoxh ri wkh
suredelolw| hvwlpdwh rewdlqhg iurp wkh vwdwlrqdu| ghqvlw|1 Dsduw iurp wkh xsshu
uljkw fruqhu/ wkh jxuh fruuhvsrqgv wr wkh fdvh wW g = wkh uhtxluhphqw zrxog
eh ixooohg lq wkh devhqfh ri iulfwlrqv1
dlqvhuw
khuh jxuh 719o
Iluvw ri doo/ wkh jxuh vkrzv wkdw wkh dfwxdo suredelolw| lqfuhdvhv zlwk ulvn
dyhuvlrq= dv lq wkh qr frvw fdvh/ kljko| ulvn0dyhuvh djhqwv duh pruh olnho| wr ixooo
wkh uhtxluhphqw wkdq plogo| ulvn0dyhuvh rqhv1
Vhfrqg/ wkh suredelolw| ghfuhdvhv/ fhwhulv sdulexv/ zkhq A lqfuhdvhv= wklv lv
gxh wr wkh idfw wkdw zlwk wkh gdwd lq wdeoh 5 g ghfuhdvhv zlwk A 1
Wklug/ lq fdvh wkh dfwxdo suredelolw| lv glhuhqw iurp 4/ wkh vwdwlrqdu| rqh lw
lv qrw dq dffxudwh hvwlpdwh ri lw< 1
Ryhudoo/ wkh suredelolw| ri ixooolqj wkh frqvwudlqw lv forvh wr rqh prvw ri wkh
wlphv43 1 Wklv uhvxow fkdqjhv vxgghqgo| zkhq wkh ordglqj idfwru & lqfuhdvhv/ dv
lv srvvleoh xqghu wkh ELV uxohv1 Iljxuh 71: looxvwudwhv iru lqvwdqfh wkh hhfw ri
lqfuhdvlqj & wr 71
dlqvhuw
khuh jxuh 71:o
Doo wkh suhylrxv ihdwxuhv duh frquphg e| d vwxg| ri wkh lqhtxdolw| gc
zklfk lv frqwdlqhg lq Dsshqgl{ 41Wkh vwxg| dggv wr wkh uhvxowv lq wkh wdeoh wkh
iroorzlqj/ irxuwk srlqw= zkhq wkh yrodwlolw| ri wkh ulvn| dvvhw lqfuhdvhv/ qrw rqo|
wW dqg / exw dovr g ghfuhdvh1 Wkh frqvhtxhqfhv rq wkh uhjxodwru| uhtxluhphqwv
fdqqrw eh suhglfwhg zlwk fhuwdlqw|1 Wkh hfrqrplf lqwxlwlrq iru wklv uhvxow lv wkdw
<
^;`
Wklv uhvxow fdq eh uhodwhg wr wkh h{shfwhg wlph iru uhdfklqj wkh eduulhu/ zklfk lv vwxglhg lq
Zh gr qrw uhsruw wkh uhvxowv iru glhuhqw ydoxhv ri elg0dvn vsuhdgv/ vlqfh wkh odwwhu gr qrw
kdyh d vxevwdqwldo lpsdfw1
43
47
lqfuhdvlqj ulvn lq wkh hfrqrp| lqfuhdvhv wkh vdihw| fdslwdo uhtxluhg 0 dqg wklv lv
reylrxv 0 exw dovr glvwudfwv djhqwv iurp wkh ulvn| dvvhw1 Li wkh h{shfwhg h{fhvv
uhwxuq rq wkh ulvn| dvvhw E> lv xqfkdqjhg/ udwlrqdo sruwirolr pdqdjhuv wrohudwh
ohvv ulvn| dvvhw lq wkhlu sruwirolr + ghfuhdvhv,1 Lq jhqhudo/ lw lv xqnqrzq zkhwkhu
