• No results found

download full version

N/A
N/A
Protected

Academic year: 2020

Share "download full version"

Copied!
10
0
0

Loading.... (view fulltext now)

Full text

(1)

2'HUHY\DQFKXN&=HQNRYD9.UDPDU%1LWVRYLFK23RWHUHMNR

'HSWRI2SWLFVDQG6SHFWURVFRS\&KHUQLYWVL1DWLRQDO8QLYHUVLW\.RWV\XE\QVN\6WU &KHUQLYWVL8NUDLQHH±PDLO]HQNRYD#LWIFYXNUWHOQHW

5HFHLYHG

$EVWUDFW

2SWLFDO VSHFWUD RI PROHFXODU FU\VWDOV ZLWK WZR PROHFXOHV LQ WKHLU HOHPHQWDU\ FHOO DUH LQYHVWLJDWHG'HSHQGLQJRQWKHSRODUL]DWLRQDQJOHRQHRUWZRH[FLWRQH[FLWDWLRQEDQGVFDQ EHUHDOL]HGLQWKHVSHFWUXP7KHEDQGVDUHVHSDUDWHGE\WKHWHUPLQDOIUHTXHQF\LQWHUYDO7KH H[FLWRQDEVRUSWLRQEDQGVJHQHVLVDQGWKHSHFXOLDULWLHVRIWKHIXQFWLRQRIWKHLUIRUPDWWKH FKDQJHRIWKHODVHUUDGLDWLRQSRODUL]DWLRQKDYHEHHQDQDO\]HGDQGWKHUHJXODULW\RIDULVLQJ ELVWDEOHVWDWHVLQRUJDQLFEHQ]ROW\SHFU\VWDOVKDVEHHQHVWDEOLVKHG

.H\ZRUGVH[FLWRQPROHFXODUFU\VWDORSWLFDOELVWDELOLW\

3$&66\

,QWURGXFWLRQ

2QH RI WKH PRVW SURVSHFWLYH ZD\V RI FUHDWLQJ RSWLFDO LQIRUPDWLRQ PHPRU\ HOHPHQWV LV WKH UHDOL]DWLRQ RI WKH RSWLFDO ELVWDELOLW\ 2% SKHQRPHQRQ >@ 7KH SULQFLSDO SRVVLELOLW\ RI REWDLQLQJ D K\VWHUHVLV ORRS LQ WKH LQFRPLQJ ±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± WKH )UHQNHO H[FLWRQ ,Q

(2)

([FLWRQ3KRQRQ,QWHUDFWLRQLQ 0ROHFXODU&U\VWDOV

7KH)UHQNHO([FLWRQFDQEHEHVWLQYHVWLJDWHGLQ PROHFXODU FU\VWDOV LQ ZKLFK WKH IRUFHV RI LQWHUDFWLRQ EHWZHHQ WKH FRPELQLQJ PROHFXOHV DUH PXFK VPDOOHU WKDQ WKH IRUFHV RI WKH LQWHUPROHFXODU LQWHUDFWLRQ $ QXPEHU RI VROLGV IRUPHGE\ELJPROHFXOHVRIRUJDQLFVXEVWDQFHV± DQWUDFHQH SHQWDFHQH QDSKWDOHQH EHQ]RO DQG RWKHUVEHORQJWRVXFKFU\VWDOV

:H VKDOO FRQVLGHU D PROHFXODU FU\VWDO ZKLFKKDVWZRPROHFXOHVLQLWVHOHPHQWDU\FHOO ZLWK GLIIHUHQW GLSROH PRPHQWV RI WUDQVIRUPLQJ LQWR DQ H[FLWHG VWDWH ,Q WKLV FDVH WKH FU\VWDO H[FLWRQ H[FLWDWLRQ HQHUJ\ ZLOO FRQVLVW RI WZR HQHUJ\EDQGV

ÃÃ

+

= N DN

H[ ( N % %

+ ar a

ZKHUH %N+%N DUH RSHUDWRUV RI FUHDWLRQ DQG DQQLKLODWLRQRIH[FLWRQVZLWKWKHTXDVLPRPHQ WXPNr6RWZRH[FLWRQH[FLWDWLRQEDQGVFRUUHV SRQG WR RQH QRQGHJHQHUDWH H[FLWHG VWDWH RI D IUHH PROHFXOH LQ D FU\VWDO 7KH HQHUJ\ RI WKHVH H[FLWRQV LQ WKH FDVH RI WZR LGHQWLFDO PROHFXOHV LQDFU\VWDOHOHPHQWDU\FHOOZLOOEH>@

N ' / N / N

( r =De+ + r + r

N ' / N / N

( r =De+ + r - r

+HUHDe ±LVWKHHQHUJ\RIPROHFXODUH[FL WDWLRQ'±LVWKHYDULDWLRQRILQWHUPROHFXODULQ WHUDFWLRQXQGHUPROHFXODUH[FLWDWLRQ/abLVWKH PDWUL[ RI WKH UHVRQDQFH LQWHUDFWLRQ DQG

