On the Informational Content of Changing Risk for Dynamic Asset Allocation

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40, Bd. du Pont d’Arve PO Box, 1211 Geneva 4 Switzerland Tel (++4122) 312 09 61 Fax (++4122) 312 10 26 http: //www.fame.ch E-mail: [email protected]

On the Informational

Content of Changing Risk

Giovanni BARONE-ADESI

Università della Svizzera Italiana, Lugano

Patrick GAGLIARDINI

Università della Svizzera Italiana, Lugano

Research Paper N° 23

FAME - International Center for Financial Asset Management and Engineering

March 2000

for Dynamic Asset

Allocation

Fabio TROJANI

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(9)

krzhyhu fodlplqj wkh wuxh xqghuo|lqj uhodwlrq ehwzhhq h{shfwhg uhwxuqv dqg yrodwlolw| wr eh hlwkhu ri wkh irup +6, ru +7, +fi1 djdlq wkh looxvwudwlrq suhvhqwhg lq Iljxuh 4,1

Iru dq| jlyhq ohyho ri frqglwlrqdo yrodwlolw| wkh rswlpdo doorfdwlrq ri d frqgl0 wlrqdo phdq yduldqfh rswlpl}hu zloo wkhq olh ehwzhhq wkh rswlpdo phdq yduldqfh doorfdwlrqv lpsolhg e| wkh k|srwkhvhv +6, dqg +7,/ zklfk duh/ uhvshfwlyho| +xvlqj wkh rswlpdolw| frqglwlrq +4,,= z4 w @4_{Y du +u} 4 w uiwmLw4, > +8, dqg z3 w @ 4s 3 Y du +uw uiwmLw4, = +9,

Lq vxfk d vlwxdwlrq d pdujlq wr ehdw ex| dqg krog vwudwhjlhv pd| wkhruhwlfdoo| h{lvw/ vlqfh zh duh lqyhvwlqj ohvv lq wkh ulvn| dvvhw zkhq wkh uhzdug wr yduldqfh udwlr lv orz/ uhvshfwlyho| pruh zkhq lw lv kljk1 Krzhyhu/ wklv jdlq pd| eh sduwldoo| rvhw e| wkh orvv ri wlph glyhuvlfdwlrq1 Lq wkh qh{w wzr vhfwlrqv wkhvh wzr edvlf k|srwkhvhv duh frpelqhg lq d Ed|hvldq iudphzrun dqg wkh lpsolhg rswlpdo wudglqj vwudwhj| ri d Ed|hvldq frqglwlrqdo phdq yduldqfh rswlpl}hu lv hydoxdwhg lq whupv ri qdqfldo ulvn dgmxvwhg xqfrqglwlrqdo shuirupdqfh rq uhdo gdwd1

515 D Vlpsoh Ed|hvldq Vwudwhj|

Zkloh wkh vshflfdwlrqv +6, dqg +7, duh hfrqrplfdoo| fohduo| lqwhusuhwdeoh k|0 srwkhvhv wkdw fdq eh hdfk hdvlo| lpsohphqwhg lq vwdqgdug hfrqrphwulf prghov/ wkh| duh gl!fxow wr glvhqwdqjoh xvlqj gdwd vhwv frqvlvwlqj ri d uhdolvwlf qxpehu ri gdlo| revhuydwlrqv1 Dv revhuyhg/ wkh hoxvlyh hpslulfdo uhodwlrq ehwzhhq h{0 fhvv uhwxuqv dqg yrodwlolwlhv pdnhv d suhflvh hvwlpdwlrq ri frqglwlrqdo h{shfwhg uhwxuqv d gl!fxow wdvn lq dssolfdwlrqv1 Dv d frqvhtxhqfh/ dq| vwdwlvwlfdo whvw0 lqj surfhgxuh wu|lqj wr dffhsw rqh ri wkhvh wzr frpshwlqj prghov ru d ixuwkhu k|srwkhvlv qhdu wr +6, ru +7, zloo kdyh d srru srzhu1

(10)

Edvhg rq wklv prwlydwlrq/ zh wuhdw +6, dqg +7, dv wzr olplw0fdvh frpshwlqj prghov wkdw fdqqrw eh uhdoo| zhoo lghqwlhg/ edvhg rq uhdolvwlf gdwd vhwv ri revhuydwlrqv/ dqg dvvljqh wr hdfk ri wkhvh wzr frpshwlqj prghov dq d sulrul suredelolw| ghwhuplqhg dw wlph 3/ lq d yhu| Ed|hvldq vslulw> fi1 dovr Mrulrq ^47`8₁

Zh frqvlghu sdudphwulf prghov iru wkh g|qdplfv ri uwuiwzkhuh frqglwlrqdo

phdqv dqg yduldqfhv fdq eh sdudphwhul}hg e| vrph fruuhvsrqglqj ixqfwlrqv pw

dqg kw hydoxdwhg dw vrph xqnqrzq sdudphwhu yhfwru & @ +> ,3=

pw+&, @ H+uw uiwmLw4, > kw+, @ Y du+uw uiwmLw4, =

Iroorzlqj wkh glvfxvvlrq lq wkh odvw vhfwlrq/ zh frqvlghu g|qdplf lqyhvwphqw srolflhv xqghu wkh wzr frpshwlqj k|srwkhvhv +6, dqg +7, rq pw=

K3 = p3w@ 3skw > kw@ kw+, > +:,

K4 = p4w@ 4 > kw@ kw+, > +;,

dqg dvvxph wkh| wr krog dw wlph }hur zlwk d jlyhq d sulrul suredelolw| dqg +4,/ uhvshfwlyho|9_{1 Dv rx wolqhg/ K}

4lv frqvlvwhqw zlwk lqyhvwruv ljqrulqj vkruw

whup xfwxdwlrqv lq yrodwlolw|/ srvvleo| ehfdxvh ri wudqvdfwlrq frvwv dqg prgho plvvshflfdwlrqv pdnlqj g|qdplf srolflhv lqhhfwlyh/ zkloh K3 fruuhvsrqgv wr

d vlwxdwlrq zkhuh lqyhvwruv duh uhzdughg iru wdnlqj ulvn e| h{fhvv uhwxuqv wkdw duh sursruwlrqdo wr wkh ulvn ehlqj wdnhq1

Rq wkh rwkhu kdqg/ ghqlqj wkh frqglwlrqdo huuru=

%w+&, @ uw uiw pw+&, > +<,

zh prgho frqglwlrqdo yduldqfhv zlwk dq dv|pphwulf JDUFK+4/4, surfhvv/ fi1 djdlq Jorvwhq/ Mdjdqqdwkdq dqg Uxqnoh ^45`=

kw@ 4. 55w4. 6kw4. 7Liw43j5w4 > +43,

8_{Lq d uhodwhg sdshu Eduehulv +^5`, vkrzv ukdw sdudphwhu xqfhuwdlqw| fdq surgxfh dq lp0}

sruwdqw lpsdfw rq wkh sruwirolr doorfdwlrq/ ghshqglqj rq wkh lqyhvwru*v wlph krul}rq1

(11)

zkhuh Liw43jlv wkh lqglfdwru ixqfwlrq ri wkh hyhqw iw4 3j1

D Ed|hvldq phdq yduldqfh rswlpl}hu pd{lpl}hv h{shfwhg txdgudwlf xwlolw| dw wlph w/ edvhg rq suhglfwlyh frqglwlrqdo ghqvlwlhv ri uw uiw> vhh djdlq Mrulrq