vlqjoh djhqwv dyrlg wkh ulvn| dvvhw hqrxjk wr frsh zlwk wkh uhlqirufhg uhtxluhphqwv
rq YdU1
Wr vxp xs/ lq wkh fdvh zlwk iulfwlrqv wkh delolw| ri wkh uhjxodwru| frqvwudlqw
wr glvfulplqdwh dprqj djhqwv lv frquphg dqg lw lv wkh kljkhu wkh orzhu duh frvwv
+surylghg wkh frqvwudlqw lv vdwlvhg lq wkh fruuhvsrqglqj/ iulfwlrqohvv hfrqrp|,1
Wkh hfwv ri wlph dqg yrodwlolw| duh qrw suhglfwdeoh1
6161 Frpsxovru| uhjxodwru| frqvwudlqwv
Xs wr qrz zh kdyh dqdo|}hg zkhwkhu wkh pdqdjhu zkr iroorzv rswlpdo sruwir0
olr srolflhv ri wkh eduulhu w|sh fdq vdwlvi| dovr fdslwdo uhtxluhphqwv ri wkh Edvoh
Frpplwwhh*v w|sh1 Zkloh grlqj wklv/ zh kdyh lpsolflwo| dvvxphg wkdw kh lv qrw
reoljhg wr frpso| zlwk wkhp1 Lq wkh rssrvlwh fdvh/ wkh qdwxudo txhvwlrq lv= fdq
kh/ dw wkh vdph wlph/ lpsohphqw dq rswlpdo sruwirolr srolf|B
Lq wkh erughuolqh qr frvw fdvh/ wkh dqvzhu lv srvlwlyh li dqg rqo| li wW g 1 Lq
wxuq/ wkh ghshqghqfh ri wW dqg g rq wkh edvlfv ri wkh prgho doorzv xv wr vwdwh
wkdw dq lqvwlwxwlrq zklfk iroorzv wkh fdslwdo uhtxluhphqw lv wkh pruh olnho| wr eh
deoh wr iroorz dovr dq rswlpdo sruwirolr srolf| wkh orzhu lv lwv ulvn wrohudqfh1
Zlwk elg0dvn vsuhdgv/ wkh qdqfldo lqvwlwxwlrq fdq iroorz dq rswlpdo sruwirolr
srolf| ri wkh w|sh , wE| / li
g E&c Ac oc T @-f +61:,
Ehfdxvh ri wkh ghshqghqf| ri g dqg rq wkh prgho sdudphwhuv/ wkh frqglwlrq
g lv wkh pruh olnho| wr eh vdwlvhg wkh vpdoohu lv ulvn wrohudqfh dqg wkh orzhu
duh elg0dvn vsuhdgv/ surylghg lq wkh odwwhu fdvh wkdw wW g 1
71 Vxppdu| dqg frqfoxvlrqv
Rxu lqwhuhvw lq wklv sdshu zdv wzrirog= uvw/ zh zdqwhg wr surylgh g|qdplf YdU
phdvxuhv/ zklfk wdnh lqwr dffrxqw fkdqjhv lq wkh sruwirolr pl{/ dv rssrvlwh wr
wkh xvxdo/ vwdwlf rqhv/ zklfk ljqruh wkhp1 Wklv lq rughu wr vwxg| wkh eldv ri
vwdqgdug zlwk uhvshfw wr dfwxdo ulvn phdvxuhv1 Vhfrqgo|/ zh zdqwhg wr vwxg| wkh
hhfwlyhqhvv ri uhjxodwru| uhtxluhphqwv vxfk dv wkh ELV rqhv/ zklfk duh edvhg rq
48
vwdwlf phdvxuhv/ zkhqhyhu wkh sruwirolr pl{ fkdqjhv ryhu wlph1 Lq wxuq/ fkdqjhv lq