/ (=

D ± LV WKH HQHUJ\ RI WKH UHVRQDQFH H[FLWRQ EDQG VSOLWWLQJ 7KHVH HQHUJ\ EDQGV FDQ EHFRUUHODWHGZLWKFRUUHVSRQGLQJVSHFLILFWUDQVL WLRQGLSROH PRPHQWV LQ FU\VWDOVGr DQGG ,Q VSLWH RI WKH IDFW WKDW H[FLWRQ H[FLWDWLRQV LQ D FU\VWDOZLWKWZRPROHFXOHVLQDQHOHPHQWDU\FHOO EHORQJ WR RQH HQHUJ\ OHYHO LQ D IUHH PROHFXOH WKH\GLIIHUQRWRQO\LQWKHLUHQHUJ\EXWLQWKHLU SRODUL]DWLRQ DV ZHOO 7KHLU SRODUL]DWLRQ LV SHU SHQGLFXODULQLWVGLUHFWLRQWKDWLVGr^Gr7KH YDOXH RI 'DY\GRY VSOLWWLQJD( LV GLIIHUHQW IRU

GLIIHUHQWFU\VWDOVDQGYDULHVEHWZHHQFPLQ

WKHDQWUDFHQHFU\VWDODQGFPLQWKHEHQ]RO

FU\VWDO

7KHHQHUJ\RIWKHFU\VWDOH[FLWRQH[FLWDWLRQ LVWKHVDPHDVZKHQWKHFU\VWDOPROHFXOHVDUH LQDVWHDG\VWDWHFRQGLWLRQLQWKHFU\VWDOODWWLFH ,Q RUGHU WR ILQG WKH HQHUJ\ RI LQWHUDFWLRQ H[FLWRQV DQG FU\VWDO ODWWLFH RVFLOODWLRQV LW LV QHFHVVDU\WRH[SDQGWKHRSHUDWRURIWKHH[FLWRQ H[FLWDWLRQHQHUJ\DVDVHULHVLQOLWWOHVKLIWVRIWKH PROHFXOHVIURPWKHLUHTXLOLEULXPVWDWH7KHQIRU WKHIXOOH[FLWDWLRQHQHUJ\RIWKHFU\VWDOLQZKLFK H[FLWRQVDQGSKRQRQVLQWHUDFWDWWKHVDPHWLPH ZHREWDLQ

LQW + + +

+= H[ + SK +

ÃÃÃ

+

-+

+

- +

=

LQW 1 )V N T%NTV%UV ETV ETV

+ rr

+HUHETV+ ETVDUHRSHUDWRUVRISURGXFWLRQ DQGGHVWUXFWLRQRISKRQRQVRIWKHVEUDQFKZLWK WKHHQHUJ\WVTDQGTXDVLPRPHQWXPTr)V±

LV WKH IXQFWLRQ ZKLFK VKRZV WKH LQWHUDFWLRQ RI WKH SKRQRQV RI WKHVEUDQFK ZLWK WKH FU\VWDO H[FLWRQ H[FLWDWLRQ+LQW RSHUDWRU GHVFULEHV WKH

HOHPHQWDU\H[FLWDWLRQVRIWKHFU\VWDOZKLFKDUH FKDUDFWHUL]HG E\ D FHUWDLQ QXPEHU RI LQGHSHQGHQWH[FLWRQVDQGSKRQRQV7KHRSHUDWRU RI LQWHUDFWLRQ GRHV QRW FKDQJH WKH WRWDO QXPEHU RI H[FLWRQV LQ WKH V\VWHP EXW RQO\ FDXVHV WKH WUDQVLWLRQ RI H[FLWRQV IURP RQH VXEOHYHORIWKH]RQHWRDQRWKHURQHDEVRUELQJRU LUUDGLDWLQJSKRQRQVDWWKHVDPHWLPH7KXVWKLV LQWHUDFWLRQ OHDGV WR WKH SURFHVV RI H[FLWRQ VFDWWHULQJ RQ SKRQRQV HPSOR\LQJ WKH ODZV RI FRQVHUYLQJHQHUJ\DQGTXDVLPRPHQWXP

7KH FDOFXODWLRQ RI WKH IXQFWLRQ RI H[FLWRQ SKRQRQERQGLQWKHJHQHUDOFDVHRIWKUHHGLPHQ VLRQDO FU\VWDOV LQYROYHV JUHDW PDWKHPDWLFDO GLIILFXOWLHV ,Q VLPSOH FXELF LVRWURSLF FU\VWDOV ZKHQ WDNLQJ LQWR FRQVLGHUDWLRQ WKH LQWHUDFWLRQ RQO\EHWZHHQWKHQHLJKERULQJPROHFXOHVZHXVH DSSUR[LPDWH H[SUHVVLRQV ,Q SDUWLFXODU IRU LQWHUDFWLRQ ZLWK DQ RSWLFDO GLVSHUVLRQ OHVV SKRQRQ>@

FRV FRV

ND TD TD I

(3)