^47` dv d edvlf uhihuhqfh1

Zh dvvxph rxu pdqdjhu kdv dw khu glvsrvdo d vhw \ @ +uw uiw,w@Q>==>4

ri revhuydwlrqv wkdw fdq eh xvhg dw wlph 3 wr hvwlpdwh wkh surfhvv iru h{fhvv pdunhw uhwxuqv xqghu K3 dqg K4/ uhvshfwlyho|1 Edvhg rq wkh uhdol}dwlrq \ /

wkh fruuhvsrqglqj d srvwhulrul suredelolwlhv iru wkhvh wzr k|srwkhvhv fdq eh frpsxwhg1

Wkh rswlpl}dwlrq sureohp ri rxu Ed|hvldq rswlpl}hu lv= pd{_z w zwH+uw uiwmLw4, ₅ z5wY du+uw uiwmLw4, > +44, zkhuh H_+mL w4, dqg Y du+mLw4, ghqrwh frqglwlrqdo h{shfwdwlrqv dqg ydul0

dqfhv edvhg rq wkh suhglfwlyh ghqvlwlhv lpsolhg e| Ed|hv irupxod/ jlyhq wkh d sulrul suredelolw| 1 Vwudljkwiruzdug frpsxwdwlrqv jlyh:₌

H_+u w uiwmLw4, @ p3w. +4 ,p4w > Y du+uw uiwmLw4, @ kw > zkhuh _@ oK3+\ , oK3+\ , . +4 ,oK4+\ , +45, lv wkh srvwhulru suredelolw| ri K3dqg o+\ , lv wkh olnholkrrg ri \ dw wlph 3 xqghu

wkh k|srwkhvlv K=1 Wkh vroxwlrq wr wkh rswlpl}dwlrq sureohp +44, lv;₌ zw@4 _p3 w. +4 ,p4w kw = +46,

Dv lqwxlwlyho| reylrxv/ wkh dvvxpswlrq ri dq lghqwlfdo yrodwlolw| surfhvv xqghu K3 dqg K4 lpsolhv dq lghqwlfdo d sulrul dqg d srvwhulrul frqglwlrqdo yrodwlo0

lw| surfhvv1 Dv d frqvhtxhqfh/ wklv dvvxpswlrq prghov d vlwxdwlrq zhuh rxu

:_{Frpsduhg wr d ixoo Ed|hvldq dssurdfk zh qhjohfw hvwlpdwlrq huuruv wkdw fdq dulvh zlwklq}

rqh ri wkh wzr frpshwlqj prgho fkrlfhv1

;_{Qrwlfh wkdw dv d sduwlfxodu fdvh rqh jhwv wkh vwdqgdug rswlpdo uxoh ri dq lqyhvwru ixoo|}

(12)

pdqdjhu lv sulpdulo| frqfhuqhg zlwk hvwlpdwlrq huuruv ghulylqj iurp wkh plv0 vshflfdwlrqv ri wkh frqglwlrqdo phdq htxdwlrq dqg ohvv zlwk wkrvh fdxvhg e| yrodwlolw| hvwlpdwhv/ zklfk duh e| vrph rughuv ri pdjqlwxgh ohvv lq xhqwldo wkdq frqglwlrqdo phdqv hvwlpdwhv<₁

Dvvxplqj frqglwlrqdoo| qrupdo huuruv k45

w %wy l=l=g Q +3> 4, >

zh fdq hdvlo| dssur{lpdwh wkh rswlpdo zhljkwv lq +45, e| frpsxwlqj wkh rgg udwlr 4 lpsolhg e| rxu frpshwlqj k|srwkhvhv= 4 @ 4 oK3+\ , oK4+\ , @ 4 e3 e4 Q > +47, zkhuh e lv wkh hvwlpdwhg vwdqgdug ghyldwlrq ri uhvlgxdov xqghu K lq wkh ROV

uhjuhvvlrqv=

uw uiw@ pw. xw > xwy Q +3> kw, = +48,

Wkh qh{w vhfwlrq whvwv rxu Ed|hvldq phwkrgrorj| rq uhdo gdwd xvlqj h{fhvv uhwxuq vhulhv iru irxu pdlq vwrfn lqglfhv1

516 Hpslulfdo Hylghqfh

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<_{Lq d vhqvh wklv dvvxpswlrq lv dovr olqnhg wr wkh frpprq sudfwlfh lq wkh olwhudwxuh ri}

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(13)

Lq nhhslqj zlwk wkh frpprq sudfwlfh lq wkh olwhudwxuh zh hvwlpdwh prgho +:, dqg +43,/ uhvshfwlyho| +;, dqg +43, +rxu wzr edvlf k|srwkhvhv, xvlqj d wzr vwhs surfhgxuh1 Lq wkh uvw vwhs wkh dv|pphwulf JDUFK+4/4, prgho +43, lv wwhg ljqrulqj wkh frqwulexwlrq ri h{shfwhg h{fhvv uhwxuqv lq ghqlqj wkh +%w,w@3>==>W

surfhvv lq +<,1 Frqglwlrqdoo| rq wkhvh yrodwlolw| hvwlpdwhv43 _{zh wkhq uhjuhvv}

lqgh{ h{fhvv uhwxuqv rq d frqvwdqw +xqghu K4,/ ru rq wkh hvwlpdwhg JDUFK

yrodwlolwlhv +xqghu K3,1

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Lqvhuw Wdeoh 4 derxw khuh

Wkh gdlo| vhulhv ri uhwxuqv dqg yrodwlolw| hvwlpdwhv lpsolhg e| wkhvh JDUFK+4/4, sdudphwhu hvwlpdwhv duh wkhq dqdo|}hg lq rughu wr lqyhvwljdwh wkh olqndjh eh0 wzhhq yrodwlolwlhv dqg h{shfwhg uhwxuqv1 D vfdwwhu sorw looxvwudwlqj wkh hpslulfdo irup ri wkh uhodwlrq revhuyhg iru wkh Grz ryhu wkh shulrg Mdqxdu| 4<<7 wr Mdqxdu| 4<<< zdv douhdg| suhvhqwhg lq Iljxuh 41 Dv qrwhg/ wkh suhglfwdeoh frpsrqhqw ri uhwxuqv hphujlqj iurp wklv suholplqdu| dqdo|vlv lv vpdoo1 Lqghhg/ li zh uhjuhvv h{fhvv uhwxuqv rq wkh vhulhv ri yrodwlolw| hvwlpdwhv wkhuh lv vrph whqghqf| wr qg srvlwlyh vorshv/ krzhyhu d qrw vljqlfdwlyh rqh44₁

Wklv zhdn hylghqfh ri frpshqvdwlrq iru yrodwlolw| ulvn lv frqvlvwhqw zlwk wkh h{lvwhqfh ri wudglqj srolflhv surgxflqj d kljkhu ulvn dgmxvwhg shuirupdqfh wkdq ex| dqg krog vwudwhjlhv1 Krzhyhu/ dq dfwlyh yrodwlolw| srolf| ri wkh irup +4, vwloo qhhgv d vshflfdwlrq ri frqglwlrqdo phdqv1 Wkh hoxvlyh uhodwlrq ehwzhhq wkh odw0 whu dqg yrodwlolwlhv vxjjhvwv xv wr frqvlghu vlpsoh vshflfdwlrqv iru zklfk d fohdu hfrqrplf lqwxlwlrq h{lvwv1 Lq wkh iroorzlqj wkh wzr +hpslulfdo, olplw0fdvh ehqfk0