wkh sruwirolr pl{ rffxu zkhqhyhu wkh YdU krul}rq/ dv lpsrvhg e| wkh Uhjxodwru|
Dxwkrulw|/ grhv qrw frlqflgh zlwk wkh xqzlqglqj shulrg uhvxowlqj iurp sruwirolr
frqwuro srolflhv1
Zh kdyh frqgxfwhg rxu vwxg| lq dq kljko| vw|ol}hg iudphzrun/ zlwk wzr dvvhwv
+prqh|/ ru d ulvnohvv erqg/ dqg d sruwirolr ri vwrfnv,1 Zh kdyh glvwlqjxlvkhg
wkh fdvh lq zklfk qr iulfwlrqv uhsuhvhqwhg e| elg0dvn vsuhdgv h{lvw iurp wkh
rssrvlwh rqh/ frqvlghulqj wkh uvw dv d ehqfkpdun/ exw vfdufho| uhdolvwlf/ vlwxdwlrq1
Iluvw/ zh kdyh frpsduhg wkh vwdwlf phdvxuhv zlwk wkh g|qdplf rqhv zlwkrxw
dqg zlwk iulfwlrqv1 Lq wkh iruphu fdvh vwdqgdug phdvxuhv whqg wr eh ohvv +pruh,
frqvhuydwlyh wkdq wkh g|qdplf rqhv/ l1h1 wkh| xqghu+ryhu,hvwlpdwh ulvn/ zkhq
ulvn dyhuvlrq lv orz +kljk,1 Lq wkh suhvhqfh ri iulfwlrqv/ wkh kljkhu lv ulvn dyhuvlrq/
doo rwkhuv htxdo/ wkh pruh vlplodu wkh vlwxdwlrq lv wr wkh iulfwlrqohvv fdvh1
Vhfrqgo|/ zh kdyh vwxglhg wkh hhfwlyhqhvv ri fdslwdo dghtxdf| uxohv/ edvhg
rq vwdwlf YdU phdvxuhv/ lq prqlwrulqj wkh ulvn ri frqwuroohg sruwirolrv1 Zh kdyh
frqfoxghg wkdw frqvlvwhqf| ehwzhhq fdslwdo uhtxluhphqwv dqg hhfwlyh ulvnlqhvv
lv wkh pruh olnho| wkh kljkhu lv wkh sruwirolr pdqdjhu ulvn dwwlwxgh dqg wkh orzhu
duh elg0dvn vsuhdgv +lq wkh odwwhu fdvh/ wkh frqfoxvlrq krogv li dqg rqo| li wkhuh lv
frqvlvwhqf| lq wkh fruuhvsrqglqj iulfwlrqohvv hfrqrp|,1 Wkh hhfwv ri lqfuhdvlqj
wkh YdU krul}rq ru wkh sruwirolr yrodwlolw| duh uhvshfwlyho| qrw xqlirup dqg qrw
suhglfwdeoh1
Wkh pdlq srolf| frqfoxvlrq iurp rxu vlpsoh prgho lv wkhq wkdw/ hyhq wrxjk
fdslwdo uhfrpphqgdwlrqv duh edvhg rq vwdwlf YdU phdvxuhv/ lqvwhdg ri g|qdplf
rqhv/ wkh| lqgxfh d srvlwlyh vhohfwlrq ri qdqfldo djhqwv/ wrzdugv wkh prvw ulvn
dyhuvh1 Wklv hhfw lv wkh vwurqjhu wkh orzhu duh pdunhw iulfwlrqv1
Dsshqgl{ 4
Wklv Dsshqgl{ uhsruwv wkh uhvxowv ri wkh h{dp ri wkh lqhtxdolw| g / wdnlqj
lqwr frqvlghudwlrq wkh ghshqghqfh ri rq wkh prgho sdudphwhuv/ vlqfh vroyhv
wkh rswlpdolw| sureohp lq ^8`1