7KH)RUP)XQFWLRQRIWKH([FLWRQ $EVRUSWLRQ%DQG

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

à Ð

<

= H[S

1 ' L W

. w j a w

Ã

=

< + p wj a

a6 1 ' GW

W % W

%4 4

+HUH WKH WLPH GHSHQGHQFH LV SXW GRZQ LQ WKH +HLVHQEHUJ 3UHVHQWDWLRQ DQG LW LV DYHUDJHG RYHUWKHIXOO+DPLOWRQLDQRIWKHFU\VWDO7KH YDOXH 6w LV FDOOHG WKH LVRODWHG H[FLWRQ DEVRUSWLRQ EDQG IRUP IXQFWLRQ 'a ± L V WKH PDWUL[ HOHPHQW RI WKH LQWHUDFWLRQ RI OLJKW DQG FU\VWDOZKRVHYDOXHGHSHQGVRQWKHDQJOHj EHWZHHQ WKH SRODUL]DWLRQ ZDYH Hr4 DQG WKH GLSROHPRPHQWGraRIWKHFU\VWDOWUDQVLWLRQLQWR WKHJLYHQH[FLWRQVWDWH

a a a

a 'H G ' FRVj

' = r4r =

+HQFHLWIROORZVWKDWWKHLQWHUDFWLRQRIOLJKW ZLWK FU\VWDO LV PD[LPXP ZKHQ WKH SKRWRQ SRODUL]DWLRQYHFWRULVSDUDOOHOWRWKHJLYHQGLSROH PRPHQW RI WKH TXDQWXP WUDQVLWLRQ ja = 7KH*UHHQIXQFWLRQPHWKRGDOORZVXVWRILJXUH RXW LQ GHWDLO WKH H[FLWRQ OLJKW DEVRUSWLRQ FRHIILFLHQWRQWKHEDVLVRIWKHGHGXFHGH[FLWRQ SKRQRQV\VWHPPDVVRSHUDWRU ` @ ^> 4 4 4 ( 4 4 6 w w w w w a a a a a G + D -G = ZKHUH0aw4=Daw4-LGaw4LVWKH PDVVRSHUDWRUZKLFKLQWKHJHQHUDOFDVHLVWKH FRPSOH[IUHTXHQF\IXQFWLRQ

7KXV WKH RXWOLQH FLUFXLW RI WKH H[FLWRQ DEVRUSWLRQ FXUYH ZLWK D ZHDN H[FLWRQSKRQRQ ERQG LV GHVFULEHG E\ D TXDVL/RUHQW] W\SH IXQFWLRQ ZLWK D ZLGWK GHWHUPLQHG E\ WKH LPDJLQDU\ SDUW RI WKH PDVV RSHUDWRU 7KH FDOFXODWLRQRIWKHPDVVRSHUDWRULVSHUIRUPHGLQ GHWDLOLQ>@ ^

> @

> @ `

TV 4 T

TV

TV 4 T

TV

0 4 ) 4 T Q 1

( 4 T L

Q 1

( 4 T L

a

a

w

w W t

w W t

+

-+

-= - í

í - + + + +

+ + + í

í - + - +

ÃÃ

+HUH 1a4 DQG QTV DUH WKH QXPEHU RI H[FLWRQ RI WKHa]RQH DQG SKRQRQV RI WKHV EUXQFKDWDJLYHQFU\VWDOWHPSHUDWXUH7 H[S -ÍÜ Í Û Ú ÍÌ Í Ë Ê -ÕÕÖ Ô ÅÅÆ Ä -= 7 N 4 ( 1 %

4 a m

a H[S -Ø Ø Ù × È È É Ç -ÕÕÖ Ô ÅÅÆ Ä W = 7 N Q % TV TV

ZKHUHN% ± LV WKH %ROW]PDQQ FRQVWDQW 7 ± LV WKHFU\VWDOWHPSHUDWXUHLQ.HOYLQXQLWVm±LV WKH H[FLWRQ FKHPLFDO SRWHQWLDO ZKRVH YDOXH LV FRQGLWLRQHGE\WKHH[FLWRQFRQFHQWUDWLRQ1H[LQD

FU\VWDODWDJLYHQWHPSHUDWXUH H[ =

Ã

1 4

9

1 a

/LJKW$EVRUSWLRQLQD%HQ]RO&U\VWDO 7KH H[SHULPHQWDO HYLGHQFH RI DULVLQJ H[FLWRQ VWDWHVLQPROHFXODUFU\VWDOVDVWKHDYDLODELOLW\RI WZR RU PRUH H[FLWRQ DEVRUSWLRQ EDQGV IRU GLI IHUHQWSRODUL]DWLRQVRIWKHLQFLGHQWOLJKWZDYH

7KXVIRULQYHVWLJDWLQJ)UHQNHOH[FLWRQVZH PXVWWDNHSRODUL]DWLRQVSHFWUDOPHDVXUHPHQWVDW ILUVW7KHVSOLWWLQJRIQRQGHJHQHUDWHGPROHFXODU WHUPV LQ SRODUL]HG OLJKW DEVRUSWLRQ VSHFWUD LV UHJLVWHUHG LQ D QXPEHU RI H[SHULPHQWDO ZRUNV DQG IRU GLIIHUHQW PROHFXODU FU\VWDOV 7KHVH DUH FU\VWDOVIRUPHGE\WKHDURPDWLFEHQ]ROPROHFXOH &+DQWUDFHQH&+QDSKWDFHQH&+

SHQWDFHQH &+ QDSKWDOHQH DQG RWKHUV 7KH

(4)