43_{Zh dovr xvhg d JDUFK lq phdq vshflfdwlrq wr hvwlpdwh doo sdudphwhuv ri lqwhuhvw vl0}

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(14)

pdunv ghqhg e| wkh k|srwkhvhv K3dqg K4lq +:, dqg +;, duh frqvlghuhg1 Vlqfh

zh fdqqrw h{shfw rx u gdwd wr surylgh d fohdu vwdwlvwlfdo hylghqfh lq idyru ri rqh ri wkh wzr k|srwkhvhv wkh| duh frpelqhg lq wkh ed|hvldq iudphzrun +46, xvlqj wkh uhodwlyh olnholkrrgv lq +47, rewdlqhg e| uxqqlqj wkh ROV uhjuhvvlrqv45_+48,1

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d srvwhulrul zhljkwv +45, iru shulrg L dqg LL duh vxppdul}hg lq Wdeoh 5 ehorz1 Lqvhuw Wdeoh 5 derxw khuh

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hvwlpdwhg h{shfwhg uhwxuq rffxuv iru wkh IWVH lq shulrg L xqghu K3zkloh wkh

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0 dv h{shfwhg 0 odujho| lqfrqfoxvlyh1 Ilqdoo|/ revhuyhg uhodwlyh olnholkrrgv duh uhodwlyho| vwdeoh ryhu wkh wzr vxevdpsohv1

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|lhogv d qhjdwlyh hvwlpdwhg lqwhufhsw iru doo lqglfhv dqdo|}hg zlwk h{hswlrq ri wkh V)S 833 erwk iru shulrg L dqg LL/ lpso|lqj yrodwlolw| udqjhv zkhuh h{shfwhg h{fhvv uhwxuqv duh qhjdwlyh1 Pruhryhu/ wkh qxphulfdo ydoxhv ri wkhvh hvwlpdwhg lqwhufhswv duh e| vrph rughuv ri pdjqlwxgh xquhdolvwlf1

(15)

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Lq Wdeoh 6 wkh ed|hvldq vwudwhj| shuirupv ehwwhu48 _{wkdq ex| dqg krog erwk lq}

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48_{Uhodwlyh wr wkh shuirupdqfh phdvxuhv xvhg1}

49_{Zkhq ghyhorslqj wkh vdph shuirupdqfh dqdo|vlv zlwk ghjhqhudwh sulruv Z ' fc / qr}

(16)

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Wkh uhvxowv lq Wdeoh 7 lqglfdwh wkdw wkh Ed|hvldq vwudwhj| grhv qrw jhqhudoo| lq0 fuhdvh wkh wklug dqg irxuwk prphqw ri wkh xqfrqglwlrqdo glvwulexwlrq ri uhwxuqv/ frpsduhg wr ex| dqg krog1 Dv d frqvhtxhqfh/ wkh vxshulru xqfrqglwlrqdo phdq yduldqfh shuirupdqfh suhvhqwhg lq Wdeoh 6 vkrx og qrw frph dw wkh h{shqvh ri rwkhu ulvn glphqvlrqv1

(17)

6 Dq Lqwhuwhpsrudo Vwrfkdvwlf Yrodwlolw| Phu0

wrq*v Prgho

Lq wklv vhfwlrq zh suhvhqw dq lqwhuwhpsrudo vwrfkdvwlf yrodwlolw| Phuwrq*v ^4<` prgho/ edvhg rq d frqwlqxrxv wlph olplw ri dq h{srqhqwldo P0JDUFK vshfl0 fdwlrq ri frqglwlrqdo h{shfwhg uhwxuqv dqg yrodwlolwlhv1 Wkh prgho doorzv iru dq h{solflw vroxwlrq ri wkh lpsolhg rswlpdo sruwirolr uxohv dqg wkh fruuhvsrqg0 lqj ydoxh ixqfwlrq/ dqg lv ulfk hqrxjk wr wdnh lqwr dffrxqw wkh pdlq hpslulfdo ihdwxuhv ri wkh uhodwlrq ehwzhhq fkdqjlqj h{shfwhg uhwxuqv dqg fkdqjlqj ulvn/ dv iru lqvwdqfh dv|pphwulhv1

Wkh h{solflw h{suhvvlrq rewdlqhg iru wkh ydoxh ixqfwlrq ri dq djhqw pd{l0 pl}lqj wkh h{shfwhg lqgluhfw srzhu xwlolw| ri hqg ri shulrg zhdowk lv xvhg wr frpsxwh wkh fhuwdlqw| htxlydohqw ri wkh g|qdplf sruwirolr uxohv xqghu lqyhvwljd0 wlrq1 Wklv |lhogv d shuirupdqfh frpsdulvrq ri g|qdplf vwudwhjlhv wkdw lv fohduo| ghqhg exw dovr/ e| frqvwuxfwlrq/ prgho ghshqghqw1

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4;_{Zkloh zh irfxv lq wklv sduw rq txdqwli|lqj wkh pdujlqv ri lpsuryhphqw ryhu ex| dqg krog/}

d wkhruhwlfdoo| lpsruwdqw dvshfw ri wkh prgho vroxwlrq lv wkh frpsohwh fkdudfwhul}dwlrq ri wkh ruljlq dqg wkh uhohydqfh ri lqwhuwhpsrudo khgjlqj lq d vwrfkdvwlf yrodwlolw| iudphzrun1

(18)

614 Wkh Edvlf Prgho

Ohw xv vwduw e| vshfli|lqj d glvfuhwh wlph h{srqhqwldo P0JDUFK surfhvv +fi1 Qhovrq ^54`, iru wkh mrlqw g|qdplfv ri uhwxuqv dqg yrodwlolwlhv4<₌

uw@ +w, 4₅5w . w}w > +49, oq 5 w @ $ . oq 5w4. &}w4. +m}w4m Hm}w4m, > +4:,

zkhuh }w y l=l=g=+3> 4, dqg +w, 4₅5w lv d srvvleoh vshflfdwlrq iru h{0

shfwhg uhwxuqv53 _{dv d ixqfwlrq ri yrodwlolw|}

w/ ghqhg e| d jlyhq ixqfwlrq

= +3> 4, $ U1 Htxdwlrq +4:, prghov wkh g|qdplfv ri yrodwlolwlhv dv dq h{srqhqwldo JDUFK surfhvv lq zklfk dv|pphwulhv lq frqglwlrqdo orj yduldqfhv duh ghvfulehg e| wkh whupv &}w4dqg +m}w4m Hm}w4m,> zkhuh w|slfdoo| & ? 3

dqg A 31 Wkh frpelqdwlrq ri wkhvh wzr sduwv lqwurgxfhv lq wkh JDUFK g|qdplfv dq dv|pphwulf uhvsrqvh ri orj yrodwlolwlhv zlwk uhvshfw wr erwk wkh vljq dqg wkh pdjqlwxgh ri xqh{shfwhg uhwxuqv/ wkhuhe| lqfrusrudwlqj d nlqg ri ohyhudjh hhfw> fi1 Eodfn ^7`1

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wlrq, d qrupdo glvwulexwlrq ri }w> zh iroorz Qhovrq ^55` lq ghulylqj wkh frqwlqxrxv