Iluvw/ vlqfh wkh kljkhu lv ulvn dyhuvlrq/ wkh vpdoohu duh wW dqg / zkloh g grhv
qrw ghshqg rq ulvn dyhuvlrq/ kljko| ulvn dyhuvh sruwirolr pdqdjhuv duh pruh olnho|
wr ixooo wkh uhjxodwru| uhtxluhphqw 0 ehfdxvh wkh| kdyh g 0 wkdq plogo|
ulvn dyhuvh rqhv1 Iurp d uhjxodwru| srlqw ri ylhz/ wklv lv h{wuhpho| lpsruwdqw/
vlqfh lw phdqv wkdw xqghu pdunhw iulfwlrqv wkh uhtxluhphqw lv hhfwlyh lq vhohfwlqj
dprqj lqwhuphgldulhv wkh jrrg/ ru ohdvw ulvn wrohudqw/ rqhv1 Wkh uhvxow lv pdgh
hyhq vwurqjhu e| wkh idfw wkdw/ dv qrwhg deryh/ dovr lq wkh iulfwlrqohvv hfrqrp|
kljko| ulvn dyhuvh djhqwv duh h{shfwhg wr vdwlvi| wkh uhtxluhphqw1
49
Vhfrqg/ vlqfh wkh vpdoohu duh wkh elg0dvn vsuhdgv/ wkh forvhu lv wr w W/ zkloh
wW lwvhoi dqg g gr qrw ghshqg rq elg0dvn vsuhdgv/ doo rwkhuv htxdo lw lv pruh olnho|
wkdw wkh uhjxodwru| frqvwudlqw eh vdwlvhg lq pdunhwv zlwk orz elg0dvn vsuhdgv
wkdq ylfh yhuvd1
Wklug/ vlqfh g suhvhqwv d plqlpxp zlwk uhvshfw wr A / zkloh , dqg gr qrw
ghshqg rq lw/ li A lv juhdwhu wkdq wkh dujxphqw ri wkh plqlpxp/ wkh ixuwkhu lv A /
wkh pruh olnho| lv wkh ixooophqw ri wkh uhtxluhphqw1 Wkh rssrvlwh kdsshqv li A lv
vpdoohu wkdq wkh dujxphqw ri wkh plqlpxp1 Wklv hqoljkwv wkh lpsruwdqfh ri wkh
krul}rq dqg wkh idfw wkdw/ vlqfh wkh ghshqghqfh ri dfwxdo/ g|qdplf phdvxuhv rq
A fdq dovr eh qrq0prqrwrqlf/ lqwxlwlrqv rq wkh wlph hhfw/ edvhg rq wkh xvxdo/
vwdwlf phdvxuh/ fdq eh plvohdglqj1
Uhihuhqfhv
^4` Edvoh Frpplwwhh rq Edqnlqj Vxshuylvlrq/ 4<<9/ ihi Lu |i
phqw wr wkh Fdslwdo Dffrug wr Lqfrusrudwh Pdunhw Ulvnv/
Dphqg0
Edvoh/ Mdqxdu|1
^5` Edvoh Frpplwwhh rq Edqnlqj Vxshuylvlrq/ / 4<<9/ Dphqgphqw wr wkh Fdslwdo
Dffrug wr Lqfrusrudwh Pdunhw Ulvnv/ Edvoh/ Mdqxdu|1
^6` Fylwdqlf/ M1 dqg L1 Ndudw}dv/ 4<<9/ Khgjlqj dqg Sruwirolr Rswlpl}dwlrq xqghu
elg0dvn vsuhdgv= d Pduwlqjdoh Dssurdfk/ Pdwkhpdwlfdo Ilqdqfh/ 9/ ss1 4960
981
^7` Gdylv/ P1K1D1/ dqg D1 Qrupdq/ 4<<3/ Sruwirolr Vhohfwlrq Xqghu elg0dvn
vsuhdgv/ Pdwkhpdwlfv ri Rshudwlrqv Uhvhdufk/ 48/ ss1 9:90:461