LQPROHFXODUFU\VWDOVLVGLIIHUHQW$QWUDFHQHDQG QDSKWDFHQH VSHFWUD KDYH ZLGH LQWHQVLYH EDQGV DQG EHQ]RO ]RQHV KDYH PXFK QDUURZHU DQG ZHDNHULQWHQVLYHEDQGV7KHQZHVKDOOGZHOORQ H[SHULPHQWDO UHVXOWV GHYRWHG WR H[FLWRQ DEVRUSWLRQLQEHQ]ROFU\VWDO>@

7KH DYDLODELOLW\ RI IRXU HOHFWURQ H[FLWDWLRQ OHYHOVLVW\SLFDOIRUEHQ]ROPROHFXOHV7KHILUVW

=

De H9± LV D SXUH HOHFWURQ WUDQVLWLRQ 7KH VHFRQG De H9 WKH WKLUG

=

De H9DQGWKHIRXUWKDe =H9 ± DUH RQHIROG WZRIROG DQG WKUHHIROG FRPELQDWLRQV RI WRWDOO\ V\PPHWULF LQWUDPROHFXODU YLEUDWLRQV H9 ZLWK DQ HOHFWURQWUDQVLWLRQ,QFU\VWDODEVRUSWLRQVSHFWUD IRXU EDQGV LQ RQH SRODUL]DWLRQ DUH REWDLQHG DFFRUGLQJO\DQGIRXUEDQGVLQWKHSHUSHQGLFXODU SRODUL]DWLRQ DUH REWDLQHG 7KH\ DUH ZLWK WKH IROORZLQJ VSHFWUDO FKDUDFWHULVWLFV WKH ILUVW RQH ZL WK WKH HQHUJ\ (=H9 DQG ZL WK WKH YDOXHRI WKH UHVRQDQW SRODUL]DEOH VSOLWWLQJ EDQG

=

D( FPWKHVHFRQGRQH

( H9DQG

(

D FP WKH WKLUG RQH

( H9

(

D FP WKH IRXUWK

( DQG

(

D FP

)RUWKHVDNHRIWKHRUHWLFDOWUHDWPHQWRIWKH H[SHULPHQWDOO\ REWDLQHG UHVXOWV ZH FKRRVH RQH RIWKHEDQGVQDPHO\WKHILUVWRQHIRUZKLFKWKH SRODUL]DWLRQ VSOLWWLQJ LV FP DQG LQ ZKRVH

VSHFWUXPWKHUHDUHWZRDEVRUSWLRQEDQGVDDQG F FRPSRQHQWV ZLWK RUWKRJRQDO SRODUL]DWLRQ 7KXV WKH FU\VWDO HOHFWURQ H[FLWDWLRQ ZLOO KDYH WZR H[FLWRQ H[FLWDWLRQ EDQGV ZKLFK DFFRUGLQJ WRDQGDUHFKDUDFWHUL]HGE\WKHHQHUJLHV

FRV

FRV

/\ (

\ /

( (

/\ ( \ /

( (

h d + + =

-+ =

+ =

-+ =

+HUH ZH LQWURGXFHG WKH IROORZLQJ GHVLJQDWLRQV ( ± LV WKH ERWWRP RI WKH H[FLWRQ

EDQG IURP ZKLFK ZH VWDUW WKH UHFNRQLQJ RI WKH FU\VWDOH[FLWDWLRQHQHUJ\/±LVWKHZLGWKRIWKH ILUVWH[FLWRQH[FLWDWLRQEDQGVd ±LVWKHH[FLWRQ UHVRQDQWVSOLWWLQJYDOXHh/±LVWKHZLGWKRIWKH

VHFRQG H[FLWRQ H[FLWDWLRQ EDQG \=NDp \ 7DNLQJ LQWR FRQVLGHUDWLRQ WKDW IRU WKH YLVXDODQGXOWUDYLROHWOLJKWLWLVLQWKLVVSHFWUDO UHJLRQWKDWH[FLWRQH[FLWDWLRQHQHUJ\LVSUHVHQW WKHSURGXFW4DKHUHLQDIWHUZHVKDOOQHJOHFW WKH SKRWRQ ZDYH YHFWRU DQG WDNH LW DV ]HUR 4

7ZR GLIIHUHQW WUDQVLWLRQ GLSROH PRPHQWV FRUUHVSRQGWRWZRH[FLWRQH[FLWDWLRQEDQGVWKXV WKH LQWHUDFWLRQ RI OLJKW ZLWK WKH H[FLWRQ H[FLWDWLRQ RI HDFK EDQG DFFRUGLQJ WR (T ZLOOEHGLIIHUHQW

'='FRVj ='FRVj j x

p

j VLQ

FRV

' '

' = - =

ZKHUH'VWDQGVIRUWKHIRUFHYDOXHRIWKHRVFLO

ODWRU RI FU\VWDO WUDQVLWLQJ LQWR WKH ILUVW H[FLWRQ VWDWHDQGx'IRUWKHIRUFHYDOXHRIWKHRVFLO ODWRURIFU\VWDOWUDQVLWLQJLQWRWKHVHFRQGH[FLWRQ VWDWHDQGWKHSRODUL]DWLRQDQJOH<j<p