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53_{Wkh prwlydwlrq iru lvrodwlqj wkh frpsrqhqw 3}

2j2| lq wklv frqglwlrqdo phdq vshflfdwlrq

ghulyhv iurp wkh idfw wkdw wklv whup zloo fdqfho rxw zkhq wdnlqj wkh frqwlqxrv wlph olplw ri +49,1 Wklv frqwlqxrxv wlph olplw surfhvv lv wkh rqh wkdw zloo eh xvhg wr ghqh dq lqwhuwhpsrudo Phuwrq*v prgho zlwk vwrfkdvwlf yrodwlolw| odwhu rq1

54_{Iru frpsohwhqhvv/ zh surylgh lq wkh Dsshqgl{ d surri ri d pruh jhqhudo vwdwhphqw wkdw}

(19)

zkhuh ] @ +]4>w> ]5>w,3w3 lv d guliwohvv Eurzqldq Prwlrq lq U5 vxfk wkdw= g]wg]w3 @ 3 C 4 4 4 D gw = +53,

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gEw

Ew @ ugw /

zkhuh u lv d frqvwdqw ulvnohvv udwh1 Ghqrwlqj e| zw wkh iudfwlrq ri fxuuhqw

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wkh lpsolhg zhdowk g|qdplfv duh=

gZw@ Zw^zw++w, u, . u`gw . Zwzwwg]4>w = +55,

Ohw eh d frqvwdqw wlph suhihuhqfh udwh/ Y +, dq lqgluhfw xwlolw| ri zhdowk dqg W d qdo gdwh1

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(20)

Wkh lqyhvwru*v remhfwlyh lv wr pd{lpl}h wkh h{shfwhg glvfrxqwhg lqgluhfw xwlolw| ri whuplqdo zhdowk=

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zlwk wkh erxqgdu| frqglwlrq M+Z> > W , @ Y +Z ,1 Wkh lpsolhg rswlpdo sruwirolr doorfdwlrq lv jlyhq e|=

zw@ _{Z M}MZ Z>Z +w, u 5 w MZ> Z MZ>Z # w = +58,

Wkh uvw whup rq wkh uljkw kdqg vlgh ri +58, lv wkh vwdqgdug frqglwlrqdoo| rswlpdo phdq yduldqfh doorfdwlrq lq Phuwrq*v prgho1 Wkh vhfrqg sduw lv dq khgjlqj srvlwlrq iru lqwhuwhpsrudo pryhphqwv lq wkh lqyhvwphqw rssruwxqlw| vhw/ gxh wr dv|pphwulhv lq wkh uhwxuq glvwulexwlrqv wkdw lqgxfh d orfdo fruuhodwlrq lq wkh frpprq g|qdplfv ri uhwxuqv dqg yduldqfhv1

Vshflfdoo|/ iru srvlwlyh57 _M

Z> dqg vlqfh # @ & zh vhh wkdw wkh khgjlqj

57_{Wklv lv wkh fdvh lq wkh h{solflw prgho suhvhqwhg ehorz jlyhq wkh vhw ri hvwlpdwhg sdudphwhuv}

(21)

615 H{solflw Vroxwlrqv iru Srzhu Lqgluhfw Xwlolw|

Wkh KME htxdwlrq +57, kdv dq h{solflwh vroxwlrq iru d srzhu lqgluhfw xwlolw| ixqfwlrq ri zhdowk=

Y +Z , @Z4s

4 s > +59, dqg d olqhdu vshflfdwlrq ri uhzdug wr yrodwlolw| udwlrv zlwk uhvshfw wr orj ydul0 dqfhv=

+w, u

w @ g3. g4w = +5:,

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Zh uhylhz eulh | wkh vroxwlrq phwkrgv1

Vxevwlwxwlqj wkhvh vshflfdwlrqv dqg wkh rswlpdo uxohv +58, lq +57,/ wkh KME htxdwlrq ri rxu rswlpl}lqj lqyhvwru orrnv h{solflwo|=

M @ Mw4₅ M 5 Z MZ Z +g3. g4, 5_#MZMZ> MZ Z +g3. g4, .uZ MZ. + ,M.# 5 5 M># 55 5 M5 Z> MZ Z > +5;, vxemhfw wr wkh erxqgdu| frqglwlrq= M+Z> > W , @ Z_{4 s}4s = +5<, Zh orrn iru d vroxwlrq ri wkh irup=

M+Z> > w, @ Z4s

4 s h{s+V+> w,, > +63, zklfk lv vxjjhvwhg e| wkh vroxwlrq iru wkh vwdqgdug frqvwdqw yrodwlolw| Phuwrq*v prgho> fi1 Phuwrq ^4<`1 Wkh sduwldo glhuhqwldo htxdwlrq +SGH, lpsolhg e| +5;, dqg +5<, iru V lv=

#5

5 V.#

5

(22)

vxemhfw wr wkh erxqgdu| frqglwlrq V +> W , @ 3> wkh frqvwdqwv D/ E/ F/ G/ H dqg I duh uhodwhg wr wkh prgho sdudphwhuv e| wkh irupxodv=

D @ 4 .+4 s,_s 5_{> E @ .}4 s

s #g3> F @ 4 s_s #g4 > G @ u+4 s, .4 s_5s g53 >

H @ 4 s_s g3g4/ I @4 s_s g54= +65,

Vroylqj wklv SGH dprxqwv wr jlylqj wkh vroxwlrq ri sureohp +56, iru wkh jlyhq vshflfdwlrqv +59, dqg +5:,1 Qrwlfh wkdw zkhq s A 4 wkhq rqh kdv

D A 3> I ? 3 > =@ F5_#5_{DI A 3 =} _+66,

Lq wkh qh{w sursrvlwlrq zh uhsruw wkh h{solflw vroxwlrq ri +56, iru wkh fdvh s A 458_{/ vlqfh wklv lqwhuydo ri ydoxhv iru s frqwdlqv wkh vhw ri ulvn dyhuvlrq sdudphwhuv}

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Sursrvlwlrq 4 Ohw s A 4> lw wkhq iroorzv= M+Z> > w, @Z_{4 s}4sh{s

d+w, . e+w, .4₅f+w,5

> +67,

zlwk dq rswlpdo doorfdwlrq +58, jlyhq e|=

zw @ g3. g4oq+ 5 w, sw . # e+w, . f+w, oq+5 w, sw @ +g3. #e+w,, . +g4. #f+w,, oq+5w, sw = +68,

58_{Wkh fdvh R : fruuhvsrqgv wr wkh qrupdo/ zhoo0ehkdyhg vroxwlrqv lq Nlp dqg}

Rpehuj^49`

(23)

zkhuh= f+w, @ s #5_D # 4 h{s+5s+W w,, 4 . h{s+5s+W w,, F s $ > e+w, @ 3 3 C4 v I . 5Ff+w, . #5Df+w,5 I 4 D . 4f+w, > d+w, @ 3+W w, . 4f+w, 34 # 5s s I D#5 h{s+s+W w,, 4 . h{s+5s+W w,, 4 $ 5oq # 4 . h{s+5s+W w,, 4 . $ =