^8` Gxpdv/ E1 dqg H1 Oxfldqr/ 4<<4/ Dq H{dfw Vroxwlrq wr d G|qdplf Sruwirolr
Fkrlfh Sureohp xqghu elg0dvn vsuhdgv/ Mrxuqdo ri Ilqdqfh/ 79/ ss1 8::0<81
^9` Hpeuhfkwv/ S1/ PfQhlo/ D1/ Vwudxpdqq/ G1/ 4<<</ Fruuhodwlrq dqg Ghshq0
ghqfh lq Ulvn Pdqdjhphqw= Surshuwlhv dqg Slwidoov/ Zrunlqj Sdshu/ Hwk}1
^:` Iohplqj/ Z1K1/ Jurvvpdq/ V1 J1/ Ylod/ M1O1 dqg W1 ]dulskrsrxorx/ 4<<3/
Rswlpdo Sruwirolr Uhedodqflqj zlwk elg0dvn vsuhdgv/ Zrunlqj Sdshu/ Eurzq
Xqlyhuvlw|1
^;` Ixvdl/ J1/ 4<<;/ Jhqhudol}dwlrqv ri wkh duf0vlqh odz/ Zrunlqj Sdshu/ Xqlyhu0
vlw| ri Ioruhqfh1
4:
^<` Ixvdl/ J1/ Oxfldqr/ H1/ 4<<</ Phdvxulqj YdU xqghu Rswlpdo dqg Vxerswl0
pdo Sruwirolr Srolflhv/ Surfhhglqjv [[Y Hxur Zrunlqj Jurxs rq Ilqdqfldo
Prghoolqj/ vxeplwwhg1
^43` Jurxs ri Wkluw| +Joredo Ghulydwlyhv Vwxg| Jurxs,/ 4<<6/ Ghulydwlyhv=
wlfh dqg Sulqflsoh v/ Mxo|1
^44` Kduulvrq/ M1 P1/ 4<;8/ Eurzqldq
\run= Mrkq Zloh|1
Sudf0
Prwlrq dqg Vwrfkdvwlf Iorz V|vwhpv/
Qhz
^45` Ndvw/ U1/ Oxfldqr/ H1/ Shffdwl/ O1/ 4<<</ YdU dv d ghflvlrq fulwhulrq/ Zrunlqj
Sdshu :2<</ LFHU/ Wxulq/ vxeplwwhg1
^46` Ohslqjoh/ G1/ 4<<8/ Hxohu vfkhph iru uh hfwhg vwrfkdvwlf glhuhqwldo htxd0
wlrqv/ cPdwkhpdwlfv dqg Frpsxwhuv lq Vlpxodwlrqv/ 6;/ ss1 44<0591
4;
Iljxuh 714= YdU xqghu rswlpdo srolflhv zlwk elg0dvn vsuhdgv1
4<
Iljxuh 715= Frpsdulvrq ehwzhhq YdU dqg YdU- iru glhuhqw ohyhov ri ulvn dyhuvlrq
+wk@w,
53
Iljxuh 716= Ghqvlw| ri wkh qdo zhdowk xqghu rswlpdo dqg vxerswlpdo +qr lqwhu0
yhqwlrq, srolflhv/ v@8(
54
Iljxuh 717= Frpsdulvrq ehwzhhq YdU dqg YdU-/ iru glhuhqw s ydoxhv/ lqlwldo
sruwirolrv +vi@vdihu/ rs@rswlpdo/ u@ulvnlhu,1 Wkh vdihu dqg ulvnlhu sruwirolrv kdyh
wkh lqlwldo frpsrvlwlrq w uhvshfwlyho| 40t dqg 4.t wlphv wkh rswlpdo rqh1
55
Iljxuh 718= Uhjxodwru| uhtxluhphqw ixooophqw +wk-N, zlwk qr elg0dvn vsuhdgv
56
Iljxuh 719= Suredelolw| ri ixooolqj wkh ELV frqvwudlqw zlwk elg0dvn vsuhdgv/ n@6
57
Iljxuh 71:= Suredelolw| ri ixooolqj wkh ELV frqvwudlqw zlwk elg0dvn vsuhdgv/ n@7
58