)RUWKHLQWURGXFHGGHVLJQDWLRQVWKHFU\VWDO DEVRUSWLRQOLJKWFRHIILFLHQWRQWKHIUHTXHQF\w ZLOOEH

>FRV VL Q @

> @

. 1 ' 6 1

' 6 1 6 1

' 1 1

a a

w j p w

p j w x j j

p a w j a w j

= =

= + =

= +

Ã

$VLWLVHYLGHQWIURP(TWKHH[FLWRQ EDQGFRQVLVWVRIWZRFRPSRQHQWVaDQGa ZKRVHXOWLPDWHIRUPIXQFWLRQRIWKHDEVRUSWLRQ FXUYH ZLOO GHSHQG RQ WKH SRODUL]DWLRQ DQJOH RI WKHLQFLGHQWZDYH:HFDQEXWDUJXHWKDWDWWKH YDOXHRIWKHDQJOHj=WKHUHZLOOEHRQO\WKH ILUVW EDQG LQ WKH VSHFWUXP WKH RQH ZLWK WKH VPDOOHUH[FLWRQHQHUJ\DQGZKHQj=p RQO\ WKHVHFRQGH[FLWRQDEVRUSWLRQEDQGZLOODSSHDU

(5)

RSWLFDO SKRQRQV 7KXV KHUHLQDIWHU ZH VKDOO UHVWULFW RXU FRQVLGHUDWLRQ RQO\ WR SURFHVVHV RI H[FLWRQVVFDWWHULQJIURPRSWLFDOSKRQRQVRIWKH (LQVWHLQ PRGHO WVT=W :H SDVV RQ LQ (T IURP WKH VXPPDWLRQ RI WKH TXDVL PRPHQWXP RI WKH SKRQRQ T WR WKH LQWHJUDWLQJ RYHUWKHLPPHDVXUDEOHSDUDPHWHU\=TD pDQG SXWGRZQDOOHQHUJ\YDOXHVLQWKHZLGWKXQLWVRI WKH ILUVW H[FLWRQ EDQG HJ W=W /

/ I I RSW

= Z=w-( / d =d /DQG VR

RQ8VLQJWKHHTXDWLRQ OLP

[ Lt 3[ Lpd [

t + =

-‰ DQGDOVRHTXDWLRQVDQGZHREWDLQ IRUWKHLPDJLQDU\SDUWRIWKHPDVVRSHUDWRUWKH IROORZLQJH[SUHVVLRQV @ > -+ + + + = G T Q T -T Q T -I Z p @ > -+ + + + = G T Q T -T Q T -I Z h p

,Q WKHVH H[SUHVVLRQV ZH LQWURGXFHG

‹ =Z‹W

-T

‹ =>Z-d ‹Wh@

-T

+DYLQJ SHUIRUPHG VLPLODU FKDQJHV IRU WKH UHDO SDUW RI WKH PDVV RSHUDWRU DV ZHOO ZH JHW WKH IROORZLQJH[SUHVVLRQV ` ^

Z =I Q , Z+W + +Q , Z-4

D ÍÜ Í Û Ú ÕÕÖ Ô ÕÕÖ Ô ÅÅÆ

Ä - -W + + + ÍÌ Í Ë Ê ÅÅÆ Ä ÕÕÖ Ô ÅÅÆ

Ä - +W = D h d h d h Z , Q Z , Q I Z H[S -Ø Ù × È É Ç -Õ Ö Ô Å Æ Ä W = q Q

ZKHUHZH LQWURGXFHG WKH IROORZLQJ DEEUHYLDWHG GHVLJQDWLRQV IRU WKH FU\VWDO WHPSHUDWXUH

/ 7 N%

=

q DQGIRUWKHLQWHJUDOV

Ð

-

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

W FPd FPh x

I 7KHFU\VWDOWHPSHUDWXUHLVFKRVHQWR

. Z f D E F

Z

(6)

EH HTXDO WR . q= DQG . q = ZH FKRRVH WKUHH YDOXHV RI SRODUL]DWLRQDQJOHVjp p )LJDQG )LJ VKRZ WKH FDOFXODWLRQ UHVXOWV RI WKH LPPHDVXUDEOH DEVRUSWLRQ FRHIILFLHQW

' I /

.LP = p IRU GLIIHUHQW DQJOHV DQG

WHPSHUDWXUHV

7KHDQDO\VLVRIWKHFDOFXODWLRQVVKRZVWKDW WKHUH LV D FRQVLGHUDEOH YDULDWLRQ LQ WKH YHFWRU GLUHFWLRQ GHSHQGHQFH RI WKH OLJKW ZDYH SRODUL]DWLRQ LQ H[FLWRQ RSWLFDO VSHFWUD ,I WKH SRODUL]DWLRQDQJOHj =WKHFXUYHDLQ)LJ LH WKH SRODUL]DWLRQ YHFWRU FRLQFLGHV ZLWK WKH GLUHFWLRQ RI WKH GLSROH PRPHQWGr WKH H[FLWRQ DEVRUSWLRQ EDQG LV FOHDUO\ UHDOL]HG DW WKH IUHTXHQF\ HTXDO WR WKH HQHUJ\ RI WKH ORZHU H[FLWRQ]RQH,QWKLVFDVHWKHEDQGVKDSHKDVD WUDGLWLRQDO VKRUWZDYH DV\PPHWU\ )RU WKH SRODUL]DWLRQ DQJOH j =p WKH FXUYH F RQ )LJ ZLWK WKH OLJKW ZDYH SRODUL]DWLRQ YHFWRU GLUHFWHG DORQJ WKH GLSROH PRPHQWGr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d.= d.= ZLWK WKH GHFUHDVH RI WHPSHUDWXUH DQG REYLRXVO\ DW WKH WHPSHUDWXUHRIDQDEVROXWH]HURUHDFKHVLWVSHDN d