Wkh sdudphwhuv / 3/ 4/ 3> 4/ 5/ 6 duh jlyhq e|=

@ 4 F s 4 ._sF > 3@ I E FH > 4@EF # 5_DH > 3 @ G . E3E_#54_DF .# 5_D 5 5 3._#54_D 54F5 #5_D 4I 53F . s F 5D . s 4 3.E F_#5_D 4 4@ # 5_D 5 54 #5D. 53 I > 5@ 4₅ 4 D . 4 3.E F_#5_D 4

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zloo qrw eh jhqhudoo| rswlpdo5:_{1 D qdwxudo txhvwlrq lv li/ jlyhq uhdolvwlf prgho}

sdudphwhuv/ khgjlqj srvlwlrqv wdnlqj dv|pphwulhv lqwr dffrxqw pdwwhu iru sudf0 wlfdo sxusrvhv1 Wkh qh{w sursrvlwlrq ghulyhv wkh h{shfwhg lqgluhfw xwlolw| ri

5:_{D fdvh zkhuh wklv wudglqj uxoh lv rswlpdo dovr zkhq i f dulvhv zkhq wkh uhzdug wr}

yrodwlolw| udwlr grhv qrw ghshqg rq j2

|/ wkdw lv zkhq _ ' f1 Wklv lv hdvlo| vhhq zkhq vhw0

wlqj _' f lq wkh frqvwdqwv / . dqg 8 lq +65, dqg wkhq vroylqj wkh lpsolhg SGH iru 71

Lqwxlwlyho|/ lq wklv fdvh frqglwlrqdo phdq yduldqfh doorfdwlrqv duh rswlpdo ehfdxvh wkh lq0 whuwhpsrudo fruuhodwlrq ri uhwxuqv dqg yduldqfhv grhv qrw dhfw wkh frqglwlrqdo rssruwxqlw| vhw dqg vshflfdoo| uhzdug wr yduldelolw| udwlrv1

(24)

whuplqdo zhdowk lq rxu prgho iru d wudglqj uxoh ri wkh irup +69, wkdw grhv qrw khgjh iru dv|pphwulhv1

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ri wkh srolf| +69, lv jlyhq e| wkh ixqfwlrqdo irup +67, iru wkh ydoxh ixqfwlrq M/ frpsxwhg vhwwlqj D @ 4 lq +65,1

D frpsdulvrq ri wkh ydoxh ixqfwlrq +68, dqg wkh lqgluhfw xwlolw| +6:, iru d jlyhq sdudphwhu fkrlfh lq wkh prgho fdq surylgh lqvljkwv derxw glhuhqfhv lq wkh fhuwdlqw| htxlydohqwv ri wkhvh wzr vwudwhjlhv/ dqg frqvhtxhqwo| rq wkh uhohydqfh ri lqwhuwhpsrudo khgjlqj iru dv|pphwulhv lq wkh suhvhqw vhwwlqj> fi1 dovr wkh qh{w vhfwlrq1

616 Fdoleudwlrq ri wkh Prgho

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Wkh sdudphwhuv ri wkh frqwlqxrxv0wlph prgho +4;, dqg +4<, lpsolhg e| wkh hvwlpdwhg sdudphwhuv ri wkh h{srqhqwldo dv|pphwulf P0JDUFK surfhvv +49, dqg +4:, duh=

@ <=8:7 > @ 3=37:3 > # @ 3=4737 > @ 3=98<3 > g3@ 3=3::3 > g4@ 3=344< =

Zh vhw wkh lqvwdqwdqhrxv lqwhuhvw udwh htxdo wr u @ 3=3335 +fruuhvsrqglqj wr dq dqqxdo lqwhuhvw udwh ri dssur{lpdwlyh 8(, dqg { d wlph suhihuhqfh udwh @ 3=33351 Ilqdoo|/ wkh ulvn dyhuvlrq sdudphwhu xvhg lv s @ 71

Wkh ghshqghqfh ehwzhhq h{shfwhg uhwxuqv dqg yrodwlolwlhv lpsolhg e| rxu sdudphwhu fkrlfh lv sorwwhg lq Iljxuh 51

(25)

D voljkw sd|phqw iru yrodwlolw| ulvn lq hvwlpdwhg h{shfwhg uhwxuqv lv revhuyhg= hvwlpdwhg h{shfwhg uhwxuqv lqfuhdvh dv d ixqfwlrq ri yrodwlolw| dqg duh ehwzhhq 313335: dqg 3133368 iru lq vdpsoh hvwlpdwhg yrodwlolwlhv udqjlqj iurp 31339 wr 313451

Iljxuh 6 suhvhqwv wkh rswlpdo doorfdwlrq zw lq +68, +lqfoxglqj khgjlqj iru

dv|pphwulhv, dqg wkh phdq yduldqfh srvlwlrq hzw +69, lq ghshqghqfh ri wkh

fxuuhqw yrodwlolw| ohyho5;₁

Dv h{shfwhg/ erwk srvlwlrqv duh ghfuhdvlqj ixqfwlrqv ri yrodwlolw|1 Iru dq| yrodwlolw| ohyho/ hzw lv ohvv wkdw zw> ehfdxvh ri wkh qhjdwlyh khgjlqj srvlwlrq

iru dv|pphwulhv lqgxfhg e| wkh qhjdwlyh hvwlpdwhg fruuhodwlrq ehwzhhq uhwxuqv dqg yrodwlolwlhv1 Krzhyhu/ wkh glhuhqfhv ehwzhhq wkh wzr srvlwlrqv wkdw duh lpsolhg e| wkh jlyhq sdudphwhu fkrlfh duh vpdoo dqg vshflfdoo| qhyhu odujhu wkdq 5(1

Iljxuh 7 sorwv hzwdqg zw dv d ixqfwlrq ri +ghfuhdvlqj, wlph wr pdwxulw| iru

dq lqyhvwphqw krul}rq ri 583 gd|v dqg d fxuuhqw yrodwlolw| ohyho 3133;81

Lw lv lqwhuhvwlqj wr qrwlfh wkdw wkh glhuhqfh lq wkh wzr srvlwlrqv lv doprvw frqvwdqw +dprxqwlqj wr derxw 5(, iru wkh uvw 493 gd|v1 Diwhu wklv gdwh/ lq0 whuwhpsrudo khgjlqj srvlwlrqv duh prqrwrqlfdoo| uhod{hg xqwlo pdwxulw|1

Zh frqfoxgh wklv vhfwlrq e| d shuirupdqfh dqdo|vlv pdnlqj xvh ri wkh fhu0 wdlqw| htxlydohqw lpsolhg e| wkh prgho vroxwlrq +67,1 Wkuhh frpshwlqj g|qdplf vwudwhjlhv duh frpsduhg= wkh rswlpdo vwudwhj| +68,/ wkh sxuh phdq yduldqfh uxoh +69, dqg d vwdwlf ex| dqg krog doorfdwlrq ryhu d wlph krul}rq ri W w @ 833 exvlqhvv gd|v1 Lqlwldo fxuuhqw zhdowk Zwlv qrupdol}hg wr 41

5;_{Uhpdun wkdw wkh jlyhq fkrlfh iru wkh sdudphwhu k lpsolhv d orqj whup h{shfwhg yrodwlolw|}

(26)

Iru wkhvh wkuhh wudglqj vwudwhjlhv zh frpsxwh wkh fruuhvsrqglqj fhuwdlqw| htxlydohqw FHwdw wlph w ghqhg e|= Y +FHw, @ Hw h+W w,_{Y +Z} W, > +6;, zkhuh ZW lv wkh whuplqdo zhdowk surgxfhg dw wlph W e| wkh vwudwhj| xqghu

vfuxwlq| dqg Y lv wkh srzhu xwlolw| ixqfwlrq1

Wkh h{shfwdwlrqv rq wkh uljkw kdqg vlgh ri +6;, zhuh douhdg| frpsxwhg lq +67, dqg +6:, iru wkh rswlpdo dqg wkh sxuh phdq yduldqfh vwudwhj|1 Iru d ex| dqg krog uxoh wklv vdph h{shfwdwlrq kdv wr eh frpsxwhg e| vlpxodwlrq5< _vlqfh

Dv h{shfwhg63_{/ rqh fdqqrw revhuyh lpsruwdqw glhuhqfhv lq fhuwdlqw| htxly0}

dohqw xqlwv ehwzhhq wkh rswlpdo vwudwhj| +68, dqg wkh sxuh phdq yduldqfh uxoh +69,1 Zh wkhuhiruh frqfhqwudwh rq d frpsdulvrq ehwzhhq ex| dqg krog dqg d sxuh phdq yduldqfh doorfdwlrq e| sorwwlqj lq Iljxuh 8 wkh uhodwlyh fhuwdlqw| htxlydohqw ri wkhvh wzr vwudwhjlhv dv d ixqfwlrq ri yrodwlolw|1

Glhuhqfhv dprxqw wr derxw 416( +rq dq dqqxdo edvlv, lq idyrxu ri wkh phdq yduldqfh doorfdwlrq dqg duh txlwh vwdeoh dv d ixqfwlrq ri yrodwlolw| +dv rqh zrxog h{shfw iru vxfk d uhodwlyho| orqj wlph krul}rq,1

7 Frqfoxvlrqv

Iru doo vwrfn lqglfhv xqghu vfuxwlq|/ zh qg d prghvw exw frqvlvwhqw hylghqfh ri d vxshulru shuirupdqfh ri g|qdplf vwudwhjlhv wdnlqj lqwr dffrxqw ydu|lqj ulvn lq d vlpsoh frqglwlrqdo phdq yduldqfh zd|1 Pruhryhu/ xvlqj h{solflw vroxwlrqv lq dq lqwhuwhpsrudo Phuwrq*v prgho lw vhhpv wkdw odujhu pdujlqv ri lpsuryhphqw

5<_{Iru hdfk ydoxh ri j}_|_{/ d Prqwh Fduor vlpxodwlrq frqvlvwlqj ri 433333 uhshwlwlrqv xvlqj wkh}

phwkrg ri dqwlwkhwlf yduldeohv zdv xvhg1

(27)

duh qrw hdvlo| dydlodeoh/ hyhq zkhq nqrzlqj wkh xqghuo|lqj gdwd jhqhudwlqj surfhvv1 Ilqdoo|/ iru sudfwlfdo sxusrvhv lqwhuwhpsrudo khgjlqj pd| eh qhjoljleoh dw ohdvw iru wkh fodvv ri prghov frqvlghuhg lq wkh sdshu1

(28)

8 Dsshqgl{

Surri1 +Wkh frqwlqxrxv wlph olplw surfhvv lq +4;, dqg +4<,, Ohw |w @

orj +Vw,1 Wkh glvfuhwh wlph surfhvv +49, dqg +4:, fdq eh wkhq uhzulwwhq dv=

|w@ |w4. +w, 4₅5w . w}w > oq+5 w, @ oq+5w4, +oq+5w4, , . j+}w4, > zkhuh= j+}, @ #} . # 4 5 4 5 4 5 +m}m , > @ H +m}m, =

Wkh qhz sdudphwhuv duh ghqhg lq dq reylrxv zd|/ dffruglqjo| wr +54,1 Uhpdun wkdw iru d qrupdoo| glvwulexwhg udqgrp yduldeoh }wrqh kdv @

t

5

+dv iru wkh

fdvh suhvhqwhg lq wkh pdlq wh{w,1 Krzhyhu/ zh gr qrw dvvxph qrupdolw| lq wklv surri1 Frqvlghu qrz/ wkh vhtxhqfh ri surfhvvhv

[k w @ +|w> oq 5w.k, = w @ 3> k> 5k> === > k $ 3 > ghqhg e|= |w@ |wk. k +w, 4₅5w . k45_w}_w > oq+5 w.k, @ oq+5w, k+oq+5w, , . k 4 5j+}_w, = Wkhq= +[k w, @ 4_kHw+[w.kk [wk, @ 3 C +w.k, 4₅5w.k +oq+5 w.k, , 4 D > dqg= +[wk, @ _k4Y duw+[w.kk [wk, @ Y duw 3 C w.k}w.k j+}w.k, 4 D @ 3 E E C 5 w.k # . #₄455 4 5 w.k # . #45 45 4 5 w.k #5 4 . 545 45 4 5 4 F F D >

(29)

zkhuh @ H+}m}m,1 Khqfh/ fryduldqfhv dprqj uhwxuqv dqg yrodwlolwlhv ruljlqdwh lq wklv prgho iurp wkh dv|pphwu| whup # dqg wkh dv|pphwulf prphqw ri wkh frqglwlrqdo glvwulexwlrq ri uhwxuqv1 Dsso|lqj d vwdqgdug Wkhruhp e| Vwurrfn dqg Ydudgkdq ^56`/ wkh vhtxhqfh ri surfhvvhv [k

w @ +|w> oq 5w.k,

frqyhujhv lq glvwulexwlrq wr wkh vroxwlrq ri wkh vwrfkdvwlf glhuhqwldo htxdwlrq= g[w@ +[w,gw . 45+[_w,g h]_w >

dv k $ 3/ zkhuh]hw

w3 lv d vwdqgdug Eurzqldq prwlrq lq U

5_{1 Wkh htxdwlrqv}

+4;, dqg +4<, iru wkh surfhvvVw> oq 5w3w3 iroorz e| Lwr*v Ohppd1

Surri1 +Sursrvlwlrq 4, D vroxwlrq ri +64, h{lvwv lq wkh irup=

V+> w, @ d+w, . e+w, .4₅f+w,5 +6<, Vxevwlwxwlqj +6<, lq +64, zh jhw d v|vwhp ri qrqolqhdu ruglqdu| glhuhqwldo htxd0 wlrqv iru d/ e dqg f=

f+w, . I . 5Ff+w, . #5_Df+w,5 _{@ 3 >}

e+w, . H . Fe+w, . Ef+w, . #5De+w,f+w, @ 3 > +73,

d+w, . G . Ee+w, .#5₅De+w,5_.#5

5 f+w, @ 3 >

zlwk wkh erxqgdu| frqglwlrqv= f+W , @ e+W , @ d+W , @ 31 Wkh lqwhjudo ri wkh uvw glhuhqwldo htxdwlrq lv jlyhq e|

] f+w, 3

gx

I . 5Fx . #5_Dx5 @ W w =

Wklv lqwhjudo lv vwdqgdug dqg iru @ F5_#5_{DI A 3 +fi1 +66,, zh rewdlq=}

4 5oq 3 C4 # 5_Df+w,.F s 4 .#5Df+w,.Fs 4 D @s+W w, . 4₅oq # 4 _sF 4 ._sF $ >

zklfk lpsolhv lpphgldwho| wkh vroxwlrq ri f1 Xvlqj wkh wuldqjxodulw| ri wkh v|vwhp +73,/ zh frxog lqvhuw wkh vroxwlrq iru f lq wkh vhfrqg htxdwlrq dqg vroyh iru e1 Krzhyhu lw lv pxfk vlpsohu wr vroyh iru e dv d ixqfwlrq ri f/ zklfk e| dex vh ri qrwdwlrq zh ghqrwh dv e+f,1 Wklv lv srvvleoh vlqfh f lv vwurqjo| lqfuhdvlqj rq