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

:KHQDQLQWHQVHOLJKWZDYHLQWHUDFWVZLWKD FU\VWDO WKH LPDJLQDU\ SDUW RI WKH PDVV RSHUDWRU ZLOOEHGHWHUPLQHGE\WKHH[SUHVVLRQ Ü Û Ú + + + + Ì Ë Ê -= G -+ + -+ T 1 Q T -T 1 Q T -I Z 1 T T p Ü Û Ú + + + + Ì Ë Ê -= G -+ + -+ T 1 Q T -T 1 Q T -I Z 1 T T h p $QGIRUWKHUHDOSDUWRIWKHPDVVRSHUDWRU LQ FRQWUDVW WR (T DQG (T ZH REWDLQ WKHIROORZLQJH[SUHVVLRQV

]

[

W -+ W + -+ + D = D Z , Z , I Z Z 1 1 1 È É Ç Ø Ù × W -+ W + -+ + D = D h d h d h Z , Z , I Z Z 1 1 1 LQZKLFKWKHYDOXHRIWKHLQWHJUDOLVGHILQHGDV

Ð

-

-=3 - \ 1 ; \ G\ ;

,1 \

(7)

H[S

Ã

-

-ØÙ × ÈÉ

Ç =

Ø Ù × È

É Ç

-=

Q $ 1

\ $ 1

Q H[

\

n pq

q

,Q RUGHU WR FDOFXODWH WKH H[FLWRQ FKHPLFDO SRWHQWLDO ZH WDNH LQWR FRQVLGHUDWLRQ WKDW WKH OHYHORIWKHFKHPLFDOSRWHQWLDOFDQQRWEHKLJKHU WKDQ WKH ERWWRP RI WKH H[FLWRQ EDQG $ ã 7KLVPHDQVWKDWWKHPD[LPXPDOORZDEOHQXPEHU RI H[FLWRQV DW WKH JLYHQ WHPSHUDWXUH ZLOO EH

H[

H[ 1

1PD[ = $ 7KXV WR GHWHUPLQHG WKH YDOXH $ ZH PD\ ZULWH GRZQ WKH IROORZLQJ HTXDWLRQ

1H[ =1µ1H[PD[ ZKHUH WKH YDOXH1 YDULHV IURP WR DQG IRU HDFK RI LWV VLJQLILFDQW ILJXUHV WKHUH LV RQO\ RQH SDUWLFXODUYDOXHRIWKHFKHPLFDOSRWHQWLDODQGRI WKHYDOXH$

,WLVHYLGHQWWKDWLQWKHFDVHRIDQH[WHUQDO H[FLWRQ SXPSLQJ WKHLU UHDO GHQVLW\ ZLOO GHSHQG RQ WKH ODVHU UDGLDWLRQ SRZHU ,Q WKLV FDVH WKH GLVWULEXWLRQRIH[FLWRQVWDWHVZLOOGLIIHUIURPWKH FRQGLWLRQ RI WKHUPRG\QDPLF HTXLOLEULXP DQG WKHLUGHQVLW\ZLOOEHWKHIXQFWLRQRIWHPSHUDWXUH DQG WKH H[FLWDWLRQ ZDYH LQWHQVLW\ ,Q RUGHU WR WDNH LQWR FRQVLGHUDWLRQ WKLV IDFWRU ZKHQ LQYHVWLJDWLQJ WKH DEVRUSWLYH SRZHU RI D FU\VWDO

ZHPXVWSXWGRZQWKHHQHUJ\EDODQFHHTXDWLRQ DV ZHOO ZKLFK GHWHUPLQHV WKH DPRXQW RI WKH DEVRUEHG HQHUJ\ IRU FUHDWLQJ WKH GHVLUDEOH GHQVLW\RIWKHH[FLWRQVWDWHV

1 Z 1 .

,µ j=

,Q WKLV H[SUHVVLRQ PD[ 51H[ ,

, = L V WKH YDOXH UHODWHG WR WKH ODVHU UDGLDWLRQ LQWHQVLW\ , >: P @ DQG WKH FRHIILFLHQW RI H[FLWRQ

UHFRPELQDWLRQ 5 >:@ 7KH H[FLWRQ DEVRUSWLRQ FRHIILFLHQW LV WKH H[FLWRQ GHQVLW\ IXQFWLRQ WKH IXQFWLRQ RI FU\VWDO WHPSHUDWXUH DQG ODVHU ZDYH SRODUL]DWLRQ IXQFWLRQ +DYLQJ IL[HG WKUHH RI WKHP IRU LQVWDQFH Zqj ZH GHWHUPLQH WKH H[FLWRQ GHSHQGHQFH RI WKH DEVRUSWLRQ ,W WXUQV RXWWKDWWKHUHLVDQH[FLWRQIUHTXHQF\UHJLRQZV

IRU ZKLFK WKH IXQFWLRQ .1 KDV D EHOO ± OLNH FKDUDFWHU DQG LQ WKLV FDVH (TXDWLRQ PD\ KDYHWKUHHVROXWLRQV>@,WVKRXOGEHQRWHGWKDW IRUSXUHSKRQRQPHFKDQLVPVRIH[FLWRQHQHUJ\ UHOD[DWLRQ WKH IUHTXHQF\ UHJLRQ ZV L V YHU\

QDUURZ )LJ SUHVHQWV WKH H[FLWDWLRQ ZDYH LQWHQVLW\ GHSHQGHQFH RI WKH H[FLWRQ GHQVLW\ VWDWHV RQ WKH IUHTXHQFLHV RI WKH ILUVW H[FLWRQ EDQG ZV DQG RI WKH VHFRQG H[FLWRQ

EDQG ZV :H VHH WKDW WKHUH LV D

K\VWHUHVLVORRSZKLFKFDXVHVWKHDULVLQJRIWZR VWDWLRQDU\ VWDWHV ZLWK DQ DVVLJQHG H[FLWRQ GHQVLW\DWWKHVDPHODVHUUDGLDWLRQLQWHQVLW\7KH

.

Z

f

D

E

F

Z

)LJ7KHH[FLWRQDEVRUSWLRQVSHFWUDDWWHPSHUDWXUH IRUWKUHHSRODUL]DWLRQDQJOHV

(8)

GHSRODUL]DWLRQRIWKHH[WHUQDOODVHUZDYHDWWKH ORZHU H[FLWRQ IUHTXHQF\ WKH LQFUHDVH RI WKH DQJOH j IURP WR p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

VKRXOG EH QRWLFHG WKDW WKH GHFUHDVH RI WKH H[FLWDWLRQ ZDYH IUHTXHQF\ FDXVHV WKH GHFUHDVH RIWKHELVWDEOHVWDWHVUHJLRQVRQWKHVFDOHRIWKH H[FLWRQGHQVLW\VWDWHV1

+DYLQJ REWDLQHG WKH H[SUHVVLRQ 1, ZH FDQ FRXQW D QHZ DEVRUSWLRQ FRHIILFLHQW DQG GHWHUPLQH WKH LQFLGHQW ZDYH LQWHQVLW\ GHSHQGHQFH ., RI WKH DEVRUSWLRQ 8VLQJ WKH FDOFXODWHG GDWD DV WKH EDVH ZH ILQG DOVR WKH DEVRUSWLRQFRHIILFLHQWRUWKHUDWLRQRIWKHRXWSXW OLJKWLQWHQVLW\WRWKHLQSXWOLJKWLQWHQVLW\IRUWKH FU\VWDOZLWKWKHWKLFNQHVVG

@ H[S>-. , µG

=

t

7KHUHVXOWVRIWKHFDOFXODWLRQVE\(T IRU WKH JLYHQ SDUDPHWHUV WKH WKLFNQHVV RI WKH FU\VWDO G í-FP DUH VKRZQ LQ )LJ

:H VHH WKDW WKH UHDOL]DWLRQ RI WKH 2%

D E F G

1

)LJ7KHH[FLWDWLRQZDYHLQWHQVLW\GHSHQGHQFHRIWKHH[FLWRQGHQVLW\VWDWHVIRUGLIIHUHQWSRODUL]DWLRQ DQJOHV E DW WKH IUHTXHQF\ DQG G DW WKH IUHTXHQF\

,

, ,

t

%

%

+

+

D

E

(9)

SKHQRPHQRQ LV SRVVLEOH LQ ERWK RUWKRJRQDO SRODUL]DWLRQV RI WKH H[FLWDWLRQ ODVHU ZDYH ,W LV DOVR FRQILUPHG WKDW ZLWK WKH GHFUHDVH RI WKH FU\VWDO DEVRUSWLYH SRZHU DW WKH XSSHU H[FLWRQ ZDYH IUHTXHQF\ WKH 2% UHDOL]DWLRQ LV RQO\ SRVVLEOH ZLWK WKH LQFUHDVH RI WKH ODVHU ZDYH LQWHQVLW\

7KH K\VWHUHVLV ORRS WUDQVIRUPDWLRQ UHJXODULWLHV LQ WKH GHSHQGHQFH t, ZL WK WKH FKDQJHRIWKHSRODUL]DWLRQDQJOHDUHWDEXODWHGLQ WKH WDEOH %DVLFDOO\ WKH\ DPRXQW WR WKH IROORZLQJ7KHGHSRODUL]DWLRQRIRSWLFDOELVWDEOH EDQGVLUUHVSHFWLYHRIWKHLUH[FLWDWLRQIUHTXHQF\ UHVXOWV LQ WKH GHFUHDVH RI WKH K\VWHUHVLV ORRS KHLJKW Dtˆ DQG WKH GHFUHDVH RI WKH LQWHUYDO YDOXHVRIWKHZHDNVWURQJDEVRUSWLRQVZLWFKLQJ LW VKLIWV WKH K\VWHUHVLV WR WKH KLJKHU LQWHQVLW\ UHJLRQ ,ˆ DQG HQODUJHV LWV ORRS D,6 ZKLFK FDXVHV WKH LQFUHDVH RI WKH UHJLRQ RI 2% UHDOL]DWLRQ DW WKH FRVW RI WKH LQFUHDVH RI WKH H[FLWDWLRQ ZDYH LQWHQVLW\ +RZHYHU WKHUH DUH YDULDWLRQV LQ WKH G\QDPLFV RI WKH QRQOLQHDU DEVRUSWLRQ GHYHORSPHQW LQ GLIIHUHQW SRODUL]DWLRQV 7KH FKDQJH RI WKH ZHDNVWURQJ DEVRUSWLRQVZLWFKLQJLQWHUYDODt–KDVGLIIHUHQW VLJQVIRUWKHORZHUH[FLWRQEDQGWKHVZLWFKLQJ LQWHUYDO LQFUHDVHV ZLWK WKH LQFUHDVH RI WKH GHSRODUL]DWLRQDQGIRUWKHXSSHUH[FLWRQEDQGLW GHFUHDVHV &OHDUO\ LW LQYROYHV GLIIHUHQW UDWLRQV RIGHFUHDVLQJWKHKHLJKWRIWKHSHDNLQGLIIHUHQW SRODUL]DWLRQV