(30)

^3> W `/ dv zh gluhfwo| ghgxfh iurp wkh glhuhqwldo htxdwlrq iru f1 Zulwlqj=

e @ ge_gff @ +D . 5Ff . # 5_Df5_, ge

gf /

dqg lqvhuwlqj wklv h{suhvvlrq lq wkh glhuhqwldo htxdwlrq iru e/ zh jhw= +I . 5Ff . #5Df5_,ge

gf+f, +F . #5Df, e+f, @ H . Ef > +74, zlwk wkh erxqgdu| frqglwlrq e+3, @ 31 Qrwlfh wkdw wklv lv d olqhdu glhuhqwldo htxdwlrq zlwk dq krprjhqhrxv sduw zklfk lv vhsdudeoh dqg dgplwv d vroxwlrq +eKRP+f,/ vd|, ri wkh irup= eKRP+f, @ frqvw v I . 5Ff . #5Df5 I > frqvw 5 U > iru doo f vxfk wkdw64₌ I . 5Ff . #5Df5 I A 3 = Ixuwkhupruh/ wkh ixqfwlrq= f :$ 3. 4f >

zlwk 3 dqg 4 dv jlyhq lq Sursrvlwlrq 4/ ghqhv d sduwlfxodu vroxwlrq ri +74,1

Xvlqj wkh erxqgdu| frqglwlrq e+3, @ 3 wklv lpsolhv= e+f, @ 3 3 C4 v I . 5Ff . #5Df5 I 4 D . 4f =

Wkh vroxwlrq iru e lv jlyhq e| e+w, @ e+f+w,,1

Wkh wklug glhuhqwldo htxdwlrq lq +73, lv vroyhg e| wkh lqwhjudo= d+w, @ G+W w, . E ] W w e+v,gv . #5D 5 ] W w e+v, 5_{gv .}#5 5 ] W w f+v,gv >

zkrvh frpsxwdwlrq lqvhuwlqj wkh vroxwlrqv iru e+w, dqg f+w, lv vwudljkwiruzdug / dowkrxjk vrphzkdw fxpehuvrph1 Wkh fkdqjh ri yduldeoh iurp w wr f lv dfrxog eh xvhixo1 Iru h{dpsoh +xvlqj wkh erxqgdu| frqglwlrq f+W , @ 3,=

] W w e+v,gv @ ] f+W , f+w, e+f, f gf @] f+w, 3 e+x,gx I . 5Fx . #5Dx5 >

64_{Qrwh wkdw wklv lv suhflvho| wkh uhohydqw grpdlq ir wkh vroxwlrq ri wkh deryh ruglqdu|}

(31)

zklfk lv vwdqgdug1 Wkh rwkhu lqwhjudov fdq eh frpsxwhg lq d vlplodu zd|1 Surri1 +Sursrvlwlrq 5, Xqghu wkh sruwirolr uxohv +69, wkh fxuuhqw zhdowk g|qdplfv duh jlyhq e|=

gZw Zw @ 4 s+g3. g4w,5. u gw . g3. g4w s g]4>w = +75, Wkh jrdo lv wkhuhiruh wr frpsxwh= h M+Zw>w> W w, @ Hw+h+W w,Z 4s W 4 s,

xqghu wkh g|qdplfv +75, dqg +4<, iru wkh vwdwh yhfwru [w @ +Zw>w,31 E| wkh

Ih|qpdq0Ndf Wkhruhp +fi1 iru lqvwdqfh ^48`/ s1 59;, hM vroyhv wkh SGH= h

M @ +D , hM

h

M+Z> > 3, @ Z_{4 s}4s +76, zkhuh =@ W w dqg D lv wkh jhqhudwru ri wkh Pdunry surfhvv +[w,_w3=

D @ +g3. g4_s,5. suZ_CZC . + ,_CC .+g3. g_5s₅4,5Z5 C5 C5_Z .#_s +g3. g4,Z C 5 CZ C . #5 5 C5 C5 = Jxhvvlqj d vroxwlrq ri +76, lq wkh irup= h M+Z> > , @Z4s 4 s h{s+ hV+> ,, > zh vhh wkdw hV vroyhv wkh SGH +64, zlwk D @ 4 lq +65,1

(32)

9 Wdeohv

WDEOH 4

GRZ V)S833 IWVH QLNNHL Shulrg L 4 4=33H 3: +5=78H3:, 7=98H 3:+6=6:H3:, 5=76H 39+4=48H39, 4=8<H 39+7=;3H3:, ₅ 3=33833 +3=347;66, 3=33376:+3=345863, 3=37976;+3=34<<5:, 3=34<;88+3=34<83, 6 3=<;9;95_+3=34<:79, 3=<974;9_+3=356957, 3=;<8335_+3=358<;<, 3=<3944<_+3=3488:5, 7 3=33<<79_+3=343447, 3=36;7;3_+3=353:85, 3=387896_+3=355:4, 3=47876:_+3=366756, Shulrg LL 4 4=3;H 39_+9=4:H3:, <=3:H 3:_+9=:7H3:, <=<7H 3:_+9=4:H3:, 6=98H 39_+5=:4H39, 5 3=337:5<_+3=348<;3, 3=394;7:_+3=3543<3, 3=3467:;_+3=347:9:, 3=346846_+3=3469:8, 6 3=<7:9<6_+3=349873, 3=<59384_+3=34<383, 3=<5;:;7_+3=34<6;;, 3=<3554<_+3=365663, ₇ 3=3959<3 +3=35;675, 3=33;663+3=358857, 3=3;8;39+3=353798, 3=467<;6+3=384;;3,

(33)

WDEOH 5

GRZ V)S833 IWVH QLNNHL Shulrg L 3 3134;<<< 31353;4; 3133<6<: 031359534 4 313334;8 31333473 041;5H08 0313338:: 31<463<6 31833333 31768<9; 3164656: Shulrg LL 3 313:53;; 31338:94 3136:<33 03133358< 4 31333856 ;1;4H038 313335:; 031333<<4 3196;<87 316<;:<8 317:4747 3175:453

(34)

WDEOH 6

GRZ V)S833 IWVH QLNNHL Ixoo Vdpsoh Shulrg

VU Ex| dqg Krog 3137495 31345:4 3134;77 03135<7<6 VU Ed|hvldq Vwudwhj| 3137;49 3134<86 313563: 3135:;6 Mhqvhq*v doskd 3133345 :133H08 <133H08 3133345 Shulrg LL VU Ex| dqg Krog 3138::; 313445< 3136387 03134;<7 VU Ed|hvldq Vwudwhj| 3138<<6 3134883 3136586 31373<; Mhqvhq*v doskd 6133H08 7133H08 5133H08 3133369

WDEOH 7

(35)