&RQFOXVLRQV

8QGHUKLJKOHYHOVRISXPSLQJWKHSURFHVVRI H[FLWRQ DEVRUSWLRQ EHFRPHV QRQOLQHDU ZLWK WKH FKDQJH RI WKH H[FLWDWLRQ ZDYH LQWHQVLW\ WKUHVKROG YDOXHV RI WKH H[FLWRQ GHQVLW\ HPHUJHDWZKLFKDMXPSOLNHWUDQVLWLRQIURP WKH ZHDN DEVRUSWLRQ VWDWH LQWR WKH VWURQJ DEVRUSWLRQVWDWHWDNHVSODFH

$K\VWHUHVLVORRSDULVHVLQWKHH[FLWDWLRQZDYH LQWHQVLW\ GHSHQGHQFH RI WKH FU\VWDO WUDQVPLVVLRQ SRZHU ZKLFK UHVXOWV LQ WKH UHDOL]DWLRQRI RSWLFDOELVWDELOLW\ LQ WKH JLYHQ FU\VWDORQWKHJLYHQH[FLWDWLRQIUHTXHQF\ 7KH ZLGWK RI WKH RSWLFDO K\VWHUHVLV WKH

GLVWDQFH LQ LQWHQVLW\ EHWZHHQ WKH WUDQVLWLRQV RI ZHDNVWURQJ DQG VWURQJZHDN DEVRUSWLRQ LVLQIOXHQFHGRQO\E\WKHRXWHUODVHUZDYH± WKH ZLGWK RI WKH K\VWHUHVLV UDQJH GHFUHDVHV VKDUSO\ ZLWK WKH IUHTXHQF\ UHDFKLQJ WKH DEVRUSWLRQEDQGPD[LPXP

7KH KHLJKW RI WKH K\VWHUHVLV ORRS WKH GLIIHUHQFH EHWZHHQ WKH VWURQJ DQG ZHDN DEVRUSWLRQ LV HVVHQWLDOO\ LQIOXHQFHG E\ WKH GHSRODUL]DWLRQ YDOXH ± WKH KHLJKW LV PD[LPXPDWVWULFWO\SRODUL]HGDEVRUSWLRQ 5HIHUHQFHV

*LEEV;2SWLFDO ELVWDELOLW\ 0RVFRZ ³0LU´ SLQ5XVVLDQ

1LWVRZLFK %0 =HQNRYD &<X 2SWLFDO (QJLQHHULQJ

'DY\GRY $6 7KH WKHRU\ RI PROHFXODU 7DEOH7KHFKDUDFWHULVWLFSDUDPHWHUVRIWKHK\VWHUHVLVORRSLQWKHGHSHQGHQFH

7KHORZHUH[FLWRQEDQGZ

V

7KHXSSHUH[FLWRQEDQGZ

V

(10)

H[FLWRQV 0RVFRZ ³1DXND´ S LQ5XVVLDQ

'DY\GRY $6 7KH WKHRU\ RI VROLG VWDWH 0RVFRZ³1DXND´SLQ5XVVLDQ %URXGH 9/ 0HGYHGHY 96 3ULNKRGNR

$3K -([SDQG 7KHRU3K\V LQ5XVVLDQ

References

Related documents

The research participants comprise of programme coordinators of Bachelor’s Degree of Social work, final year students within the programme and teachers teaching courses relevant

We use a combination of Labor Force Survey data on employment characteristics and register data on demographic outcomes to investigate how the labor market status and stability

For cereals Crowther (1945) showed that higher yields could be obtained by combine-drilling rather than by broadcasting dressings of superphosphate, but little information is

The data attributes a and

The Com/PC Embedded Gateway Linux (EGL/2) operating system comes with a pre-installed Apache Web server.. Please see also mHT-CPC1L-07.pdf: How to use the Apache Web

Biodiversity &amp; Ecosystem Services; Comparative Studies in Regenerative Food Systems; Comparative Studies in Sustainable Living; Complexity Theory &amp; Systems

In Proceedings of the IEEE Computer Society Conference on Com- puter Vision and Pattern Recognition , volume 1 , pages 285 – 292 , San Diego, California, June 2005.. Speech

With- out weight compression and using 16-bit parameters, the DRNN achieved an effective throughput of 1.2 TOp/s and MAC computa- tion efficiency of 623.96% with negligible