Uhihuhqfhv

^4` Edqvdo/ U1/ dqg F1 U1 Kduyh|= Shuirupdqfh Hydoxdwlrq lq wkh Suhvhqfh ri G|qdplf Wudglqj Vwudwhjlhv/ kwws=22zzz1gxnh1hgx2fkduyh|2Ylwdh1kwpo ^5` Eduehulv/ Q1 / 5333= Lqyhvwlqj iru wkh Orqj Uxq Zkhq Uhwxuqv duh Suh0

glfwdeoh/ Mrxuqdo ri Ilqdqfh 88/ 55805971

^6` Elhohfnl/ W1 U1/ V1 U1 Solvnd/ dqg P1 Vkhuulv= Ulvn Vhqvlwlyh Dvvhw Doorfd0 wlrq/ wr dsshdu lq Mrxuqdo ri Hfrqrplf G|qdplfv dqg Frqwuro1

^7` Eodfn/ I1/ 4<:9= Vwxglhv lq Vwrfn Sulfh Yrodwlolw| Fkdqjhv/ Surfhhglqjv ri wkh 4<:9 Exvlqhvv Phhwlqj ri wkh Exvlqhvv Hfrqrplf Vwdwlvwlfv vhfwlrq/ DVD/ 4::04;41

^8` Eroohuvohy/ W1/ 4<;9= Jhqhudol}hg Dxwruhjuhvvlyh Frqglwlrqdo Khw0 hurvnhgdvwlflw|/ Mrxuqdo ri Hfrqrphwulfv/ 64/ 63:065:1

^9` Eroohuvohy/ W1/ dqg M1 P1 Zrrogulgjh/ 4<<5= Txdvl Pd{lpxp Olnholkrrg Hvwlpdwlrq dqg Lqihuhqfh lq G|qdplf Prghov zlwk Wlph Ydu|lqj Frydul0 dqfhv/ Hfrqrphwulf Uhylhzv/ 44/ 47604:51

^:` Euhqqdq/ P1 M1/ H1 V1 Vfkzduw}/ dqg U1 Odjqdgr/ 4<<:= Vwudwhjlf Dvvhw Doorfdwlrq/ Mrxuqdo ri Hfrqrplf G|qdplfv dqg Frqwuro/ 54/ 46::047361 ^;` Fdpsehoo/ M1 \1/ 4<;:= Vwrfn Uhwxuqv dqg wkh Whup Vwuxfwxuh/ Mrxuqdo

ri Ilqdqfldo Hfrqrplfv/ 4;/ 6:607331

^<` Hqjoh/ U1/ dqg Y1 Qj / 4<<6= Phdvxulqj dqg Whvwlqj wkh Lpsdfw ri Qhzv rq Yrodwlolw|/ Mrxuqdo ri Ilqdqfh/ 7;/ 4:7<04::;1

^43` Hydqv/ P1 G1/ 4<<7= H{shfwhg Uhwxuqv/ Wlph Ydu|lqj Ulvn dqg Ulvn Suhpld/ Mrxuqdo ri Ilqdqfh/ 7</ 98809:<1

^44` Ihuvrq/ Z1 H1/ dqg F1 U1 Kduyh|/ 4<<4= Wkh Yduldwlrq ri Hfrqrplf Ulvn Suhpld/ Mrxuqdo ri Srolwlfdo Hfrqrp|/ <</ 6;807481

(36)

^45` Jorvwhq/ O1/ U1 Mdjdqqdwkdq/ dqg G1 Uxqnoh/ 4<<6= Rq wkh Uhodwlrq eh0 wzhhq wkh H{shfwhg Ydoxh dqg wkh Yrodwlolw| ri wkh Qrplqdo H{fhvv Uhwxuq rq Vwrfnv/ Mrxuqdo ri Ilqdqfh/ 7;/ 4::<04;341

^46` Kh/ M1/ U1 Ndq/ O1 Qj/ dqg F1 ]kdqj/ 4<<9= Whvwv ri Uhodwlrqv dprqj Pdunhwzlgh Idfwruv/ Ilup0vshflf Yduldeohv/ dqg Vwrfn Uhwxuqv Xvlqj d Frqglwlrqdo Dvvhw Sulflqj Prgho/ Mrxuqdo ri Ilqdqfh/ 84/ 4;<404<3;1 ^47` Mrulrq/ S1/ 4<;9= Ed|hv0Vwhlq Hvwlpdwlrq iru Sruwirolr Dqdo|vlv/ Mrxuqdo

ri Ilqdqfldo dqg Txdqwlwdwlyh Dqdo|vlv/ 54/ 5:<05<51

^48` Ndudw}dv/ L1/ dqg V1H1 Vkuhyh/ 4<<4= Eurzqldq Prwlrq dqg Vwrfkdvwlf Fdo0

fxoxv1 Vsulqjhu0Yhuodj/ Qhz \run1

^49` Nlp/ W1 V1/ dqg H1 Rpehuj/ 4<<9= G|qdplf Qrqp|rslf Sruwirolr Ehkdy0 lru/ Wkh Uhylhz ri Ilqdqfldo Vwxglhv/ Yro1 </ Qr1 4/ 47404941

^4:` Ol/ \1/ 4<<;= Wlph Yduldwlrq lq Ulvn Suhpld/ Yrodwlolw| dqg Uhzdug0wr0 Yrodwlolw|/ Wkh Mrxuqdo ri Ilqdqfldo Uhvhdufk/ 54/ 76407791

^4;` Ol/ \1/ 4<<;= H{shfwhg vwrfn uhwxuqv/ ulvn suhpld/ dqg yrodwlolwlhv ri hfr0 qrplf idfwruv/ Mrxuqdo ri Hpslulfdo Ilqdqfh/ 8/ 9<0<:1

^4<` Phuwrq/ U1/ 4<9<= Olihwlph Sruwirolr Vhohfwlrq Xqghu Xqfhuwdlqw|= wkh Frqwlqxrxv Wlph Fdvh/ Uhylhz ri Hfrqrplfv dqg Vwdwlvwlfv/ 84/ 57:58:1 ^53` Phuwrq/ U1/ 4<:4= Rswlpxp Frqvxpswlrq dqg Sruwirolr Uxohv lq d

Frqwlqxrxv0Wlph Prgho/ Mrxuqdo ri Hfrqrplf Wkhru|/ 6/ 6:67461 ^54` Qhovrq/ G1 E1/ 4<<3= DUFK Prghov dv Glxvlrq Dssur{lpdwlrqv/ Mrxu0

qdo ri Hfrqrphwulfv/ 78/ :06;1

^55` Qhovrq/ G1 E1/ 4<<4= Frqglwlrqdo Khwhurvnhgdvwlflw| lq Dvvhw Uhwxuqv= d qhz Dssurdfk/ Hfrqrphwulfd/ 8</ 67:06:31

^56` Vwurrfn/ G1 Z1/ dqg V1 U1 V1 Ydudgkdq/ 4<:<= Pxowlglphqvlrqdo Glxvlrq

(37)

^57` Zklwhodz/ U1/ 4<<7= Wlph Yduldwlrq dqg Fryduldwlrqv lq wkh H{shfwdwlrq dqg Yrodwlolw| ri Vwrfn Pdunhw Uhwxuqv/ Mrxuqdo ri Ilqdqfh/ 7</ 4<<7/ 84808741

(38)

-0.10 -0.05 0.00 0.05 0.000 0.005 0.010 0.015 0.020 0.025 VOL_DOW E X D O W Iljxuh 4=

(39)

(40)

(41)

(42)